1 Introduction
The operation of semidirect product of finite semigroups on the pseudovariety level has been the subject of extensive study in the recent past. See Chapter 10 in the book [Reference Almeida1] by Almeida for a wide sample of results obtained as yet in this direction. New prospects in this area have been opened later on by Almeida and Weil in [Reference Almeida and Weil3]. The collection of all pseudovarieties of finite semigroups forms a monoid under the mentioned operation of semidirect product. Considerable effort has been exerted in order to identify the idempotents of this monoid. These idempotents can be characterized alternatively as the semidirectly closed pseudovarieties of finite semigroups. That is to say, a pseudovariety of finite semigroups is said to be semidirectly closed if it is closed under the formation of semidirect products of its members. In the same manner, semidirectly closed pseudovarieties of finite monoids can also be introduced. Semidirectly closed pseudovarieties of either semigroups or monoids have been treated for the first time in Chapter V of the monograph [Reference Eilenberg4] by Eilenberg. For further significant developments in this area, consult the overview of the respective results provided by Almeida in Section 10.10 of his book [Reference Almeida1].
For arbitrary pseudovarieties
$\mathbf {V}$
and
$\mathbf {W}$
of finite semigroups or monoids, the semidirect product of
$\mathbf {V}$
and
$\mathbf {W}$
is denoted by
$\mathbf {V}*\mathbf {W}$
. It is defined as the pseudovariety of finite semigroups or monoids generated by the family of all semidirect products of individual semigroups or monoids of the form
$S*T$
, where S is a member of
$\mathbf {V}$
and T is a member of
$\mathbf {W}$
. For every pseudovariety
$\mathbf {V}$
of finite monoids, it is further possible to consider its localization
$\mathbf {LV}$
. By definition,
$\mathbf {LV}$
is the class of all finite semigroups all of whose local submonoids belong to
$\mathbf {V}$
. It is easy to verify that then
$\mathbf {LV}$
is itself a pseudovariety of finite semigroups. It is also conceivable to extend this definition of
$\mathbf {LV}$
in verbally the same way from pseudovarieties
$\mathbf {V}$
of finite monoids to any pseudovarieties
$\mathbf {V}$
of finite semigroups.
It has been proved by Eilenberg in [Reference Eilenberg4] that, for arbitrary pseudovarieties
$\mathbf {V}$
and
$\mathbf {W}$
of finite semigroups or monoids, the inclusion of pseudovarieties
$\mathbf {V}*\mathbf {LW} \subseteq \mathbf {L}(\mathbf {V}*\mathbf {W})$
holds. See Proposition 5.6 in Chapter V of [Reference Eilenberg4] in this respect. In this context, the question has been posed in [Reference Eilenberg4] whether, for arbitrary pseudovarieties
$\mathbf {V}$
and
$\mathbf {W}$
of finite monoids, the inclusion of pseudovarieties of finite semigroups
$\mathbf {LV}*\mathbf {LW}\subseteq \mathbf {L}(\mathbf {V}*\mathbf {W})$
holds. This question has been formulated by Eilenberg as an open problem at the very end of Section 7 in Chapter V of his monograph [Reference Eilenberg4]. As a particular case, one may ask, if
$\mathbf {V}$
is any semidirectly closed pseudovariety of finite monoids, that is, if the equality
$\mathbf {V}*\mathbf {V}=\mathbf {V}$
holds, whether then the inclusion of pseudovarieties
$\mathbf {LV}*\mathbf {LV}\subseteq \mathbf {LV}$
holds. Observe that, if this is the case, then, in fact, the equality
$\mathbf {LV}*\mathbf {LV}=\mathbf {LV}$
holds, and so
$\mathbf {LV}$
becomes a semidirectly closed pseudovariety of finite semigroups. Thus, all in all, one may ask, if
$\mathbf {V}$
is any semidirectly closed pseudovariety of finite monoids, whether then
$\mathbf {LV}$
also is a semidirectly closed pseudovariety of finite semigroups. Note that these questions have been taken over later on by Almeida into the list of open problems posed in his book [Reference Almeida1]. In any case, these questions seem to have been open for more than four decades already. In the present paper, we shall answer the mentioned questions in the negative. Furthermore, we shall characterize precisely those nontrivial semidirectly closed pseudovarieties
$\mathbf {V}$
of finite monoids, for which their localizations
$\mathbf {LV}$
are semidirectly closed pseudovarieties of finite semigroups.
In order to achieve this objective, we shall need some concepts from the seminal paper [Reference Tilson10] by Tilson. In that paper, Tilson has introduced the notion of pseudovarieties of finite categories as classes of finite categories that are closed under the formation of finitary direct products of finite categories and under the division of finite categories. Every finite monoid can be viewed as a finite category having only one vertex. Thus, when speaking of the division of finite categories, we also have in our mind what it means that a finite category divides a finite monoid. Furthermore, every pseudovariety
$\mathbf {V}$
of finite monoids can thus be regarded as a class of finite categories. With this pseudovariety
$\mathbf {V}$
of finite monoids, two special pseudovarieties of finite categories are intimately connected. They are the pseudovariety
$g\mathbf {V}$
of finite categories generated by the pseudovariety
$\mathbf {V}$
, which can be described as the class of all finite categories that divide some monoid from
$\mathbf {V}$
, and the pseudovariety
$\ell \mathbf {V}$
of all finite categories all of whose local monoids belong to
$\mathbf {V}$
. Then, of course, we have the inclusion
$g\mathbf {V}\subseteq \ell \mathbf {V}$
. Now, following Tilson, as in Section 13 of [Reference Tilson10], we say that our pseudovariety
$\mathbf {V}$
of finite monoids is local, if, in place of the mentioned inclusion, the equality
$g\mathbf {V}=\ell \mathbf {V}$
holds. Many important pseudovarieties of finite monoids are known to be local. From among the numerous papers written on this subject, we quote here just the paper [Reference Jones and Trotter5] by Jones and Trotter, the paper [Reference Almeida2] by Almeida, and also the author’s paper [Reference Kad’ourek6].
We begin our deliberations in the present paper by showing that, for arbitrary pseudovarieties
$\mathbf {V}$
and
$\mathbf {W}$
of finite monoids, it is sufficient to assume that the pseudovariety
$\mathbf {V}$
is local, in order to assure that the abovementioned inclusion of pseudovarieties of finite semigroups
$\mathbf {LV}*\mathbf {LW}\subseteq \mathbf {L}(\mathbf {V}*\mathbf {W})$
holds. For this purpose, we apply the celebrated Derived Category Theorem of Tilson from his notable paper [Reference Tilson10]. Next, we turn to a certain consequence of the presumption that the inclusion
$\mathbf {LV}*\mathbf {LW}\subseteq \mathbf {L}(\mathbf {V}*\mathbf {W})$
should hold, for any pseudovarieties
$\mathbf {V}$
and
$\mathbf {W}$
of finite monoids. That is to say, we exhibit a necessary condition involving the pseudovarieties
$\mathbf {V}$
and
$\mathbf {W}$
, which must be valid, if the inclusion
$\mathbf {LV}*\mathbf {LW}\subseteq \mathbf {L}(\mathbf {V}*\mathbf {W})$
has to hold. These two conditions—the sufficient one requiring the pseudovariety
$\mathbf {V}$
to be local, and the just mentioned necessary one involving both pseudovarieties
$\mathbf {V}$
and
$\mathbf {W}$
—will be of such a kind that they will make it possible to infer, hence, the following consequence regarding semidirectly closed pseudovarieties of finite monoids. Namely, we shall see that, for any nontrivial semidirectly closed pseudovariety
$\mathbf {V}$
of finite monoids, its localization
$\mathbf {LV}$
also is a semidirectly closed pseudovariety of finite semigroups if, and only if, the pseudovariety
$\mathbf {V}$
in question is local. Note that the fact that, for any semidirectly closed pseudovariety
$\mathbf {V}$
of finite monoids, it is sufficient to assume that
$\mathbf {V}$
is local in order to guarantee that then the localization
$\mathbf {LV}$
is a semidirectly closed pseudovariety of finite semigroups can be documented using a very simple argument based on well-known results on pseudovarieties of finite semigroups and monoids—see the comment on it in the final section of this paper. Thus, the true novelty of the latter consequence consists in the announcement that the request that
$\mathbf {V}$
should be local is also necessary, if the localization
$\mathbf {LV}$
of a nontrivial semidirectly closed pseudovariety
$\mathbf {V}$
of finite monoids has to be itself semidirectly closed.
In a previous paper [Reference Kad’ourek8], we have produced a continuum of semidirectly closed pseudovarieties of finite monoids which are not local. Thus, we may conclude that the localization
$\mathbf {LV}$
of our pseudovariety
$\mathbf {V}$
of finite monoids need not be semidirectly closed, even if the pseudovariety
$\mathbf {V}$
itself is semidirectly closed. Returning eventually to our initial reasonings, we hence also deduce that the inclusion of pseudovarieties
$\mathbf {LV}*\mathbf {LW} \subseteq \mathbf {L}(\mathbf {V}*\mathbf {W})$
need not hold, provided that
$\mathbf {V}$
and
$\mathbf {W}$
are arbitrary pseudovarieties of finite monoids.
