2.1 Quandles

Recall that a magma is a pair $(X,\ast )$ , where X is a set and $\ast \colon X^2\to X$ is a binary operation. If $(X,\ast )$ and $(Y,\cdot )$ are magmas, then a homomorphism is a function $f\colon X\to Y$ such that $f(w\ast x)=f(w)\cdot f(x)$ for all $w,x\in X$ . For all $x\in X$ , define the corresponding right multiplication map $R_x\colon X\to X$ by $y\mapsto y\ast x$ and define the left multiplication map $L_x\colon X\to X$ by $y\mapsto x\ast y$ .

In the following, let $(X,\ast )$ be a magma. We consider several important varieties of magmas.

• • We say that $(X,\ast )$ is idempotent if $x\ast x=x$ for all $x\in X$ .

• • We say that $(X,\ast )$ is a rack if, for all $x\in X$ , the right multiplication map $R_x$ is an automorphism of X. In particular, $\ast $ distributes over itself on the right.

• • Quandles are idempotent racks. Kei are quandles such that, for all $x\in X$ , the right multiplication map $R_x$ is an involution.

• • Equip $X^2$ with the product magma structure. We say that $(X,\ast )$ is medial if ${\ast \colon X^2\to X}$ is a magma homomorphism. Several authors prefer the terms abelian or entropic.

• • We say that $(X,\ast )$ is Latin or a left quasigroup if, for all $x\in X$ , the left multiplication map $L_x$ is a permutation.

• • We say that $(X,\ast )$ is commutative or symmetric if $x\ast y=y\ast x$ for all $x,y\in X$ ; that is, $L_x=R_x$ for all $x\in X$ .

The following is well-known and straightforward.

2.1.1 Main examples

Two auxiliary results of this article (Proposition 4.1 and Corollary 4.2) state that all medial Latin quandles and medial commutative quandles fall into the following two classes of examples.

Example 2.2 [Reference Jedlička, Pilitowska, Stanovský and Zamojska-Dzienio10, Reference Nelson16]

Alexander quandles, also called affine quandles, form an important class of medial quandles. Given an automorphism $\varphi $ of an abelian group A, define the Alexander quandle ${\mathrm {Alex}}(A,\varphi ):= (A,\ast )$ by

(2.1) $$ \begin{align} x\ast y:= \varphi(x)+(\mathrm{id}-\varphi)(y). \end{align} $$

Note that ${\mathrm {Alex}}(A,\varphi )$ is Latin if and only if $\mathrm {id}-\varphi $ is an automorphism of A (cf. [Reference Jedlička, Pilitowska, Stanovský and Zamojska-Dzienio10]). Actually, every medial Latin quandle is isomorphic to an Alexander quandle of this form (see Proposition 4.1).

Example 2.3 Our prototypical examples of commutative quandles generalize the commutative quandles constructed in [Reference Bardakov and Elhamdadi1, Example 5.1]; we call these midpoint quandles. Let R be a unital ring in which $2$ is invertible, and let M be a left R-module. (Note that M can also be viewed as a module over $\mathbb {D}$ .) We define the midpoint quandle $M_{\mathrm {mid}}:= (M,\ast )$ by

$$\begin{align*}x\ast y:= \frac{1}{2}(x+y), \end{align*}$$

which is commutative. In particular, let $n\geq 0$ , let $R:= \mathbb {D}$ , and let $M:= \mathbb {Z}/(2n+1)$ be the cyclic group of order $2n+1$ . Then we call $C_{2n+1}:= M_{\mathrm {mid}}$ the cyclic midpoint quandle of order $2n+1$ . Note that this definition of $C_{2n+1}$ coincides with the one in [Reference Bardakov and Elhamdadi1] because $2^{-1}=n+1$ in M.

Remark 2.4 Midpoint quandles are a special class of so-called weighted average quandles (see [Reference Burrows and Tuffley3]), which are themselves a special class of Alexander quandles. Indeed, if $M_{\mathrm {mid}}$ is a midpoint quandle, then multiplication by $1/2$ is an R-module automorphism of M whose corresponding Alexander quandle is precisely $M_{\mathrm {mid}}$ . The converse also holds. In particular, all midpoint quandles are medial.

