1 Introduction
The interplay between symbolic dynamics, directed graphs, and the
$C^*$
-algebras of directed graphs has long been a fruitful area of research, revealing deep structural connections across these domains. In this paper, we explore the relationship between the dynamical notion of insplitting and the structure of the resulting
$C^*$
-algebras in two dimensions. This lays the groundwork for a
$C^*$
-algebraic framework for studying one-sided conjugacy for higher-dimensional dynamical systems.
While the research presented in this article was motivated by
$C^*$
-algebraic concerns, our main theorems are phrased purely in dynamical terms; we relegate the
$C^*$
-algebraic considerations to Appendix A and future papers.
In both one and two dimensions, shifts of finite type (SFTs) are precisely those shifts that possess a graphical description. The representation of one-dimensional (1D) SFTs via directed graphs has been known since the early development of the field (see [Reference Lind and MarcusLM95, Theorem 2.3.2]), and this correspondence has been highly successful in describing conjugacy, flow equivalence and related invariants of dynamical systems. In the two-dimensional (2D) setting, textile systems (consisting of a pair of directed graphs ‘woven’ together) serve a similar purpose. Introduced by Nasu in [Reference NasuNas95], textile systems were shown to model all 2D SFTs by Johnson and Madden in [Reference Johnson and MaddenJM99, Proposition 2.3] (see also [Reference AsoAso00, Theorem 4.1]). They also characterized conjugacy for SFTs via textile systems in [Reference Johnson and MaddenJM99, Corollary 3.10] and [Reference AsoAso00, Theorem 3.1]. As in the 1D case, two 2D SFTs are conjugate—that is, dynamically isomorphic—if and only if their textile systems can be transformed into one another via the graphical operations of insplitting, outsplitting, and inversion (and their inverses). (The operation of inversion interchanges the horizontal and vertical directions in the 2D SFT, and has no analogue in the 1D setting.)
Directed graphs and their higher-dimensional generalizations have also led to significant advances in the theory of
$C^*$
-algebras, in which
$C^*$
-algebraic versions of insplitting and outsplitting have played a major role. Directed graphs (or, equivalently, 1D SFTs) yield key examples of
$C^*$
-algebras, which are tractable because their internal algebraic structure mirrors the structure of the underlying graph (see [Reference Kumjian, Pask and RaeburnKPR98, Reference Bates, Hong, Raeburn and SzymańskiBHRS02, Reference Raeburn and SzymańskiRS04]). As
$C^*$
-algebras are analytic objects with very few a priori structural restrictions, the tight structural link between a directed graph and its
$C^*$
-algebra makes graph
$C^*$
-algebras a key source of examples for
$C^*$
-algebraists (see [Reference Hong and SzymańskiHS04, Reference Mann, Raeburn and SutherlandMRS92]). Indeed, the program of classifying
$C^*$
-algebras up to isomorphism (or the weaker invariant of Morita equivalence) has seen spectacular success for graph
$C^*$
-algebras [Reference SørensenSø13, Reference Eilers, Restorff, Ruiz and SørensenERRS17, Reference Eilers, Restorff, Ruiz and SørensenERRS21, Reference Arklint, Eilers and RuizAER22]: two graph
$C^*$
-algebras are isomorphic if and only if their underlying directed graphs can be converted into each other via a finite list of moves, including insplitting (and outsplitting in the case of Morita equivalence). Graph
$C^*$
-algebras and related constructions have also led to new insights in symbolic dynamics, by providing new characterizations of dynamical invariants such as flow equivalence [Reference Cuntz and KriegerCK80, Reference MatsumotoMat10], continuous orbit equivalence [Reference Giordano, Putnam and SkauGPS95, Reference Matsumoto and MatuiMM14, Reference Arklint, Eilers and RuizAER18, Reference Carlsen, Eilers, Ortega and RestorffCEOR19], one- and two-sided conjugacy [Reference Carlsen and RoutCR17, Reference Brix and CarlsenBC20b], and eventual (one-sided) conjugacy [Reference Brix and CarlsenBC20a, Reference BrixBri22].
The tight structural link between a directed graph and its
$C^*$
-algebra also means that many interesting
$C^*$
-algebras, such as the rotation algebras [Reference Evans and SimsES12] or any
$C^*$
-algebra with torsion
$K_1$
group [Reference CuntzCun81, Reference Raeburn and SzymańskiRS04], cannot arise as graph
$C^*$
-algebras. Higher-rank graphs, or
$k$
-graphs, were introduced by Kumjian and the fifth-named author in [Reference Kumjian and PaskKP00] to provide combinatorial models for such
$C^*$
-algebras. Like their 1D cousins the directed graphs,
$k$
-graphs also have a strong link with dynamical systems, see [Reference Kumjian and PaskKP03]. In [Reference TangTan13], Tang established a bijection between 2-graphs (or 2D higher-rank graphs) and left-resolving (LR) textile systems. More generally, Carlsen and Rout initiated in [Reference Carlsen and RoutCR17] the
$C^*$
-algebraic analysis of two-sided conjugacy, eventual one-sided conjugacy, and continuous orbit equivalence for the dynamical systems arising from
$k$
-graphs.
Unlike textile systems,
$k$
-graphs are categories, by definition [Reference Kumjian and PaskKP00]. That is, in a
$k$
-graph, concatenation of ‘paths’ is really a composition of morphisms in a category. For consistency, therefore, we think of paths in a
$k$
-graph as pointing right-to-left: two paths
$p,q$
are composable with product
$pq$
if
$s(p) = r(q)$
. This convention unfortunately leaves us at odds with the literature on textile systems; we discuss the translation in Remark 3.3.
Both graph
$C^*$
-algebras and
$k$
-graph
$C^*$
-algebras are built from the underlying one-sided SFT [Reference Kumjian, Pask, Raeburn and RenaultKPRR97, Reference Kumjian and PaskKP00]. This underlies the fact (discovered by Bates and the fifth-named author in [Reference Bates and PaskBP04]) that insplitting yields an isomorphism of graph
$C^*$
-algebras, while outsplitting yields only a Morita equivalence. (In [Reference Bates and PaskBP04], as in much of the dynamics literature, the roles of insplitting and outsplitting are reversed from the discussion in this paper. This is due to our view of
$k$
-graphs as categories, which leads to a different choice of convention between [Reference Bates and PaskBP04] and the current paper (which follows [Reference Kumjian and PaskKP00]) in the definition of the graph
$C^*$
-algebra.) Indeed, the asymmetry between the
$C^*$
-algebraic implications of insplitting and outsplitting persists for
$k$
-graph
$C^*$
-algebras [Reference Eckhardt, Fieldhouse, Gent, Gillaspy, Gonzales and PaskEFG+22, Reference ListhartkeLis24]. This is the main reason why we concern ourselves with one-sided shifts.
However, for 2D dynamical systems, Theorem 5.6 shows that these natural
$C^*$
-algebraic formulations of insplitting and outsplitting do not agree with the textile-system insplitting and outsplitting of [Reference Johnson and MaddenJM99]. However, in the 1D setting, a dynamically constructed isomorphism of
$C^*$
-algebras frequently implies that the underlying one-sided shift spaces are conjugate (see [Reference Cuntz and KriegerCK80, Reference Brix and CarlsenBC20b, Reference Armstrong, Brix, Carlsen and EilersABCE23]). This raises the natural question of whether, in the 2D setting as well, the
$C^*$
-algebraic definition of insplitting should yield a conjugacy of the associated one-sided SFTs. Equivalently, are the 2D versions of insplitting from [Reference Eckhardt, Fieldhouse, Gent, Gillaspy, Gonzales and PaskEFG+22] and [Reference Johnson and MaddenJM99] related by more than mere nomenclature?
Theorem 6.1 gives an affirmative answer to this question. More precisely, suppose
$T$
is an LR textile system (Definition 3.1); then Theorem 4.6 shows that
$T$
determines a 2-graph
$\Lambda _T$
(Definition 4.1) and, hence, a
$C^*$
-algebra by [Reference Kumjian and PaskKP00]. Theorem 6.1 says that a 2-graph insplit on
$\Lambda _T$
(as in [Reference Eckhardt, Fieldhouse, Gent, Gillaspy, Gonzales and PaskEFG+22]) can be recreated via textile-system insplits and inversions on
$T$
. In particular, the 2-graph insplitting of [Reference Eckhardt, Fieldhouse, Gent, Gillaspy, Gonzales and PaskEFG+22] yields a conjugacy of one-sided, 2D shifts of finite type. In addition, we present various constructions in Section 7 that recover the textile-system insplitting [Reference Johnson and MaddenJM99] from a combination of directed graph insplitting and 2-graph insplitting.
1.1 Structure of the paper
As one-sided 2D dynamics have not received as much attention in the literature as their two-sided 1D counterparts, we begin in Section 2 with definitions and basic results about these shifts. In particular, we define conjugacy and higher block presentations for one-sided 2D SFTs and show that any conjugacy is induced by a factor map.
Section 3 recalls the definition of a textile system and in Section 4, 2-graphs are briefly introduced, as well as their associated dynamical systems. The link between 2-graphs and textile systems is established in Theorem 4.6. We review the definitions of insplitting for textile systems and for 2-graphs in Section 5.
Our main results lie in Sections 6 and 7. In addition to establishing, in Theorem 6.1, that 2-graph insplitting can be understood via textile system insplitting, we provide several perspectives on the relationship between 2-graph insplitting and textile system insplitting. Theorem 7.3 identifies when a single textile-system insplitting could alternately be interpreted as a 2-graph insplitting, and Theorem 7.1 shows how to interpret 2-graph insplitting as a textile insplit together with an extra insplit of the ‘bottom’ graph
$E$
from the textile system. We also demonstrate that certain insplittings of the base graph simultaneously yield both types of insplitting. Indeed, these different perspectives are equivalent, as we establish in Section 7.4. We conclude the paper with a brief discussion of the
$C^*$
-algebraic consequences of 2-graph insplitting and outsplitting, as well as the dynamical consequences of 2-graph outsplitting, in Appendix A.
2 Two-dimensional noninvertible symbolic dynamics
We use the convention that
$0 \in \mathbb {N}$
(see Remark 2.4). The standard generators of
$\mathbb {N}^2$
(respectively
$\mathbb {Z}^2$
) are denoted
$\varepsilon _1,\varepsilon _2$
and we write
$n_i$
for the
$i\,{\textrm {th}}$
coordinate of
$n \in \mathbb {N}^2$
. We define a partial order on
$\mathbb {N}^2$
by
$m\leq n$
if
$m_i \leq n_i$
for
$i=1, 2$
. For
$m,n \in \mathbb {N}^2$
, we write
$m \vee n$
for their coordinatewise maximum and
$m \wedge n$
for their coordinatewise minimum.
For completeness, we include a brief overview of 2D one-sided shifts of finite type, as systematic treatments of this topic are scarce in the literature. We assume familiarity with the basic theory of 1D shifts of finite type, as found in [Reference Lind and MarcusLM95, Reference KitchensKit98].
To fix notation, we begin with a discussion of 1D dynamical systems.
2.1 One-dimensional preliminaries
Definitions 2.1. A directed graph is a quadruple
$E= (E^0,E^1,r,s)$
, where
$E^0, E^1$
are sets of vertices and edges, respectively, and
$r, s: E^1 \to E^0$
are maps giving the range and source of each edge, respectively. We say that
$E$
is source-free or has no sources if
$r$
is onto; it is essential if both
$r,s$
are onto.
Example 2.2. From a finite set
$X$
, we construct an essential (directed) graph
$B_X$
, the ‘bouquet of
$X$
loops’. This graph has one vertex and
$B_X^1 = \{ e_x\mid x \in X\}$
; the range and source maps are uniquely defined.
For a directed graph
$E$
and
$n \in \mathbb {N}$
,
$E^n$
denotes the directed paths of length
$n$
in
$E$
:
We denote the collection of finite directed paths in
$E$
by
$E^* := \bigcup _{n \ge 0} E^n$
. The range and source maps
$r, s: E^1 \to E^0$
extend naturally to
$E^*$
, where
$r(v)=s(v)=v$
for all
$v \in E^0$
. Given
$v,w \in E^0$
and
$F \subseteq E^*$
, define
$vF := r^{-1}(v) \cap F$
,
$Fw := s^{-1}(w) \cap F$
, and
$vFw := vF \cap Fw$
.
Remark 2.3. The set
$E^*$
becomes a category where the objects are the vertices
$E^0$
and the domain of a morphism (path) is its source and the range of a morphism (path) is its categorical range. In [Reference Mac LaneML75], this is called the free (or path) category generated by E since there are no relations required.
Remark 2.4. One can identify
$\mathbb {N}$
with the path category of the directed graph
$B_X$
, where
$X$
has one element.
Let
$E=(E^0,E^1,r,s)$
be an essential directed graph. Then, as in [Reference Lind and MarcusLM95, Definition 2.2.5 and Section 13.8], we define
to be the one-sided, 1D shift of finite type associated to the directed graph
$E$
.
2.2 One-sided, 2D shifts of finite type
The following definition is adapted from [Reference SchmidtSch98, Section 2].
Let
$\mathcal {A}$
be a finite set (alphabet). Define
$\mathcal {A}^{\mathbb {N}^2} = \{ x : \mathbb {N}^2 \to \mathcal {A}\}$
. A typical point
$x~\in ~\mathcal {A}^{\mathbb {N}^2}$
is written as
$x= ( x_n : n \in \mathbb {N}^2)$
, where
$x_n \in \mathcal {A}$
denotes the value of
$x$
at
$n \in \mathbb {N}^2$
.
We equip
$\mathcal A^{\mathbb {N}^2}$
with the product topology. That is, the basic open sets of
$\mathcal A^{\mathbb {N}^2}$
are the cylinder sets
$Z(x[0,m])$
for
$x \in \mathcal A^{\mathbb {N}^2}$
and
$m \in \mathbb {N}^2$
, defined by
There is an action
$\sigma $
of
$\mathbb {N}^2$
on
$\mathcal {A}^{\mathbb {N}^2}$
defined by
for every
$x=(x_n) \in \mathcal {A}^{\mathbb {N}^2}$
and
$m \in \mathbb {N}^2$
.
Definitions 2.5. A subset
$X \subset \mathcal {A}^{\mathbb {N}^2}$
is called shift-invariant if
$\sigma ^m(X) =X$
for every
$m \in \mathbb {N}^2$
, and a closed shift-invariant subset
$X \subset \mathcal {A}^{\mathbb {N}^2}$
is called a (one-sided) subshift. If
$X \subset \mathcal {A}^{\mathbb {N}^2}$
is a subshift, then we write
$\sigma = \sigma _X$
for the restriction of the shift action (2-2) to
$X$
and we say that
$(X,\sigma )$
is a shift space.
For any subset
$S \subset \mathbb {N}^2$
, we denote by
$\pi _S : \mathcal {A}^{\mathbb {N}^2} \to \mathcal {A}^S$
the projection map that restricts every
$x \in \mathcal {A}^{\mathbb {N}^2}$
to
$S$
. A subshift
$X \subset \mathcal {A}^{\mathbb {N}^2}$
is a one-sided SFT if there exist a finite set
$Z \subset \mathbb {N}^2$
and a finite collection
$\mathcal F$
of functions
$f: Z \to \mathcal A$
such that
This definition resembles the ‘allowed blocks’ model of shift spaces described in [Reference Lind and MarcusLM95, Section 2.1]: a configuration
$x \in \mathcal {A}^{\mathbb {N}^2}$
belongs to the shift space
$X$
if whenever
$x$
is restricted to some translate of
$Z$
the values of
$\mathcal {A}$
found there match those allowed in
$Z$
.
2.3 Sliding-block codes, conjugacy, inversion
Definition 2.6 (Blocks, block map).
Let
$m \in \mathbb {N}^2$
and
$X \subset \mathcal {A}^{\mathbb {N}^2}$
be a one-sided SFT. The collection of
$m$
-blocks in
$X$
is defined by
Without loss of generality,
$B_0(X)= \mathcal {A}$
.
The following is adapted from Kitchens [Reference KitchensKit98, page 25]. Let
$X$
be an SFT over the alphabet
$\mathcal A$
and
$Y$
be an SFT over the alphabet
$\mathcal B$
. Fix
$\ell \in \mathbb {N}^2$
. An
$\ell $
-block map
$\phi : (X, \sigma _X) \to (Y, \sigma _Y)$
is determined by a map
$\Phi : B_\ell (X) \to B_0 (Y)$
that is consistent with
$X$
and
$Y$
in the following sense: for every
$x \in X, a \in \mathbb {N}^2$
, and
$i=1,2$
,
the associated
$\ell $
-block map
$\phi : (X,\sigma _X) \to (Y,\sigma _Y)$
is given by
Lemma 2.7. Every block map
$\phi : (X, \sigma _X) \to (Y, \sigma _Y)$
is continuous and shift commuting, that is,
$\sigma _Y \circ \phi = \phi \circ \sigma _X.$
Proof. Suppose
$\phi : X \to Y$
is an
$\ell $
-block map. Fix
$n \in \mathbb {N}^2$
and consider a basic open set
$Z(y[0, n])$
in
$Y$
. Note that
being a union of cylinder sets, is open. Hence,
$\phi $
is continuous.
To see that
$\sigma _Y \circ \phi = \phi \circ \sigma _X,$
let
$x\in X$
,
$n \in \mathbb {N}^2$
, and
$i= 1,2.$
Then,
while
$\phi (\sigma ^{\varepsilon _i}_X(x))_n = \phi ((x_{k+\varepsilon _i})_k)_n = \Phi (x[n+\varepsilon _i, n+\varepsilon _i +\ell ]).$
The argument of [Reference KitchensKit98, Theorem 1.4.9] generalizes to show that any continuous shift commuting map between shift spaces is a block map. In particular, every conjugacy can be viewed as a block map.
Standing assumption. In the same way that Lind and Marcus reduce arguments to using
$1$
-block maps for shifts of finite maps in [Reference Lind and MarcusLM95, Section 2], we may do the same here. By considering the sizes of all the forbidden blocks and then applying a suitable higher block presentation (see [Reference KitchensKit98, page 27]), we may assume that
in (2-3).
Definition 2.8 (Conjugacy).
The shift spaces
$(X,\sigma _X)$
and
$(Y,\sigma _Y)$
are conjugate if there is a homeomorphism
$h : X \to Y$
satisfying
In two dimensions, the choice of horizontal and vertical directions is somewhat arbitrary. Hence (following [Reference Johnson and MaddenJM99]), we define an operation on a shift space which swaps these directions.
Definition 2.9 (Inversion).
Fix an alphabet
$\mathcal A$
. The inversion of the shift space
$(X, \sigma )$
is
$(\theta (X), \sigma ')$
, where
We write
$x = \theta (y)$
. The shift map is also inverted in
$\theta (X)$
; that is,
$\sigma '$
is given by
For
$x\in \theta (X)$
, we write
$x= \theta (y)$
for some
$y\in X$
, and observe that
$X$
is a SFT if and only if
$\theta (X)$
is.
Lemma 2.10. Let
$(X, \sigma )$
be a shift space and
$( \theta (X), \sigma ' )$
be the inversion of
$X$
. Then,
$\theta : X \to \theta (X)$
is a conjugacy.
Proof. Evidently,
$\theta ^{-1} $
and
$\theta $
are given by the same formula; in particular,
$\theta $
is a bijection. Moreover,
$\theta \circ \sigma = \sigma ' \circ \theta ,$
since for any
$m, n \in \mathbb {N}^2,$
It is straightforward to see that
$\theta $
is continuous; for instance, the inverse image of the cylinder set
$Z(x[(0,(m_1, m_2))])$
is
which is
$Z(\theta (x)[(0, (m_2, m_1))])$
, another cylinder set. We conclude that
$\theta $
is a homeomorphism, which completes the proof.
3 Textile systems and their dynamical systems
Just as every 1D SFT can be realized as the edge shift of a directed graph [Reference Lind and MarcusLM95, Theorem 2.3.2], every 2D SFT can be realized using a pair of graphs woven together into the form of a textile system. Johnson and Madden establish in [Reference Johnson and MaddenJM99, Proposition 2.3] that every (two-sided) 2D shift of finite type is conjugate to a textile system SFT, by using a higher block argument similar to that given in Section 2.3.
