Free Medial Quandles

Home , Dedekind domain, Domain (ring theory), Racks and quandles

Premyslˇ Jedlicka,ˇ Agata Pilitowska, and Anna Zamojska-Dzienio

Abstract. This paper gives the construction of free medial quandles as well as free n-symmetric medial quandles and free m-reductive medial quandles.

1. Introduction

A binary algebra (Q, ) is called a rack if the following conditions hold, for · every x, y, z Q: ∈ x(yz)=(xy)(xz) (we say Q is left distributive), • the equation xu = y has a unique solution u Q (we say Q is a left • ∈ quasigroup). An idempotent rack is called a quandle (we say Q is idempotent if xx = x for every x Q). A quandle Q is medial if, for every x, y, u, v Q, ∈ ∈ (xy)(uv)=(xu)(yv). An important example of a medial quandle is an abelian group A with an operation defined by a b = (1 h)(a)+h(b), where h is an automorphism ∗ ∗ − of A. This construction is called an affine quandle (or sometimes an Alexander quandle) and denoted by Aff(A, h). In the literature [6, 7], the group A is 1 usually considered to be a Z[t, t− ]-module, where t a = h(a), for each a A. · ∈ We shall adopt this point of view here as well and we usually write Aff(A, r) instead, where r is a ring element. Note that in universal algebra terminology, an algebra is said to be affine if it is polynomially equivalent to a module. A subreduct of an affine algebra is called a quasi-affine algebra, see e.g., [13]. Clearly, affine quandles are quasi- affine algebras. Medial quandles lie in the intersection of the class of quandles and the class of modes [16]. Recent development in quandle theory is motivated by knot theory (see e.g., [1, 4]). The knot quandle is a very powerful knot invariant. Quandles also have applications in differential geometry [14] and graph theory [3]. Modes are generally idempotent and entropic algebras (algebras with a commutative clone of term operations). Mediality is another name for en- tropicity in the binary case. For a more detailed history of medial quandles, we refer to [9].

Presented by P. Dehornoy. Received April 14, 2016; accepted in ﬁnal form July 15, 2016. 2010 Mathematics Subject Classiﬁcation: Primary: 08B20; Secondary: 15A78, 20N02. Key words and phrases: quandles, medial quandles, binary modes, free algebras. 442 P. Jedlicˇka,P. Jedliˇcka, A. Pilitowska, A. Pilitowska, and A. and Zamojska A. Zamojska AlgebraAlgebra Univers. univers.

Medial quandles do not form a variety of binary algebras, unless we intro- duce an additional binary operation and the identities \ x (x y) y and x (x y) y \ ∗ ≈ ∗ \ ≈ to define the left quasigroup property equationally. The structure of free medial quandles remained open for a long time. There were only results about more general classes, i.e., general free modes were investigated by Stronkowski [18] and general free quandles by Joyce [11] and Stanovský[17]. In [2], racks and quandles were studied under the names LD-quasigroups and LDI-quasigroups, respectively. Among others, the con- structions of free racks and free quandles based on free groups were provided [2, Propositions V.1.17 and X.4.8]. Just recently in [5], such free algebras were again described but the characterization does not give any useful information about their structure. There were also some special cases investigated, like involutory medial quandles, which means medial quandles satisfying additionally x (x y) y. ∗ ∗ ≈ Theorem 1.1 ([11, Theorem 10.5]). Let n N. Denote by F the subset of the ∈ quandle Aff(Zn, 1) consisting of those n-tuples where at most one coordinate − is odd. Then (F, ) is a free (n + 1)-generated involutory medial quandle over ∗ (0,...,0), (1, 0,...,0), (0, 1, 0,...,0),...,(0,...,0, 1) . { } Here we generalize the result of Joyce but not directly. We choose the path started in [9] instead and we study certain permutation groups acting on quandles, called displacement groups. It turns out that, in the case of free 1 medial quandles, these groups are free Z[t, t− ]-modules and we can construct the free medial quandles based on these modules. Another important result is that the free medial quandles embed into affine quandles. This shows that the variety of medial quandles is generated by affine quandles. Next we focus on two special classes: n-symmetric and m-reductive medial quandles which play a significant role within the class of finite medial quandles. A quandle (Q, ) is n-symmetric if it satisfies the identity ∗ x (x (x y) ) y. ∗ ∗···∗ ∗ ··· ≈ n times − We construct here free n -symmetric medial quandles and we prove that free n-symmetric quandles embed into products of affine quandles over modules over Dedekind domains. This is useful especially when studying finite medial quandles since each finite left quasigroup is n-symmetric, for some natural number n. A quandle (Q, ) is m-reductive if it satisfies the identity ∗ ( (x y) y) y y. ··· ∗ ∗···∗ ∗ ≈ m times − A quandle is called reductive if it is m-reductive, for some m N. Reductivity ∈ turns out to be a very important notion in the study of medial quandles as 2 P. Jedliˇcka, A. Pilitowska, and A. Zamojska Algebra univers. Vol. 00, XX FreeFree medialmedial quandlesquandles 453