2.2 Affine modules

Given a ring R, we define the category $\mathsf {AffMod}_R$ of affine modules over R (also called affine R-modules) as follows. The objects of $\mathsf {AffMod}_R$ are left R-modules along with the empty set.Footnote ¹ The morphisms are sums of R-module homomorphisms and constant maps, which we call affine transformations.

The following is straightforward.

In other words, two nonempty affine R-modules are isomorphic if and only if they are isomorphic as left R-modules in the usual sense.

Given a ring homomorphism $R\to S$ , we can define the restriction of scalars functor $\text {res}\colon \mathsf {AffMod}_S\to \mathsf {AffMod}_R$ in exactly the same way as in ordinary module theory. In particular, we make the following observation.

Remark 2.7 Our definition of $\mathsf {AffMod}_R$ is one of many equivalent ways to define the category of affine R-modules. The usual definition (as in [Reference Csákány4, Reference Lawvere14]) is as the category $\mathcal {A}$ of set-theoretic models of the Lawvere theory whose n-ary operations are precisely the n-tuples $(r_1,\dots ,r_n)\in R^n$ such that $\sum ^n_{i=1}r_i=1$ ; such an operation represents linear combinations $(x_1,\dots ,x_n)\mapsto \sum ^n_{i=1}r_ix_i$ . In particular, the empty set is an object in $\mathcal {A}$ due to the lack of a nullary operation.

For fun, we encourage the reader to verify that the forgetful functor $\mathsf {AffMod}_R\to \mathcal {A}$ is an equivalence of categories. As a hint, essential surjectivity is shown as follows. Given a nonempty object $M\in \mathcal {C}$ , fix a basepoint $0\in M$ . Define the following left R-module structure $(\oplus ,\cdot )$ on M, where $0$ plays the role of the identity element of ${(M,\oplus )}$ as an abelian group:

$$\begin{align*}x\oplus y:= 1x+1y-1(0), \qquad r\cdot x:= rx+(1-r)0. \end{align*}$$

3.1 Non-mediality

The first part of [Reference Bardakov and Elhamdadi1, Question 7.1] asks whether every finite commutative quandle can be written as a direct sum of cyclic midpoint quandles $C_{2n+1}$ . This holds for all finite medial commutative quandles (see Theorem 6.4) but not for all finite commutative quandles.

To give a negative answer to the first part of [Reference Bardakov and Elhamdadi1, Question 7.1], it suffices to find a finite commutative quandle that is not medial. This is because cyclic midpoint quandles are medial (see Remark 2.4), and the direct sum of medial quandles is also medial.

Indeed, Kepka and Němec [Reference Kepka and Němec13, Corollary 12.5] constructed such two such quandles in 1981.Footnote ² They called these quandles $D(1)$ and $D(3)$ ; these quandles are the first and third entries of [Reference Stanovský21, Example 3.4]. We refer the reader to these two articles or the author’s blog post [Reference Ta24, Section 3.1] for a detailed description of these quandles and their failure to be medial.

It is worth noting that $D(1)$ and $D(3)$ are quandles of order 81, and no other non-medial commutative quandles of order up to 81 exist [Reference Kepka and Němec13, Corollary 12.5]. Therefore, Theorem 6.4 provides a positive answer to the first part of [Reference Bardakov and Elhamdadi1, Question 7.1] for all commutative quandles up to order 80.

3.2 Cocommutative quandles

The second part of [Reference Bardakov and Elhamdadi1, Question 7.1] seeks a description of commutative quandles whose duals are commutative. Recall that if $(X,\ast )$ is a rack, then the dual rack $X^{\mathrm {op}}:= (X,\overline {\ast })$ is the rack defined by

$$\begin{align*}x\mathop{\overline{\ast}}y:= R_y^{-1}(x). \end{align*}$$

See [Reference Burrows and Tuffley3] for a proof that $X^{\mathrm {op}}$ is a rack.