The definition that follows is an adaptation of the original definition given by [Reference NasuNas95].
Definition 3.1. A textile system
$T=(p,q:F \to E)$
consists of two directed graphs
$F, E$
and two graph homomorphisms
$p, q: F \to E$
such that the function
$A : F^1 \to F^0 \times E^1 \times F^0 \times E^1$
given by
is injective.
Remark 3.2. In [Reference Johnson and MaddenJM99], Johnson and Madden restrict their attention to textile systems where
$E=B_X$
for some set
$X$
. We find this condition too restrictive for our aims and do not enforce this restriction in this paper.
A square in the textile system
$T=(p,q :F \to E)$
is a four-sided object, whose edges are labeled using the edges of
$E$
and vertices of
$F$
as
Let
$W_T = \{ T_f \mid f \in F^1 \}$
denote the collection of squares associated to
$T$
.
Remark 3.3. Here, our conventions for labeling the squares associated to a textile system are different from those in [Reference NasuNas95, Reference Johnson and MaddenJM99]. This is the first visible sign of our categorical preferences mentioned in Section 1. To be precise, we use the convention that
$r(f)$
lies on the left of the square denoting the edge
$f$
and
$s(f)$
on the right so that (with concatenation of paths as composition of morphisms)
$F^*$
is a category. To maintain consistency with Nasu’s definition of LR textile systems (in particular, so that we measure the path lifting properties of
$p$
with respect to
$r$
and not
$s$
in Definition 3.4), this choice forces us to place
$p(f)$
on the top of the square
$T_f$
and
$q(f)$
on the bottom.
Definitions 3.4. A graph homomorphism
$p: F \to E$
is said to have (unique) r-path lifting if, for all
$v \in F^0,$
whenever
$p(v) = r(e) \in E^0$
for some
$e\in E^1$
, there exists (a unique)
$f \in F^1$
with
$p(f) = e$
and
$r(f) = v.$
Having unique
$r$
-path lifting is sometimes referred to as being left resolving in the dynamics literature. Similarly,
$p$
has (unique) s-path lifting if, for all
$v\in F^0$
, whenever
$p(v) = s(e) \in E^0$
for some
$e\in E^1$
, there exists (a unique)
$f \in F^1$
with
$p(f) = e$
and
$s(f) = v$
. Having unique
$s$
-path lifting is sometimes referred to as being right resolving in the dynamics literature.
A textile system
$T= (p, q: F\to E)$
is called LR if
$p$
has unique
$r$
-path lifting and
$q$
has unique
$s$
-path lifting.
Example 3.5. Consider the directed graphs shown as follows:
Define
$p,q : F \to E$
by
$p(c_1)= b_1, p(c_2)= p(l_2) = b_2$
and
$q(c_1)= q(l_2) = b_1, q(c_2)=b_2$
. Then,
$T = (p,q:F \to E)$
is a non-LR textile system since
$p(l_2)=p(c_2)=b_2$
and
$r(l_2)=r(c_2)=a_2$
. Worse yet,
$p$
fails to have
$s$
-path lifting, since there is no edge in
$F^1$
with
$s(f)= a_1$
and
$p(f)= b_1$
, and
$q$
fails to have
$r$
-path lifting since there is no edge
$f \in F^1$
with
$r(f) = a_1$
and
$q(f) = b_2$
.
Lemma 3.6. Let
$E$
and
$F$
be directed graphs and
$p,q:F\to E$
be graph morphisms. If
$p$
has unique
$r$
- or unique
$s$
-path lifting, then
$(p,q:F \to E)$
is a textile system for any
$q$
. Similarly, if
$q$
has unique
$r$
- or unique
$s$
-path lifting, then
$(p,q:F \to E)$
is a textile system for any
$p$
.
Proof. Suppose that
$p$
has unique
$r$
-path lifting. We will show that the function
$f\mapsto (r(f),p(f),s(f)$
,
$q(f))$
is injective. Let
$f , g \in F^1$
be such that
$(r(f),p(f),s(f),q(f)) = (r(g),p(g),s(g)$
,
$q(g))$
. Then,
$p(f) = p(g)$
and
$r(f)=r(g)$
; hence,
$f=g$
. The other cases are proved mutatis mutandis.
However, Example 3.5 shows that the converse of Lemma 3.6 does not hold.
The following is adapted from [Reference SchmidtSch98, Section 4].
Definition 3.7 (Textile tiling).
Let
$T = ( p,q : F \to E)$
be a textile system, with the associated collection
$W_T = \{ T_f \mid f \in F^1 \}$
of squares. A (one-sided) textile weaved by T is a covering of
$\mathbb {R}_{\ge 0}^2$
by translating copies of elements of
$W_T$
with nonoverlapping interiors such that the following conditions are satisfied:
-
(i) every corner of each square lies in $\mathbb {N}^2 \subset \mathbb {R}_{\ge 0}^2 $
; -
(ii) two squares are only allowed to touch along edges in the following sense:
-
(T1) $s(f) = r(f^{\prime })$
whenever
$T_f, T_{f^{\prime }}$
are horizontally adjacent squares with
$T_f$
to the left of
$T_{f^{\prime }}$
; or -
(T2) $p(f) = q(f^{\prime })$
if
$T_f, T_{f^{\prime }}^{\prime }$
are vertically adjacent with
$T_{f^{\prime }}$
above
$T_f$
.
-
We write
$\textsf {X}_T^+ $
for the subshift of
$W_T^{\mathbb N^2}$
(with the usual horizontal and vertical shift maps) consisting of all textiles weaved by
$T$
. That is,
Remark 3.8. It is straightforward to show that
$\textsf {X}_{T}^+ \subset W_T^{\mathbb {N}^2}$
is closed and shift invariant. When the graph
$F$
is finite,
$\textsf {X}_T^+$
is a shift of finite type as it is of the form (2-3) for
$Z=\{0,1\}^2$
: the restrictions of (3-1) specify which
$2\times 2$
blocks of tiles are allowed.
For simplicity, we identify
$W_T$
with
$F^1$
, that is, the alphabet of the shift space
$\textsf {X}_T^+$
is
$F^1$
.
We now proceed to give sufficient conditions for
$\textsf {X}_T^+$
to be nonempty.
Theorem 3.9 (See [Reference TangTan13, Proposition 4.6]).
Let
$T = (p,q : F \to E)$
be a textile system in which
$F$
is source-free,
$q|_{F^0}$
is onto, and
$q$
has
$r$
-path lifting. Then,
$\textsf X_T^+$
is nonempty.
Proof. We first show that for any infinite path
$x^0 \in \textsf X^+_F$
in
$F$
, there is
$x^1 \in \textsf X_F^+$
such that
$q(x^1) = p(x^0).$
As
$F$
is source-free,
$\textsf {X}^+_F$
defined in (2-1) is nonempty, so we may choose
${x^0 = (x_i^0 )_{i \in \mathbb {N}} \in \textsf {X}^+_F}$
. Consider
$x_0^0 \in F^1$
and let
$u_0 = r( p(x_0^0))$
. Since
$q$
has
$r$
-path lifting and
$q|_{F^0}$
is onto, we may choose
$w_0 \in F^0$
with
$q (w_0) =u_0$
and
$x^1_0 \in F^1$
such that
$r ( x^1_0) = w_0$
and
$q (x^1_0 ) =p ( x_0^0 )$
. Let
$w_1 = s(x^1_0) $
; as
$q$
is a graph homomorphism,
The fact that
$q$
has
$r$
-path lifting now implies the existence of
$x^1_1 \in F^1$
with
$q(x^1_1) = p(x^0_1)$
and
$r(x^1_1) = w_1$
. Let
$w_2 = s(x^1_1)$
. Continuing inductively, we construct
${x^1 = (x_i^1)_{i\in \mathbb N} \in X^+_F}$
with
$q(x^1_i) = p(x_i^0)$
for all
$i$
. The construction is indicated in the figure following the proof.
Following the same construction, we can create
$(x^n)_{n \in \mathbb N} \subseteq \textsf X_F^+$
such that
$p(x^n) = q(x^{n+1})$
for all
$n\in \mathbb N$
. That is, by construction,
$x = (x^i_j)_{(j,i)\in \mathbb N^2} \in \textsf {X}_T^+.$
The following definition follows the notation of [Reference Johnson and MaddenJM99]; the textile system
$\widehat {T}$
is called the dual textile system in [Reference NasuNas95, page 15].
Definition 3.10 (Inverted textile system).
Let
$T=(p,q:F \to E)$
be a textile system. Define directed graphs
$\widehat {E} = ( E^0 , F^0 , q , p )$
and
$\widehat {F} = ( E^1 , F^1 , q , p )$
. Then,
$\hat {p} = ( s_E , s_F)$
,
${\hat {q} = ( r_E , r_F )}$
are graph morphisms from
$\widehat {F}$
to
$\widehat {E}$
and
$\widehat {T}= ( \hat {p} , \hat {q} : \widehat {F} \to \widehat {E} )$
is a textile system, called the inverted textile system.
We record the following lemma, whose proof is straightforward.
Lemma 3.11. For any textile system
$T$
, we have
$(\widehat {T})\widehat {\vphantom {T}} = T$
, and
$\theta (\mathsf {X}_T^+) = \mathsf {X}_{\widehat {T}}^+$
, in the notation of Definition 2.9. Furthermore, if
$T=(p,q:F \to E)$
is LR, then the inverted system
$\widehat {T} = ( \hat {p},\hat {q}: \widehat {F}\to \widehat {E} )$
is LR.
4 2-graphs and their dynamical systems
Theorem 4.6 shows that all LR textile systems can be described as higher-rank graphs of rank two, or
$2$
-graphs. As we mentioned in Section 1, graphical moves such as insplitting have been developed in [Reference Eckhardt, Fieldhouse, Gent, Gillaspy, Gonzales and PaskEFG+22] for 2-graphs (and indeed for higher-rank graphs in general), and a main goal of this paper is to clarify the relationship between 2-graph insplitting and textile system insplitting. Thus, this section defines 2-graphs and establishes the equivalence between
$2$
-graphs and LR textile systems.
We restrict our attention to the 2D setting (in this section, and throughout the paper) because, while
$k$
-graphs of arbitrary rank
$k$
have been considered since the inception of the study of higher-rank graphs [Reference Kumjian and PaskKP00], higher-dimensional analogs of textile systems have not yet been developed in the literature.
4.1 Rank-two graphs and two-colored graphs
In what follows, we view
$\mathbb N^2$
as a category with one object, namely 0; composition of morphisms is given by coordinate-wise addition. Thus, the standard notation ‘
$n \in \mathbb N^2$
’ indicates that
$n$
is a morphism in
$\mathbb N^2$
. For consistency with this perspective, for a general category
$\Lambda $
, we write ‘
$\lambda \in \Lambda $
’ to indicate that
$\lambda $
is a morphism in
$\Lambda $
. We identify the objects in
$\Lambda $
with the identity morphisms.
Definition 4.1 [Reference Kumjian and PaskKP00, Definition 1.1].
A rank-two graph (or a 2-graph) is a countable category
$\Lambda $
with a degree functor
$d:\Lambda \to \mathbb {N}^2$
satisfying the factorization property: if
$\lambda \in \Lambda $
and
$m,n\in \mathbb {N}^2$
are such that
$d(\lambda )=m+n$
, then there are unique
$\mu ,\nu \in \Lambda $
with
$d(\mu )=m$
,
$d(\nu )=n$
, and
$\lambda =\mu \nu $
.
Given
$m \in \mathbb {N}^2$
, we define
$\Lambda ^m := d^{-1}(m)$
.
The factorization property then allows us to identify
$\Lambda ^0$
(the morphisms of degree 0) with the identity morphisms (equivalently, the objects
$\operatorname {Obj}(\Lambda )$
) of
$\Lambda $
; given any
$\lambda \in \Lambda $
, there are unique
$v, w \in \Lambda ^0$
with
$\lambda = v \lambda w$
. We write
$v =: r(\lambda )$
and
$w = :s(\lambda )$
, and we call
$\Lambda ^0$
the vertices of
$\Lambda $
. A vertex
$v \in \Lambda ^0$
is a sink if
$s^{-1} (v) = \{ v\} $
, and
$w \in \Lambda ^0$
is a source if
$r^{-1} (w) = \{w\}$
.
For
$v,w \in \Lambda ^0$
and
$F \subseteq \Lambda $
, define
$vF := r^{-1}(v) \cap F$
,
$Fw := s^{-1}(w) \cap F$
, and
${vFw := vF \cap Fw}$
. If
$v \Lambda ^m$
is finite for all
$m \in \mathbb {N}^2$
and
$v \in \Lambda ^0$
, then
$\Lambda $
is called row-finite. If
$v \Lambda ^m$
and
$\Lambda ^m v$
are nonempty for all
$m\in \mathbb {N}^2$
and
$v \in \Lambda ^0$
, then
$\Lambda $
is called essential.
By the factorization property, for each
$\lambda \in \Lambda $
and
$m \leq n \leq d(\lambda )$
, we may write
${\lambda =\lambda '\lambda "\lambda "'}$
, where
$d(\lambda ')=m, d(\lambda ") = n-m$
and
$d(\lambda "')=d(\lambda )-n$
; then
$\lambda (m,n) :=\lambda "$
. We write
For more information about
$k$
-graphs, see [Reference Kumjian and PaskKP00, Reference Raeburn, Sims and YeendRSY03] for example.
Example 4.2. Following [Reference Kumjian and PaskKP00], let
$\Omega _2 $
be the category with morphisms
$\operatorname {Mor}(\Omega _2)= \{ (m,n) \in \mathbb {N}^2 \times \mathbb {N}^2\mid 0 \leq m \le n \}$
; composition of morphisms is given by
$ ( \ell ,m) (q,n)= \delta _{m,q}(\ell , n).$
Set
$d ( m , n ) = n - m$
; then
$d: \Omega _2 \to \mathbb {N}^2$
is a functor and
$( \Omega _2 , d )$
is a row-finite
$2$
-graph. The vertices
$\Omega _2^0 = \{ (m,m) \mid m \in \mathbb {N}^2 \}$
may be identified with
$\mathbb {N}^2 = \operatorname {Obj}(\Omega _2)$
.
One can also profitably think of a 2-graph as a quotient of the path category of a directed graph with two colors of edges. To be precise, for each
$i= 1, 2$
, we call
$d^{-1}(\varepsilon _i) =\Lambda ^{\varepsilon _i} $
the edges of color i in
$\Lambda $
and call
$\Lambda ^1 := \Lambda ^{\varepsilon _1} \sqcup \Lambda ^{\varepsilon _2}$
the edges in
$\Lambda $
. Then, every element of
$\Lambda $
can be written as a composition of colored edges: given
$\lambda \in \Lambda $
with
$d(\lambda ) = n = (n_1, n_2)$
, writing
and the factorization property tells us that
$\lambda = e_1 \cdots e_{n_1} f_1 \cdots f_{n_2}$
for a unique collection of edges
$e_i \in \Lambda ^{\varepsilon _1}$
of color 1 and
$ f_i \in \Lambda ^{\varepsilon _2}$
of color 2.
To see why
$\Lambda $
is a nontrivial quotient of the path category of the edge-colored directed graph
$(\Lambda ^0, \Lambda ^{\varepsilon _1} \sqcup \Lambda ^{\varepsilon _2}, r, s)$
, observe that the representation
depends on the order in which we write
$n = (n_1, n_2)$
as a sum of generators of
$\mathbb {N}^2$
. For example, since
$ \varepsilon _1 + \varepsilon _2 = \varepsilon _2 + \varepsilon _1$
, the factorization property implies that for any
$\lambda \in \Lambda ^{\varepsilon _1 + \varepsilon _2}$
, there are unique
$e, e' \in \Lambda ^{\varepsilon _1}$
and
$f, f' \in \Lambda ^{\varepsilon _2}$
satisfying
Thus, to obtain
$\Lambda $
from the path category of
$(\Lambda ^0, \Lambda ^{\varepsilon _1} \sqcup \Lambda ^{\varepsilon _2}, r, s)$
, we need to take the quotient with respect to the equivalence relation
$\sim $
generated by
The following makes this idea precise.
Definition 4.3. A 2-colored graph
$(G, d)$
is a directed graph
$G=(G^0,G^1,r,s)$
along with a degree map
$d : G^1 \to \{\varepsilon _1,\varepsilon _2\}$
. We think of
$\varepsilon _1, \varepsilon _2$
as the colors of the edges in
$G$
.
We extend the degree map to a map
$d: G^* \to \mathbb {N}^2$
on the path category of
$G$
by
Let
$\sim $
be an equivalence relation on
$G^*$
that preserves the degree, range, and source of paths. Then, by [Reference Hazlewood, Raeburn, Sims and WebsterHRSW13, Theorems 4.4 and 4.5], the quotient
$\Lambda = G^*/\sim $
of the path category of
$G$
by an equivalence relation
$\sim $
is a
$2$
-graph if and only if
$\sim $
satisfies the following conditions.
-
(KG0) Unique factorization: if $\lambda \in G^*$
is a path such that
$\lambda = \lambda _2 \lambda _1$
, then the equivalence class
$[\lambda ] = [p_2] [p_1]$
whenever
$p_i\in [\lambda _i]$
for
$i=1,2$
. -
(KG1) Non-degeneracy: if $f, g\in G^1$
are edges, then
$f \sim g \iff f = g.$
-
(KG2) Completeness: whenever $fg \in G^2$
is a path consisting of one edge of each color, there is a unique path
$g' f' \sim fg$
with
$d(g') = d(g)$
and
$d(f') = d(f)$
.
Remark 4.4. If
$\Lambda = G^*/\sim $
is a 2-graph, we call
$(G,d)$
the 1-skeleton of
$\Lambda $
. As
$\sim $
respects the degree, range, and source maps from
$G$
, these descend to maps on
$\Lambda $
. The fundamental building blocks of
$\Lambda $
are the commuting squares of
$\Lambda = G^*/\sim $
:
Indeed (see [Reference Pask, Quigg and RaeburnPQR04, Proposition 4.3]), fixing a set
$\mathcal C_\Lambda $
of commuting squares satisfying Condition (KG2) leads to a uniquely defined equivalence relation
$\sim $
on
$G^*$
which respects degree, source, and range, and also satisfies Conditions (KG0) and (KG1) (that is, an equivalence relation making
$G^*/\sim $
into a 2-graph). To be precise, we guarantee Condition (KG1) by defining
$[e_1] = \{ e_1\}$
if
$e_1 \in G^0 \cup G^1,$
and we ensure Condition (KG0) by extending
$\sim $
to be a symmetric and transitive relation on
$G^*$
.
4.2 2-graphs and textile systems
The perspective of 2-colored graphs enables us to articulate the link between 2-graphs and (LR) textile systems. The following definition was first given in [Reference TangTan13].
Definition 4.5 (
$2$
-colored graph associated to a textile system).
Let
${T = (p,q:F \to E)}$
be a textile system. Define the 2-colored graph
$(G_T,d)$
as follows: let
$(G_T)^0 = E^0$
,
$(G_T)^1 = E^1 \sqcup F^0$
. For
$e \in E^1$
, define
$s ( e ) = s_E (e)$
,
$r (e) = r_E (e)$
, and
$d (e) = \varepsilon _1$
; and for
$w \in F^0$
, define
$s (w) = p(w)$
,
$r(w) = q(w)$
, and
$d (w) = \varepsilon _2$
.
We define an equivalence relation
$\sim $
on
$G_T^*$
by setting
$[\alpha ] = \{ \alpha \}$
for any
${\alpha \in G_T^0 \cup G_T^1}$
; defining
$\sim $
on 2-colored paths of length 2 by
and extending
$\sim $
to paths of other lengths inductively by defining
$[\lambda \mu ] = [\lambda ] [\mu ]$
.
That is, the commuting squares in
$G_T$
correspond to the edges in
$F$
as follows:
The next result, which is a major foundation for the research in this paper, is a generalization of [Reference TangTan13, Theorem 3.8].