Medial quandles do not form a variety of binary algebras, unless we intro- each finite medial quandle embeds into a product of a reductive quandle and duce an additional binary operation and the identities a quasigroup [10]. \ x (x y) y and x (x y) y The paper contents four Sections. In Section 2, we recall and present some \ ∗ ≈ ∗ \ ≈ facts about general medial quandles. Section 3 contains the main results. to define the left quasigroup property equationally. Theorem 3.3 gives a description of free medial quandles and Theorem 3.5 a The structure of free medial quandles remained open for a long time. There construction of affine quandles into which the free quandles embed. Section were only results about more general classes, i.e., general free modes were in- 4 is devoted to n-symmetric and m-reductive free medial quandles. In both vestigated by Stronkowski [18] and general free quandles by Joyce [11] and cases, the displacement group of the free algebra turns out to be a free Z[t]/(f)- Stanovský[17]. In [2], racks and quandles were studied under the names module, for a suitable polynomial f. The description of the free quandles in LD-quasigroups and LDI-quasigroups, respectively. Among others, the con- these varieties is analogous to that for general medial quandles. structions of free racks and free quandles based on free groups were provided Note that when studying left quasigroups, important tools are the mappings [2, Propositions V.1.17 and X.4.8]. Just recently in [5], such free algebras were Le : x e x, called the left translations. We use also the right translations again described but the characterization does not give any useful information → ∗ Re : x x e. The idempotency and the mediality imply that both Le and about their structure. → ∗ Re are endomorphisms. The left quasigroup property means that Le is an There were also some special cases investigated, like involutory medial quan- automorphism. dles, which means medial quandles satisfying additionally x (x y) y. ∗ ∗ ≈ Theorem 1.1 ([11, Theorem 10.5]). Let n N. Denote by F the subset of the ∈ quandle Aff(Zn, 1) consisting of those n-tuples where at most one coordinate 2. Preliminaries − is odd. Then (F, ) is a free (n + 1)-generated involutory medial quandle over ∗ (0,...,0), (1, 0,...,0), (0, 1, 0,...,0),...,(0,...,0, 1) . This section recalls some important notions from [9] where the structure of { } Here we generalize the result of Joyce but not directly. We choose the medial quandles was described. Key ingredients are two permutation groups path started in [9] instead and we study certain permutation groups acting acting on quandles. on quandles, called displacement groups. It turns out that, in the case of free 1 Definition 2.1. Let Q be a quandle. medial quandles, these groups are free Z[t, t− ]-modules and we can construct The left multiplication group of Q is the group LMlt(Q)= Lx; x Q . the free medial quandles based on these modules. Another important result 1 ∈ The displacement group is the group Dis(Q)= LxL− ; x, y Q . is that the free medial quandles embed into affine quandles. This shows that y ∈ the variety of medial quandles is generated by affine quandles. It was proved in [8, Proposition 2.1] that the actions of both groups on Q Next we focus on two special classes: n-symmetric and m-reductive medial have the same orbits. We use, in the sequel, the word orbit plainly without quandles which play a significant role within the class of finite medial quandles. explicitly mentioning the acting groups. The orbit of Q containing x is de- A quandle (Q, ) is n-symmetric if it satisfies the identity ∗ noted by Qx and the stabilizer subgroup of x is denoted by Dis(Q)x. For β 1 x (x (x y) ) y. two permutations α, β, we write α = βαβ− . The commutator is defined by ∗ ∗···∗ ∗ ··· ≈ β 1 [β,α]=α α− . The identity permutation is denoted by 1. n times − It is also useful to understand the structure of the displacement group. We construct here free n -symmetric medial quandles and we prove that free n-symmetric quandles embed into products of affine quandles over modules Lemma 2.2 ([8, Proposition 2.1]). Let Q be a quandle. over Dedekind domains. This is useful especially when studying finite medial quandles since each finite left quasigroup is n-symmetric, for some natural ε1 ε2 εk Dis(Q)= Lx1 Lx2 Lxk xi Q, εi = 1, εi =0 . number n. { ··· | ∈ ± } A quandle (Q, ) is m-reductive if it satisfies the identity From this lemma, we can clearly see that Dis(Q) is a normal subgroup ∗ ( (x y) y) y y. of LMlt(Q). Moreover, in our context, the group is commutative. ··· ∗ ∗···∗ ∗ ≈ m times − Proposition 2.3 ([12]). Let Q be a quandle. Then Q is medial if and only A quandle is called reductive if it is m-reductive, for some m N. Reductivity if Dis(Q) is abelian. ∈ turns out to be a very important notion in the study of medial quandles as 464 P. Jedlicˇka,P. Jedliˇcka, A. Pilitowska, A. Pilitowska, and A. and Zamojska A. Zamojska AlgebraAlgebra Univers. univers.