2 Preliminaries
As it has been clarified already in the introduction, we shall have to deal in this paper with semidirect products of pseudovarieties of both finite semigroups and finite monoids. Therefore, we shall have to work henceforth with semidirect products of individual objects of the form
$S*T$
, where each of S and T is either a finite semigroup or a finite monoid. We shall retain here the custom introduced by Almeida at the beginning of Section 10.1 in his book [Reference Almeida1], which requires that, even if T is viewed as a semigroup, if this T happens coincidentally to be a monoid, then the underlying left action of T on S giving rise to the semidirect product
$S*T$
should be left unitary. Now, if
$\mathbf {V}$
and
$\mathbf {W}$
are any pseudovarieties of finite semigroups, then the semidirect product
$\mathbf {V}*\mathbf {W}$
of these two pseudovarieties is defined to be the pseudovariety of finite semigroups generated by the class of all semidirect products of the form
$S*T$
, where S is a semigroup from
$\mathbf {V}$
and T is a semigroup from
$\mathbf {W}$
, observing the custom regarding the underlying left action of T on S mentioned above. It is easy to verify that then this semidirect product
$\mathbf {V}*\mathbf {W}$
consists, in fact, of all divisors of the semidirect products
$S*T$
just described. If, on the other hand,
$\mathbf {V}$
and
$\mathbf {W}$
are any pseudovarieties of finite monoids, then the semidirect product
$\mathbf {V}*\mathbf {W}$
of these two pseudovarieties is defined to be the pseudovariety of finite monoids generated by the class of all semidirect products of the form
$S*T$
, where S is a monoid from
$\mathbf {V}$
and T is a monoid from
$\mathbf {W}$
, and where, this time, the underlying left action of T on S giving rise to the semidirect product
$S*T$
is assumed to be both left and right unitary, which ensures that
$S*T$
is again a monoid. It is also possible to consider mixed semidirect products
$\mathbf {V}*\mathbf {W}$
, where one of
$\mathbf {V}$
and
$\mathbf {W}$
is a pseudovariety of finite semigroups, while the other is a pseudovariety of finite monoids. But we shall not meet with such mixed semidirect products in this paper, except for some remarks in the last section of this paper.
Along with the semidirect products of finite semigroups, we shall encounter also the wreath products of finite semigroups. Let S and T be any finite semigroups. We denote by
$T^1$
the finite monoid which is equal to T, if T already is a monoid, and which is obtained from T by adjoining an identity to it, otherwise. We further denote by
$S^{T^1}$
the semigroup of all functions of
$T^1$
into S. Then, the wreath product
$S\circ T$
of the given semigroups is defined to be the semidirect product
$S^{T^1}*T$
of the semigroups
$S^{T^1}$
and T determined by the left action of T on
$S^{T^1}$
given by the following formula. For every
$t\in T$
and for every function
$f:T^1\to S$
, the corresponding function
${}^t\!f:T^1\to S$
is defined by the rule
$t'({}^t\!f)=(t't)f$
, for every
$t'\in T^1$
, where functions are written on the right of their arguments. Note that if T is by coincidence a monoid, then this left action of T on
$S^{T^1}$
is left unitary. Furthermore, if S is monoid, then this left action of T on
$S^{T^1}$
is right unitary. Thus, if both S and T are monoids, then the wreath product
$S\circ T$
is again a monoid. It is a familiar fact that the wreath product
$S\circ T$
of the finite semigroups S and T enjoys the following significant property. For every left action of T on S, which is left unitary if T happens to be a monoid, the semidirect product
$S*T$
determined by this left action can be embedded into the wreath product
$S\circ T$
. Hence, it also follows that, for any pseudovarieties
$\mathbf {V}$
and
$\mathbf {W}$
of finite semigroups, the semidirect product
$\mathbf {V}*\mathbf {W}$
of these pseudovarieties can be obtained as the pseudovariety of finite semigroups generated by the class of all wreath products of the form
$S\circ T$
, where S is any semigroup from
$\mathbf {V}$
and T is any semigroup from
$\mathbf {W}$
. An analogous statement holds if
$\mathbf {V}$
and
$\mathbf {W}$
are any pseudovarieties of finite monoids.
In the introduction already, we have introduced, for every pseudovariety
$\mathbf {V}$
of finite monoids, its localization
$\mathbf {LV}$
. It is a certain pseudovariety of finite semigroups derived from
$\mathbf {V}$
. Now, we look at this notion in more detail. For every finite semigroup S and for every idempotent
$e\in S$
, the subset
$eSe$
of S forms a subsemigroup in S, which is itself a monoid whose identity is the idempotent e itself, and which is called a local submonoid of S. Now, the localization
$\mathbf {LV}$
of the pseudovariety
$\mathbf {V}$
is defined to be the class of all finite semigroups all of whose local submonoids belong to
$\mathbf {V}$
. Notice, however, that the adjective “local” in this definition can be dropped. Indeed, every submonoid T of a finite semigroup S, that is, every subsemigroup T of S which is itself a monoid, can be seen to be a submonoid of the local submonoid
$eSe$
of S, where e is the identity of T. Thus, the localization
$\mathbf {LV}$
of the pseudovariety
$\mathbf {V}$
is just the class of all finite semigroups all of whose submonoids belong to
$\mathbf {V}$
. It can be checked straightforwardly that then this localization
$\mathbf {LV}$
is itself a pseudovariety of finite semigroups.
We have also seen in the introduction that, aside from finite semigroups and finite monoids, we shall have to deal in this paper also with finite categories. Roughly speaking, by a finite category, we mean a finite directed graph endowed with an associative partial binary operation of composition of consecutive edges admitting local identities. In more detail, a finite graph
$\varGamma $
consists of a finite set
$V(\varGamma )$
of vertices and a finite set
$E(\varGamma )$
of edges together with two mappings
$\alpha ,\omega :E(\varGamma )\to V(\varGamma )$
assigning to every edge
$e\in E(\varGamma )$
its beginning
$\alpha (e)$
and its end
$\omega (e)$
. For any vertices
$u,v\in V(\varGamma )$
, we write
$\varGamma (u,v)$
for the set of all edges
$e\in E(\varGamma )$
such that
$\alpha (e)=u$
and
$\omega (e)=v$
. By a finite category, we mean a finite graph C endowed with an associative partial binary operation of multiplication of edges which, for arbitrary vertices
$u,v,w\in V(C)$
, assigns to any edges
$e\in C(u,v)$
and
$f\in C(v,w)$
an edge
$ef\in C(u,w)$
and which has, for every vertex
$v\in V(C)$
, an identity edge
$1_v\in C(v,v)$
acting as an identity element with respect to the mentioned operation of multiplication. In this context, we also call the sets of edges of the form
$C(u,v)$
the hom-sets of C. We also write shortly
$C(v)$
instead of
$C(v,v)$
. For every vertex
$v\in V(C)$
, the set
$C(v)$
together with the multiplication from C restricted to
$C(v)$
forms a monoid. This monoid is called the local monoid of C at v.
If
$\varGamma $
and
$\varDelta $
are finite graphs, then a graph mapping
$\varphi :\varGamma \to \varDelta $
consists of two mappings
$V(\varGamma ) \to V(\varDelta )$
and
$E(\varGamma )\to E(\varDelta )$
, also denoted by
$\varphi $
, such that, for every edge
$e\in E(\varGamma )$
, one has
$\alpha (\varphi (e))=\varphi (\alpha (e))$
and
$\omega (\varphi (e))=\varphi (\omega (e))$
. If C and D are finite categories, then a homomorphism
$\psi :C\to D$
of these categories is a graph mapping such that, for any vertices
$u,v,w\in V(C)$
and any edges
$e\in C(u,v)$
and
$f\in C(v,w)$
, the equality
$\psi (ef)=\psi (e)\psi (f)$
holds and, in addition, for any vertex
$v\in V(C)$
, also the equality
$\psi (1_v)=1_{\psi (v)}$
holds. Furthermore, we say that the mentioned homomorphism
$\psi :C\to D$
is faithful if it is injective on hom-sets of C, that is, if for arbitrary vertices
$u,v\in V(C)$
, the restriction of
$\psi $
to
$C(u,v)$
is injective. We say that the homomorphism
$\psi :C\to D$
is a quotient homomorphism if it is bijective on the vertices of C, that is, if it determines a bijection of
$V(C)$
onto
$V(D)$
, and if it is surjective on hom-sets, that is, if for arbitrary vertices
$u,v\in V(C)$
, the restriction of
$\psi $
to
$C(u,v)$
is a surjection of
$C(u,v)$
onto
$D(\psi (u),\psi (v))$
. Having all that at hand, we say that a category C divides a category D if there exists a category B, a faithful homomorphism
$\varphi :B\to D$
, and a quotient homomorphism
$\psi :B\to C$
. Then, we write
$C\prec D$
. Note that then the underlying category B is isomorphic to a certain subcategory of the direct product of categories
$C\times D$
. Thus, if we have assumed previously that C and D are finite categories, then the mentioned category B must also come out finite. Recall in this context that finite monoids can be viewed as finite categories having only one vertex. For finite monoids, we have the standard notion of divisibility—a finite monoid S is usually said to divide a finite monoid T if there exists a submonoid U of T and a surjective homomorphism of monoids
$\psi :U\to S$
. It is easy to check that the notion of divisibility of finite categories that we have introduced above subsumes as a particular case this common notion of divisibility of finite monoids.