Theorem 4.6. Let
$T=(p,q:F \to E)$
be a textile system with
$p,q$
surjective. Let
$(G_T,d)$
be the associated
$2$
-colored graph and
$\sim $
be the equivalence relation on
$G_T^*$
from Definition 4.5. Then,
$ \Lambda _T:= G_T^*/\sim $
is a 2-graph if and only if
$T$
is LR (that is,
$p$
has unique
$r$
-path lifting and
$q$
has unique
$s$
-path lifting).
Proof. First, suppose that
$p$
has unique
$r$
-path lifting and
$q$
has unique
$s$
-path lifting. Let
$\alpha \beta $
be a
$2$
-colored path in
$G_T^2$
. Suppose without loss of generality that
$d(\alpha ) = \varepsilon _1$
and
$d(\beta ) = \varepsilon _2$
. By Definition 4.5, this implies
$\alpha \in E^1$
and
$\beta \in F^0$
, and the fact that
$\alpha \beta $
is a path in
$G_T$
implies that
$s_E(\alpha ) = s_{G_T}(\alpha ) = r_{E_T}(\beta ) = q(\beta )$
.
Since
$q$
has unique
$s$
-path lifting, there is a unique
$f\in F^1\beta $
with
$q(f) = \alpha $
. By definition of
$\sim $
, we have
$r(f) p(f) \sim \alpha \beta $
; the uniqueness of
$f$
guarantees that
$\sim $
satisfies Condition (KG2). If
$\alpha $
has color 2 and
$\beta $
has color 1, the same argument applies, using unique
$r$
-path lifting of
$p$
in place of unique
$s$
-path lifting of
$q$
. Thus,
$\Lambda _T = G_T^*/\sim $
is a 2-graph.
Now, suppose that
$G_T^*/\sim $
is a 2-graph, that is,
$\sim $
satisfies Condition (KG2). We show that
$p$
has unique
$r$
-path lifting. The argument for
$q$
having unique
$s$
-path lifting is similar. Fix
$e \in E^1$
,
$w \in F^0$
with
$r_E(e) = p(w)$
, that is, with
$r_{G_T}(e) = s_{G_T}(w)$
. Then,
$we$
is a 2-colored composable path in
$G_T$
, so by Condition (KG2), there is a unique 2-colored path
$e' v$
such that
$d(e') = d(e)$
,
$d(v) = d(w)$
, and
$we \sim e'v$
. That is, there exists a unique
$f \in F^1$
with
$r_F(f) = w$
and
$p(f) =e$
. So,
$p$
has unique
$r$
-path lifting.
In other words, an LR textile system
$T = (p, q: F\to E)$
yields a 2-graph
$\Lambda _T$
by identifying
$F^1$
with
$\Lambda _T^{\varepsilon _1 + \varepsilon _2}.$
Conversely, given a 2-graph
$\Lambda $
, we can define a textile system
$T_\Lambda = (p, q: F_\Lambda \to E_\Lambda )$
by
The uniqueness guaranteed by the factorization property implies that
$T_\Lambda $
is an LR textile system, and we have
$\Lambda _{T_\Lambda } = \Lambda $
and
$T_{\Lambda _T} = T$
.
Remark 4.7. By Definition 4.5, if
$\Lambda = G^*/\sim $
is a 2-graph with associated textile system
$T = (p,q: F \to E)$
, then
$\Lambda ^0 = E^0$
and
$\Lambda ^{ 1} = E^1 \sqcup F^0.$
For any
$v \in F^0$
,
whereas
$s_\Lambda (e) = s(e), r_\Lambda (e) = r(e)$
for
$e \in E^1.$
Next, we show that the notion of essentiality translates well between LR textile systems and rank-2 graphs. Recall (Definition 4.1) that a 2-graph
$\Lambda $
is called essential if
$v \Lambda ^m$
and
$\Lambda ^m v$
are nonempty for all
$m \in \mathbb {N}^2$
and
$ v \in \Lambda ^0$
. We now introduce an analogous notion of essential textile systems.
Definition 4.8 (Essential textile system).
A textile system
$T= (p,q: F \to E)$
is said to be essential if
$F$
is essential (as in Definition 2.1), and
$p$
and
$q$
are surjective maps.
Note that if
$T$
is an essential textile system, it follows immediately that
$E$
is also essential. We extensively use this fact in the following proposition to prove that essential LR textile systems are equivalent to essential rank-2 graphs.
Proposition 4.9. An LR textile system
$T= (p,q: F \to E)$
is essential if and only if its corresponding 2-graph
$\Lambda = \Lambda _T$
is essential.
Proof. First, assume that the textile system
$T= (p,q: F \to E)$
is essential. Since
$p,q,r_E,s_E, r_F, s_F$
are all surjective, we have that for each
$z \in \Lambda ^0 = E^0$
,
It follows that
$z \Lambda ^m$
and
$\Lambda ^m z$
are nonempty for all
$m \in \mathbb {N}^2$
. Thus,
$\Lambda $
is essential when
$T$
is essential.
Conversely, assume that
$\Lambda $
is essential. Then, for each
$z \in E^0$
,
Similarly, if
$e \in E^1$
with
$s_E(e) =z$
, the fact that
$z\Lambda ^{(0,1)}\not = \emptyset $
implies that there exists
$w \in F^0$
with
$q(w) = z = s_E(e). $
As
$T$
is LR, there exists (a unique)
$f \in F^1$
with
$q(f) = e$
; that is,
$q$
is surjective. An analogous argument, using
$r$
-path lifting of
$p$
, shows that
$p$
is surjective.
We have shown that
$p$
and
$q$
are surjective and
$E$
is essential. Now, suppose
$v \in F^0$
and
$z = p(v)$
. Since
$E$
is essential, there exists
$e \in E^1$
such that
$r_E(e) = z = p(v)$
. By
$r$
-path lifting of
$p$
, there exists
$f \in F^1$
such that
$p(f) = e$
and
$r_F(f) = v$
. So,
$r_F^{-1}(v) \ne \emptyset $
for any
$v \in F^0$
and, hence,
$r_F$
is surjective. Similarly, suppose
$v \in F^0$
and
$z = q(v)$
. Since
$E$
is essential, there exists
$e \in E^1$
such that
$s_E(e) = z = q(v)$
. By
$s$
-path lifting of
$q$
, there exists
$f \in F^1$
such that
$q(f) = e$
and
$s_F(f) = v$
. So,
$s_F^{-1}(v) \ne \emptyset $
for any
$v \in F^0$
and, hence,
$s_F$
is surjective. Thus,
$F$
is also essential.
4.3 The shift space associated to a rank-two graph
Definitions 4.10. Let
$\Lambda $
be a
$2$
-graph. The one-sided infinite path space of
$\Lambda $
is
For each
$p \in \mathbb {N}^2$
, define
$\sigma ^p : \Lambda ^\infty \rightarrow \Lambda ^\infty $
by
$\sigma ^p (x) (m,n) = x(m+p,n+p)$
for
$x \in \Lambda ^\infty $
and
$(m,n) \in \Omega $
. Note that
$\sigma ^{p+q} = \sigma ^p \circ \sigma ^q$
.
For
$\lambda \in \Lambda $
, define
We endow
$\Lambda ^\infty $
with the topology generated by the basic open sets
$\{ Z(\lambda )\mid \lambda \in \Lambda \}$
. With this topology, the shift maps
$\sigma ^p$
are local homeomorphisms (see [Reference Kumjian and PaskKP00, Remark 2.5]).
Moreover, if we associate to
$\Lambda $
the textile system
$T_\Lambda $
as in (4-1) and construct the associated shift of finite type
$\mathsf {X}_{T_\Lambda }^+$
as in (3-1), we obtain the following result.
Lemma 4.11. Let
$\Lambda $
be a 2-graph with
$ \Lambda ^{\varepsilon _1} \sqcup \Lambda ^{\varepsilon _2}$
finite. Then, the shift space
$( \Lambda ^\infty , \sigma )$
is of finite-type. In particular, we can identify
$\Lambda ^\infty $
with
$\mathsf {X}^+_{T_\Lambda }$
.
Proof. Recall that
$\mathsf {X}_{T_\Lambda }^+$
consists of all tilings of
$\mathbb {R}^2_{\geq 0}$
, using squares from
$F_\Lambda ^1 = \Lambda ^{\varepsilon _1 + \varepsilon _2}$
, that satisfy Definition 3.7. In particular, thanks to the factorization property in
$\Lambda $
and the fact that adjacent squares in a tiling must coincide on their common edges, whenever
$m \leq n$
in
$\mathbb N^2$
and
$x \in \mathsf {X}_{T_\Lambda }^+$
, the squares
$\{ x_{m+v}\}_{0\leq v \leq n-m}$
identify a unique morphism in
$\Lambda $
of degree
$n-m$
. We denote this morphism by
$x(m,n)$
for coherence with the definition of
$\Lambda ^\infty .$
Indeed, we can define
$\psi : \mathsf {X}_{T_\Lambda }^+ \to \Lambda ^\infty $
by
$\psi (x)(m,n) = x(m,n)$
whenever
$m \leq n$
, and the inverse map
$\varphi : \Lambda ^\infty \to \mathsf {X}_{T_\Lambda }^+$
is given by (for
$n \in \mathbb N^2$
)
It is straightforward to check that
$\psi $
and
$\varphi $
are shift commuting and take cylinder sets to cylinder sets. That is,
$\psi $
is a conjugacy, giving the desired identification of
$\Lambda ^\infty $
with
$\mathsf {X}_{T_\Lambda }^+$
as dynamical systems.
Remarks 4.12 (Dynamics results concerning higher-rank graphs).
We briefly summarize here some results in the literature related to applications of higher-rank graphs to dynamics.
-
(i) In [Reference Skalski and ZachariasSZ08], the authors compute the topological entropy of the shift associated to a higher-rank graph. The factorization property ensures that it is zero. They also compute the entropy induced in the associated $C^*$
-algebra (which is also zero). -
(ii) In [Reference Kumjian and PaskKP03], the authors study the dynamical system arising from a higher-rank graph. They show that the action of shift map on the path space of a higher-rank graph is an expansive action. In addition, they show that if the higher-rank graph is primitive, then the shift map is topologically mixing. The existence of the Parry measure described in [Reference Kumjian and PaskKP03, Proposition 4.2] led to the work by others on the KMS states on higher-rank graphs (see [Reference Huef, Laca, Raeburn and SimsHLRS15, Reference Huef, Laca, Raeburn and SimsHLRS14, Reference Farsi, Gillaspy, Kang and PackerFGKP15] and [Reference Farsi, Gillaspy, Larsen and PackerFGLP21]).
-
(iii) In [Reference Pask, Raeburn and WeaverPRW09], the authors use a set of basic data coming from algebraic information (inspired by the work of Schmidt) to create a set of 2D tiles. Using a standard overlap condition from dynamics, they construct a $2$
-graph from such tiles and show that, in many situations, the associated
$C^*$
-algebras are purely infinite and simple. Moreover, their shift spaces tally with the one coming from the data applied to the Schmidt examples (see [Reference SchmidtSch98]). In [Reference Lewin and PaskLP10], the authors verify that the entropy of these shifts is zero using different techniques to [Reference Skalski and ZachariasSZ08]. -
(iv) In [Reference Carlsen and RoutCR21, Reference Brownlowe, Carlsen and WhittakerBCW17, Reference Carlsen, Eilers, Ortega and RestorffCEOR19], the authors consider orbit equivalence of rank-one and higher-rank dynamical systems.
5 Insplitting for textile systems and rank-2 graphs
In the literature, one can find both a definition for insplitting a textile system [Reference Johnson and MaddenJM99] and for insplitting a higher-rank graph [Reference Eckhardt, Fieldhouse, Gent, Gillaspy, Gonzales and PaskEFG+22]. Unfortunately, when the higher-rank graph is the 2-graph associated to an LR textile system as in Theorem 4.6, these definitions do not agree. We briefly review their definitions before moving on to the main results of this paper in Sections 6 and 7, which address the problem of reconciling the two notions of insplitting.
5.1 Insplitting of textile systems
Definition 5.1 [Reference Lind and MarcusLM95, Definition 2.4.7].
For a directed graph
$F$
, an insplitting partition of
$F$
is a partition of
$r^{-1}(v)\subseteq F^1$
, for each
$v \in F^0$
, into
$m(v)$
(nonempty) subsets
$\mathcal F_v^1, \ldots , \mathcal F_v^{m(v)}$
, for some
$m(v) \geq 1$
. The insplit graph
$F_I$
has
with
$s(f^j) = s(f)^j$
and
$r(f^j) = r(f)^k$
if
$f \in \mathcal F_{r(f)}^k$
.
In [Reference Johnson and MaddenJM99, Section 3], Johnson and Madden give the definition of insplitting for textile systems with
$E = B_X$
being a one-vertex graph. Here, we have adapted their definition to textile systems with no restrictions on
$E$
.
Definition 5.2 (Textile insplitting).
Let
$T = (p,q : F \to E )$
be a textile system. An insplit textile system is
$T_I = (p_I, q_I: F_I \to E_I)$
, where
$E_I = E$
and
$F_I$
is an insplit of
$F$
as in Definition 5.1.
We have
$p_I(f^j) = p(f), p_I(v^i) = p(v)$
, and
$q_I$
is similarly induced from
$q$
.
We sometimes call
$T_I$
a Johnson–Madden insplit of
$T$
to distinguish from the 2-graph insplits of Definition 5.8.
It is straightforward to check that
$p_I, q_I$
are graph homomorphisms, and if
$T$
is a textile system, then so is
$T_I$
.
Remarks 5.3 (The effect of insplitting on squares).
-
(i) Let $T=(p,q:F \to E)$
be a textile system. Suppose square
$T_f$
is below square
$T_g$
, that is,
$p(f)=q(g)$
. Then, this connection persists in any textile insplit
$T_I = ( p_I , q_I : F_I \to E )$
of
$T$
, that is,
$p_I ( f^j) = p(f) = q(g) = q_I (g^k)$
regardless of the values of
$j,k$
. -
(ii) Let $T=(p,q:F \to E)$
be a textile system. Consider the insplitting
$T_I$
given by a fixed partition
$\{ \mathcal {F}_v^i \mid 1 \leq i \leq m(v), v \in F^0 \}$
. Suppose
$f,g \in F^1$
are such that
$s(f)=r(g)$
, so squares
$T_f$
and
$T_g$
are adjacent, and suppose
$g \in \mathcal {F}_{r(g)}^\ell $
. In order that
$f^j , g^k \in F_I^1$
be consecutive edges in
$T_I$
, we must have $$ \begin{align*} s(f)^j= s( f^j )\stackrel{!}{=} r( g^k )= r(g)^\ell; \end{align*} $$that is, $j=\ell $
.
Example 5.4 (Textile system insplitting).
Given the directed graph
$F$
shown on the left in (5-2), let
$X=\{a,b\}$
and form the textile system
$T_{JM} = ( p,q: F \to B_X )$
with squares
Set
$m(u)=1,m(v)=2$
, and
$\mathcal {F}_v^1=\{ g \}$
,
$\mathcal {F}_v^2=\{f \}$
. Then,
$F_I$
is as shown here on the right.
Definition 5.2 tells us that the squares associated to
$T_{JM_I}$
are as follows:
Remark 5.5. Looking at the tiles in (5-1), one can see that the original textile system is LR. However, (5-3) shows that in
$T_{JM_I}$
, there are two different squares (those labeled
$g^1, g^2$
), which both have left edge
$v^1$
and top edge
$a$
. Hence,
$T_{JM_I}$
is not LR. Nonetheless, the shift spaces associated to
$T$
and
$T_{JM_I}$
are conjugate by Theorem 5.7.
The problem encountered in Example 5.4 occurs in general.
Theorem 5.6 (Insplitting always removes LR).
Let
$T = ( p,q : F \to E )$
be a textile system that is LR. Fix a partition
$\{ \mathcal {F}_v^i \mid 1 \le i \le m(v), v \in F^0 \}$
with
$m(v) \ge 2$
for some
$v$
. Then, the insplit textile system
$T_I$
is not LR.
Proof. Let
$v$
be a vertex with
$m(v) \ge 2$
. Let
$1 \le i < j \le m(v)$
and fix an edge
$f \in \mathcal {F}^k$
with source
$v$
. Consider the edges
$f^i \neq f^j$
in
$F_I$
. By definition,
$r_I ( f^i) = r_I ( f^j ) = r(f)^k$
. However,
$p_I( f^i )= p(f)= p_I( f^j )$
and
Hence,
$p_I$
is not left-resolving.
Recall from Definition 3.7 a textile system
$T = ( p,q : F \to E )$
gives rise to a textile tiling
$\textsf {X}_T^+$
. By Remark 3.8, we may consider this as a one-sided 2D shift of finite type with alphabet
$\mathcal {A}=F^1$
. The following result was stated in [Reference Johnson and MaddenJM99, Lemma 3.4] for textile systems with
$E = B_X$
; for completeness, we include a proof of the general result here.
Theorem 5.7 (Textile insplitting induces conjugacy).
Let
$T= (p,q : F \to E )$
be a textile system. Fix a partition
$\{ \mathcal {F}_v^i \mid 1 \le i \le m(v), v \in F^0 \}$
to form an insplitting
$T_I$
. Then,
$\textsf {X}_T^+$
is conjugate to
$\textsf {X}_{T_I}^+$
.
Proof. Define a
$0$
-block map
$\Phi : B_{0}( \textsf {X}_{T_{I}}^+ ) \to B_0 ( \textsf {X}_T^+ )$
by
$\Phi ( f^j)=f$
for
$f \in F^1$
,
$1 \le j \le m(s(f))$
. Let
$\phi : \textsf {X}_{T_I}^+ \to \textsf {X}_T^+$
be the induced map. To define the inverse map
${\psi : \textsf {X}_T^+ \to \textsf {X}_{T_I}^+}$
, we use an
$\varepsilon _1$
block map. That is, we define
$\Psi : B_{\varepsilon _1}(X_T^+) \to B_0(X_{T_I}^+)$
by
To see that
$\psi \circ \phi = \text {id}$
, let
$x \in \textsf {X}^+_{T_I}$
. Then, for each
$m \in \mathbb {N}^2, \ x_m = f^{j_m}$
for some
${f\in F^1, 1 \leq j_m \leq m(s(f))}$
. As
$s(f^{j_m}) = s(f)^{j_m}$
, we must have
$r(x_{m+\varepsilon _1}) = s(f)^{j_m},$
that is,
Consequently, for all
$m$
,
$ \Psi (\Phi (x_m) \Phi (x_{m +\varepsilon _1})) = \Phi (x_m)^{j_m} = x_m,$
that is,
$\psi \circ \phi = \text {id}.$
To see that
$\phi \circ \psi = \text {id},$
let
$y \in \textsf {X}_T^+.$
For each
$m \in \mathbb N^2,$
we have
$y_{m+\varepsilon _1} \in \mathcal F^{j_m}_{s(y_m)}$
for a unique
$j_m$
. We then have
and so
$(\phi (\psi (y)))_m = y_m;$
that is,
$\phi \circ \psi = \text {id}.$
As both
$\phi , \psi $
are block maps, Lemma 2.7 and Definition 2.8 imply that
$\phi , \psi $
are conjugacies, as claimed.
5.2 Insplitting for rank-two graphs
Following [Reference Eckhardt, Fieldhouse, Gent, Gillaspy, Gonzales and PaskEFG+22, Section 3], a
$2$
-graph insplitting is defined using the
$1$
-skeleton of
$\Lambda $
. In order that the result is a
$2$
-graph, the authors of [Reference Eckhardt, Fieldhouse, Gent, Gillaspy, Gonzales and PaskEFG+22] impose the pairing condition on the insplitting partition. For ease of notation, in [Reference Eckhardt, Fieldhouse, Gent, Gillaspy, Gonzales and PaskEFG+22], insplitting is defined at a single vertex
$v$
and the insplitting partition consists of exactly two nonempty sets. To capture the full generality of insplitting, we extend the definition from [Reference Eckhardt, Fieldhouse, Gent, Gillaspy, Gonzales and PaskEFG+22] to the following.