Since Dis(Q) is abelian, conjugations by elements from the same coset of Dis(Q) yield the same results.

Lemma 2.4. Let Q be a medial quandle. Let α Dis(Q) and x, y Q. Then ∈ ∈ αLx = αLy .

Lx 1 1 1 1 1 Ly Proof. α = LxαLx− = LxαLx− LyLy− = LxLx− LyαLy− = α due to the abelianess of Dis(Q).

From now on, by writing αL, we mean αLx for an arbitrary x Q, since ∈ conjugation does not depend on the element x. α It is easy to see that for α Aut(Q) and x Q, we have Lα(x) = Lx . In ∈ Ly ∈ particular, for α = Ly, we obtain Ly x = Lx . This implies that LMlt(Q) has ∗ only few generators. On the other hand, Dis(Q), in spite of being a subgroup of LMlt(Q), has usually more generators than LMlt(Q).

Proposition 2.5. Let Q be a medial quandle generated by X Q and choose ⊂ z X. Then ∈ the group LMlt(Q) is generated by Lx x X ; • { | 1 ∈Lk } the group Dis(Q) is generated by (LxL− ) x X, k Z . • { z | ∈ ∈ }

Proof. The generating set for LMlt(Q) is obtained by induction using Lx y = 1 1 ∗ LxLyL−x and Lx y = Lx− LyLx. \ Suppose now α Dis(Q). By Lemma 2.2 and the previous observation, we ∈ can suppose α = Lε1 Lε2 Lεn , where x X and ε = 1 with ε = 0 for x1 x2 ··· xn i ∈ i ± i 1 i n. We prove the claim by an induction on n. For n = 2, the claim is ≤ ≤ true. Let the induction hypothesis hold for words of length n 2. If ε = ε , ≤ − 1 n then w = w1w2 with wi Dis(Q) and we use the induction hypothesis. Let ∈ 1 now ε1 = 1 and εn = 1. Then w = Lx1 wLx−n and w, by the induction − 1 Lk hypothesis, is a product of elements from (LxL− ) x X, k Z . But { z | ∈ ∈ } 1 1 1 L 1 1 1 w = Lx1 wLx−1 Lx1 Lz− LzLx−n =(w) (Lx1 Lz− )(Lxn Lz− )− , proving the claim. The argument is similar for ε = 1 and ε = 1. 1 − n This result cannot be much improved—it is shown in Proposition 3.2 that the displacement group of a free medial quandle is not finitely generated. The abelian group Dis(Q) can be easily endowed with the structure of a 1 Z[t, t− ]-module; it suffices to pick an automorphism of Dis(Q). A natural choice is the inner automorphism α αL. Hence, from now on, the group → Dis(Q) is treated, depending on the situation, either as a permutation group 1 acting on Q or as an R-module, where R is a suitable image of Z[t, t− ], with t L 1 the action of t defined by α = α . Note that for f Z[t, t− ], we have ∈ αf = αf(L). 4 P. Jedliˇcka, A. Pilitowska, and A. Zamojska Algebra univers. Vol. 00, XX FreeFree medialmedial quandlesquandles 475

Since Dis(Q) is abelian, conjugations by elements from the same coset Example 2.6. Let f = (1 t)2. Then − of Dis(Q) yield the same results. f 1 2t+t2 f(L) 1 2L+L2 L 2 L2 α = α − = α = α − = α(α )− α . Lemma 2.4. Let Q be a medial quandle. Let α Dis(Q) and x, y Q. Then ∈ ∈ αLx = αLy . It was proved in [9, Proposition 3.2] that for any x Q, the orbit Qx ∈ is affine over Dis(Q)/Dis(Q)x and we can naturally identify the sets Qx and Proof. αLx = L αL 1 = L αL 1L L 1 = L L 1L αL 1 = αLy due to the x x− x x− y y− x x− y y− Dis(Q)/Dis(Q)x by defining the group operation on Qx as abelianess of Dis(Q). 1 α(x)+β(x)=αβ(x) and α(x)=α− (x). − From now on, by writing αL, we mean αLx for an arbitrary x Q, since ∈ conjugation does not depend on the element x. The group so defined is denoted by OrbQ(x) and called the orbit group for α Qx. Moreover, Dis(Q) is a submodule of Dis(Q): suppose α(x)=x; then It is easy to see that for α Aut(Q) and x Q, we have Lα(x) = Lx . In x ∈ ∈ t 1 1 Ly α (x)=LxαL− (x)=x. This means that Dis(Q)/Dis(Q)x is a [t, t− ]- particular, for α = Ly, we obtain Ly x = Lx . This implies that LMlt(Q) has x Z ∗ only few generators. On the other hand, Dis(Q), in spite of being a subgroup module and we can call OrbQ(x) the orbit module for Qx. of LMlt(Q), has usually more generators than LMlt(Q).