We will say that a rack $(X,\ast )$ is cocommutative if $X^{\mathrm {op}}$ is commutative. We completely characterize cocommutative racks and, in particular, commutative cocommutative quandles. This solves [Reference Bardakov and Elhamdadi1, Question 7.1].

Proof First, note that condition (3) is equivalent to the statement that, for all $x,y\in X$ ,

(3.1) $$ \begin{align} x\ast (x\ast y)=y. \end{align} $$

(1) $\implies $ (2): If X is cocommutative, then Lemma 2.1 implies that $X^{\mathrm {op}}$ is a quandle. That is, $R_x^{-1}(x)=x$ for all $x\in X$ . Applying $R_x$ to both sides shows that X is also a quandle.

(2) $\implies $ (3): Suppose that X is a cocommutative quandle. Given $x,y\in X$ , let ${z:= R_y^{-1}(x)}$ , so $z=R_x^{-1}(y)$ by assumption. Since X is a quandle, it follows that

$$\begin{align*}x\ast(x\ast y)=(z\ast y)\ast(x\ast y)=(z\ast x)\ast y=y\ast y=y, \end{align*}$$

which verifies (3.1).

(3) $\implies $ (1): Given $x,y\in X$ , let $z:= R_y^{-1}(x)$ . We have to show that $z=R_x^{-1}(y)$ . Indeed,

$$\begin{align*}R_x(z)=R_{R_y(z)}(z)=z\ast (z\ast y)=y. \end{align*}$$

In the last equality, we have used (3.1). Since $R_x$ is invertible, the proof is complete.

4.1 Essential surjectivity

Recall that a quasigroup is a magma such that all multiplication maps $L_x$ and $R_x$ are permutations. In particular, all Latin quandles and (by Lemma 2.1) commutative quandles are quasigroups.

In the medial case, this observation allows us to appeal to the classical Bruck–Murdoch–Toyoda theorem, which states the following: for every medial quasigroup $(Q,\ast )$ , there exists an abelian group A, a fixed element $z\in A$ , and two commuting automorphisms $\varphi $ and $\psi $ of A such that $(Q,\ast )$ is isomorphic to the medial quasigroup $(A,\cdot )$ defined by

(4.1) $$ \begin{align} x\cdot y:=\varphi(x)+\psi(y)+z. \end{align} $$

Proof By Proposition 4.1, it suffices to show that every commutative Alexander quandle $ {\mathrm {Alex}}(A,\varphi )$ is a midpoint quandle. Indeed, commutativity implies that

$$\begin{align*}\varphi(x)=x\ast 0=0\ast x=x-\varphi(x) \end{align*}$$

for all $x\in A$ ; that is, $2\varphi =\mathrm {id}$ . Since $\varphi $ is invertible, it follows that multiplication by $2$ is invertible. Hence, A is a module over $\mathbb {D}$ , and $\varphi $ is multiplication by $1/2$ , so $ {\mathrm {Alex}}(A,\varphi )=A_{\mathrm {mid}}$ .

5.1 Medial Latin quandles

Henceforth, let $\mathsf {MLQnd}$ denote the category of medial Latin quandles and quandle homomorphisms, and recall that $\mathbb {L}$ denotes the ring of integral Laurent polynomials in t and $1-t$ . We show that $\mathsf {MLQnd}$ and $\mathsf {AffMod}_{\mathbb {L}}$ are equivalent.

5.1.1 Functor construction

As noted in [Reference Burrows and Tuffley3, Reference Jedlička, Pilitowska, Stanovský and Zamojska-Dzienio10, Reference Nelson16], the data of a nonempty object M in $\mathsf {AffMod}_{\mathbb {L}}$ are equivalent to the data of an abelian group automorphism $\varphi \in {\mathrm {Aut}}(M)$ such that the map $\mathrm {id}-\varphi $ is invertible.Footnote ³ So, define a functor $ {\mathrm {Alex}}\colon \mathsf {AffMod}_{\mathbb {L}}\to \mathsf {MLQnd}$ on objects by sending M to the induced Alexander quandle $ {\mathrm {Alex}}(M,\varphi )$ . Let $ {\mathrm {Alex}}$ fix all morphisms as set-theoretic maps. We begin with two auxiliary results.