Definitions 5.8 (
$2$
-graph insplittings, pairing property).
Let
$\Lambda = G^*/\sim $
be a row-finite
$2$
-graph with 1-skeleton
$(G, d)$
.
An insplitting partition of
$G$
is a partition
$\mathcal G = \{ \mathcal G_v \mid v \in G^0 \}$
of
$G^1$
, where
${\mathcal G_v = \{ \mathcal G^1_v , \ldots , \mathcal G^{m(v)}_v\}}$
is a partition of
$v G^1$
into
$m(v)$
nonempty sets for each
$v \in G^0$
, which satisfies the pairing condition: whenever
$a, f \in v G^1$
for some
$v \in G^0$
and there exist edges
$g ,b\in G^1$
such that
$ ag \sim fb$
, then
$f \in \mathcal G^j_v$
if and only if
$a \in \mathcal G^j_v$
.
An insplitting partition of a 2-graph gives rise to a new 2-graph (the insplit 2-graph
$\Lambda _I$
of
$\Lambda $
), as follows. Given an insplitting partition
$\mathcal G$
of
$G$
, we define a
$2$
-colored graph
$G_I = ((G^0_I, G^1_I, r_I, s_I),d_I)$
, where
with
$d_I(f^i)= d(f)$
for all
$1 \le i \le m(v)$
.
The range and source maps in the directed graph
$G_I$
are defined as follows:
The equivalence relation in
$G_I$
is given by
$f^i g^k \sim _I a^j b^k$
if and only if
$g \in \mathcal G_{s(f)}^i$
,
$b \in \mathcal G_{s(a)}^j$
, and
$ab \sim fg \in \Lambda .$
(The fact that the two paths in a commuting square are required to have the same source forces the superscripts of g and b to coincide.)
Remark 5.9. Requiring that the partition
$\mathcal G$
satisfies the pairing condition is equivalent to specifying that the edges
$\lambda (0, \varepsilon _1)$
and
$\lambda (0, \varepsilon _2)$
lie in the same partition set
$\mathcal G^j_{r(\lambda )}$
for each
$\lambda \in \Lambda ^{\varepsilon _1 + \varepsilon _2}$
.
The fact that
$\Lambda _I := G_I^*/\sim _I$
is indeed a 2-graph is established in the following theorem.
Theorem 5.10 (
$2$
-graph insplitting produces
$2$
-graph).
Let
$\Lambda $
be an essential row-finite
$2$
-graph with associated
$2$
-colored graph
$(G,d)$
. Let
$\mathcal G$
be an insplitting partition of
$(G,d)$
. Then, the associated quotient
$G_I^*/\sim _I$
is a 2-graph
$\Lambda _I$
.
Proof. Recall from Remark 4.4 that we denote by
$\sim _I$
the equivalence relation satisfying Conditions (KG0) and (KG1) that is generated by the set of commuting squares specified in Definitions 5.8. To see that
$G_I^*/\sim _I$
is a 2-graph, it suffices to show that every 2-colored path
$f^i g^k$
in
$G_I$
is equivalent to a unique
$2$
-colored path
$a^j b^k$
. If
$f^i g^k \in G_I^2$
, then
$g \in \mathcal G_{s(f)}^i$
; in particular,
$s(f) = r(g)$
, so
$fg \in G^2$
. Since
$\Lambda $
is a 2-graph, there is a unique path
$ab \in G^2$
with
$d(a) = d(g), d(b) = d(f)$
, and
$ab \sim fg$
. In particular,
$s(b) = s(g) $
and
$r(b) = s(a)$
. There is a unique
$j$
such that
$b \in \mathcal G_{s(a)}^j$
; therefore,
$a^j b^k$
is the unique 2-colored path with
$d(a^j) = d(g^k)$
and
$a^j b^k \sim f^i g^k$
.
We prove in Appendix A that Definitions 5.8 yields the same
$C^*$
-algebraic properties as the original definition of 2-graph insplitting from [Reference Eckhardt, Fieldhouse, Gent, Gillaspy, Gonzales and PaskEFG+22].
Remark 5.11. Thanks to the definition of the equivalence relation
$\sim _I$
given at the end of Definitions 5.8, every commuting square in
$\Lambda _I$
arises from a (unique) commuting square in
$\Lambda $
. In fact, each commuting square
$\lambda = a b \sim f g$
in
$\Lambda $
yields
${m(s(g)) = m(s_\Lambda (\lambda ))}$
commuting-square ‘children’ in
$\Lambda _I$
.
Example 5.12 (Failing the pairing property).
Consider the following
$2$
-colored graph
$G$
with commuting squares as shown. Observe that each 2-colored path in
$G$
occurs in exactly one commuting square, that is,
$\sim $
satisfies Condition (KG2) and
$G^*/\sim $
is a 2-graph.
Since
$av\sim vc$
and
$du \sim vb$
, the pairing condition implies that any insplitting partition
$\mathcal G$
has only one set in the partition
$\mathcal G_s$
. Similarly, since
$ua \sim cu$
and
$ud \sim bv$
, there can be only one set in
$\mathcal G_t$
. That is, there is no nontrivial 2-graph insplitting of this 2-graph.
6 2-graph insplitting in terms of Johnson–Madden moves
If we wish to insplit an LR textile system
$T$
, we have two options. Of course, we can perform a textile-system insplitting, which (thanks to Theorem 5.6) does yield a non-LR textile system. However, we can alternatively form the associated 2-graph
$\Lambda = \Lambda _T$
as in Theorem 4.6; perform a 2-graph insplitting on
$\Lambda _T$
to yield a 2-graph
$\Lambda _I$
; and construct the associated textile system
$T_I$
as in (4-1). Theorem 4.6 tells us that
$T_I$
is again LR, that is, it cannot arise from
$T$
via a textile insplitting. Indeed, this discrepancy was our inspiration for the research contained here.
Despite this inconsistency, we show in Theorem 6.1 that we can reconstruct 2-graph insplitting from a sequence of conjugacy-preserving moves (textile-system insplitting and inversion) on the associated textile system.
Theorem 6.1. Let
$T=(p, q: F\to E)$
be an LR textile system and
$\Lambda _T$
be the associated
$2$
-graph. Let
$\mathcal G = \{ \mathcal G^i_z\mid 1\leq i \leq m(z) , z \in \Lambda ^0 \}$
be an insplitting partition of
$\Lambda _T$
, and let
$\Lambda _I$
and
$T_I= ( p_I, q_I : F_I \to E_I )$
be the resulting insplit 2-graph and associated textile system. Then, by performing a Johnson–Madden insplit on
$T$
; inverting this textile system; performing a second Johnson–Madden insplit; and inverting again, we construct a textile system
$T_D$
such that
$T_I$
and
$T_D$
give rise to identical tilings of
$\mathbb R_{\geq 0}^2$
. In particular,
$2$
-graph insplitting gives a conjugacy of one-sided dynamical systems.
Proof. We first perform four conjugacy-preserving textile-system moves on the textile system
$T$
and then compare the resulting shift space with that associated to the textile system
$T_I$
associated to the
$2$
-graph insplitting on
$\Lambda _T$
.
Step 1. Recall
$\Lambda _T^0 = E^0$
. By hypothesis, we can insplit the associated 2-graph
$\Lambda _T$
using the partition
$\{\mathcal {G}^i_z\mid 1\leq i\leq m(z) , \ z \in E^0 \}$
of
$ \Lambda _T^1= E^1 \sqcup F^0$
. We use this partition to create a partition of
$F^1 = \Lambda _T^{\varepsilon _1 + \varepsilon _2}$
as follows. Fix
$v\in F^0$
and write
$z = p(v)$
. For each
$1\leq i\leq m(z)$
, we define
The collection
$\{\mathcal {F}_v^i\mid v\in F^0, 1\leq i\leq m(p(v))\}$
gives a partition of
$F^1$
. Use this partition to perform an insplitting of
$T$
as in Definition 5.2, resulting in the textile system
$T_A:= (p_A , q_A : F_A \to E_A= E )$
, where
We have
$p_A ( \lambda ^i ) = p( \lambda )$
and
$q_A ( \lambda ^i ) = q ( \lambda )$
, with
$s_A ( \lambda ^i ) = s (\lambda )^i$
and
$r_A ( \lambda ^i )=r(\lambda )^k= v^k$
, where
$\lambda \in \mathcal {F}_{v}^k$
.
Step 2. Recall the definition of an inverted textile system from Definition 3.10. The inverted textile system
$T_B:= (p_B, q_B: F_B \to E_B )$
from
$T_A$
has
$F_B^0= E_A^1= E^1$
,
$E_B^0= E_A^0= E^0$
,
with
$r_{F_B}( {\lambda ^i} )= q_A({\lambda ^i} ) =q(\lambda )$
,
$s_{F_B}({\lambda ^i} )= p_A({\lambda ^i} )=p(\lambda )$
,
$s_{E_B}(u^i) = p_A(u^i)=p(u)$
, and
$r_{E_B}(u^i)= q_A(u^i)= q(u)$
. Similarly,
$p_B ({\lambda ^i} )= s_A({\lambda ^i} ) = s (\lambda )^i, \ q_B ({\lambda ^i} ) = r_A ( {\lambda ^i} ) = r (\lambda )^\ell $
if
$\lambda \in \mathcal {F}^\ell _{v}$
. We also have
$p_B(e) = s(e)$
and
$q_B(e) = r(e).$
Step 3. Recall the partition
$\{\mathcal {G}^i_z\mid z \in E^0 , \ 1\leq i\leq m(z)\}$
of
$\Lambda _T^{1} = E^1 \sqcup F^0$
from Step 1. In a similar way to Step 1, we use this partition to create a partition of
$F_B^1$
as follows. Fix
$z \in E^0$
and
$e \in F_B^0 = E^1 $
with
$p_B(e)= z$
and
$1\le j \le m(z)$
. Parallel to Step 1, we want to define
$\mathcal {H}^j_e = \{\lambda ^i \in F_B^1 \mid \ e = r_{F_B} ( \lambda ^i ), \ p_B ( \lambda ^i) \in \mathcal G^j_z\}$
; however,
$p_B(\lambda ^i) = s(\lambda )^i$
is not in
$\mathcal G^j_z \subseteq E^1 \sqcup F^0.$
Hence, we define
We observe that
$\mathcal H_e^j$
is always nonempty, since
$T$
is LR and
$\mathcal G$
is a 2-graph insplitting partition. In particular, in each
$\mathcal G^j_z$
, we have at least one
$w \in F^0$
and one
$\widetilde {e} \in E^1.$
Thus, given
$e \in E^1$
with
$p_B(e) = s(e) = z$
and
$1 \leq j\leq m(z),$
choose
$w \in \mathcal G^j_z \cap F^0.$
Then,
$q(w) = z = s(e)$
(see Definition 4.5). Consequently, the unique
$s$
-path lifting of
$q$
implies that there is a unique
$\lambda \in F^1$
with
$q(\lambda ) = e$
and
$s(\lambda ) = w,$
and we have
$\lambda ^i \in \mathcal H^j_e$
for all
$1\leq i\leq m(s(p(\lambda ))).$
For every
$\lambda \in F^1$
,
$s(\lambda ) $
lies in a unique
$\mathcal G^j_{s(q(\lambda ))}$
, so
$\{\mathcal {H}^j_e\mid e\in F_B^0, 1 \leq j \leq m(p_B(e))\}$
forms a partition of
$F_B^1$
. We thus perform an insplitting using this partition and get the textile system
$T_C:= (p_C, q_C : F_C\to E_C$
). Then,
and
$E_C= E_B$
given in Step 2. We have
$s_{C} (( \lambda ^i )^j ) = s_{F_{B}}(\lambda ^i)^j$
,
$r_C((\lambda ^i )^j )= r_{F_{B}}( \lambda ^i )^{\ell }$
, where
$\lambda ^i \in \mathcal {G}^\ell _{e}$
, and
$p_{C}((\lambda ^i )^j )= p_B( \lambda ^i)$
,
$q_C(( \lambda ^i )^j)= q_B(\lambda ^i) $
.
Step 4. Now, we invert the textile system
$T_C$
to get
$T_D:= (p_D, q_D: F_D \to E_D)$
, where
$F_D^0 = E_C^1 = E_B^1= F_A^0$
,
and
$E_D^1 = F_C^0$
. In the graph
$F_D$
, we have
$r_{F_D}((\lambda ^i )^j) = q_C((\lambda ^i)^j) = r(\lambda )^\ell $
if
$p(\lambda ) \in \mathcal {G}^\ell _{p(r(\lambda ))}$
,
$s_{F_D}(( \lambda ^i)^j) = p_C ((\lambda ^i)^j) = s(\lambda )^i$
; and on
$E_D$
, we have
$r_{E_D} (e^j) = q_C (e^j) = q_B(e) = r(e)$
,
$s_{E_D}(e^j) = p_C (e^j) = p_B(e) = s(e)$
. Finally, we have the graph homomorphisms
$p_D((\lambda ^i)^j )= s_C(( \lambda ^i)^j ) = (s_B(\lambda ^i))^j = p(\lambda )^j$
and
$q_D((\lambda ^i)^j)= r_C((\lambda ^i)^j) = (r_B(\lambda ^i))^k= q(\lambda )^k$
if
$s(\lambda ) \in \mathcal {G}^k_{s(q(\lambda ))}$
.
To compare the resulting textile system
$T_D$
with the 2-graph
$\Lambda _I$
, recall that in a textile system, the quadruple
$(p, q, r, s)$
is injective. Thus, we can visualize the textile system
$T_D$
by describing the boundary of each square
$(\lambda ^i)^j \in F_D^1$
, for each
$1 \leq i\ \mathrm{and}\ j \leq m(p(s(\lambda )))$
. Unpacking the previous definitions, we see that
Pictorially, we have the square
Now, consider the result of performing 2-graph insplitting on
$\Lambda _T$
with respect to the partition
$\{ \mathcal G^i_z\mid z \in \Lambda ^0 = E^0, 1\leq i \leq m(z)\}$
.
Given
$\lambda \in F^1 = \Lambda ^{\varepsilon _1 + \varepsilon _2}$
, write
$\lambda = q(\lambda ) s_F(\lambda ) \sim r_F(\lambda ) p(\lambda )$
with
$s_F(\lambda ) \in \mathcal G_{s(q(\lambda ))}^k$
and
$p(\lambda ) \in \mathcal G_{r(p(\lambda ))}^\ell $
. Recall from Remark 5.11 that every commuting square in
$\Lambda _I$
arises from a commuting square in
$\Lambda $
, and each
$\lambda \in \Lambda ^{\varepsilon _1 + \varepsilon _2}$
yields
$m(s(p(\lambda )))$
‘children’ in
$\Lambda _I^{\varepsilon _1 + \varepsilon _2}$
. Indeed, the children of the aforementioned
$\lambda $
are
That is, in
$\Lambda _I$
, we have the commuting squares
for
$1 \leq i \leq m(s(p(\lambda )))$
.
While all of the squares from the 2-graph insplitting appear in
$T_D$
constructed earlier, we also have many ‘extra’ squares in
$T_D$
, namely
$\{ (\lambda ^i)^{j}\mid i\not = j\}$
. However, we claim that none of these extra squares show up in any
$2\times 2$
block; in particular, they do not appear in the infinite path space
$X_D$
. To establish the claim, suppose that we have a
$2\times 2$
block in
$\mathsf {X}_D^+$
of the form
where
$i \neq j$
. Then, the superscript on the edge
$q(\eta )= p(\lambda )$
in the diagram above must be
$j$
, that is,
$\eta ^a \in \mathcal H^j_{q(\eta )}$
, and so
$s(\eta )\in \mathcal G_z^j$
for some
$z\in E^0$
. Similarly, the superscript on the edge
$r(\mu )= s(\lambda )$
in the diagram above must be
$i$
, that is,
$\mu \in \mathcal F_{r(\mu )}^i$
, or in other words,
$p(\mu ) \in \mathcal G_{z'}^i$
for some
$z'\in E^0$
. Since the sets
$\mathcal G_z^i$
satisfy the pairing condition, if
$i \neq j$
, there is no square in
$\Lambda ^{\varepsilon _1 + \varepsilon _2} = F^1$
whose left edge lies in
$\mathcal G^j$
and whose bottom edge lies in
$\mathcal G^i$
. Consequently, there is no edge in
$F_D^1$
that fills in the unlabeled top-right square of this
$2\times 2$
block. In other words, no tiling in
$\mathsf {X}_D^+$
contains any of the squares
$\lambda ^{i^j}$
with
$i\neq j$
; so the map
$\lambda ^{i^i} \mapsto \lambda ^i$
gives the desired relabeling that identifies
$\mathsf {X}_D^+$
with
$\mathsf {X}_I^+$
.
We can also view this relabeling in the setting of textile systems, as follows. Define
$F_D' = (F_D^0, \{ \lambda ^{i^i}\mid \lambda \in F^1, 1 \leq i \leq m(p(s(\lambda )))\}, r_D, s_D).$
Then, the restrictions of
$p_D, q_D$
to
$F_D'$
are still graph homomorphisms, so
$T':= (p_D, q_D : F^{\prime }_D \to E_D)$
is a textile system. Moreover, as we observed above,
$\mathsf {X}_D^+ = \mathsf {X}_{T'}^+ = \mathsf { X}_{T_I}^+$
.
Remark. Theorem 6.1 can be summarized as the following commuting diagram. Note that we do not claim that the textile systems
$T_D$
and
$T_I$
are identical, but the corresponding infinite path spaces are, in the sense that the diagram commutes at the level of the infinite path spaces.
Example 6.2. Consider the LR textile system
$T= (p, q: F \to E)$
, where the graphs
$F, E$
are depicted as follows.
The graph homomorphisms
$p$
and
$q$
are given by
$p(\lambda _1)= p(\lambda _2)= e_1$
,
$p(\lambda _3)= e_2$
,
$p(\lambda _4)= e_3$
,
$p(f_1)= p(f_2)= v$
, and
$p(f_3)= w$
;
$q(\lambda _1)= e_1$
,
$q(\lambda _2)= e_2$
,
$q(\lambda _3)= q(\lambda _4)= e_3$
, and
$q(f_1)= v$
,
$q(f_2)= q(f_3)= w$
. The associated 2-graph
$\Lambda = \Lambda _T$
was described in [Reference Eckhardt, Fieldhouse, Gent, Gillaspy, Gonzales and PaskEFG+22, Example 6.1] and has the following 1-skeleton:
Observe that if we define
then the partitions
$\{\mathcal {G}_v^1 \}, \{\mathcal {G}_w^1, \mathcal {G}_w^2\}$
of
$v\Lambda ^1, w\Lambda ^1$
satisfy the pairing condition. Insplitting
$\Lambda _T$
using this partition yields the 2-graph
$\Lambda _I$
with 1-skeleton
with the commuting squares
$\lambda _j^i$
for
$i\leq 2$
and
$j= 1, 2, 3, 4$
:
We may deduce the formulas for
$p_I,q_I$
from the above squares. The graphs
$F_{\Lambda _I}$
and
$E_{\Lambda _I}$
are as follows:
As indicated in the proof of Theorem 6.1, we can also iteratively perform the textile-system moves of insplitting (Definition 5.2) and inversion (Definition 3.10) on
$T$
. The resulting textile system (which is non-LR) gives rise to the identical infinite path space as the infinite path space from the system
$T_{\Lambda _I}$
associated to
$\Lambda _I$
.
Step 1. Johnson–Madden insplitting at all
$f_i$
,
$i= 1, 2, 3$
, on T using the partition
$\mathcal {F}_{f_i}^l$
,
$l= 1, 2$
. For
$f_i\in F^0$
,
$i= 1, 2, 3$
, the partitions
$\mathcal F^l_{f_i}$
of (6-1) become
The resulting textile system
$T_A= (p_A, q_A : F_A\to E_A)$
has
$E_A = E$
and
$F_A$
the directed-graph insplit of
$F$
using the partition (6-5); the graph homomorphisms
$p_A, q_A$
are inherited from
$p, q$
, respectively.