Proposition 2.5. Let Q be a medial quandle generated by X Q and choose ⊂ 3. Free medial quandle z X. Then ∈ the group LMlt(Q) is generated by Lx x X ; In this section, we present the free medial quandles. Regarding the gener- • { | 1 ∈Lk } ating set, we see that in any quandle Q, for all x, y Q, we have y x Qx the group Dis(Q) is generated by (LxLz− ) x X, k Z . ∈ ∗ ∈ • { | ∈ ∈ } as well as y x Qx. Hence, each orbit has to contain at least one generator. \ ∈ Proof. The generating set for LMlt(Q) is obtained by induction using Lx y = 1 1 ∗ Lemma 3.1. Let Q be a quandle generated by X Q. Then the set X Qx LxLyLx− and Lx y = Lx− LyLx. \ ⊂ ∩ Suppose now α Dis(Q). By Lemma 2.2 and the previous observation, we is nonempty for each x Q. ∈ ∈ε1 ε2 εn can suppose α = L L L , where xi X and εi = 1 with εi = 0 for x1 x2 ··· xn ∈ ± The following proposition characterizes the free medial quandles. Formally, 1 i n. We prove the claim by an induction on n. For n = 2, the claim is ≤ ≤ it is proved to be a sufficient condition only, but we can see from Theorem 3.3 true. that such an object exists, making the condition necessary too. Let the induction hypothesis hold for words of length n 2. If ε = ε , ≤ − 1 n then w = w1w2 with wi Dis(Q) and we use the induction hypothesis. Let Proposition 3.2. Let F be a medial quandle generated by a set X F . ∈ 1 ⊂ now ε1 = 1 and εn = 1. Then w = Lx1 wLx−n and w, by the induction − k Choose z X arbitrarily. Then F is free over X if the following conditions 1 L ∈ hypothesis, is a product of elements from (LxL− ) x X, k Z . But { z | ∈ ∈ } are satisfied: 1 1 1 L 1 1 1 (1) each two elements of X lie in different orbits; w = Lx1 wLx−1 Lx1 Lz− LzLx−n =(w) (Lx1 Lz− )(Lxn Lz− )− , 1 1 (2) Dis(F ) is a free Z[t, t− ]-module with LxL− x X z as a free { z | ∈ { }} proving the claim. The argument is similar for ε1 = 1 and εn = 1. basis; − (3) the action of Dis(F ) on F is free. This result cannot be much improved—it is shown in Proposition 3.2 that the displacement group of a free medial quandle is not finitely generated. Proof. First, for any y F , there exists exactly one x X and exactly ∈ ∈ The abelian group Dis(Q) can be easily endowed with the structure of a one α Dis(F ) such that y = α(x). Indeed, the existence of x comes from 1 ∈ Z[t, t− ]-module; it suffices to pick an automorphism of Dis(Q). A natural Lemma 3.1, and its uniqueness from (1). The uniqueness of α is due to (3). choice is the inner automorphism α αL. Hence, from now on, the group Let Q be a medial quandle and let Y Q. Let ψ be a mapping X Y . We → ⊂ → Dis(Q) is treated, depending on the situation, either as a permutation group prove that ψ can be extended to a homomorphism Ψ: F Q. We define first 1 1 → acting on Q or as an R-module, where R is a suitable image of Z[t, t− ], with a Z[t, t− ]-module homomorphism Φ: Dis(F ) Dis(Q) on the basis of Dis(F ) t L 1 1 1 → L t t the action of t defined by α = α . Note that for f Z[t, t− ], we have by setting Φ(LxLz− )=Lψ(x)L− . Note that Φ(α ) = Φ(α ) = Φ(α) = ∈ ψ(z) αf = αf(L). Φ(α)L. Now set Ψ(α(x))=Φ(α)(ψ(x)) for all α Dis(F ) and x X. ∈ ∈ 486 P. Jedlicˇka,P. Jedliˇcka, A. Pilitowska, A. Pilitowska, and A. andZamojska A. Zamojska AlgebraAlgebra Univers. univers.

The mapping Ψ is well deﬁned since every element of F has a unique representation by α and x.

Ψ(α(x)) Ψ(β(y))=Φ(α)(ψ(x)) Φ(β)(ψ(y)) = L Φ(β)(ψ(y)) ∗ ∗ Φ(α)(ψ(x)) Φ(α) 1 1 1 = Lψ(x) Φ(β)(ψ(y))=Φ(α)Lψ(x)Φ(α− )Φ(β)Lψ−(x)Lψ(x)Lψ−(y)(ψ(y)) 1 L 1 1 L 1 = Φ(α)(Φ(α− β)) Lψ(x)Lψ−(y)(ψ(y))=Ψ(α(α− β) LxLy− (y)) 1 = Ψ(αL α− β(y)) = Ψ(L β(y)) = Ψ(α(x) β(y)), x α(x) ∗ and Ψ is a homomorphism that extends ψ.