As it has been mentioned already in the introduction, a class
$\mathbf {W}$
of finite categories is said to be a pseudovariety of finite categories if it is closed under the formation of finitary direct products of finite categories and under the division of finite categories. We have already noted above that every finite monoid can be viewed as a finite category having a single vertex. Namely, such a finite monoid can be identified with the local monoid of the finite category in question at its unique vertex. Thus, if we have introduced in the previous paragraph the notion of divisibility of finite categories, we have said at the same time what it means that a finite category C divides a finite monoid S. If this is the case, then we write
$C\prec S$
. Furthermore, in this manner, various classes of finite monoids can be viewed as classes of finite categories. For instance, every pseudovariety
$\mathbf {V}$
of finite monoids can be viewed as a class of finite categories. Having this in mind, with the given pseudovariety
$\mathbf {V}$
of finite monoids, two pseudovarieties of finite categories can be naturally associated. One of them is the pseudovariety
$g\mathbf {V}$
of finite categories generated by the given pseudovariety
$\mathbf {V}$
. This pseudovariety
$g\mathbf {V}$
consists exactly of all finite categories C such that
$C\prec S$
for some finite monoid S from
$\mathbf {V}$
. The other of the mentioned two pseudovarieties is the pseudovariety
$\ell \mathbf {V}$
of all finite categories all of whose local monoids belong to
$\mathbf {V}$
. Then, of course, the inclusion
$g\mathbf {V}\subseteq \ell \mathbf {V}$
holds. Following the terminology introduced by Tilson in Section 13 of [Reference Tilson10], we say that our pseudovariety
$\mathbf {V}$
of finite monoids is local if the equality
$g\mathbf {V}=\ell \mathbf {V}$
of the mentioned pseudovarieties of finite categories holds.
In the next section, we shall need the famous Derived Category Theorem established by Tilson in [Reference Tilson10]. In order to state this theorem here, we have to adopt from Section 4 of [Reference Tilson10] the notion of the derived category of a relational morphism of finite monoids. Recall that a relational morphism
$\varphi :\,S\,{\mathrel {{\longrightarrow \kern-12pt}{\ominus }\hspace {5pt}}}\, T$
od finite monoids S and T is defined to be a submonoid of the direct product
$S\times T$
of these monoids which projects onto S in the first component.
Thus, let
$\varphi :\,S\,{\mathrel {{\longrightarrow \kern-12pt}{\ominus }\hspace {5pt}}}\, T$
be a relational morphism of finite monoids. We define the derived category
$D_\varphi $
of this relational morphism
$\varphi $
in the following way. The set
$V(D_\varphi )$
of vertices of this category is just the set T. The set
$E(D_\varphi )$
of edges of this category can be described as follows. For any elements
$t_1\in T$
and
$s\in S$
,
$t\in T$
such that
$t\in \varphi (s)$
, consider the function
$[t_1,(s,t)]:\,\varphi ^{-1}(t_1)\to \varphi ^{-1}(t_1t)$
given by the following rule. For every element
$s_1\in S$
such that
$t_1\in \varphi (s_1)$
, the function
$[t_1,(s,t)]$
sends the element
$s_1$
to the element
$s_1s$
. Then, the set
$E(D_\varphi )$
of edges of the category
$D_\varphi $
consists of all functions
$[t_1,(s,t)]$
of this form, with the supplement that such an edge
$[t_1,(s,t)]$
is directed from the vertex
$t_1$
to the vertex
$t_1t$
. The multiplication of edges of this category is given by the following rule. For any elements
$t_1\in T$
and
$s,s'\in S$
,
$t,t'\in T$
such that
$t\in \varphi (s)$
and
$t'\in \varphi (s')$
, the product of the edges
$[t_1,(s,t)]$
and
$[t_1t,(s',t')]$
is the edge
$[t_1,(ss',tt')]$
. It can be verified straightforwardly that this rule says precisely that the product of the edges
$[t_1,(s,t)]$
and
$[t_1t,(s',t')]$
is just the composition of the functions
$[t_1,(s,t)]$
and
$[t_1t,(s',t')]$
. The identity edge at the vertex
$t\in T$
then is the identity function on the set
$\varphi ^{-1} (t)$
, which can be represented as the edge
$[t,(1,1)]$
.
Now, we are ready to evoke the Derived Category Theorem from the paper [Reference Tilson10] by Tilson. We shall not need in the sequel this theorem in its primary form, as it is given in Section 5 of [Reference Tilson10]. Instead, we shall work with a consequence of this theorem, which could be named as the pseudovarietal form of the Derived Category Theorem and which is stated as Theorem 13.2 in [Reference Tilson10].
Theorem 2.1 Let
$\mathbf {V}$
and
$\mathbf {W}$
be pseudovarieties of finite monoids, and let S be a finite monoid. Then,
$S\in \mathbf {V}*\mathbf {W}$
if, and only if, there exists a relational morphism
$\varphi :\,S\,{\mathrel {{\longrightarrow \kern-12pt}{\ominus }\hspace {5pt}}}\, T$
such that
$T\in \mathbf {W}$
and
$D_\varphi \in g\mathbf {V}$
.
Later on in this paper, we shall need yet another concept introduced by Tilson in [Reference Tilson10]. Namely, we shall need the notion of the consolidation of a finite category introduced in Section 3 of [Reference Tilson10]. According to the definition given in [Reference Tilson10], the consolidation of a finite category C is the semigroup S whose set of elements consists of the set
$E(C)$
of all edges of C together with a new element
$0$
. This new element
$0$
is a multiplicative zero in S, and the product of any two edges from
$E(C)$
is as in C when it is defined and it is equal to
$0$
otherwise. It has been demonstrated by Tilson in Section 3 of [Reference Tilson10] that every finite category C divides the finite monoid
$S^1$
obtained from the consolidation S of C by adjoining an identity to it, provided that S does not already contain one. Indeed, if C has only one vertex, then S is obtained by adjoining a new zero element
$0$
to the only local monoid of C, and
$S^1$
is equal to S. Thus, the mapping
$\varepsilon :C\to S$
acting as an identity on the set
$E(C)$
is a faithful homomorphism of finite categories. Therefore,
$C\prec S^1$
in this case. If C contains several vertices, then the consolidation S of C is merely a semigroup, and the monoid
$S^1$
arises by adjoining a new identity to S. In that case, consider the finite category
$C^\bullet $
obtained from C by adjoining a completely new identity to every local monoid of C. Then, the mapping
$\varphi :C^\bullet \to S^1$
acting as an identity on the set
$E(C)$
and sending every new identity in
$C^\bullet $
to the new identity appearing in the monoid
$S^1$
is a faithful homomorphism of finite categories, and the mapping
$\psi :C^\bullet \to C$
acting as an identity on each of the sets
$V(C)$
and
$E(C)$
and sending every new identity in each of the local monoids of
$C^\bullet $
to the original identity of the respective local monoid of C is a quotient homomorphism of finite categories. Hence, it ensues that
$C\prec S^1$
in this case, as claimed above.
3 A sufficient condition
In this section, we show that, for arbitrary pseudovarieties
$\mathbf {V}$
and
$\mathbf {W}$
of finite monoids, it is sufficient to assume that the pseudovariety
$\mathbf {V}$
is local, in order to make it sure that the inclusion of pseudovarieties of finite semigroups
$\mathbf {LV} *\mathbf {LW}\subseteq \mathbf {L}(\mathbf {V}*\mathbf {W})$
holds. That is, we are going to verify the following statement.
Theorem 3.1 Let
$\mathbf {V}$
and
$\mathbf {W}$
be any pseudovarieties of finite monoids. If the pseudovariety
$\mathbf {V}$
is local, then the inclusion of pseudovarieties
$\mathbf {LV}*\mathbf {LW}\subseteq \mathbf {L}(\mathbf {V}*\mathbf {W})$
holds.
Proof In order to verify the mentioned inclusion, assume that S is an arbitrary semigroup in the pseudovariety
$\mathbf {LV}*\mathbf {LW}$
. Then, S must divide a semidirect product of semigroups of the form
$P*Q$
where P is some semigroup in
$\mathbf {LV}$
and Q is some semigroup in
$\mathbf {LW}$
. Thus, for all idempotents
$e\in P$
, the local submonoids
$ePe$
of P belong to
$\mathbf {V}$
, and, for all idempotents
$f\in Q$
, the local submonoids
$fQf$
of Q belong to
$\mathbf {W}$
. We wish to show that then the semigroup S belongs to the pseudovariety
$\mathbf {L}(\mathbf {V}*\mathbf {W})$
. Because S divides the semigroup
$P*Q$
, it will be sufficient to verify that
$P*Q$
belongs to
$\mathbf {L}(\mathbf {V}*\mathbf {W})$
. It means that it will be enough to check that, for all idempotents
$g\in P*Q$
, the local submonoids
$g(P*Q)g$
of
$P*Q$
belong to the pseudovariety
$\mathbf {V}*\mathbf {W}$
. In order to achieve this goal, we shall take advantage of the Derived Category Theorem of Tilson from [Reference Tilson10].