The graph
$F_A$
is
and the commuting squares are
Step 2. Invert
$T_A$
to get
$T_B= (p_B, q_B: F_B \to E_B)$
, where
$F_B^0= E_A^1= E^1$
,
$F_B^1= F_A^1= \{\lambda _1^1, \lambda _2^1, \lambda _3^1, \lambda _4^i\mid i= 1, 2\}$
,
$E_B^0= E_A^0= E^0$
, and
$E_B^1= F_A^0= \{f_1^1, f_2^1, f_3^i\mid i= 1, 2\}$
. The corresponding graphs are
The commuting squares are the transposes of the commuting squares from Step 1.
Step 3. Johnson–Madden insplit
$T_B$
at all
$e_i$
,
$i= 1, 2, 3$
, using the partition
$\mathcal {H}^\ell _{e_i}= \{ \lambda ^\ell \mid \lambda \in F^1 , \ e_i = q( \lambda ) , \ s ( \lambda ) \in \mathcal G_{s(e_i)}^\ell \} $
for
$\ell \leq 2$
. This definition yields
The textile system
$T_C= (p_C, q_C : F_C \to E_C)$
consists of the graph
$F_C$
with
and the graph
$E_C= E_B$
. The graph
$F_C$
is as follows:
The commuting squares are
Step 4. Invert
$T_C$
from Step 3 to get the final textile system
$T_D:= (p_D, q_D: F_D \to E_D)$
, where
$E_D^0= E_C^0= \{v, w\}$
, and
$E_D^1= F_C^0= \{e_1^i, e_2^i, e_3^i\mid i \leq 2\}$
. That is,
and
$E_D \cong E_C = E_B$
by relabeling each edge
$f_i^j \in E_B^1$
as
$e_i^j$
. We obtain the commuting squares for
$T_D$
by flipping the 7 commuting squares obtained in Step 3 along the diagonal.
To recover the infinite path space associated to
$T_{\Lambda _I}$
from the above four Johnson–Madden moves, we delete the edges
$\lambda _{k}^{j^i} \in F_D$
where
$i\neq j$
. Note that this procedure converts
$F_D$
into
$F_{\Lambda _I}$
.
As further evidence that the conjugacy
$\mathsf {X}_{T_I}^+ \cong \mathsf {X_D}^+$
guaranteed by Theorem 6.1 does not arise from an isomorphism of textile systems, however, note that we do not have
$E_D \cong E_{\Lambda _I}$
, as
$E_{\Lambda _I}$
has three vertices and
$E_D$
has two.
7 When textile-system insplitting yields a 2-graph insplitting
In the previous section, we showed in Theorem 6.1 how to reconstruct a 2-graph insplitting via four Johnson–Madden moves. In this section, we identify several other perspectives on the relationship between 2-graph insplitting and Johnson–Madden insplitting. We first show (Theorem 7.1) that, alternatively, the 2-graph insplitting may be reconstructed via a single Johnson–Madden insplit, together with an insplitting of the base graph
$E$
of the textile system. Then, Theorem 7.3 shows that certain Johnson–Madden insplits yield 2-graph insplits, and Theorem 7.12 establishes that certain insplits of the base graph
$E$
of a textile system yield both Johnson–Madden insplits and 2-graph insplits. Finally, Section 7.4 reveals the compatibility of these perspectives.
These three theorems can be summarized by the following diagrams. In all of the diagrams,
$T = (F, E, p, q)$
denotes an LR textile system and
$\widetilde T = (\widetilde F, \widetilde{E}, \widetilde p, \widetilde q)$
is the textile system resulting from a textile-system insplit of
$T$
, while
$\widetilde E$
is a directed-graph insplit of
$E$
. Similarly,
$\Lambda $
denotes the 2-graph associated to
$T$
and
$\Lambda _I$
denotes the 2-graph resulting from a 2-graph insplit of
$\Lambda $
. The solid arrows indicate relationships that are assumed (or known in general) between the various objects, and the dotted arrows indicate relationships that are constructed in the theorem.
The results in Section 7.4 confirm that the diagrams above can be superimposed on each other; or, equivalently, that any one of the horizontal arrows in the diagram determines the others. For example, the dotted arrow
$F \to \widetilde F$
in the first diagram (which is constructed in Theorem 7.1) satisfies the hypothesis in Theorem 7.3.
7.1 Textile insplitting from 2-graph insplitting
While Theorem 6.1 shows that a 2-graph insplit can be recovered by four Johnson–Madden moves, we establish in the following theorem that we could alternatively recover the 2-graph insplitting by combining a single Johnson–Madden insplitting with an insplit of the base graph
$E$
of the textile system.
Theorem 7.1. Let
$T = (p,q:F\to E)$
be an LR textile system. Let
$\Lambda $
denote the 2-graph associated to
$T$
and assume that
$\Lambda $
can be (2-graph) insplit using the partition
$\{ \mathcal {G}^j_z \mid z \in E^0, 1\leq j \leq m(z)\}$
of
$\Lambda ^1$
. For each
$v \in F^0$
, define
and for
$ z \in E^0$
, define
Denote by
$\widetilde {F}$
the directed graph resulting from directed-graph insplitting on
$F$
at every vertex
$v \in F^0$
, using the partition
$\{ \mathcal F_v^i\mid 1 \leq i \leq m(p(v)), v \in F^0\}.$
Let
$\widetilde {E}$
denote the directed graph resulting from directed-graph insplitting on
$E$
at each vertex
$z \in E^0$
, using the partition
$\{ \mathcal E^i_z\mid 1 \leq i \leq m(z), z \in E^0\}.$
Define maps
$\widetilde {p}, \widetilde {q}: \widetilde {F} \to \widetilde {E}$
by
where
$\lambda \in F^1$
,
$v \in F^0$
. Then, we have the following.
-
(1) The maps $\widetilde {p}, \widetilde {q}: \widetilde {F} \to \widetilde {E}$
are graph homomorphisms. -
(2) $\widetilde {T} := (\widetilde {p}, \widetilde {q} : \widetilde {F} \to \widetilde {E} )$
is an LR textile system. -
(3) The 2-graph associated to $\widetilde {T}$
(denoted by
$\widetilde {\Lambda }$
) is identical to the 2-graph
${\Lambda }_I$
resulting from insplitting
$\Lambda $
using the partition
$\{ \mathcal G^j_z\mid z \in E^0, 1 \leq j \leq m(z)\}$
.
Proof. To see statement (1), we will check that
$r(\widetilde {p}(\lambda ^i)) = \widetilde {p}(r(\lambda ^i))$
and that
$r (\widetilde {q}(\lambda ^i)) = \widetilde {q}(r(\lambda ^i))$
for all
$\lambda ^i \in \widetilde {F}^1$
. (The checks that
$s\circ \widetilde {p} = \widetilde {p} \circ s$
and
$s \circ \widetilde {q} = \widetilde {q} \circ s$
are straightforward computations that we leave to the reader.) For the first computation, if
$\lambda \in \mathcal F^j_{r(\lambda )}$
, then
$r(\lambda ^i) = r(\lambda )^j$
and
$p(\lambda ) \in \mathcal G^j_{p(r(\lambda ))}$
. If
$r(p(\lambda )) (=p(r(\lambda ))) = z \in E^0$
, then when we insplit
$E$
, we have
$r(p(\lambda )^k) = z^j$
for all
$k$
. Therefore,
as desired.
To see that
$r(\widetilde {q}(\lambda ^i)) = \widetilde {q}(r(\lambda ^i))$
, observe that since
$\mathcal G$
is a 2-graph insplitting partition, we have
$r(\lambda )$
and
$q(\lambda )$
in the same partition set
$\mathcal G^n_{r(q(\lambda ))}$
. Therefore, regardless of which
$m$
satisfies
$s(\lambda ) \in \mathcal G^m_{q(s(\lambda ))}$
,
and regardless of which
$\ell $
satisfies
$\lambda \in \mathcal F^\ell _{r(\lambda )}$
, we have
since
$q$
is a graph homomorphism. We conclude that
$\widetilde {p}, \widetilde {q}$
are graph homomorphisms, as claimed.
For statement (2), to see that
$\widetilde {T}$
is an LR textile system, we first observe that the function
$\widetilde {F}^1 \ni \lambda ^i \mapsto (r(\lambda ^i), \widetilde {p}(\lambda ^i), s(\lambda ^i), \widetilde {q}(\lambda ^i)) \in \widetilde {F}^0 \times \widetilde {E}^1 \times \widetilde {F}^0 \times \widetilde {E}^1$
is injective, because if
$r(\lambda ^i) = r(\mu ^j)$
, then
$r(\lambda ) = r(\mu ) =: v$
and
$\lambda , \mu \in \mathcal F^n_v$
lie in the same partition set. Furthermore, if
$\widetilde {p}(\lambda ^i) = \widetilde {p}(\mu ^j)$
, then
$p(\lambda )^i = p(\mu )^j$
, so we must have
$p(\mu ) = p(\lambda )$
and
$i=j$
. Since
$T$
is LR,
$r(\lambda ) = r(\mu )$
, and
$p(\lambda ) = p(\mu )$
, we conclude that
$\lambda = \mu $
. As
$i=j$
, we also have
$\lambda ^i = \mu ^j$
, so the function is injective as claimed and
$\widetilde {T}$
is a textile system.
To see that
$\widetilde {T}$
is LR, observe that if
$\widetilde {p}(v^i)= r(e^j)\in \widetilde {E}^0$
, then the definitions of the insplitting
$\widetilde {E} $
and the homomorphism
$\widetilde {p}$
imply that we have
$p(v)^i= r(e)^n$
, where
${e \in \mathcal E^n_{r(e)}}$
. In particular, we have
$p(v) = r(e)$
and
$i=n$
. Since the textile system
$T$
is LR, there exists a unique
$\lambda \in F^1$
with
$p(\lambda )= e$
and
$r(\lambda )= v$
. As
$p(\lambda ) = e \in \mathcal G^i_{p(v)}$
, we have
$\lambda \in \mathcal F^i_v$
and so
$r(\lambda ^j) = v^i$
. The uniqueness of
$\lambda $
with
$p(\lambda ) = e$
and
$r(\lambda ) = v$
implies that
$\lambda ^j \in \widetilde {F}^1$
is the unique edge with
$\widetilde {p}(\lambda ^j) = e^j$
and
$r(\lambda ^j) = v^i$
. That is,
$\widetilde {p}$
is left resolving. To check that
$\widetilde {q}$
is right resolving, suppose that
$\widetilde {q}(v^i)= s(e^m)\in \widetilde {E}^0$
for some
$e^m\in \widetilde {E}^1$
. As
$s(e^m) = s(e)^m$
in
$\widetilde {E}^0$
, the definition of
$\widetilde {q}$
implies that
$s(e) = q(v)$
and, moreover, that
$v \in \mathcal G^m_{q(v)}$
. Since
$T$
is LR, that is,
$q$
is right resolving, there exists a unique
$\lambda \in F^1$
such that
$q(\lambda )= e$
and
$s(\lambda )= v$
. For any
$1 \leq j \leq m(p(r(\lambda )))$
, we then have
$\widetilde {q}(\lambda ^j)= q(\lambda )^m= e^m$
and
$s(\lambda ^j)= s(\lambda )^j= v^j$
. That is, there is a unique edge
$f$
in
$\widetilde {F}^1$
with source
$v^i$
and
$q(f) =e^m$
, namely,
$f = \lambda ^i$
. We conclude that
$\widetilde {q}$
is right resolving and so
$\widetilde {T}$
is LR.
For statement (3), we claim that the 2-graph
$\Lambda _I$
resulting from 2-graph insplitting
$\Lambda $
according to the partition
$\{ \mathcal G^i_z\mid z \in E^0, 1\leq i \leq m(z)\}$
is identical to the graph
$\widetilde {\Lambda }$
associated to the textile system
$\widetilde {T}$
. To see this, we first recall that since each set
$\mathcal G^j_z$
contains at least one edge of each color, each set
$\mathcal E^i_z$
is nonempty. Thus,
Similarly,
$ \Lambda _I^0 = \{ z^i\mid z \in E^0, 1 \leq i \leq m(z)\},$
so
$\Lambda _I^0 = \widetilde {\Lambda }^0.$
Next, note that
$\widetilde {\Lambda }^{\varepsilon _1} = \widetilde {E}^1 = \{ e^j\mid e \in E^1, 1 \leq j \leq m(s(e))\},$
while
As
$s_\Lambda (v) = p(v)$
, we also have
$\Lambda _I^{\varepsilon _2} = \{ v^i\mid v \in F^0, 1 \leq i \leq m(s_\Lambda (v))\} $
, while
Moreover, the range and source maps are the same in
$\Lambda _I$
as in
$\widetilde {\Lambda }$
. To see this, we first observe that for each
$e^i\in \widetilde {E}^1 = \widetilde {\Lambda }^{\varepsilon _1} = \Lambda _I^{\varepsilon _1},$
we have
$r_{\widetilde {\Lambda }}(e^i)= r_{\widetilde {E}}(e)= r(e)^j$
if
$e \in \mathcal E^j_{r(e)} \subseteq \mathcal G^j_{r(e)}$
, while
$r_{\Lambda _I}(e^i) = r(e)^j$
. Similarly,
$s_{\widetilde {\Lambda }}(e^i)= s_{\widetilde {E}}(e^i)= s_E(e)^i$
and
$s_{\Lambda _I}(e^i) = s_\Lambda (e)^i = s_E(e)^i$
. For
$v^i\in \widetilde {F}^0$
, we also have
$s_{\widetilde {\Lambda }}(v^i)= \widetilde {p}(v^i)= p(v)^i$
, whereas
$ s_{\Lambda _I}(v^i) = s_\Lambda (v)^i = p(v)^i$
. Finally, if
$v \in \mathcal G^m_{q(v)}$
,
$r_{\widetilde {\Lambda }}(v^i)= \widetilde {q}(v^i)= q(v)^m$
and
$ r_{\Lambda _I}(v^i) = r_\Lambda (v)^m = q(v)^m$
since
$v\in \mathcal {G}^m_{q(v)}$
.
It remains to check that the factorization rules are the same in
$\Lambda _I$
as in
$\widetilde {\Lambda }$
. To that end, suppose that
$w^k f^\ell $
is a composable 2-color path in
$\Lambda _I$
with
$w^k \in \Lambda ^{\varepsilon _2}_I, f^\ell \in \Lambda ^{\varepsilon _1}_I$
. As
$\Lambda _I$
is a 2-graph by Theorem 5.10, there is a unique composable path
$e^i v^j$
with
$e^i v^j \sim _{\Lambda _I} w^k f^\ell .$
Since
$s_{\Lambda _I}(w^k) = s_\Lambda (w)^k = p(w)^k$
, we must have
$f \in \mathcal {G}^k_{r(f) = p(w)}$
. Since
$r(f) = p(w)$
and
$p$
is left resolving, there is a unique
$\lambda \in F^1$
with
$p(\lambda ) = f, r(\lambda ) = w$
. As shown, if
$e^i v^j \sim _{\Lambda _I} w^k f^\ell $
, then, in particular,
$ev \sim _\Lambda w f$
. That is, the unique
$\lambda \in F^1$
that satisfies
$p(\lambda ) = f, r(\lambda ) =w$
also has
$q(\lambda ) = e, s(\lambda ) = v$
(and vice versa). Moreover, as
$s_{\Lambda _I}(v^j) = p(v)^j$
and
$s_{\Lambda _I}(f^\ell ) = s(f)^\ell $
, we must have
$j = \ell .$
Now, consider
$\sim _{\widetilde {\Lambda }}$
. We show that we also have
$w^k f^\ell \sim _{\widetilde {\Lambda }} e^i v^\ell $
. Since the source and range maps are the same in
$\widetilde {\Lambda }$
and in
$\Lambda _I$
, we know that
$w^k f^\ell $
is composable in
$\widetilde {\Lambda }$
, that is,
$r_{\widetilde {\Lambda }} (f^\ell )= s_{\widetilde {\Lambda }}(w^k) = \widetilde {p}(w^k)$
. In other words,
$f \in \mathcal G^k_{p(w)}$
. As we established when we checked (as part of statement (2)) that
$\widetilde p$
was left resolving, there is a unique
$\lambda ^\ell \in \widetilde {F}^1$
with
$\widetilde {p}(\lambda ^\ell ) = f^\ell = p(\lambda )^\ell $
and
$r(\lambda ^\ell )= w^k.$
In particular,
$\lambda \in F^1$
satisfies
$p(\lambda ) = f$
and
$r(\lambda ) = w$
, so
$q(\lambda ) = e$
and
$s(\lambda ) = v$
for the
$e, v$
discussed in the previous paragraph. That is, we have
$v \in \mathcal {G}^i_{s(e)}.$
Thus,
$w^k f^\ell \sim _{\widetilde {\Lambda }} e^i v^\ell $
, since
$v \in \mathcal G^i_{s(e)}$
implies that
$ r_{\widetilde {\Lambda }}(v^\ell ) = \widetilde {q}(v^\ell ) = q(v)^i = s(e)^i = s(e^i).$
As both
$\widetilde {\Lambda }$
and
$\Lambda _I$
are 2-graphs (thanks to Theorems 4.6 and 5.10, respectively), every blue-red path is equivalent to a unique red-blue path. The fact that any 2-color path
$w^k f^\ell $
is equivalent to the same path
$e^i v^\ell $
under both equivalence relations implies that
$\widetilde {\Lambda } = \Lambda _I$
, as claimed.
Example 7.2. Consider again the 2-graph
$\Lambda $
associated to the textile system from Example 6.2:
The commuting squares are
As we observed in (6-3) of Example 6.2, if we define
then the partition
$ \mathcal G = \{\mathcal {G}_v^1, \mathcal {G}_w^1, \mathcal {G}_w^2\}$
of
$\Lambda ^{\varepsilon _1} \sqcup \Lambda ^{\varepsilon _2}$
satisfies the pairing condition. Applying the construction of Theorem 7.1 to this partition, we see that
Let
$\widetilde {F}$
be the directed graph insplitting of
$F$
at every vertex
$f_j\in F^0$
using the partition
$\{\mathcal {F}^i_{f_j}\}_{i\leq 2}$
, and
$\widetilde {E}$
be the directed graph insplitting of
$E$
at every vertex
$z\in E^0$
using the partition
$\{\mathcal {E}_z^i\}_{i\leq 2}$
. Since the partition
$\{\mathcal {F}^i_{f_j}\}_{i\leq 2}$
is precisely that of (6-5), we have
$\widetilde {F} = F_A$
. Similarly,
$\widetilde {E} = E_{\Lambda _I}.$
The definition of
$\widetilde {p}, \widetilde {q}: \widetilde {F} \to \widetilde {E}$
from Theorem 7.1 tells us that the commuting squares of
$\widetilde {T}$
are
From the given commuting squares, we see that the textile system
$\widetilde {T}$
is LR. Moreover, the associated 2-graph
$\Lambda _{\widetilde {T}}$
is precisely the 2-graph
$\Lambda _I$
(described in (6-4) from Example 6.2) arising from insplitting
$\Lambda $
using the partition
$\mathcal G.$
7.2 Which textile insplits yield 2-graph insplits?
The following theorem explains how to identify when a Johnson–Madden insplitting of an LR textile system could alternatively be obtained from a 2-graph insplitting. In other words, while we know from Theorem 5.6 that Johnson–Madden insplitting never yields an LR textile system, Theorem 7.3 identifies which Johnson–Madden insplittings of an LR textile system can be combined with a directed-graph insplitting of the base graph
$E$
to yield an LR textile system. Thus, the insplittings described in the following theorem take an LR textile system to an LR textile system.
Theorem 7.3. Let
$T = (p,q: F \to E)$
be an LR textile system in which
$p$
is surjective and
$F$
is source-free, and let
$\Lambda = \Lambda _T$
be the corresponding 2-graph. Suppose for each
$v \in F^0$
we have a partition
$\mathcal F_v = \{ \mathcal {F}_v^1, \ldots , \mathcal {F}_v^{m(v)}\}$
of
$vF^1$
and let
$\mathcal {F} = \{\mathcal {F}_v\}_{v\in F^0}$
. Suppose that
$\mathcal {F}$
satisfies:
-
(1) if $v, w \in F^0$
and
$p (\mathcal F^i_v) \cap p(\mathcal F^j_w) \not = \emptyset $
, then
$p(\mathcal F^i_v) = p(\mathcal F^j_w);$
and -
(2) for all $v \in F^0$
, there exists
$w \in F^0$
and
$1 \leq j \leq m(w)$
such that
$q(v F^1) \subseteq p(\mathcal F^j_w)$
.