In the sequel, we use the following notation. Let X be a set. We choose z X arbitrarily and denote by X− the set X z . We often do not specify ∈ { } the element z since we actually rarely need it explicitly. Now let R be a ring and consider the free R-module of rank X− , i.e., M = x X R. We then | | ∈ − choose a free basis of M, let us say e i X− , and by deﬁning e =0 M, { i | ∈ } z ∈ we have deﬁned e as an element of M, for each i X. i ∈ Theorem 3.3. Let X be a set and let z X. Denote by X− the set X z . 1 ∈ { } Let M = Z[t, t− ]. Let ei i X− be a free basis of M. Moreover, x X− { | ∈ } let e =0 M∈ . Let us denote by F the set M X equipped with the operation z ∈ × (a, i) (b, j)=((1 t) a + t b + e e ,j). ∗ − · · i − j Then (F, ) is a free medial quandle over (0,i) i X . ∗ { | ∈ } Proof. Idempotency is evident. Mediality is proved by the observation that

((a, i) (b, j)) ((c, k) (d, n)) = ((1 t)2 a +(t t2) (b + c) ∗ ∗ ∗ − · − · + t2 d + (1 t) e + t (e + e ) (1 + t) e ,n). · − · i · j k − · n The left-quasigroup operation is given by the formula

1 1 (a, i) (b, j) = ((1 t− ) a + t− (b + e e ),j). \ − · · j − i Hence, F is a medial quandle. We know now that F is a medial quandle and we want to prove its freeness by Proposition 3.2. We start with analyzing the structure of Dis(F ).

1 L L− ((c, k))=(a, i) ((b, j) (c, k)) (a,i) (b,j) ∗ \ 1 1 =(a, i) ((1 t− ) b + t− (c + e e ),k) ∗ − · · k − j = ((1 t) a +(t 1) b + c + e e + e e ,k) − · − · k − j i − k =(c + (1 t) (a b)+e e ,k). − · − i − j 1 In particular, L(0,i)L(0−,z)((c, k)) = (c + ei,k). Now we prove by induction that n 1 L n L(0,i)L− ((c, j)) = (c + t ei,j) for each i X−, j X, and n Z. (0,z) · ∈ ∈ ∈ 6 P. Jedliˇcka, A. Pilitowska, and A. Zamojska Algebra univers. Vol. 00, XX Free medial quandles 497

The mapping Ψ is well deﬁned since every element of F has a unique rep- The case n = 0 was already proved. Now suppose n>0.