Recall at first that every idempotent
$g\in P*Q$
is of the form
$(a,h)$
, where
$a\in P$
and
$h\in Q$
are such elements that
$(a,h)(a,h)=(a,h)$
, that is,
$(a{}^h\!a,hh)=(a,h)$
, which entails that
$a{}^h\!a=a$
and
$hh=h$
. Thus, h is an idempotent in Q, and therefore also
${}^h\!a {}^h\!a={}^h\!a$
, so that
${}^h\!a$
is an idempotent in P. Consider next an arbitrary element in the local submonoid
$g(P*Q)g$
of
$P*Q$
with g equal to
$(a,h)$
, that is, in the local submonoid
$(a,h)(P*Q)(a,h)$
. Such an element must be of the form
$(a,h)(p,q)(a,h)$
for some
$p\in P$
and
$q\in Q$
, that is, it must be of the form
$(a{}^h\!p{}^{hq}\!a,hqh)$
for these
$p\in P$
and
$q\in Q$
. Here,
$hqh$
belongs to the local submonoid
$hQh$
of Q. Therefore, it is possible to think of the mapping
$\vartheta :(a,h)(P*Q)(a,h)\to hQh$
given by the formula
$(a,h)(p,q)(a,h) \mapsto hqh$
. This mapping
$\vartheta $
obviously is a homomorphism of finite monoids. Hence, we may consider the derived category
$D_\vartheta $
of this homomorphism
$\vartheta $
. Note in passing that the local submonoid
$hQh$
of Q belongs to the pseudovariety
$\mathbf {W}$
. We proceed by verifying that the local monoids of the derived category
$D_\vartheta $
belong to the pseudovariety
$\mathbf {V}$
.
The vertices of the derived category
$D_\vartheta $
are just the elements of the local submonoid
$hQh$
of Q, that is, they are the elements of the form
$hq_1h$
, for any
$q_1\in Q$
. We continue by realizing how the edges of the derived category
$D_\vartheta $
look like. By the definition of
$D_\vartheta $
, these edges are exactly the functions of the form
$[hq_1h,((a,h)(p,q)(a,h),hqh)]:\vartheta ^{-1}(hq_1h)\to \vartheta ^{-1} (hq_1hqh)$
, for any
$q_1\in Q$
and any
$p\in P$
and
$q\in Q$
. Such an edge is directed from the vertex
$hq_1h$
to the vertex
$hq_1hqh$
. For any
$q_1\in Q$
, the elements of the set
$\vartheta ^{-1}(hq_1h)$
are of the form
$(a,h)(p_2,q_2)(a,h)$
, for any
$p_2\in P$
and
$q_2\in Q$
having the property that
$\vartheta $
maps
$(a,h)(p_2,q_2)(a,h)$
to
$hq_1h$
, which means that
$hq_2h=hq_1h$
holds. The function
$[hq_1h,((a,h)(p,q) (a,h),hqh)]$
then sends such an element to the element
$(a,h)(p_2,q_2) (a,h)(p,q)(a,h)$
. Notice that
$\vartheta $
maps the latter element to
$hq_2hqh$
, and that we actually have
$hq_2hqh=hq_1hqh$
, because
$hq_2h =hq_1h$
holds.
Next, consider the local monoid of the derived category
$D_\vartheta $
at its vertex
$hq_1h$
, where
$q_1\in Q$
. The edges that are elements of this local monoid are directed from the vertex
$hq_1h$
to the vertex
$hq_1h$
itself. Thus, having in view the description of arbitrary edges of
$D_\vartheta $
given in the previous paragraph, we conclude that the edges of the local monoid of
$D_\vartheta $
at
$hq_1h$
must be the functions of the form
$[hq_1h,((a,h)(p,q)(a,h),hqh)]:\vartheta ^{-1} (hq_1h)\to \vartheta ^{-1}(hq_1h)$
, for any
$p\in P$
and
$q\in Q$
such that
$hq_1hqh=hq_1h$
holds.
From what has been said in the last paragraph but one it readily follows that such a function
$[hq_1h,((a,h)(p,q)(a,h),hqh)]$
sends every element of the set
$\vartheta ^{-1}(hq_1h)$
, which is of the form
$(a,h)(p_2,q_2) (a,h)$
, for any
$p_2\in P$
and
$q_2\in Q$
such that
$hq_2h=hq_1h$
, to the element
$(a,h)(p_2,q_2)(a,h)(p,q)(a,h)$
. The former element
$(a,h)(p_2,q_2)(a,h)$
is equal to
$(a{}^h\!p_2{}^{hq_2}\!a,hq_2h)$
, and as we have
$hq_2h=hq_1h$
, it is further equal to
$(a{}^h\!p_2{}^{h q_2}\!a,hq_1h)$
. The latter element
$(a,h)(p_2,q_2)(a,h)(p,q)(a,h)$
is equal to
$(a{}^h\!p_2{}^{hq_2}\!a{}^{hq_2h}\!p{}^{hq_2hq}\!a,hq_2hqh)$
, and as we have
$hq_1hqh=hq_1h$
and
$hq_2h=hq_1h$
, whence it ensues that
$hq_2hqh=hq_2h$
, this element is further equal to
$(a{}^h\!p_2{}^{hq_2} \!a{}^{hq_2h}\!p{}^{hq_2hq}\!a,hq_2h)$
. Furthermore, we have seen above that
$a{}^h\!a=a$
, which entails that
${}^{hq_2}\!a{}^{hq_2h}\!a={}^{h q_2}\!a$
and
${}^{hq_2hq}\!a{}^{hq_2hqh}\!a={}^{hq_2hq}\!a$
. Because
$hq_2hqh=hq_2h$
, the second of these equalities yields that
${}^{hq_2h q}\!a{}^{hq_2h}\!a={}^{hq_2hq}\!a$
. Therefore, the latter element shown above can be rewritten in the form
$(a{}^h\!p_2{}^{hq_2}\!a{}^{hq_2h}\!a {}^{hq_2h}\!p{}^{hq_2hq}\!a{}^{hq_2h}\!a,hq_2h)$
. Because we have
$hq_2h =hq_1h$
, this last element can be still rewritten in the form
$(a{}^h\!p_2{}^{hq_2}\!a{}^{hq_1h}\!a{}^{hq_1h}\!p{}^{hq_1hq}\!a{}^{hq_1 h}\!a,hq_1h)$
. All in all, we see in this manner that the function
$[hq_1h,((a,h)(p,q)(a,h),hqh)]$
sends the element
$(a{}^h\!p_2{}^{h q_2}\!a,hq_1h)$
to the element
$(a{}^h\!p_2{}^{hq_2}\!a{}^{hq_1h}\!a {}^{hq_1h}\!p{}^{hq_1hq}\!a{}^{hq_1h}\!a,hq_1h)$
, for any
$p_2\in P$
and
$q_2\in Q$
such that
$hq_2h=hq_1h$
. Thus, the function
$[hq_1h,((a,h) (p,q)(a,h),hqh)]$
acts on the set
$\vartheta ^{-1}(hq_1h)$
by multiplying the first components
$a{}^h\!p_2{}^{hq_2}\!a$
of the elements of this set, for any
$p_2\in P$
and
$q_2\in Q$
such that
$hq_2h=hq_1h$
, with the element
${}^{hq_1h}\!a{}^{hq_1h}\!p{}^{hq_1hq}\!a{}^{hq_1h}\!a$
on the right. In addition, we have also seen above that
${}^h\!a{}^h\!a ={}^h\!a$
, which entails that
${}^{hq_1h}\!a{}^{hq_1h}\!a={}^{hq_1 h}\!a$
, so that also the element
${}^{hq_1h}\!a$
is an idempotent in P. Therefore, the abovementioned element
${}^{hq_1h}\!a{}^{hq_1 h}\!p{}^{hq_1hq}\!a{}^{hq_1h}\!a$
belongs to the local submonoid
${}^{hq_1h}\!aP{}^{hq_1h}\!a$
of P.
Consider furthermore the partial mapping
$\psi $
of the last-named local submonoid
${}^{hq_1h}\!aP{}^{hq_1h}\!a$
of P into the local monoid of the derived category
$D_\vartheta $
at the vertex
$hq_1h$
sending, for every
$p\in P$
and
$q\in Q$
such that
$hq_1hqh=hq_1h$
, the element
${}^{hq_1h}\!a{}^{hq_1h}\!p{}^{hq_1hq}\!a{}^{hq_1h}\!a$
to the function
$[hq_1h,((a,h)(p,q)(a,h),hqh)]$
. From what has just been said in the previous paragraph, it follows that this partial mapping
$\psi $
is well defined. It is also clear that
$\psi $
is surjective. We proceed to show that the domain of this partial mapping
$\psi $
is a submonoid in the local submonoid
${}^{hq_1h}\!aP{}^{hq_1h}\!a$
of P and that the mapping
$\psi $
itself is a homomorphism of this submonoid onto the local monoid of
$D_\vartheta $
at
$hq_1h$
.