Let
$T_{JM}= (p_{JM}, q_{JM}:F_{JM} \to E_{JM})$
be the result of performing a Johnson–Madden insplitting with respect to the partition
$\mathcal F$
. Then, there is a directed graph insplit
$\widetilde E$
of
$E_{JM}$
, together with graph homomorphisms
$\widetilde {p}, \widetilde {q}: F_{JM} \to \widetilde E$
, so that the textile system
$\widetilde T := (\widetilde {p}, \widetilde {q}: F_{JM}\to \widetilde E)$
is LR.
Moreover,
$\widetilde T$
can be identified as the textile system
$T_{\Lambda _I}$
of a 2-graph
$\Lambda _I$
that arises from 2-graph insplitting on
$\Lambda $
using a partition
$\mathcal G$
(described in Proposition 7.9) of
$\Lambda ^{1}$
that arises from
$\mathcal F.$
Theorem 7.3 is proved in a series of propositions.
Proposition 7.4. Let
$T=(p,q:F\to E)$
be an LR textile system and, for each
$v \in F^0$
,
$\mathcal {F}_v = \{ \mathcal {F}_v^i \mid i=1, \ldots , m(v)\}$
be a partition of
$vF^1$
. Then:
-
(1) if $v \in F^0$
and
$p(\mathcal {F}_v^i) \cap p(\mathcal {F}_v^j) \neq \emptyset $
, then
$i=j$
; and -
(2) if $v, w \in F^0$
and
$p(v) = p(w)$
, then for every
$i \in \{1, \ldots , m(v)\}$
, there exists
$j \in \{1, \ldots , m(w)\}$
such that
$p(\mathcal {F}_v^i) \cap p(\mathcal {F}_w^j) \neq \emptyset $
.
Proof. (1): Let
$e \in p(\mathcal {F}_v^i) \cap p(\mathcal {F}_v^j)$
. Then, there are
$\lambda \in \mathcal {F}_v^i$
and
$\mu \in \mathcal {F}_v^j$
such that
$p(\lambda ) = p(\mu ) = e$
. Because
$p$
is a graph homomorphism,
$p(r(\lambda )) = r(p(\lambda )) = r(e) = p(r(\mu ))$
, so by unique
$r$
-path lifting of
$p$
(since
$T$
is LR),
$\lambda = \mu $
. Thus,
$\mathcal {F}_v^i \cap \mathcal {F}_v^j \not = \emptyset $
. Since
$\{\mathcal {F}_v^k \mid k=1, \ldots , m(v)\}$
is a partition of
$vF^1$
, this implies
$i=j$
.
(2): Let
$e \in p(\mathcal {F}_v^i)$
. Then,
$r(e) = p(v) = p(w)$
, so by unique
$r$
-path lifting for
$p$
, there exists a unique
$\eta \in F^1$
such that
$r(\eta ) = w$
and
$p(\eta ) = e$
. We have
$\eta \in \mathcal {F}_w^j$
for some
${j \in \{1, \ldots , m(w)\}}$
. Hence,
$e \in p(\mathcal {F}_v^i) \cap p(\mathcal {F}_w^j)$
.
Proposition 7.5. Let
$T=(p,q:F\to E)$
be an LR textile system with a partition
$\mathcal {F}$
of
$F^1$
satisfying Condition (1) of Theorem 7.3. If
$v, w \in F^0$
with
$p(v) = p(w)$
, then
$m(v) = m(w)$
.
Proof. By Proposition 7.4(2) and Condition (1) of Theorem 7.3, if
$p(v) = p(w)$
, then for each
$i \in \{1, \ldots , m(v)\}$
, there exists
$j_i \in \{1, \ldots , m(w)\}$
such that
$p(\mathcal {F}_v^i) = p(\mathcal {F}_w^{j_i})$
. If
$i, k \in \{1, \ldots , m(v)\}$
and
$J:= j_i = j_k$
, then
$p(\mathcal {F}_v^i) = p(\mathcal {F}_w^J) = p(\mathcal {F}_v^k)$
, so
$i = k$
; hence,
${i \mapsto j_i}$
is injective. The map is surjective because for each
$k \in \{1, \ldots , m(w)\}$
, there is
$i_k \in \{1, \ldots , m(v)\}$
such that
$p(\mathcal {F}_w^k) = p(\mathcal {F}_v^{i_k})$
(again, by combining Proposition 7.4(2) and Condition (1) of Theorem 7.3) so
$k = j_{i_k}$
. Hence,
$m(v) = m(w)$
.
Remark 7.6. We can thus assume without loss of generality that for each
$z \in E^0$
, for all
$v, w \in p{}^{-1}(z)$
and
$i=1,\ldots ,m(v)$
,
$p(\mathcal {F}_v^i) = p(\mathcal {F}_w^i)$
, and we are justified in setting
$m(z) = m(v)$
.
Proposition 7.7. Under the assumptions of Theorem 7.3, the collection
is a partition of
$zE^1$
for each
$z \in E^0$
. If
$\widetilde E$
is the insplit of
$E$
with respect to this partition, then the maps
$\widetilde {p}, \widetilde {q} : F_{JM} \to \widetilde E$
defined, for
$v \in F^0$
,
$\lambda \in F^1$
, by
and, if
$s_F(\lambda ) = v$
and
$q(vF^1) \subseteq p(\mathcal {F}_w^j),$
are graph homomorphisms and so
$\widetilde T := (\widetilde {p}, \widetilde {q}: F_{JM} \to \widetilde E)$
is a textile system.
Proof. To ease the burden of notation in this proof, we abuse notation and use the same symbols
$r,s$
for the range and source maps in every graph that appears in the proof. We trust that the context suffices to indicate the domain and range of each occurrence of
$r, s$
.
Since
$p$
is surjective, every
$e \in zE^1$
is in some
$\mathcal E_z^{i} = p(\mathcal {F}_v^i)$
. Moreover, Remark 7.6 and Condition (1) of Theorem 7.3 indicate that
$\mathcal E^i_z \cap \mathcal E^j_z= \emptyset $
if
$i\not = j$
, so
$\mathcal E_z$
is indeed a partition of
$z E^1.$
To see that
$\widetilde {p}$
is well defined, suppose that
$e = p(\lambda ) = p(\mu )$
. Then,
so Remark 7.6 implies that insplitting
$E$
and
$F$
creates the same number of ‘copies’ of
$e, \lambda ,$
and
$\mu $
. It follows that
$\widetilde {p}$
is well defined as claimed. To see the same for
$\widetilde {q}$
, we suppose
$q(vF^1) \subseteq p(\mathcal {F}_w^j) \cap p(\mathcal {F}_u^k)$
. Since
$F$
is assumed to be source-free,
$v F^1 \not = \emptyset $
and, consequently,
$q(vF^1) \neq \emptyset $
. We then have
$p(\mathcal {F}_w^j) \cap p(\mathcal {F}_u^k) \neq \emptyset $
and so Condition (1) of Theorem 7.3 yields
$p(\mathcal {F}_w^j) = p(\mathcal {F}_u^k)$
. Remark 7.6 implies
$j = k$
and, hence,
$\widetilde {q}$
is well defined.
Now, we check that
$\widetilde {p}$
and
$\widetilde {q}$
are graph homomorphisms. First, suppose
$\lambda \in \mathcal {F}_u^k$
, with
$s(\lambda ) = v$
,
$q(vF^1) \subseteq p(\mathcal {F}_w^j)$
, and
$q(uF^1) \subseteq p(\mathcal {F}_{\hat u}^n)$
. Let
$e = p(\lambda )$
,
$f = q(\lambda )$
,
$z = p(u)$
,
$x = q(u) = p(\hat u)$
,
$y = p(v),$
and
$t = q(v) = p(w)$
.
Since
$p, q$
are graph homomorphisms,
and similarly,
$s(e) = y$
,
$r(f) = x$
, and
$s(f) = t$
. Now,
so
$\widetilde {p}(s(\lambda ^i)) = y^i = s(\widetilde {p}(\lambda ^i))$
.
Next, since
$\lambda \in \mathcal F^k_u$
, we have
$ \widetilde {p}(r(\lambda ^i)) = \widetilde {p}(u^k) = p(u)^{k} = z^{k}. $
However, since
${e=p(\lambda ) \in p(\mathcal {F}_u^k) = \mathcal {E}_z^k}$
and
$ r(\widetilde {p}(\lambda ^i)) = r(p(\lambda )^i)= r(e^i)= z^k. $
That is,
$r(\widetilde {p}(\lambda ^i)) = z^{k}= \widetilde {p}(r(\lambda ^i))$
and we conclude that
$\widetilde {p}$
is a graph homomorphism.
Now, we check that
$\widetilde {q}$
is a graph homomorphism. First, since
$\lambda \in \mathcal F^k_u$
and
${q(uF^1) \subseteq p(\mathcal F^n_{\hat u}),}$
while the facts that
$s(\lambda ) = v$
and
$q(v F^1) \subseteq p(\mathcal F^j_w)$
imply
$ r(\widetilde {q}(\lambda ^i)) = r(q(\lambda )^j) = r(f^j) = x^n,$
since
$f \in q(u F^1) \subseteq p(\mathcal {F}_{\hat u}^n) = \mathcal {E}_x^n$
. So,
$\widetilde {q}(r(\lambda ^i)) = x^n = r(\widetilde {q}(\lambda ^i))$
.
Finally, since
$q(vF^1) \subseteq p(\mathcal {F}_w^j)$
, we have
$ \widetilde {q}(s(\lambda ^i)) = \widetilde {q}(v^i) = q(v)^{j} = t^{j},$
and
Thus,
$\widetilde {q}(s(\lambda ^i)) = t^j = s(\widetilde {q}(\lambda ^i))$
and
$\widetilde {q}$
is also a graph homomorphism, as claimed.
We have the following observation, which is needed for the next proof.
Lemma 7.8. Under the conditions of Theorem 7.3, if
$u, v \in F^0$
and
$\mu \in vF^1$
such that
$q(\mu ) \in p(\mathcal {F}_u^i)$
, then
$q(vF^1) \subseteq p(\mathcal {F}_u^i)$
.
Proof. By Proposition 7.4(2), there exist
$w \in F^0$
and
$j \in \{1, \ldots , m(w)\}$
such that
$q(vF^1) \subseteq p(\mathcal {F}_w^j)$
. So,
$q(\mu ) \in p(\mathcal {F}_w^j) \cap p(\mathcal {F}_u^i)$
. Hence, by Condition (1) in Proposition 7.4,
$p(\mathcal {F}_w^j) = p(\mathcal {F}_u^i)$
.
Proposition 7.9. Under the assumptions of Theorem 7.3, let
$\Lambda = \Lambda _T$
be the 2-graph of the LR textile system
$T$
, and for each
$z \in \Lambda ^0 = E^0$
and each
$i \in \{1, \ldots , m(z)\}$
, set
Then, the collection
${\mathcal {G} = \{\mathcal {G}_z^{i} \mid z \in \Lambda ^0, i=1, \ldots , m(z)\}}$
is a partition of
$\Lambda ^{\varepsilon _1} \sqcup \Lambda ^{\varepsilon _2}$
, which satisfies the pairing condition for 2-graph insplitting.
Proof. First, we show that
$\mathcal {G}$
is a partition of
$\Lambda ^{\varepsilon _1} \sqcup \Lambda ^{\varepsilon _2} (= E^1 \sqcup F^0)$
. By Proposition 7.7, the collection
$\mathcal {E} = \{\mathcal {E}_z^i \mid z \in E^0 = \Lambda ^0, i=1, \ldots , m(z)\}$
is a partition of
$E^1$
. Hence, it suffices to show that the collection
is a partition of
$F^0$
.
Suppose
$v \in F^0$
. Since
$F$
is source-free,
$v=r(\lambda )$
for some
$\lambda \in F^1$
. Hence,
$q(\lambda ) \in \mathcal {E}_z^{i}$
for some
$z \in E^0, i\in \{1, \ldots , m(z)\}$
and, therefore,
$v = r(\lambda ) \in r_F(q^{-1}(\mathcal {E}_z^{i})) $
lies in some set in
$\mathcal C$
. To see that the sets in
$\mathcal {C}$
are pairwise disjoint, observe first that the hypotheses of Theorem 7.3 guarantee that
$q(vF^1) \subseteq p(\mathcal F^j_w) $
for a unique set
$p(\mathcal F^j_w).$
Thus, fix
${v \in F^0}$
and suppose that there exist
$z, x \in \Lambda ^0 = E^0$
such that
$v \in r_F(q^{-1}(\mathcal {E}_z^i)) \cap r_F(q^{-1}(\mathcal {E}_x^j))$
. That is, there exist
$\mu , \nu \in F^1$
such that
$v = r(\mu ) = r(\nu )$
,
$q(\mu ) \in \mathcal {E}_z^{i} = p(\mathcal F^i_w)$
for some
$w \in F^0$
with
$p(w) = z$
, and
$q(\nu ) \in \mathcal {E}_x^{j} = p(\mathcal F^j_u)$
for some
$u \in F^0$
with
$p(u) =x$
. As
${\mu , \nu \in vF^1}$
, Lemma 7.8 forces
$p(\mathcal F^j_u) = p(\mathcal F^i_w)$
, and hence,
$z=x$
and
$i=j$
. In other words,
$ r_F(q^{-1}(\mathcal {E}_z^i)) \cap r(q^{-1}(\mathcal {E}_x^j))\not = \emptyset $
implies
$z=x$
and
$i=j$
, so
$\mathcal C$
is indeed a partition of
$F^0$
, as claimed.
For the partition
$\mathcal G$
to satisfy the pairing condition, we must have that for all
$\lambda \in F^1$
,
$q(\lambda ) \in \mathcal {G}_z^i$
if and only if
$r_F(\lambda ) \in \mathcal {G}_z^i$
. To see this, fix
$\lambda \in F^1$
and suppose
$q(\lambda ) = e \in E^1 \cap \mathcal {G}_z^{i} = \mathcal {E}_z^{i}$
. Then,
$r_F(\lambda ) \in r_F(q{}^{-1}(\mathcal {E}_z^i)) \subseteq \mathcal {G}_z^i$
, as desired. Now, suppose
$r_F(\lambda ) = v \in F^0 \cap \mathcal {G}_z^{i} = r_F(q{}^{-1}(\mathcal {E}_z^{i}))$
. Thus, there exists
$\mu \in vF^1$
such that
$q(\mu ) \in \mathcal {E}_{z}^{i} = p(\mathcal {F}_u^i)$
for some
$u \in F^0$
with
$p(u) = z$
. Since
$q(\mu ) \in p(\mathcal {F}_u^i), $
Lemma 7.8 implies that
$p(\mathcal F^i_u) \supseteq q(vF^1) \ni q(\lambda ).$
As
$\mathcal E^i_z = p(\mathcal F^i_u)$
, we have
$q(\lambda ) \in \mathcal E^i_z \subseteq \mathcal G^i_z$
whenever
$r_F(\lambda ) \in \mathcal G^i_z.$
We conclude that
$\mathcal G$
satisfies the pairing condition.
Thus, we can perform 2-graph insplitting on
$\Lambda _T$
with respect to the partition
$\mathcal G$
. Recall from Definitions 4.5 and 5.8 that the resulting 2-graph
$\Lambda _I$
has
with
$s_I(e^i) = s(e)^i, s_I(v^i) = p(v)^i, r_I(e^i) = r(e)^j$
if
$e \in \mathcal E^j_z$
, and
$r_I(v^i) = q(v)^k$
if
${v \in r_F(q^{-1}(\mathcal E^k_z)),}$
that is, if
$q(vF^1) \subseteq p(\mathcal F^k_u)$
for some
$u$
with
$p(u) = q(v) = z.$
We have
That is,
$v^j e^i \sim _I f^\ell w^i$
if and only if:
-
• there is $\lambda \in F^1$
with
$r_F(\lambda ) = v, s_F(\lambda ) = w, p(\lambda ) = e, q(\lambda ) = f;$
-
• we have $q(wF^1) \subseteq p(\mathcal F^\ell _{u})$
for some
$u \in F^0$
with
$p(u) = q(w)$
; and -
• $\lambda \in \mathcal F^j_v$
(so that
$e = p(\lambda ) \in \mathcal E^j_{r(e)} = \mathcal {E}^j_{p(v)}$
).
In other words, each
$\lambda = ve \sim _\Lambda f w \in F^1$
yields commuting squares
in
$\Lambda _I$
(note that
$m(s(e)) = m(p(w))$
); the indices
$j, \ell $
are determined by
$\lambda $
and are the same for each
$\lambda ^i$
.
Theorem 7.10. Under the assumptions of Theorem 7.3, let
$\Lambda _I$
be the 2-graph insplitting of
$\Lambda _T$
with respect to the partition
$ \{\mathcal {G}_z^i\mid z\in \Lambda ^0, i= 1, \ldots , m(z)\}$
from Proposition 7.9 and let
$T_I = (p_I, q_I: F_I \to E_I)$
be the textile system built from
$\Lambda _I$
as in (4-1). Then,
$T_I \cong \widetilde T = (\widetilde {p}, \widetilde {q}: \widetilde {F} := F_{JM} \to \widetilde {E})$
, the textile system given in Proposition 7.7.
Proof. By (4-1) and Remark 7.6, the LR textile system
$T_{I} = (p_I, q_I: F_{I} \to E_{I})$
has
$s_{F_1}(\lambda ^i) = s_F(\lambda )^i$
, and
$ \ r_{F_I}(\lambda ^i) = r_F(\lambda )^j \text { if } \lambda \in \mathcal F^j_v.$
When we compare
$F_I$
with the construction of
$F_{JM}$
as in Definition 5.2, the fact that
$m(p(v)) = m(v)$
for all
$v \in F^0$
implies that
$F_I = F_{JM}.$
Similarly, we observe that the definitions of
$\widetilde E, \widetilde {p}, \widetilde {q}$
in Proposition 7.7 exactly match the definitions of
$E_I, p_I, q_I$
from (4-1). The fact that
$\widetilde E = E_I$
follows from the fact that the insplitting partition
$\mathcal E$
used to create
$\widetilde E$
satisfies
$\mathcal E^i_z = \Lambda ^{\varepsilon _1}_I \cap \mathcal G^i_z ,$
and the source and range maps in
$\widetilde E$
coincide with the 2-graph insplitting source and range maps. Moreover,
$\widetilde {p} = p_I$
and
$\widetilde {q} = q_I$
, since for any
$\lambda ^i = r_F(\lambda )^k p(\lambda )^i \sim _I q(\lambda )^\ell s_F(\lambda )^i \in F_I^1$
and any
$v^i \in F_I^0 = \Lambda _I^{\varepsilon _2}$
,
if
$v \in \mathcal {G}^k_{q(v)}$
, or equivalently,
$v \in r_F(q^{-1}(\mathcal E^k_{q(v)}).$
That is,
$q_I(v^i) = \widetilde q(v^i).$
Finally, to compare
$q_I(\lambda ^i)$
and
$\widetilde {q}(\lambda ^i)$
, recall that
$\lambda ^i = r_F(\lambda )^k p(\lambda )^i \sim _I q(\lambda )^\ell s_F(\lambda )^i \in F_I^1$
, where
${p(\lambda ) \in \mathcal E^k_{p(r_F(\lambda )))}}$
and
$s_F(\lambda ) \in \mathcal G^\ell _{q(s_F(\lambda ))}$
. By construction, then,
$q_I(\lambda ^i) = q(\lambda )^\ell $
. However,
${s_F(\lambda ) \in \mathcal G^\ell _{q(s_F(\lambda ))}}$
implies that
$s_F(\lambda ) \in r_F(q^{-1}(\mathcal E^\ell _z))$
, so (thanks to Lemma 7.8)
$q(s_F(\lambda ) F^1) \subseteq p(\mathcal F^\ell _w)$
for some
$w \in F^0$
. Consequently,
$\widetilde {q}(\lambda ^i) = q(\lambda )^\ell = q_I(\lambda ^i)$
, as desired.