n n 1 resentation by α and x. 1 L 1 L − 1 L(0,i)L(0−,z) ((c, j)) = L(0,z) L(0,i)L(0−,z) L(0−,z)((c, j)) Ψ(α(x)) Ψ(β(y))=Φ(α)(ψ(x)) Φ(β)(ψ(y)) = L Φ(β)(ψ(y)) n 1 Φ(α)(ψ(x)) 1 L − 1 ∗ ∗ = L(0,z) L(0,i)L− ((t− (c + ej),j)) Φ(α) 1 1 1 (0,z) · = Lψ(x) Φ(β)(ψ(y))=Φ(α)Lψ(x)Φ(α− )Φ(β)Lψ−(x)Lψ(x)L−ψ(y)(ψ(y)) 1 n 1 n = L(0,z) ((t− (c + ej)+t − ei,j)) = (c + t ei,j). 1 L 1 1 L 1 · · · = Φ(α)(Φ(α− β)) Lψ(x)L− (ψ(y))=Ψ(α(α− β) LxL− (y)) ψ(y) y The case n<0 is analogous. Moreover, from this we see that 1 = Ψ(αL α− β(y)) = Ψ(L β(y)) = Ψ(α(x) β(y)), x α(x) 1 f ∗ (L L− ) ((c, j))=(c + f e ,j), (0,i) (0,z) · i and Ψ is a homomorphism that extends ψ. 1 for any c M, f Z[t, t− ], and i, j X. ∈ ∈ ∈ In the sequel, we use the following notation. Let X be a set. We choose Let (fi)i X ,j be in F . We now prove that this element lies in the ∈ − z X arbitrarily and denote by X− the set X z . We often do not specify subquandle generated by (0,i) i X . But it is not difficult to see that ∈ { } { | ∈ } the element z since we actually rarely need it explicitly. Now let R be a ring 1 fi − (fi)i X− ,j = L(0,i)L(0,z) ((0,j)). and consider the free R-module of rank X− , i.e., M = R. We then ∈ x X− i X | | ∈ ∈− choose a free basis of M, let us say ei i X− , and by defining ez =0 M, { | ∈ } ∈ The product is finite since only finitely many fi are non-zero, and hence we have defined e as an element of M, for each i X. i ∈ (0,i) i X = F . Moreover, we see that different generators lie in differ- { | ∈ } Theorem 3.3. Let X be a set and let z X. Denote by X− the set X z . ent orbits. Ln 1 ∈ { } 1 Let M = Z[t, t− ]. Let ei i X− be a free basis of M. Moreover, Since the set L(0,i)L(0−,z) i X, n Z generates Dis(F ) due to Pro- x X− { | ∈ } | ∈ ∈ let e =0 M∈ . Let us denote by F the set M X equipped with the operation position 2.5, we see that Dis(F ) acts freely on every orbit of F . That means z ∈ × also that Dis(F ) is isomorphic to M and