In order to verify that the domain of
$\psi $
is a subsemigroup in the local submonoid
${}^{hq_1h}\!aP{}^{hq_1h}\!a$
of P, let us consider any
$p,p'\in P$
and
$q,q'\in Q$
such that
$hq_1hqh=hq_1h$
and
$hq_1hq'h=hq_1h$
, and let us examine the corresponding elements
${}^{h q_1h}\!a{}^{hq_1h}\!p{}^{hq_1hq}\!a{}^{hq_1h}\!a$
and
${}^{hq_1h}\!a {}^{hq_1h}\!p'{}^{hq_1hq'}\!a{}^{hq_1h}\!a$
in
${}^{hq_1h}\!aP{}^{h q_1h}\!a$
. The first of these elements is such that the function
$[hq_1h,((a,h)(p,q)(a,h),hqh)]$
acts on the set
$\vartheta ^{-1}(hq_1 h)$
, which consists of all elements of the form
$(a{}^h\!p_2{}^{hq_2} \!a,hq_1h)$
, for all
$p_2\in P$
and
$q_2\in Q$
such that
$hq_2h=hq_1h$
, by multiplying the first components
$a{}^h\!p_2{}^{hq_2}\!a$
of these elements with the element
${}^{hq_1h}\!a{}^{hq_1h}\!p{}^{hq_1hq}\!a {}^{hq_1h}\!a$
on the right. Likewise, the function
$[hq_1h,((a,h) (p',q')(a,h),hq'h)]$
acts on the same set
$\vartheta ^{-1}(hq_1h)$
, by multiplying the first components
$a{}^h\!p_2{}^{hq_2}\!a$
of its elements with the element
${}^{hq_1h}\!a{}^{hq_1h}\!p'{}^{hq_1hq'}\!a {}^{hq_1h}\!a$
on the right. The composition
$[hq_1h,((a,h)(p,q)(a,h) (p',q')(a,h),hqhq'h)]$
of the mentioned functions
$[hq_1h,((a,h)(p,q) (a,h),hqh)]$
and
$[hq_1h,((a,h)(p',q')(a,h),hq'h)]$
then acts on the same set
$\vartheta ^{-1}(hq_1h)$
by multiplying the first components
$a {}^h\!p_2{}^{hq_2}\!a$
of its elements consecutively with the mentioned elements
${}^{hq_1h}\!a{}^{hq_1h}\!p{}^{hq_1hq}\!a{}^{hq_1h}\!a$
and
${}^{hq_1h}\!a{}^{hq_1h}\!p'{}^{hq_1hq'}\!a{}^{hq_1h}\!a$
on the right. However, the product of the latter two elements, which is equal to the element
${}^{hq_1h}\!a{}^{hq_1h}\!p{}^{hq_1hq}\!a{}^{hq_1h}\!a{}^{hq_1 h}\!p'{}^{hq_1hq'}\!a{}^{hq_1h}\!a$
, can be seen to be itself of the form
${}^{hq_1h}\!a{}^{hq_1h}\!p"{}^{hq_1hq"}\!\!a{}^{hq_1h}\!a$
, if we put
$p"=p{}^q\!aap'$
and
$q"=q'$
. Note also that
$hq_1hq"h=hq_1h q'h=hq_1h$
. Consequently, this product of the two elements in question belongs to the domain of the partial mapping
$\psi $
, and the mapping
$\psi $
sends this product to the composition
$[hq_1h,((a,h)(p,q)(a,h) (p',q')(a,h),hqhq'h)]$
of the abovementioned two functions. Therefore, the domain of
$\psi $
forms a subsemigroup in the local submonoid
${}^{hq_1h}\!aP{}^{hq_1h}\!a$
of P, and the mapping
$\psi $
itself is a homomorphism of semigroups. It remains to add that the identity on the set
$\vartheta ^{-1}(hq_1h)$
, which can clearly be represented as the function
$[hq_1h,((a,h),h)]$
, corresponds through the mapping
$\psi $
to the element
${}^{hq_1h}\!a{}^{hq_1h}\!a{}^{hq_1h}\!a{}^{hq_1 h}\!a$
, which is equal to the idempotent
${}^{hq_1h}\!a$
representing the identity of the local submonoid
${}^{hq_1h}\!aP{}^{hq_1h}\!a$
of P. Thus, this idempotent
${}^{hq_1h}\!a$
belongs to the domain of
$\psi $
. In this way, it turns out that the domain of
$\psi $
actually forms a submonoid in the local submonoid
${}^{hq_1h}\!aP{}^{hq_1h}\!a$
of P and that the mapping
$\psi $
itself is, in fact, a homomorphism of monoids.
The conclusions made at the end of the previous paragraph show that the local monoid of the derived category
$D_\vartheta $
at the vertex
$hq_1h$
divides the local submonoid
${}^{hq_1h}\!aP{}^{hq_1h}\!a$
of P. But we have mentioned above already that the local submonoids of P belong to the pseudovariety
$\mathbf {V}$
. Hence, it ensues that the local monoid of
$D_\vartheta $
at
$hq_1h$
also belongs to
$\mathbf {V}$
. Therefore, we may conclude that the local monoids of the derived category
$D_\vartheta $
belong to the pseudovariety
$\mathbf {V}$
, which we had to show.
The fact that the local monoids of
$D_\vartheta $
belong to
$\mathbf {V}$
means that the derived category
$D_\vartheta $
itself belongs to the pseudovariety
$\ell \mathbf {V}$
of finite categories. Now, we assume that the pseudovariety
$\mathbf {V}$
is local. That is, we assume that the equality
$g\mathbf {V}=\ell \mathbf {V}$
of pseudovarieties of finite categories holds. Consequently, the derived category
$D_\vartheta $
belongs to the pseudovariety
$g\mathbf {V}$
of finite categories generated by the pseudovariety
$\mathbf {V}$
. Recall also that the local submonoid
$hQh$
of Q belongs to
$\mathbf {W}$
. Now, we are in a position to apply the Derived Category Theorem to the homomorphism of monoids
$\vartheta :(a,h)(P*Q)(a,h)\to hQh$
. By this theorem, that is, by Theorem 2.1, we may conclude that the local submonoid
$(a,h)(P*Q)(a,h)$
of
$P*Q$
belongs to the semidirect product of pseudovarieties
$\mathbf {V}*\mathbf {W}$
. All in all, this shows that the local submonoids of
$P*Q$
belong to the pseudovariety
$\mathbf {V} *\mathbf {W}$
. Consequently, the semigroup
$P*Q$
itself belongs to the pseudovariety
$\mathbf {L}(\mathbf {V}*\mathbf {W})$
, as desired. ▪
Remark The condition that the pseudovariety
$\mathbf {V}$
of finite monoids should be local, in order to ensure that the inclusion of pseudovarieties of finite semigroups
$\mathbf {LV}*\mathbf {LW} \subseteq \mathbf {L}(\mathbf {V}*\mathbf {W})$
holds, appearing in Theorem 3.1, is thus sufficient, but it need not be necessary. Indeed, consider the pseudovariety
$\mathbf {J}$
of all finite
$\mathcal {J}$
-trivial monoids and the pseudovariety
$\mathbf {G}$
of all finite groups, for example. It is well known that the pseudovariety
$\mathbf {J}$
is not local—see Example 15.8 in Tilson’s paper [Reference Tilson10]. But it can be shown that the inclusion
$\mathbf {LJ}*\mathbf {LG}\subseteq \mathbf {L}(\mathbf {J}*\mathbf {G})$
still holds. In order to achieve this, we can proceed as in the proof of Theorem 3.1 until the last paragraph of this proof where the assumption that
$g\mathbf {V}=\ell \mathbf {V}$
is used. Now, in our present situation, we have
$g\mathbf {J}\subsetneqq \ell \mathbf {J}$
. However, it has been proved in [Reference Kad’ourek7] that, for every prime number r, the inclusion
$\ell \mathbf {J}\subseteq g(\mathbf {J}*\mathbf {Ab}_r)$
holds, where
$\mathbf {Ab}_r$
is the pseudovariety of all elementary abelian r-groups. Consequently, the derived category
$D_\vartheta $
now belongs to the pseudovariety
$g(\mathbf {J}*\mathbf {Ab}_r)$
. The local submonoid
$hQh$
of Q belongs to
$\mathbf {G}$
. Thus, as in the above proof, we can apply the Derived Category Theorem to the homomorphism
$\vartheta :(a,h)(P*Q)(a,h)\to hQh$
in order to show that the local submonoid
$(a,h)(P*Q)(a,h)$
of
$P*Q$
now belongs to
$(\mathbf {J}*\mathbf {Ab}_r)*\mathbf {G}$
. But we have
$(\mathbf {J} *\mathbf {Ab}_r)*\mathbf {G}=\mathbf {J}*(\mathbf {Ab}_r*\mathbf {G}) =\mathbf {J}*\mathbf {G}$
. Thus, the local submonoid
$(a,h)(P*Q)(a,h)$
of
$P*Q$
actually belongs to
$\mathbf {J}*\mathbf {G}$
. This shows that the local submonoids of
$P*Q$
belong to the pseudovariety
$\mathbf {J} *\mathbf {G}$
, and so the semigroup
$P*Q$
itself belongs to the pseudovariety
$\mathbf {L}(\mathbf {J}*\mathbf {G})$
, which accomplishes our task.