In other words,
$T_I = \widetilde T;$
since
$T_I$
is LR, being a textile system arising from a 2-graph, it follows that
$\widetilde T$
is also LR.
The above theorem completes the proof of Theorem 7.3.
Example 7.11. Consider again the LR textile system
$T= (p, q: F\to E)$
in Example 6.2 and its corresponding
$2$
-graph
$\Lambda $
, pictured in (6-2). Note that for each
$f_k\in F^0$
,
$k= 1, 2, 3$
, the partition
$\mathcal {F}_{f_k}= \{\mathcal {F}^i_{f_k}\mid i \leq 2\}$
of (6-5) satisfies the conditions given in Theorem 7.3. That is, the graph
$F_A$
of Example 6.2 is the graph
$F_{JM}$
of Theorem 7.3. Observe that
As
$p(f_1) = p(f_2) = v$
and
$q(f_3) = w,$
It follows that the directed graph insplitting
$\widetilde E$
of
$E$
with this partition is precisely the directed graph
$E_{\Lambda _I}$
shown in Example 6.2. Thus, applying the formulas from Proposition 7.7 in this case, we obtain maps
$\widetilde {p}, \widetilde {q}: F_{JM} = F_A \to \widetilde E = E_{\Lambda _I}$
that are given by
and
$\widetilde {p}(\lambda _j^i) = p(\lambda _j)^i$
for all
$i, j$
. These formulas agree with the commuting squares of
$\Lambda _I$
, as asserted by Theorem 7.10. Indeed, note that
so the partition
$\mathcal G$
of Proposition 7.9 is given by
In other words, the 2-graph insplit induced by the partition
$\mathcal F$
of (6-5) is precisely the initial 2-graph insplit of (6-3).
7.3 Textile and 2-graph insplits from E-insplits
Although Johnson–Madden insplitting focuses on the top graph
$F$
of a textile system, to understand the connection between Johnson–Madden insplitting and 2-graph insplitting, it is perhaps more natural to start by insplitting the bottom graph
$E$
, as we now explain. This leads us to an alternate perspective to Theorem 7.3.
Let
$T = (p,q:F\to E)$
be an LR textile system and let
$\Lambda _T$
denote the 2-graph associated to
$T$
. The following theorem shows that by starting with a suitable partition of
$E$
, one can construct an LR textile system
$\widetilde {T}$
that yields a 2-graph
$\Lambda _I$
coinciding with the 2-graph insplitting of
$\Lambda _T$
.
Theorem 7.12. Let
$T = (p, q: F \to E)$
be an LR textile system with associated 2-graph
$\Lambda $
. Assume
$p$
is surjective and
$F$
is source free. Suppose that, for each
$z \in E^0$
, we have a partition
$\{ \mathcal E^1_z, \mathcal E^2_z, \ldots , \mathcal E^{m(z)}_z\}$
of
$zE^1$
such that for each
$u \in F^0$
, there exists
$ z \in E^0$
and
$j \in \{ 1, 2, \ldots , m(z)\}$
such that
$q(uF^1) \subseteq \mathcal E^j_z$
.
We construct a partition
$\{ \mathcal F^i_v\mid v \in F^0, 1\leq i \leq m(p(v))\}$
of
$F^1$
and a partition
$\{ \mathcal {G}^i_z\mid z \in \Lambda ^0 = E^0, 1 \leq i \leq m(z)\}$
of
$\Lambda ^{1}$
as follows. For each
$v \in F^0$
and
$ z \in E^0$
, define
Let
$\widetilde {F}$
denote the directed graph resulting from directed-graph insplitting
$F$
using the partition
$\{\mathcal {F}^i_v\}_{i,v}$
. Let
$\widetilde {E}$
denote the directed graph resulting from directed-graph insplitting
$E$
using the partition
$\{\mathcal {E}^i_z\}_{i,z}$
. Define maps
$\tilde {p}, \tilde {q}: \widetilde {F} \to \widetilde {E}$
by
where
$\lambda \in F^1$
,
$v \in F^0$
,
$s(\lambda ) = v$
, and
$j$
satisfies
$q(vF^1) \subseteq \mathcal E^j_{q(v)}$
. Then, we have the following.
-
(1) The maps $\tilde {p}, \tilde {q}: \widetilde {F} \to \widetilde {E}$
are graph homomorphisms and
$\widetilde {T} := (\tilde {p}, \tilde {q} : \widetilde {F} \to \widetilde {E} )$
is an LR textile system. -
(2) The collection $\{ \mathcal {G}^i_z := \mathcal E^i_z \sqcup r(q^{-1}(\mathcal E_z^i)) \mid z \in E^0, 1\le i \le m(z) \}$
satisfies the pairing condition. -
(3) The 2-graph associated to $\widetilde {T}$
(denoted by
$\widetilde {\Lambda }$
) is identical to the 2-graph
${\Lambda }_I$
resulting from insplitting
$\Lambda $
using the partition
$\{ \mathcal {G}^j_z\}_{j,z}.$
Proof. We begin by observing that if
$q(uF^1) \subseteq \mathcal E^j_z$
, then
$z = q(u)$
and
$j$
is unique, since
$q$
is a graph homomorphism and
$\{ \mathcal E^j_z\}_j$
is a partition of
$z E^1$
. Thus,
$\tilde q$
is well defined.
Next, we show that
$\{ \mathcal F_v^i\mid v \in F^0, 1 \leq i \leq m(p(v)) \}$
is a well-defined partition of
$F^1$
. First, note that if
$v \in F^0$
and
$p(v) = z$
, then
This is because there exists some
$e \in \mathcal {E}^i_z$
(partition sets are nonempty by convention) and since
$p(v) = z = r(e)$
, unique
$r$
-path lifting of
$p$
implies that there exists a unique
$\lambda \in F^1$
such that
$r_F(\lambda ) = v$
and
$p(\lambda ) = e$
. Hence,
$\lambda \in vF^1 \cap p^{-1}(\mathcal E^i_z)$
. It follows that
${\mathcal {F}^i_v \ne \emptyset} $
for all
$v \in F^0$
and
$ 1 \le i \le m(p(v))$
. Consequently,
$\{ \mathcal F_v^i\mid v \in F^0, 1 \leq i \leq m(p(v)) \}$
is a well-defined partition of
$F^1$
:
and
$ \mathcal {F}^k_u \cap \mathcal {F}^l_v \ne \emptyset \implies ( v = u \text { and } \mathcal E^k_{p(u)} \cap \mathcal E^l_{p(v)} \ne \emptyset ) \implies v= u \text { and } k=l.$
In fact, the partition
$\{ \mathcal F^i_v\}_{i,v}$
satisfies Hypotheses (1) and (2) of Theorem 7.3. To see this, first observe that the sets
$\mathcal E^i_z$
and
$\mathcal F^i_w$
of the current theorem satisfy
$\mathcal E^i_z = p(\mathcal F^i_v)$
for any
$v$
with
$p(v) = z$
. (The inclusion
$\supseteq $
is immediate; to see equality, choose
$e \in \mathcal E^i_z$
and
$v \in F^0$
with
$p(v) = z$
. Then,
$r$
-path lifting yields
$f \in F^1$
with
$r(f) = v$
and
$p(f) = e$
, that is,
$f \in p^{-1} (\mathcal E^i_{p(v)}) \cap v F^1 = \mathcal F^i_v$
and
$e = p(f)$
.) Thus, Statement (1) holds since
$\{ \mathcal E^i_z\}_{i,z}$
is a partition of
$E^1$
and Statement (2) holds by our hypothesis on the partition
$\{ \mathcal E^i_z\}_{i,z}$
.
Consequently, the graph
$\widetilde {E}$
of the current theorem is the same as the graph
$\widetilde {E}$
of Theorem 7.3 and our
$\widetilde {F}$
is the graph
$F_{JM}$
of Theorem 7.3. Moreover, the maps
$\tilde {p}, \tilde {q}: \widetilde {F} \to \widetilde {E}$
defined in the statement of the current theorem are precisely the maps discussed in Proposition 7.7. Hence, Proposition 7.7 establishes that
$\tilde {p}, \tilde {q}: \widetilde {F} \to \widetilde {E}$
are well-defined graph homomorphisms. The fact that
$\widetilde {T} := (\tilde {p}, \tilde {q} : \widetilde {F} \to \widetilde {E} )$
is an LR textile system follows from the identification
$\widetilde {T} = T_I$
of Theorem 7.10.
Finally, observe that the sets
$\mathcal G^i_z$
of the current theorem are precisely the same as those of Proposition 7.9, so that proposition yields Statement (2) of the theorem; and the proof of Theorem 7.10 yields Statement (3).
7.4 Uniting the approaches
As mentioned at the beginning of this section, the relationships between 2-graph insplitting and Johnson–Madden insplitting established in Theorems 7.1, 7.3, and 7.12 are compatible. That is, the coherent choice of notation in Theorems 7.1, 7.3, and 7.12 was neither accidental nor an abuse of notation, because all the constructions in these results are equivalent. To obtain a clean statement of the equivalences, we assume throughout this section that the top graph
$F$
is source-free and the graph homomorphism
$p$
is surjective, as is done in Theorem 7.3.
Remark 7.13. While it is sufficient to assume that
$p$
is surjective and
$F$
is source-free, we note that the standard dynamical assumption of an essential system (essential textile system
$T$
or equivalently essential 2-graph
$\Lambda _T$
) also implies all the results in this section.
We begin by showing that the setup in Theorem 7.1 leads to the results in Theorem 7.3.
Theorem 7.14 (Theorem 7.1
$\longrightarrow $
Theorem 7.3).
Let
$T = (p, q: F \to E)$
be an LR textile system and let
$\Lambda $
be its associated 2-graph. Further, assume that
$p$
is surjective and
$F$
is source free. Let
$\{ \mathcal {G}^i_z\mid z \in E^0, 1 \le i \le m(z)\}$
be a 2-graph insplitting partition of
$\Lambda $
. Then, we have the following.
-
(1) The partition $\{\mathcal {F}^i_v\mid v \in F^0, 1 \le i \le m(v)\}$
of
$F^1$
defined in Theorem 7.1 as $$ \begin{align*} \mathcal{F}^i_v := \{ \lambda \in F^1\mid r(\lambda) = v,\ p(\lambda) \in \mathcal{G}^i_{r(p(\lambda)) = p(v)} \} \end{align*} $$satisfies the hypotheses of Theorem 7.3, namely:
-
(a) if $v, w \in F^0$
such that
$p(\mathcal {F}^i_v) \cap p(\mathcal {F}^j_w) \ne \emptyset $
, then
$p(\mathcal {F}^i_v) = p(\mathcal {F}^j_w)$
; and -
(b) for each $v \in F^0$
, there exists
$w \in F^0$
and
$1 \le j \le m(w)$
such that
${q(vF^1) \subseteq p(\mathcal {F}^j_w)}$
.
-
-
(2) The partition $\{\mathcal {E}^i_z\mid z \in E^0, 1 \le i \le m(z)\}$
of
$E^1$
defined in Theorem 7.1 by (7-1) $$ \begin{align} \mathcal{E}^i_z := \mathcal{G}^i_z \cap z E^1 \end{align} $$satisfies $\mathcal {E}^i_z = p(\mathcal {F}^i_v)$
for any
$v\in F^0$
such that
$z= p(v)$
and, hence, agrees with the corresponding partition of
$E^1$
in Theorem 7.3.
-
(3) The original 2-graph partition $\{ \mathcal G^i_z\mid z \in E^0, 1 \leq i \leq m(z)\}$
satisfies $$ \begin{align*} \mathcal{G}^i_z = \mathcal E^i_z \sqcup r_F(q^{-1}(\mathcal E^i_z)) \end{align*} $$for all $v$
such that
$z = p(v) \in E^0$
and for all
$1\le i\le m(v)$
. In particular, the 2-graph insplitting partition constructed in Theorem 7.3 agrees with the original partition
$\{\mathcal {G}^i_z\mid z\in E^0, 1\leq i\leq m(z) \}$
.
-
(4) The constructions of the textile system $\widetilde {T} = (\widetilde {p}, \widetilde {q}: \widetilde {F}\to \widetilde {E})$
in Theorems 7.1 and 7.3 coincide.
Proof. We first observe that for any
$v \in F^0$
and
$1 \leq i \leq m(p(v)),$
The containment
$\subseteq $
follows from the definition and the reverse containment follows from
$r$
-path lifting for
$p$
. It immediately follows that (2) holds. Furthermore,
$p(\mathcal F^i_v) = p(\mathcal F^i_w)$
whenever
$p(v) = p(w).$
To see Statement (1a), recall from Theorem 7.1 that
$\{ \mathcal E^i_z\}_{i,z}$
is a partition of
$E^1$
. Thus, Statement (1a) follows from (7-2).
For Statement (1b), suppose
$\lambda \in vF^1$
has
$q(\lambda )\in p(\mathcal F^j_w) = \mathcal E^j_{p(w)} \subseteq \mathcal {G}^j_{p(w)}$
. For each
$v$
, such a
$\lambda $
exists since
$F$
is source free. As
$\{ \mathcal G^j_z\}_{j,z}$
satisfies the pairing condition, we have
$v \in \mathcal {G}^j_z$
as well. Thus, the pairing condition, together with the fact that
$p(\mathcal F^i_u) = p(\mathcal F^i_w)$
whenever
$p(u) = p(w)$
, implies that for any
$\mu \in vF^1$
,
$q(\mu ) \in \mathcal {G}^j_z \cap E^1 = p(\mathcal F^j_w).$
That is, Statement (1b) holds.
For Statement (3), Equation (7-1) implies that
$\mathcal {G}^i_z \cap E^1 = \mathcal E^i_z$
. To see that
$\mathcal {G}^i_z \cap F^0 = r_F(q^{-1}(\mathcal E^i_z))$
, we use the fact that
$F$
is source free, the pairing condition, and the definition of
$r_\Lambda $
. From these, we conclude that
$w \in F^0 \cap \mathcal {G}^i_z$
if and only if
$q(w) = r_\Lambda (w) = z$
, and there is
$\lambda \in F^1$
with
$r_F(\lambda ) = w$
and
$q(\lambda ) \in \mathcal {G}^i_z \cap E^1 = \mathcal E^i_z .$
That is,
$w \in F^0 \cap \mathcal {G}^i_z$
if and only if
$w \in r_F(q^{-1}(\mathcal E^i_z))$
, so Statement (3) holds.
As we have now proved that the partitions
$\mathcal E$
and
$\mathcal F$
constructed in Theorem 7.1 have the same properties as the partitions of Theorem 7.3, and the formulas for
$\tilde p, \tilde q$
are the same in both theorems, Statement (4) holds.
We next show that starting from the setting in Theorem 7.3, one can obtain the results in Theorem 7.12.
Theorem 7.15 (Theorem 7.3
$\longrightarrow $
Theorem 7.12).
Let
$T= (p,q: F \to E)$
be an LR textile system with associated 2-graph
$\Lambda $
. Further, assume that
$p$
is surjective and
$F$
is source free. Let
$\{\mathcal {F}^i_v\mid v \in F^0, 1 \le i \le m(v)\}$
be a partition of
$F^1$
as in Theorem 7.3, namely:
-
(a) if $v, w \in F^0$
such that
$p(\mathcal {F}^i_v) \cap p(\mathcal {F}^j_w) \ne \emptyset $
, then
$p(\mathcal {F}^i_v) = p(\mathcal {F}^j_w)$
; and -
(b) for each $v \in F^0$
, there exists
$w \in F^0$
and
$1 \le j \le m(w)$
such that
$q(vF^1) \subseteq p(\mathcal {F}^j_w)$
.
Then, we have the following.
-
(1) The partition $\{\mathcal {E}^i_z\mid z \in E^0, 1 \le i \le m(v)\}$
of
$E^1$
defined in Theorem 7.3 as
${\mathcal {E}^i_z := p(\mathcal {F}^i_v)}$
, where
$p(v) = z$
, satisfies the hypothesis of Theorem 7.12, that is, for each
$u \in F^0$
, there is a unique
$j \in \{ 1, 2, \ldots , m(q(u))\}$
such that
$q(uF^1) \subseteq \mathcal E^j_{q(u)}$
. -
(2) The partition $\{\mathcal {F}^i_v\mid v \in F^0, 1 \le i \le m(v)\}$
of
$F^1$
constructed from
$\{ \mathcal E^i_z\}_{i,z}$
in Theorem 7.12 agrees with the original partition
$\{\mathcal {F}^i_v \}$
.
Consequently, the 2-graph insplitting partitions and the textile systems
$\widetilde {T}$
constructed in the two theorems coincide.
Proof. Let
$\{\mathcal {F}^i_v\mid v \in F^0, 1 \le i \le m(v)\}$
be a partition of
$F^1$
as in Theorem 7.3.
-
(1) Recall from Proposition 7.7 that $\{ \mathcal {E}^i_z := p(\mathcal {F}^i_v)\}_{i,z }$
, where
$z = p(v)$
, is a well-defined partition of
$E^1$
. Fix
$u \in F^0$
. Then by Hypothesis (b), there exists
$w \in F^0$
and
${1 \le j \le m(w)}$
such that
$q(uF^1) \subseteq p(\mathcal {F}^j_w) = \mathcal E^j_{p(w)}$
. As
$\mathcal E^j_{p(w)} \subseteq p(w)E^1$
and
$r_E(q(uF^1)) = q(r_F(uF^1)) = q(u)$
, we must have
$q(u) = p(w)$
. As
$\mathcal E$
is a partition and hence
$\mathcal E^k_{q(u)} \cap \mathcal E^j_{q(u)} \ne \emptyset $
implies
$k=j$
, Statement (1) holds. -
(2) It is sufficient to prove that for all $v \in F^0$
and
$1\le i\le m(v)$
, $$ \begin{align*} \mathcal{F}^i_v = vF^1 \cap p^{-1}(p(\mathcal{F}^i_v)). \end{align*} $$
Clearly, $\mathcal {F}^i_v \subseteq vF^1 \cap p^{-1}(p(\mathcal {F}^i_v))$
. To prove the opposite containment, suppose that
$f \in vF^1 \cap p^{-1}(p(\mathcal {F}^i_v))$
but
$f \in \mathcal {F}^k_v$
for some
$k$
. Then,
$p(f) = p(\lambda )$
for some
$\lambda \in \mathcal {F}^i_v$
, that is,
$p(f) \in p(\mathcal {F}^i_v) \cap p(\mathcal {F}^k_v)$
, so by Hypothesis (a),
$p(\mathcal {F}^i_v) = p(\mathcal {F}^k_v)$
. Proposition 7.4 now tells us that
$i = k$
, as desired.
The 2-graph insplitting partitions constructed in both theorems are the same, by definition:
Finally, as we have already observed that the partitions used to construct
$\widetilde {F}$
and
$\widetilde {E}$
in Theorems 7.3 and 7.12 coincide, the graphs
$\widetilde {F}, \widetilde {E}$
resulting from the two theorems are the same. Moreover, the two theorems use the same definitions of
$\tilde p, \tilde q$
; thus, the two textile systems coincide.
Lastly, we show that starting from the setting of Theorem 7.12, we recover the results in Theorem 7.1.
Theorem 7.16 (Theorem 7.12
$\longrightarrow $
Theorem 7.1).
Let
$T= (p,q: F \to E)$
be an LR textile system with associated 2-graph
$\Lambda $
. Further, assume that
$p$
is surjective and
$F$
is source free. Let
$\{ \mathcal E^i_z\mid z \in E^0, 1 \le i \le m(z)\}$
be a partition of
$E^1$
as in Theorem 7.12, so that for each
$u \in F^0$
, there exist
$ z = q(u) \in E^0$
and a unique
$j \in \{ 1, 2, \ldots , m(z)\}$
such that
$q(uF^1) \subseteq \mathcal E^j_z$
.