Proposition 5.1 The assignment $ {\mathrm {Alex}}\colon \mathsf {AffMod}_{\mathbb {L}}\to \mathsf {MLQnd}$ is a functor.

Proof We only have to show that every affine transformation of $\mathbb {L}$ -modules $f\colon M\to N$ is also a quandle homomorphism $f\colon {\mathrm {Alex}}(M,\varphi )\to {\mathrm {Alex}}(N,\psi )$ . If M is empty, then the claim is trivial. Otherwise, f has the form $f=T+z$ for some $\mathbb {L}$ -linear map $T\colon M\to N$ and some constant $z\in N$ . In particular, $T\circ \varphi = \psi \circ T$ , so

$$ \begin{align*} f(x\ast y)&= f(\varphi(x-y)+y) \\ &= (T\circ\varphi)(x-y)+T(y)+z\\ &= (\psi\circ T)(x-y) + f(y)\\ &= \psi(f(x)-f(y)) + f(y)\\ &= f(x)\ast f(y) \end{align*} $$

for all $x,y\in M$ . Hence, f is a quandle homomorphism.

Lemma 5.2 Let M and N be $\mathbb {L}$ -modules. Then every $\mathbb {Z}[t]$ -linear map $T\colon M\to N$ is $\mathbb {L}$ -linear.

Proof Left to the reader; use the fact that T commutes with t and $1-t$ and the fact that t and $1-t$ are invertible.

5.1.2 Main theorem

We prove the main result of this section.

Theorem 5.3 The functor $ {\mathrm {Alex}}\colon \mathsf {AffMod}_{\mathbb {L}}\to \mathsf {MLQnd}$ is an equivalence of categories.

Proof By Propositions 4.1 and 5.1 and the discussion preceding Proposition 5.1, $ {\mathrm {Alex}}$ is an essentially surjective functor. Clearly, $ {\mathrm {Alex}}$ is also faithful. Therefore, it suffices to show that $ {\mathrm {Alex}}$ is full. To that end, let $f\colon {\mathrm {Alex}}(M,\varphi )\to {\mathrm {Alex}}(N,\psi )$ be a homomorphism of Latin Alexander quandles, and view M and N as $\mathbb {L}$ -modules as in the discussion preceding Proposition 5.1.

We have to show that f is a morphism in $\mathsf {AffMod}_{\mathbb {L}}$ . Let $z:= f(0)$ , and define $T\colon M\to N$ by $T:= f-z$ . Since $f=T+z$ , we only have to show that T is $\mathbb {L}$ -linear. By Lemma 5.2, it suffices to show that T is a homomorphism of abelian groups such that $T\circ \varphi =\psi \circ T$ . Clearly, $T(0)=0$ .

Since f is a quandle homomorphism, the reader can verify that T is also a quandle homomorphism. Equivalently,

(5.1) $$ \begin{align} T(\varphi(x-y)+y)=\psi(T(x))+(\mathrm{id}-\psi)(T(y)) \end{align} $$

for all $x,y\in M$ . In particular, given $m\in M$ , take $(x,y):= (\varphi ^{-1}(m),0)$ in (5.1) to deduce that $T=\psi \circ T\circ \varphi ^{-1}$ . Therefore, $T\circ \varphi =\psi \circ T$ , as desired.

It remains to show that T distributes over addition. To that end, define $g\colon M^2\to N$ by

$$\begin{align*}g(x,y):= T(x+y)-T(x)-T(y).\end{align*}$$

We have to show that $g\equiv 0$ . Since $\varphi $ is an endomorphism of abelian groups and T is a quandle homomorphism, we first compute

$$ \begin{align*} T((a\ast b)+(c\ast d)) &= T(\varphi(a)+(\mathrm{id}-\varphi)(b)+\varphi(c)+(\mathrm{id}-\varphi)(d)) \\ &= T(\varphi(a+c)+(\mathrm{id}-\varphi)(b+d)) \\ &= T((a+c)\ast(b+d)) \\ &= T(a+c)\ast T(b+d) \end{align*} $$