4 A necessary condition
It follows from the remark at the close of the previous section that it makes sense to look for necessary conditions which must be satisfied, if the inclusion of pseudovarieties of finite semigroups
$\mathbf {LV} *\mathbf {LW}\subseteq \mathbf {L}(\mathbf {V}*\mathbf {W})$
has to hold. In this section, we provide one such necessary condition, which will prove to be sufficiently strong for our later purposes. We denote by
$\mathbf {G}$
the pseudovariety of all finite groups and by
$\mathbf {R}$
the pseudovariety of all finite
$\mathcal {R}$
-trivial monoids. We are now going to verify the following statement:
Proposition 4.1 Let
$\mathbf {V}$
and
$\mathbf {W}$
be any pseudovarieties of finite monoids such that
$\mathbf {V}\nsubseteq \mathbf {G}$
and
$\mathbf {W} \nsubseteq \mathbf {R}$
. If the inclusion of pseudovarieties
$\mathbf {LV} *\mathbf {LW}\subseteq \mathbf {L}(\mathbf {V}*\mathbf {W})$
holds, then the inclusion of pseudovarieties of finite categories
$\ell \mathbf {V} \subseteq g(\mathbf {V}*\mathbf {W})$
holds.
Proof We first decipher what it means that
$\mathbf {W}\nsubseteq \mathbf {R}$
. If this condition has to hold, then the pseudovariety
$\mathbf {W}$
must contain some finite monoid which is not
$\mathcal {R}$
-trivial. Such a finite monoid must therefore contain some nontrivial regular
$\mathcal {R}$
-class. Then, there are several cases which may occur. Either this monoid contains some nontrivial regular
$\mathcal {H}$
-class, and therefore it contains some finite nontrivial subgroup. Or else this monoid contains no nontrivial regular
$\mathcal {H}$
-class, and hence it is an aperiodic monoid. Then, it may happen that some nontrivial regular
$\mathcal {R}$
-class of this monoid contains several idempotents. In this case, our monoid contains some finite nontrivial right zero subsemigroup, and hence it contains the two-element right zero semigroup
$R_2$
as a subsemigroup. Finally, our monoid may be aperiodic, and it may contain no nontrivial right zero subsemigroup. But still this monoid contains some nontrivial regular
$\mathcal {R}$
-class. Such a regular
$\mathcal {R}$
-class then must contain some idempotent e and some nonidempotent element s lying in different one-element
$\mathcal {H}$
-classes. The inverse t of s then belongs to another regular
$\mathcal {R}$
-class of our monoid, we have
$e=st$
, and the idempotent
$f=ts$
also belongs to the latter regular
$\mathcal {R}$
-class. Again, the idempotent f and the inverse t of s lie in different one-element
$\mathcal {H}$
-classes. Because our monoid contains no nontrivial right zero subsemigroup, the element t must also be nonidempotent. Thus, the elements
$e,f,s,t$
constitute the nontrivial
$\mathcal {J}$
-class of the five-element aperiodic Brandt semigroup
$B_2$
. More precisely, this five-element aperiodic Brandt semigroup
$B_2$
can be seen to divide our finite monoid. Thus, all in all, we see that we will have to treat separately three cases: either the pseudovariety
$\mathbf {W}$
contains some finite nontrivial group, or
$\mathbf {W}$
contains the monoid
$R_2^1$
obtained by adjoining an identity to the two-element right zero semigroup
$R_2$
, or else
$\mathbf {W}$
contains the monoid
$B_2^1$
obtained by adjoining an identity to the five-element aperiodic Brandt semigroup
$B_2$
.
We wish by now to verify the inclusion of pseudovarieties of finite categories
$\ell \mathbf {V}\subseteq g(\mathbf {V}*\mathbf {W})$
. Thus, let C be any finite category lying in the pseudovariety
$\ell \mathbf {V}$
. We clearly may assume that C has several vertices. Then, all local monoids of C at all vertices of C belong to the pseudovariety
$\mathbf {V}$
. Because
$\mathbf {V}\nsubseteq \mathbf {G}$
, the pseudovariety
$\mathbf {V}$
must contain a nontrivial monoid which is not a group. Such a monoid must contain the two-element semilattice
$Y_2$
as a submonoid. Therefore, the two-element semilattice
$Y_2$
belongs to
$\mathbf {V}$
. Consequently, the pseudovariety
$\mathbf {V}$
contains with each of its monoids M also the monoid
$M^0$
obtained by adjoining a new zero
$0$
to M, because
$M^0$
is clearly isomorphic to a Rees quotient of the direct product
$M\times Y_2$
. Consider now the consolidation S of the category C constructed toward the end of Section 2. Then, every submonoid of S is contained as a subsemigroup in one of the monoids of the form
$M^0$
obtained by adjoining the new zero
$0$
of the consolidation S to a local monoid M of C. Thus, every submonoid of S divides a monoid
$M^0$
of this form as a monoid. Now, each local monoid M of C belongs to
$\mathbf {V}$
, and hence, as we have just seen, also the monoid
$M^0$
belongs to
$\mathbf {V}$
. This entails that each submonoid of S belongs to the pseudovariety
$\mathbf {V}$
, and hence S itself is a finite semigroup lying in the pseudovariety
$\mathbf {LV}$
. Let further
$S^1$
be the monoid obtained by adjoining an identity to the semigroup S. We have seen at the end of Section 2 that then the category C divides the monoid
$S^1$
. Thus, in order to show that the category C belongs to the pseudovariety
$g(\mathbf {V}*\mathbf {W})$
, it suffices to document that the monoid
$S^1$
belongs to
$\mathbf {V}*\mathbf {W}$
. This is equivalent to saying that the monoid
$S^1$
belongs to
$\mathbf {L}(\mathbf {V}*\mathbf {W})$
. Because we assume that the inclusion of pseudovarieties
$\mathbf {LV} *\mathbf {LW}\subseteq \mathbf {L}(\mathbf {V}*\mathbf {W})$
holds, we will be done if we show that the monoid
$S^1$
belongs to the semidirect product of pseudovarieties
$\mathbf {LV}*\mathbf {LW}$
.
Now, we have to distinguish the three cases discussed in the first paragraph of this proof. Thus, assume first that the pseudovariety
$\mathbf {W}$
contains some finite nontrivial group G. Then, of course, this group G belongs to
$\mathbf {LW}$
. Because
$\mathbf {W}$
is a pseudovariety, we may further suppose that this group G has sufficiently many elements, at least as many as there are vertices in the category C. In this situation, let
$\iota :V(C)\to G$
be an injective mapping. Recall that the semigroup S belongs to
$\mathbf {LV}$
. Consider further the wreath product
$S\circ G$
. Then, this wreath product belongs to
$\mathbf {LV}*\mathbf {LW}$
. Recall also that this wreath product
$S\circ G$
is just the semidirect product
$S^G*G$
determined by the left action of G on
$S^G$
assigning to every element
$g\in G$
and every function
$f:G\to S$
the function
${}^g\!f:G\to S$
given by the formula
$g'({}^g\!f)=(g'g)f$
, for all
$g'\in G$
. Now, for any vertices
$v,w\in V(C)$
and for every edge
$e\in E(C)$
such that
$\alpha (e)=v$
and
$\omega (e)=w$
, consider the function
$\bar {e}:G\to S$
defined by the rules
$\iota (v)\bar {e}=e$
and
$h\bar {e}=0$
for all other values
$h\in G$
. Let further
$o:G\to S$
be the zero function assigning the zero
$0$
of S to all elements of G. Then, the set
$I=\{(o,h): h\in G\}$
can be seen to be an ideal in the semigroup
$S\circ G$
. Thus, we may consider the corresponding Rees quotient semigroup
$(S\circ G)/I$
. Then, it can be routinely verified that the mapping
$\eta :S\to (S\circ G)/I$
given, for all
$e\in E(C)$
with
$\alpha (e)=v$
and
$\omega (e)=w$
, by the rules
$\eta (e)=(\bar {e},\iota (v)^{-1}\iota (w))$
and
$\eta (0)=I$
, is an injective homomorphism of semigroups. Thus, the semigroup S is embedded into the semigroup
$(S\circ G)/I$
. In addition, consider the mapping
$\lambda :G\to S$
assigning, for all vertices
$u\in V(C)$
, the identity of the local monoid of C at u to the element
$\iota (u)$
, and assigning the zero
$0$
of S to all other elements of G. Then, the pair
$(\lambda ,1)$
, where
$1$
is the identity of the group G, is an idempotent of
$S\circ G$
, and hence of
$(S\circ G)/I$
, which acts as an identity element with respect to all pairs
$(\bar {e},h)$
which may appear in the subsemigroup
$\eta (S)$
of
$(S\circ G)/I$
. Thus, the set
$\eta (S)$
together with the pair
$(\lambda ,1)$
forms a monoid which is isomorphic to the monoid
$S^1$
. This monoid is inserted in
$(S\circ G)/I$
as a subsemigroup, and so it belongs to the pseudovariety
$\mathbf {LV}*\mathbf {LW}$
. Consequently, also the monoid
$S^1$
belongs to
$\mathbf {LV}*\mathbf {LW}$
, as desired.