Define partitions of
$F^1$
and
$\Lambda ^{1}$
as in Theorem 7.12, namely: for each
$v \in F^0$
and
$ z \in E^0$
, let
Then, we have the following.
-
(1) The partition of $E^1$
constructed in Theorem 7.1 from the 2-graph partition (7-4) agrees with the original partition
$\{\mathcal {E}^i_z \}$
, that is, $$ \begin{align*} \mathcal{E}^i_z = \mathcal{G}^i_z \cap z E^1. \end{align*} $$
-
(2) The partition of $F^1$
constructed in Theorem 7.1 from
$\{\mathcal {G}^i_z\}_{i,z}$
agrees with the partition of (7-3), that is, $$ \begin{align*} \mathcal{F}^i_v = \{ \lambda \in vF^1 \mid p(\lambda) \in \mathcal{G}^i_{ p(v)} \}. \end{align*} $$
-
(3) The constructions of the textile system $\widetilde {T} = (\tilde {p}, \tilde {q}: \widetilde {F}\to \widetilde {E})$
in Theorems 7.1 and 7.12 coincide.
Proof. Following the fact that
$\mathcal {G}^i_z \cap z E^1 = (\mathcal E^i_z \sqcup r(q^{-1}(\mathcal E_z^i))) \cap z E^1 = \mathcal E^i_z \cap z E^1 = \mathcal {E}^i_z$
, we obtain Statement (1). To establish Statement (2), we note that
$f \in \mathcal {F}^i_v$
if and only if
$f \in \{ \lambda \in vF^1 \mid p(\lambda ) \in \mathcal E^i_{p(v)}\}$
, which is equivalent to
$f \in \{ \lambda \in vF^1 \mid p(\lambda ) \in \mathcal {G}^i_{p(v)}\}$
as
$p(f)\in E^1$
, but
$p(f) \notin r_F(q^{-1}(\mathcal E_z^i))$
.
For Statement (3), it is sufficient to show that the maps
$\tilde {q}$
defined in Theorem 7.1 and Theorem 7.12 are the same. To this end, we note that for
$v \in F^0$
,
where the last equivalence follows from the hypothesis. So, the definition of
$\tilde {q}$
in Theorem 7.1 and Theorem 7.12 coincide, and hence, the textile systems are the same.
Appendix
A
$C^*$
-algebraic results
In this appendix, we discuss the
$C^*$
-algebraic implications of our work. We exclusively discuss 2-graphs and their
$C^*$
-algebras in this appendix, because although 2-graphs (and
$k$
-graphs in general) were developed in [Reference Kumjian and PaskKP00] to construct
$C^*$
-algebras, there is no established procedure for associating
$C^*$
-algebras directly to textile systems.
First, we show in Theorem A.3 that the
$C^*$
-algebra of a 2-graph is unchanged under a 2-graph insplitting; this result was established in [Reference Eckhardt, Fieldhouse, Gent, Gillaspy, Gonzales and PaskEFG+22, Theorem 3.9] for the one-vertex-at-a-time version of 2-graph insplitting discussed in that paper. We also show that this isomorphism is gauge-invariant and diagonal-preserving. Thus, [Reference Carlsen and RoutCR21, Theorem 3.2] shows that 2-graph insplitting yields an eventual one-sided conjugacy. (In fact, as we have expressed 2-graph insplitting as a composition of conjugacy-preserving moves on textile systems in Theorem 6.1, we have already proved the stronger result that 2-graph insplitting is a one-sided conjugacy.) We then define outsplitting for 2-graphs, as introduced in [Reference ListhartkeLis24], and explain why this paper has focused on linking the textile-system and 2-graph definitions of insplitting rather than outsplitting.
To simplify our calculations, we use an edge-based definition of
$2$
-graph algebras that has seen some light in the literature, see [Reference Eckhardt, Fieldhouse, Gent, Gillaspy, Gonzales and PaskEFG+22, Definition 2.1.1], [Reference Raeburn, Sims and YeendRSY04, Theorem C.1], [Reference Kumjian, Pask and SimsKPS12, Definition 7.4], and alluded to in [Reference Kumjian and PaskKP00, Remark 1.6].
Definition A.1 (Edge definition of a
$2$
-graph
$C^*$
-algebra).
Let
$\Lambda $
be a row-finite
$2$
-graph. An edge Cuntz–Krieger
$\Lambda $
-family in a
$C^*$
-algebra
$A$
is a function
$s : \Lambda ^0 \cup \Lambda ^{\varepsilon _1} \cup \Lambda ^{\varepsilon _2} \to \text {PIsom}(A)$
, denoted
$x \mapsto s_x$
, that assigns a partial isometry
$s_x$
to each
$x \in \Lambda ^0 \cup \Lambda ^{\varepsilon _1} \cup \Lambda ^{\varepsilon _2} $
such that:
-
(ECK1) $\{s_v \mid v\in \Lambda ^0\}$
is a collection of mutually orthogonal projections; -
(ECK2) $s_e s_f = s_{f'} s_{e'}$
whenever
$e,e' \in \Lambda ^{\varepsilon _1}$
,
$f,f' \in \Lambda ^{\varepsilon _2}$
, and
$e f \sim f' e'$
; -
(ECK3) $s^*_f s_f = s_{s(f)}$
for all
$\lambda \in \Lambda ^{\varepsilon _1 + \varepsilon _2}$
; and -
(ECK4) $s_v = \sum _{f \in v\Lambda ^{\varepsilon _i}} s_f s_f^*$
for all
$v\in \Lambda ^0$
and
$i=1,2$
.
The
$C^*$
-algebra
$C^*(\Lambda )$
is the universal
$C^*$
-algebra generated by an edge Cuntz–Krieger
$\Lambda $
-family. We write
$\{ t_\lambda \mid \lambda \in \Lambda ^0 \cup \Lambda ^{\varepsilon _1} \cup \Lambda ^{\varepsilon _2}\}$
for the generators of the universal
$C^*$
-algebra
$C^*(\Lambda )$
. That is, for any edge Cuntz–Krieger family
${s: \Lambda ^0 \cup \Lambda ^{\varepsilon _1} \cup \Lambda ^{\varepsilon _2} \to A}$
, there is a unique surjective
$*$
-homomorphism
$\pi _s$
from
$C^*(\Lambda ) $
onto the
$C^*$
-algebra generated by the image of
$s$
, which satisfies
$\pi _s(t_\lambda ) = s_\lambda $
for all generators
$t_\lambda $
. (The existence of
$C^*(\Lambda )$
can be proved by following for instance [Reference RaeburnRae05, Proposition 1.21].)
For general elements
$\lambda \in \Lambda $
, if
$e_1 \cdots e_n$
is a representative of
$\Lambda $
with each
$e_j \in \Lambda ^{\varepsilon _i}$
for some
$i=1,2$
, we define
$t_\lambda = t_{e_1} \cdots t_{e_n}.$
Condition (ECK2) guarantees that
$t_\lambda $
is independent of the choice of representative of
$\lambda $
.
Because of the universal property of
$C^*(\Lambda )$
, there is a canonical action
$\gamma $
of
${\mathbb T}^2$
on
$C^*(\Lambda )$
, called the gauge action, which satisfies
for all
$z \in {\mathbb T}^2, e \in \Lambda ^{\varepsilon _1} \sqcup \Lambda ^{\varepsilon _2}$
, and
$ v \in \Lambda ^0$
. A standard
$\varepsilon /3$
argument shows that this action is strongly continuous.
Remarks A.2 (Switching between edge-based and path-based models for
$C^* ( \Lambda )$
).
One can extract from [Reference Kumjian, Pask and SimsKPS15, Reference Kumjian, Pask and SimsKPS12] a proof that Definition A.1 and the original definition [Reference Kumjian and PaskKP00, Definition 1.5] of
$C^*(\Lambda )$
agree. To be precise, Definition A.1 corresponds to the definition of Cuntz–Krieger
$\iota $
-family given in [Reference Kumjian, Pask and SimsKPS12, Definition 7.4], where
$\iota $
denotes the trivial cubical cocycle
$\iota \in Z^2 ( \Lambda , \mathbb {T} )$
, so the universal
$C^*$
-algebra of Definition A.1 agrees with the
$C^*$
-algebra
$C^*_\iota (\Lambda )$
of [Reference Kumjian, Pask and SimsKPS12, Definition 7.5]. Then, [Reference Kumjian, Pask and SimsKPS15, Theorem 3.16] identifies
$\iota $
with the trivial categorical cocycle
$c_\iota \in \underline {Z}^2 ( \Lambda , \mathbb {T} )$
. Next, [Reference Kumjian, Pask and SimsKPS15, Theorem 5.3] gives a gauge-equivariant isomorphism
$\phi : C_\iota ^* ( \Lambda ) \to C^* ( \Lambda , c_\iota )$
. Finally, one checks that
$C^* ( \Lambda , c_\iota ) $
defines the same object as the original
$ C^* ( \Lambda )$
from [Reference Kumjian and PaskKP00, Definition 1.5], as both are universal
$C^*$
-algebras with the same generators and relations.
To link our work with that of Carlsen and Rout [Reference Carlsen and RoutCR21], we recall that the diagonal subalgebra
$\mathcal D(\Lambda )$
of
$C^*(\Lambda )$
is the Abelian subalgebra densely spanned by
$\{ t_\lambda t_\lambda ^*\mid \lambda \in \Lambda \}.$
Theorem A.3 (
$2$
-graph insplitting induces gauge-invariant isomorphism).
Let
$\Lambda $
be an essential row-finite 2-graph with associated 1-skeleton
$(G_\Lambda ,c_\Lambda )$
. Let
$\mathcal E$
be an insplitting partition of
$(G_\Lambda ,c_\Lambda )$
satisfying Definition 5.8. Let
$\Lambda _I$
be the 2-graph resulting from insplitting
$\Lambda $
with this partition. Then, there is a gauge-equivariant diagonal-preserving isomorphism
$\pi _S : C^*(\Lambda ) \to C^*(\Lambda _I)$
.
Proof. Let
$\{ s_v , s_f \mid v \in \Lambda ^0 , f \in \Lambda ^{\varepsilon _1} \sqcup \Lambda ^{\varepsilon _2} \}$
be an edge Cuntz–Krieger
$\Lambda $
-family generating
$C^* ( \Lambda )$
and
$\{ t_{v^j} , t_{f^j} \mid v^j \in \Lambda _I^0, f^j \in \Lambda _I^{\varepsilon _1} \sqcup \Lambda _I^{\varepsilon _2} \}$
be an edge Cuntz–Krieger
$\Lambda _I$
-family generating
$C^* ( \Lambda _I )$
. For
$v \in \Lambda ^0$
and
$f \in \Lambda ^{\varepsilon _1} \sqcup \Lambda ^{\varepsilon _2}$
, set
We claim that the family
$\{ S_v , S_f \mid v \in \Lambda ^0 , f \in \Lambda ^{\varepsilon _1} \sqcup \Lambda ^{\varepsilon _2} \}$
is an edge Cuntz–Krieger
$\Lambda $
-family in
$C^* ( \Lambda _I )$
. To do this, we first observe that each
$S_f$
is a partial isometry since it is a finite sum of partial isometries with different source projections (see [Reference Maloney, Pask and RaeburnMPR14, Lemma 2.1]).
Next, we check that this family
$\{ S_f, S_v\}$
satisfies the relations (ECK1)–(ECK4).
We begin with Condition (ECK1). The
$S_v$
are nonzero mutually orthogonal projections since they are sums of projections satisfying the same properties. For Condition (ECK3), let
$f \in \Lambda ^{\varepsilon _i}$
,
$i=1,2$
. Since the generators
$t_{v^j}, t_{f^j}$
of
$C^*(\Lambda _I)$
satisfy Conditions (ECK1) and (ECK3),
For Condition (ECK2), let
$g , g'\in \Lambda ^{\varepsilon _1}$
and
$h, h' \in \Lambda ^{\varepsilon _2}$
be such that
$gh \sim h'g'$
in
$\Lambda $
. Suppose that
$h \in \mathcal E_{s(g)}^\ell $
and
$g' \in \mathcal E_{s(h')}^m$
. Then, the fact that
$T := \{ t_x\mid x \in \Lambda _I^0 \cup \Lambda _I^{\varepsilon _1} \cup \Lambda _I^{\varepsilon _2}\}$
is a family of partial isometries satisfying Condition (ECK1) allows us to conclude that
is zero unless
$j= \ell $
. Moreover, the definition of
$\sim _I$
, together with the fact that
$T$
satisfies Condition (ECK2), implies that
$t_{g^\ell } t_{h^k} = t_{(h')^m} t_{(g')^k}. $
Since
$gh \sim h'g'$
implies in particular that
$s(h) = s(g')$
,
For Condition (ECK4), recall that every
$g \in \Lambda _I^{\varepsilon _i}$
is of the form
$g = f^k$
for some
$f \in \Lambda ^{\varepsilon _i}$
and that
$r_{\Lambda _I}(f^k) = r_\Lambda (f)^j$
if
$f \in \mathcal E^j_{r(f)}$
. Therefore, for any
$v \in \Lambda ^0$
and
$i=1,2$
, we can write
Moreover, since
$t_{f^k} = t_{f^k} t_{s(f)^k}$
by Condition (ECK3), we have
$t_{f^k} t_{f^\ell }^* = \delta _{\ell ,k} t_{f^k} t_{f^\ell }^*$
by Condition (ECK1). It follows that
so
$\{ S_v, S_f \}$
satisfies Condition (ECK4). By the universal property of
$C^* ( \Lambda )$
, there is a
$*$
-homomorphism
$\pi _{S} : C^* ( \Lambda ) \to C^* ( \Lambda _I )$
taking
$s_x$
to
$S_x$
for all
$x \in \Lambda ^0 \cup ( \Lambda ^{\varepsilon _1} \sqcup \Lambda ^{\varepsilon _2})$
. Moreover, the definition of
$S_x$
ensures that
$\pi _S$
is gauge-equivariant.
We claim that
$\pi _S$
is onto. To do this, we show that the generators of
$C^* ( \Lambda _I )$
lie in the image
$C^* ( \{ S_x \mid x \in \Lambda ^0 \cup (\Lambda ^{\varepsilon _1} \sqcup \Lambda ^{\varepsilon _2})\} )$
of
$\pi _S$
.
Fix
$v^j$
, where
$1 \le j \le m(v)$
. Recall that, for any
$i = 1,2$
,
Using the fact that the generators of
$C^*(\Lambda _I)$
satisfy Condition (ECK4), we conclude that for each
$i=1,2,$
so each generator
$t_{v^j}$
of
$C^*(\Lambda _I)$
lies in the image of
$\pi _S$
.
Applying Condition (ECK1) now implies that, for any
$1\leq j \leq m(s(f))$
,
also lies in the image of
$\pi _S$
. We conclude that
$\pi _S: C^*(\Lambda ) \to C^*(\Lambda _I)$
is onto.
To see that
$\pi _S$
is injective, we invoke the gauge-invariant uniqueness theorem [Reference Kumjian and PaskKP00, Theorem 3.4]. Since
$\pi _S$
is gauge-equivariant, it is injective because
$S_v$
is nonzero for any
$v \in \Lambda ^0$
, thanks to the universal property of
$C^*(\Lambda _I)$
. That is,
$\pi _S$
is a
$*$
-isomorphism.
Finally, we check that
$\pi _S(\mathcal D(\Lambda )) = \mathcal D(\Lambda _I).$
Given
$\lambda \in \Lambda $
, choose a representative
$e_1 \cdots e_n$
of
$\Lambda $
with each
$e_j \in \Lambda ^{\varepsilon _i}$
for some
$i=1,2$
. As
$t_{e_j^k} t_{e_{j+1}^m} = 0$
unless
$e_{j+1} \in \mathcal E_{s(e_j)}^k,$
where
$e_j \in \mathcal E^{m_{j-1}}_{r(e_j)}$
for all
$2 \leq j \leq n.$
By Conditions (ECK1) and (ECK3),
$t_{e_n^\ell } t_{e_n^k}^* = \delta _{\ell , k} t_{e_n^\ell } t_{e_n^\ell }^*$
. Thus,
lies in
$\mathcal D(\Lambda _I)$
, being a finite sum of elements of the form
$t_\lambda t_\lambda ^*.$
As every
$*$
-isomorphism is norm-preserving, we conclude that
$\pi _S(\mathcal D(\Lambda )) \subseteq \mathcal D(\Lambda _I).$
To show that
$\pi _S(\mathcal D(\Lambda ))= \mathcal D(\Lambda _I)$
, since
$\pi _S$
is norm-preserving, it suffices to show that
$t_\lambda t_\lambda ^* \in \pi _S(\mathcal D(\Lambda ))$
for all
$\lambda \in \Lambda _I$
. So, fix
$\lambda = e_1^{j_1} e_2^{j_2} \cdots e_n^{j_n} \in \Lambda _I$
and observe that since
$s(e_n^{j_n}) = s(e_n)^{j_n},$
Therefore, by Condition (ECK4),
As above, recall that for all
$1 \leq i \leq n-1$
, there is a unique
$1 \leq j_i \leq m(s(e_i))$
such that
$t_{e_1}^{j_1} \cdots t_{e_n}^{j_n} \not = 0$
(namely,
$j_i = m$
if and only if
$e_{i+1} \in \mathcal E^m_{s(e_i)}$
). Consequently, using the fact that
$s(e_n^k) = s(e_n)^k,$
We conclude that
$\pi _S$
is diagonal preserving, as claimed.
A.1 What about outsplitting?
The following definition was introduced in [Reference ListhartkeLis24, Definition 1.5.1].
Definition A.4 (2-graph outsplitting).
Let
$\Lambda $
be a row-finite source-free 2-graph. Let
$G= (\Lambda ^0, \Lambda ^1, r, s)$
be its 1-skeleton. An outsplitting partition of
$G$
is a partition
$\mathcal {E}= \{\mathcal {E}_w\mid w\in \Lambda ^0\}$
of
$\Lambda ^{\epsilon _1}\sqcup \Lambda ^{\epsilon _2}$
, where
$\mathcal {E}_w = \{ \mathcal E_w^1, \ldots , \mathcal E_w^{m(w)}\} $
is a partition of
$s^{-1}(w)\cap \Lambda ^1$
satisfying the following analog of the pairing condition:
We then define a 2-colored graph
$((\Lambda _O^0, \Lambda _O^1, r_O, s_O), d)$
, where:
-
• $\Lambda _O^0= \{v^j\mid v\in \Lambda ^0, 1 \leq j \leq m(v)\};$
-
• $\Lambda _O^{\epsilon _i}= \{f^j\mid f\in \Lambda ^{\epsilon _i}, 1\leq j \leq m(r(f)) \}; $
-
• $s_{\Lambda _O}(f^j)= s(f)^k$
if
$f \in \mathcal E_{s(f)}^k;$
-
• $r_{\Lambda _O}(f^j)= r(f)^j$
; -
• $d(f^j) = d(f).$
The commuting squares in
$\Lambda _O$
are given by
An argument analogous to Theorem 5.10 (see also [Reference ListhartkeLis24, Claim 1.5.1]) shows that
$\Lambda _O$
is a 2-graph.
In general, by [Reference ListhartkeLis24, Theorem 4.0.1], 2-graph outsplitting results in a stable isomorphism of
$C^*$
-algebras, but not an on-the-nose isomorphism
$C^*(\Lambda ) \cong C^*(\Lambda _O)$
. Therefore, by [Reference Carlsen and RoutCR21, Theorem 3.2], outsplitting (even for directed graphs; see [Reference Bates and PaskBP04, Example 5.1]) does not give rise to an eventual conjugacy and, in particular, does not give rise to a conjugacy. In addition, we cannot expect to describe 2-graph outsplitting in terms of Johnson–Madden outsplitting or other related moves.
Acknowledgements
The fifth author would like to thank Aidan Sims, Yuxiang Tang, and Samuel Webster for interesting conversations about textile systems. This research was initiated at the Women in Operator Algebras II workshop at BIRS and continued during a Research in Groups stay at ICMS. We thank both of these organizations for the congenial research atmosphere they provided.
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