for all $a,b,c,d\in M$ . Since T is a quandle homomorphism and $\psi $ is an endomorphism of abelian groups, it follows that

$$ \begin{align*} g((a\ast b),(c\ast d))&= T((a\ast b)+(c\ast d)) - T(a\ast b) - T(c\ast d) \\ &= [T(a+c)\ast T(b+d)] - [T(a)\ast T(b)] - [T(c)\ast T(d)] \\ &= \psi(T(a+c)-T(a)-T(c)) \\ &\quad +(\mathrm{id}-\psi)(T(b\ast d)-T(b)-T(d)) \\ &= g(a,c)\ast g(b,d) \end{align*} $$

for all $a,b,c,d\in M$ .

In particular, given $x,y\in M$ , let

$$\begin{align*}a:=\varphi^{-1}(x),\quad b:= 0,\quad c:=0,\quad d:=(\mathrm{id}-\varphi)^{-1}(y).\end{align*}$$

Then $x=a\ast b$ and $ y=c\ast d$ , so the previous calculation yields

$$\begin{align*}g(x,y)=g((a\ast b),(c\ast d))=g(a,c)\ast g(b,d)=g(a,0) \ast g(0,d)=0\ast 0=0\end{align*}$$

because $g(-,0)\equiv 0$ and $g(0,-)\equiv 0$ . Hence, T is $\mathbb {L}$ -linear, so f is a morphism in $\mathsf {AffMod}_{\mathbb {L}}$ .

5.2 Medial commutative quandles

Let $\mathsf {MCQnd}$ denote the category of medial commutative quandles and quandle homomorphisms, and recall that $\mathbb {D}=\mathbb {Z}[1/2]$ denotes the ring of dyadic rationals. Motivated by the study of $\mathsf {MCQnd}$ in [Reference Even, Gran and Montoli6], we show that $\mathsf {MCQnd}$ and $\mathsf {AffMod}_{\mathbb {D}}$ are equivalent.

Proof By Lemma 2.6, the surjective ring homomorphism

$$\begin{align*}\mathbb{L}\twoheadrightarrow\mathbb{L}/(2t-1)\xrightarrow{\sim} \mathbb{D} \end{align*}$$

induces a fully faithful embedding $\text {res}\colon \mathsf {AffMod}_{\mathbb {D}}\hookrightarrow \mathsf {AffMod}_{\mathbb {L}}$ . By Theorem 5.3 and Remark 2.4, the composite functor $ {\mathrm {Alex}}\circ \text {res}$ is an equivalence of categories from $\mathsf {AffMod}_{\mathbb {D}}$ to the category of midpoint quandles and quandle homomorphisms. By Corollary 4.2, the inclusion functor $\iota $ from the latter category into $\mathsf {MCQnd}$ is also an equivalence of categories. On the other hand, Remark 2.4 shows that $\text {mid}\colon \mathsf {AffMod}_{\mathbb {D}}\to \mathsf {MCQnd}$ is the composite functor

$$\begin{align*}\text{mid}=\iota\circ {\mathrm{Alex}}\circ \text{res}. \end{align*}$$

Hence, $\text {mid}$ is an equivalence of categories.

6.1 Free affine modules

First, we describe free affine modules over a ring R. Since affine modules over R form a variety in the sense of universal algebra (see Remark 2.7), free objects in $\mathsf {AffMod}_R$ exist and satisfy the usual universal property.

Namely, the free affine R-module generated by a set X is an object $ {\mathrm {FAM}}_R(X)\in \mathsf {AffMod}_R$ along with an injective map $\iota \colon X\hookrightarrow {\mathrm {FAM}}_R(X)$ satisfying the following universal property: for all affine R-modules M and all set-theoretic functions ${f\colon X\to M}$ , there exists a unique affine transformation $\tilde {f}\colon {\mathrm {FAM}}_R(X)\to M$ such that $\tilde {f}\circ \iota =f$ .