Assume further that the pseudovariety
$\mathbf {W}$
contains the monoid
$R_2^1$
obtained by adjoining an identity to the two-element right zero semigroup
$R_2$
. Because
$\mathbf {W}$
is a pseudovariety, it contains arbitrary finitary direct powers of the monoid
$R_2^1$
. Consequently, for every positive integer n,
$\mathbf {W}$
contains the monoid
$R_n^1$
obtained by adjoining an identity to the n-element right zero semigroup
$R_n$
. Of course, this monoid
$R_n^1$
then belongs to
$\mathbf {LW}$
. Suppose further that the number n is at least as large as the number of vertices of the category C. In this situation, let
$\iota :V(C)\to R_n$
be an injective mapping. Recall that the semigroup S belongs to
$\mathbf {LV}$
. Consider further the wreath product
$S \circ R_n^1$
. Then, this wreath product belongs to
$\mathbf {LV}*\mathbf {LW}$
. Recall again that this wreath product
$S\circ R_n^1$
is just the semidirect product
$S^{R_n^1}*R_n^1$
determined by the left action of
$R_n^1$
on
$S^{R_n^1}$
assigning to every element
$r\in R_n^1$
and every function
$f:R_n^1\to S$
the function
${}^r\!f:R_n^1\to S$
given by the formula
$r'({}^r\!f)=(r'r)f$
, for all
$r'\in R_n^1$
. Now, for any vertices
$v,w\in V(C)$
and for every edge
$e\in E(C)$
such that
$\alpha (e)=v$
and
$\omega (e)=w$
, consider the function
$\bar {e}:R_n^1 \to S$
defined by the rules
$\iota (v)\bar {e}=e$
and
$\rho \bar {e}=0$
for all other values
$\rho \in R_n^1$
. Let further
$o:R_n^1\to S$
be the zero function assigning the zero
$0$
of S to all elements of
$R_n^1$
. Then, the set
$I=\{(o,\rho ):\rho \in R_n^1\}$
can be seen to be an ideal in the semigroup
$S\circ R_n^1$
. Thus, we may consider the corresponding Rees quotient semigroup
$(S\circ R_n^1)/I$
. Then, it can be routinely verified that the mapping
$\eta :S\to (S\circ R_n^1)/I$
given, for all
$e\in E(C)$
with
$\alpha (e)=v$
and
$\omega (e)=w$
, by the rules
$\eta (e)=(\bar {e},\iota (w))$
and
$\eta (0)=I$
, is an injective homomorphism of semigroups. Thus, the semigroup S is embedded into the semigroup
$(S\circ R_n^1)/I$
. In addition, consider the mapping
$\lambda :R_n^1 \to S$
assigning, for all vertices
$u\in V(C)$
, the identity of the local monoid of C at u to the element
$\iota (u)$
, and assigning the zero
$0$
of S to all other elements of
$R_n^1$
. Then, the pair
$(\lambda ,1)$
is an idempotent of
$S\circ R_n^1$
and hence of
$(S\circ R_n^1)/I$
, which acts as an identity element with respect to all pairs
$(\bar {e},\rho )$
which may appear in the subsemigroup
$\eta (S)$
of
$(S\circ R_n^1)/I$
. Thus, the set
$\eta (S)$
together with the pair
$(\lambda ,1)$
forms a monoid which is isomorphic to the monoid
$S^1$
. This monoid is inserted in
$(S\circ R_n^1)/I$
as a subsemigroup, and so it belongs to the pseudovariety
$\mathbf {LV}*\mathbf {LW}$
. Consequently, also the monoid
$S^1$
belongs to
$\mathbf {LV}*\mathbf {LW}$
, as desired.
Finally, assume that the pseudovariety
$\mathbf {W}$
contains the monoid
$B_2^1$
obtained by adjoining an identity to the five-element aperiodic Brandt semigroup
$B_2$
. For every positive integer n, denote by
$B_n$
the aperiodic Brandt semigroup containing
$n^2+1$
elements. Furthermore, denote by
$B_n^1$
the monoid obtained by adjoining an identity to
$B_n$
. As usual, we may represent the nontrivial
$\mathcal {J}$
-class of
$B_n$
as the direct product
$\{1,2,\dots ,n\}\times \{1,2,\dots ,n\}$
, where the multiplication of pairs
$(i,j)$
from this direct product is given by the standard rule: the product
$(i,j)(i',j')$
is equal to
$(i,j')$
if
$j=i'$
, and it is equal to the zero
$0$
of
$B_n$
otherwise. Because
$\mathbf {W}$
is a pseudovariety containing the monoid
$B_2^1$
, it contains, in fact, the monoids
$B_n^1$
, for all positive integers n, because these monoids
$B_n^1$
can be seen to divide the direct products of a suitable number of copies of the monoid
$B_2^1$
. Of course, these monoids
$B_n^1$
then belong to
$\mathbf {LW}$
. Suppose further that the number n is at least as large as the number of vertices of the category C. In this situation, let
$\pi :V(C)\to \{1,2,\dots ,n\}$
be an injective mapping. Consider further the set
$\{0,1,2,\dots ,n\}$
and the semidirect product
$S^{\{0,1,2,\dots ,n\}}*B_n^1$
determined by the following left action of
$B_n^1$
on
$S^{\{0,1,2,\dots ,n\}}$
. For any function
$g:\{0,1,2,\dots ,n\}\to S$
and for any element
$b\in B_n^1$
, let
${}^b\!g:\{0,1,2,\dots ,n\}\to S$
be the function given by the formula
$i({}^b\!g)=(ib)g$
, for all
$i\in \{0,1,2,\dots ,n\}$
, where the product
$ib$
is determined by the following rules: if
$i=0$
, then
$ib=0$
, if
$i\in \{1,2,\dots ,n\}$
, then
$i1=i$
,
$i(i,j)=j$
, for all
$j\in \{1,2,\dots ,n\}$
, and
$ib=0$
otherwise. In this way, a left action of
$B_n^1$
on
$S^{\{0,1,2,\dots ,n\}}$
is indeed obtained. This left action is left unitary. Because the semigroup S lies in
$\mathbf {LV}$
and the monoid
$B_n^1$
lies in
$\mathbf {LW}$
, the semidirect product
$S^{\{0,1,2,\dots ,n\}}*B_n^1$
belongs to
$\mathbf {LV}*\mathbf {LW}$
. Now, for any vertices
$v,w\in V(C)$
and for every edge
$e\in E(C)$
such that
$\alpha (e)=v$
and
$\omega (e)=w$
, consider the function
$\bar {e}: \{0,1,2,\dots ,n\}\to S$
defined by the rules
$\pi (v)\bar {e}=e$
and
$j\bar {e}=0$
for all other values
$j\in \{0,1,2,\dots ,n\}$
. Let further
${\scriptstyle \mathcal {O}}:\{0,1,2,\dots ,n\}\to S$
be the zero function assigning the zero
$0$
of S to all numbers in
$\{0,1,2,\dots ,n\}$
. Then, the set
$\Im =\{({\scriptstyle \mathcal {O}},c):c\in B_n^1\}$
can be seen to be an ideal in the semigroup
$S^{\{0,1,2,\dots ,n\}}*B_n^1$
. Thus, we may consider the corresponding Rees quotient semigroup
$(S^{\{0,1,2,\dots ,n\}}*B_n^1)/\Im $
. Then, it can be routinely verified that the mapping
$\mu :S\to (S^{\{0,1,2,\dots ,n\}}*B_n^1)/\Im $
given, for all
$e\in E(C)$
with
$\alpha (e)=v$
and
$\omega (e)=w$
, by the rules
$\mu (e)=(\bar {e},(\pi (v),\pi (w)))$
and
$\mu (0)=\Im $
, is an injective homomorphism of semigroups. Thus, the semigroup S is embedded into the semigroup
$(S^{\{0,1,2,\dots ,n\}}*B_n^1)/\Im $
. In addition, consider the mapping
$\delta :\{0,1,2,\dots ,n\}\to S$
assigning, for all vertices
$u\in V(C)$
, the identity of the local monoid of C at u to the number
$\pi (u)$
, and assigning the zero
$0$
of S to all other numbers from
$\{0,1,2,\dots ,n\}$
. Then, the pair
$(\delta ,1)$
is an idempotent of
$S^{\{0,1,2,\dots ,n\}}*B_n^1$
and hence of
$(S^{\{0,1,2, \dots ,n\}}*B_n^1)/\Im $
, which acts as an identity element with respect to all pairs
$(\bar {e},c)$
which may appear in the subsemigroup
$\mu (S)$
of
$(S^{\{0,1,2,\dots ,n\}}*B_n^1)/\Im $
. Thus, the set
$\mu (S)$
together with the pair
$(\delta ,1)$
forms a monoid which is isomorphic to the monoid
$S^1$
. This monoid is inserted in
$(S^{\{0,1,2,\dots ,n\}}*B_n^1) /\Im $
as a subsemigroup, and so it belongs to the pseudovariety
$\mathbf {LV}*\mathbf {LW}$
. Consequently, also the monoid
$S^1$
belongs to
$\mathbf {LV}*\mathbf {LW}$
, as desired. ▪
5 Conclusions
As it has been advised already in the introduction, in this concluding section of the present paper, we deduce consequences from the results obtained up to now in the previous two sections of this paper. But prudence is in place in this regard. In the corollary stated below, which includes the main message furnished in this section, it is claimed that the condition that the pseudovariety
$\mathbf {V}$
should be local is necessary and sufficient for a nontrivial semidirectly closed pseudovariety
$\mathbf {V}$
of finite monoids to have the property that its localization
$\mathbf {LV}$
is also semidirectly closed. However, the fact that this condition is sufficient can be derived using a direct argument based on very well-known results on pseudovarieties of finite semigroups and monoids. We are going to indicate now how this can be done.