More explicitly, we can construct $ {\mathrm {FAM}}_R(X)$ as follows. If X is empty, then $ {\mathrm {FAM}}_R(X)$ is also empty. Otherwise, fix a basepoint $z\in X$ , and let $ {\mathrm {FAM}}_R(X):= R^{(X\setminus \{z\})}$ be the free left R-module generated by the set $X\setminus \{z\}$ . Define $\iota \colon X\hookrightarrow {\mathrm {FAM}}_R(X)$ by $z\mapsto 0$ and $x\mapsto x$ for all $x\in X\setminus \{z\}$ . The reader can verify that $ {\mathrm {FAM}}_R(X)$ satisfies the desired universal property with $\tilde {f}:= T+z$ , where $T\colon {\mathrm {FAM}}_R(X)\to M$ is the unique R-linear map obtained by linearly extending the assignment $x\mapsto f(x)-z$ for all $x\in X$ .

In particular, if X is a set of cardinality $1\leq n<\infty $ , then $ {\mathrm {FAM}}_R(X)\cong R^{n-1}$ .

6.2 Free medial Latin quandles

Since equivalences of categories preserve universal properties, Theorem 5.3 and Corollary 5.4 allow us to construct all free objects in $\mathsf {MLQnd}$ and $\mathsf {MCQnd}$ .

Given a set X, we construct the free medial Latin quandle $ {\mathrm {FML}}(X)$ and the free medial commutative quandle $ {\mathrm {FMC}}(X)$ generated by X as follows. If X is empty, then these quandles are also empty. Otherwise, fix a basepoint $z\in X$ , and consider the associated free affine modules $ {\mathrm {FAM}}_{\mathbb {L}}(X)$ and $ {\mathrm {FAM}}_{\mathbb {D}}(X)$ . We define

$$\begin{align*}{\mathrm{FML}}(X):= {\mathrm{Alex}}({\mathrm{FAM}}_{\mathbb{L}}(X),\varphi),\qquad {\mathrm{FMC}}(X):= {\mathrm{FAM}}_{\mathbb{D}}(X)_{\mathrm{mid}}, \end{align*}$$

where $\varphi $ denotes multiplication by t. By the above discussion, $ {\mathrm {FML}}(X)$ and $ {\mathrm {FMC}}(X)$ satisfy the desired universal properties.

In particular, since $ {\mathrm {FAM}}_R(X)\cong R^{n-1}$ for all sets X of cardinality $1\leq n<\infty $ , we obtain the following description of nonempty finitely generated free objects in $\mathsf {MLQnd}$ and $\mathsf {MCQnd}$ .

Proposition 6.1 Let X be a set of cardinality $1\leq n<\infty $ . Then the free medial Latin quandle and the free medial commutative quandle generated by X are given by

$$\begin{align*}{\mathrm{FML}}(X)\cong {\mathrm{Alex}}(\mathbb{L}^{n-1},\varphi),\qquad{\mathrm{FMC}}(X)\cong (\mathbb{D}^{n-1})_{\mathrm{mid}}, \end{align*}$$

where $\varphi $ denotes multiplication by t.

In the following corollary, the $n=2$ case provides a positive answer to [Reference Bardakov and Elhamdadi1, Question 7.3].

6.3 Medial commutative rack decomposition

As one last application of Corollary 5.4, we obtain a structure theorem for finitely generated commutative medial quandles. This completely characterizes quandles for which the first part of [Reference Bardakov and Elhamdadi1, Question 7.1] has a positive answer.

Recall that $C_{2n+1}=(\mathbb {Z}/(2n+1))_{\mathrm {mid}}$ denotes the cyclic midpoint quandle of order $2n+1$ .

Theorem 6.4 (cf. [Reference Bardakov and Elhamdadi1, Question 7.1])

Let X be a nonempty finitely generated rack. Then X is medial and commutative if and only if there exists a rack isomorphism

$$\begin{align*}X\cong (\mathbb{D}^r)_{\mathrm{mid}} \oplus\bigoplus^n_{i=1} C_{2m_i+1}\end{align*}$$

for some nonnegative integers $r,n,m_1,\dots ,m_n\geq 0$ .

In particular, every nonempty finite medial commutative rack is isomorphic to a direct sum of cyclic midpoint quandles $C_{2m_i+1}$ .