It has been shown by Eilenberg in [Reference Eilenberg4] that, for every pseudovariety
$\mathbf {U}$
of finite monoids, the inclusion
$\mathbf {U}*\mathbf {D}\subseteq \mathbf {LU}$
holds, where
$\mathbf {D}$
is the pseudovariety of definite semigroups—see Section 12 in Chapter V of [Reference Eilenberg4]. Furthermore, Tilson has proved in Section 17 of his paper [Reference Tilson10] that here the equality
$\mathbf {U}*\mathbf {D} =\mathbf {LU}$
holds in place of the mentioned inclusion if, and only if, the pseudovariety
$\mathbf {U}$
is local. In addition, Stiffler has shown in Section 3 of [Reference Stiffler9] that the pseudovariety
$\mathbf {D}$
is equal to the semidirectly closed pseudovariety
$[\kern-1.5pt[R_2]\kern-1.5pt]$
generated by the two-element right zero semigroup
$R_2$
. Thus, in particular,
$\mathbf {D}$
is a semidirectly closed pseudovariety. Moreover, it has been observed by Almeida in Section 10.10 of his book [Reference Almeida1] that, from the results established by Stiffler in Section 2 of [Reference Stiffler9], it follows that, for every pseudovariety
$\mathbf {U}$
of finite monoids, the inclusion of semidirect products
$\mathbf {D} *\mathbf {U}\subseteq \mathbf {U}*\mathbf {D}$
holds.
Therefore, if now our pseudovariety
$\mathbf {V}$
of finite monoids is semidirectly closed and local, then we can draw from the previous remarks the following sequence of relationships:
$$\begin{align*}\mathbf{LV}*\mathbf{LV}=\mathbf{V}*\mathbf{D}*\mathbf{V} *\mathbf{D}\subseteq\mathbf{V}*\mathbf{V}*\mathbf{D}*\mathbf{D} =\mathbf{V}*\mathbf{D}=\mathbf{LV}.\end{align*}$$
But this shows that
$\mathbf {LV}$
is a semidirectly closed pseudovariety of finite semigroups. This simple reflection confirms the sufficiency of the condition that the pseudovariety
$\mathbf {V}$
should be local in the corollary given below. In its proof, however, we offer another argument based on our present Theorem 3.1.
Corollary 5.1 Let
$\mathbf {V}$
be any nontrivial semidirectly closed pseudovariety of finite monoids. Then, the pseudovariety
$\mathbf {LV}$
of finite semigroups is also semidirectly closed if, and only if, the given pseudovariety
$\mathbf {V}$
is local.
Proof We start by considering the case when our nontrivial semidirectly closed pseudovariety
$\mathbf {V}$
of finite monoids satisfies the inclusion
$\mathbf {V}\subseteq \mathbf {R}$
. Then, every monoid in
$\mathbf {V}$
is
$\mathcal {R}$
-trivial, and hence it is aperiodic. But still our pseudovariety
$\mathbf {V}$
is nontrivial, and so it contains some nontrivial monoid. This monoid contains only trivial subgroups, and therefore it must contain the two-element semilattice
$Y_2$
as a submonoid. Thus,
$\mathbf {V}$
contains this semilattice
$Y_2$
. Because the pseudovariety
$\mathbf {V}$
is semidirectly closed, it must contain the entire semidirectly closed pseudovariety
$[\kern-1.5pt[Y_2]\kern-1.5pt]$
generated by this semilattice
$Y_2$
. However, it has been shown by Stiffler in Section 3 of [Reference Stiffler9] that this semidirectly closed pseudovariety
$[\kern-1.5pt[Y_2]\kern-1.5pt]$
is already equal to the whole pseudovariety
$\mathbf {R}$
. Thus,
$\mathbf {R}$
is a semidirectly closed pseudovariety. In addition, if our nontrivial semidirectly closed pseudovariety
$\mathbf {V}$
satisfies
$\mathbf {V} \subseteq \mathbf {R}$
, then we inevitably have
$\mathbf {V}=\mathbf {R}$
. Thus, in this case, we have to treat only the sole pseudovariety
$\mathbf {R}$
of all
$\mathcal {R}$
-trivial monoids.
It has been verified by Eilenberg in [Reference Eilenberg4] that the mentioned pseudovariety
$\mathbf {R}$
satisfies the equality
$\mathbf {R}*\mathbf {D}=\mathbf {LR}$
—see again Section 12 in Chapter V of [Reference Eilenberg4]. Hence, in view of the remarks given prior to this corollary, we may conclude that the pseudovariety
$\mathbf {R}$
is local. Recall also from the above remarks that, in particular, the inclusion
$\mathbf {D}*\mathbf {R}\subseteq \mathbf {R}*\mathbf {D}$
holds. From this finding and from the fact that both pseudovarieties
$\mathbf {R}$
and
$\mathbf {D}$
are semidirectly closed, it ensues that also the pseudovariety
$\mathbf {R}*\mathbf {D}$
is semidirectly closed. That is, the pseudovariety
$\mathbf {LR}$
is semidirectly closed. Thus, all in all, both pseudovarieties
$\mathbf {R}$
and
$\mathbf {LR}$
are semidirectly closed, and
$\mathbf {R}$
is local. This shows that, for the pseudovariety
$\mathbf {R}$
, the statement of the above corollary holds.
Thus, we may further assume that our nontrivial semidirectly closed pseudovariety
$\mathbf {V}$
of finite monoids satisfies
$\mathbf {V} \nsubseteq \mathbf {R}$
. If we incidentally have
$\mathbf {V}\subseteq \mathbf {G}$
, then
$\mathbf {V}$
is a nontrivial pseudovariety of finite groups, and hence it is local by Proposition 13.8 in Tilson’s paper [Reference Tilson10]. The assumption that
$\mathbf {V}$
is semidirectly closed means that the equality
$\mathbf {V}*\mathbf {V}=\mathbf {V}$
holds. Now, if the pseudovariety
$\mathbf {LV}$
of finite semigroups is also semidirectly closed, that is, if the inclusion
$\mathbf {LV}*\mathbf {LV} \subseteq \mathbf {LV}$
holds, or, yet otherwise stated, if the inclusion
$\mathbf {LV}*\mathbf {LV}\subseteq \mathbf {L}(\mathbf {V}*\mathbf {V})$
holds, and if we have
$\mathbf {V}\nsubseteq \mathbf {G}$
, then, by Proposition 4.1, the inclusion of pseudovarieties of finite categories
$\ell \mathbf {V}\subseteq g(\mathbf {V}*\mathbf {V})$
holds, that is, the inclusion
$\ell \mathbf {V}\subseteq g\mathbf {V}$
holds. But this means precisely that the pseudovariety
$\mathbf {V}$
is local, even if
$\mathbf {V}\nsubseteq \mathbf {G}$
.
On the other hand, if the pseudovariety
$\mathbf {V}$
is local, then, by Theorem 3.1, the inclusion
$\mathbf {LV}*\mathbf {LV}\subseteq \mathbf {L}(\mathbf {V}*\mathbf {V})$
holds, that is, the inclusion
$\mathbf {LV}*\mathbf {LV}\subseteq \mathbf {LV}$
holds. But this means that
$\mathbf {LV}$
is a semidirectly closed pseudovariety of finite semigroups. This finding verifies the statement of the above corollary in the case when the pseudovariety
$\mathbf {V}$
satisfies
$\mathbf {V} \nsubseteq \mathbf {R}$
.▪
In a previous paper [Reference Kad’ourek8], we have produced a continuum of semidirectly closed pseudovarieties of finite monoids which are not local. More concretely, we have shown in [Reference Kad’ourek8] that, for every prime number p, there is a continuum of semidirectly closed pseudovarieties of finite monoids whose intersection with the pseudovariety
$\mathbf {G}$
of all finite groups is equal to the pseudovariety
$\mathbf {G}_p$
of all finite p-groups. Moreover, we have also shown in [Reference Kad’ourek8] that there is a continuum of semidirectly closed pseudovarieties of finite monoids whose intersection with the pseudovariety
$\mathbf {G}$
of all finite groups is equal to the pseudovariety
$\mathbf {G}_{\mathrm {sol}}$
of all finite solvable groups. In order to document these two assertions, we have indeed constructed in [Reference Kad’ourek8] a continuum of such pseudovarieties of finite monoids, in each of these two cases. In the last section of [Reference Kad’ourek8], we have eventually shown that all of these pseudovarieties so constructed are not local. Thus, according to Corollary 5.1, we obtain a continuum of semidirectly closed pseudovarieties of finite monoids whose localizations are pseudovarieties of finite semigroups that are not semidirectly closed. This finding answers in the negative the question posed by Eilenberg in [Reference Eilenberg4] that has been addressed in the introduction to the present paper.
Acknowledgment
I would like to thank the referee whose eagle eyes have spotted a mistake left in Proposition 4.1 in the preprint version of this paper.