L L i X− is clearly its { (0,i) (0,z) | ∈ } (a, i) (b, j)=((1 t) a + t b + ei ej,j). free basis. According to Proposition 3.2, F is free over (0,i) i X . ∗ − · · − { | ∈ } Then (F, ) is a free medial quandle over (0,i) i X . ∗ { | ∈ } In [9], the structure of medial quandles was represented using a heteroge- Proof. Idempotency is evident. Mediality is proved by the observation that neous structure called the indecomposable affine mesh. We do not recall the definition here as it is not needed, we just remark that the free medial quandle ((a, i) (b, j)) ((c, k) (d, n)) = ((1 t)2 a +(t t2) (b + c) now constructed is the sum of the affine mesh ∗ ∗ ∗ − · − · 2 1 + t d + (1 t) ei + t (ej + ek) (1 + t) en,n). Z[t, t− ] i X ; (1 t)i,j X ;(ei ej)i,j X . · − · · − · ∈ − ∈ − ∈ x X− The left-quasigroup operation is given by the formula ∈ Recall that subquandles of affine quandles are quasi-affine. Every (both 1 1 (a, i) (b, j) = ((1 t− ) a + t− (b + e e ),j). sided) cancellative medial quandle is quasi-affine—to see this we can either \ − · · j − i use a result by Kearnes [13] for idempotent cancellative algebras having a Hence, F is a medial quandle. central binary operation or a result by Romanowska and Smith (see e.g., [16]) We know now that F is a medial quandle and we want to prove its freeness for cancellative modes. Nevertheless, a direct proof is simple. by Proposition 3.2. We start with analyzing the structure of Dis(F ). Proposition 3.4. Let Q be a cancellative medial quandle. Then Q embeds

1 into any of its orbits. L L− ((c, k))=(a, i) ((b, j) (c, k)) (a,i) (b,j) ∗ \ 1 1 Proof. Rx is an endomorphism of Q for each x Q. The right cancellativity =(a, i) ((1 t− ) b + t− (c + ek ej),k) ∈ ∗ − · · − ensures that Rx is injective. Hence, for each x Q, Rx embeds Q into Qx. = ((1 t) a +(t 1) b + c + ek ej + ei ek,k) ∈ − · − · − − The free medial quandle that we have constructed is cancellative, and there- =(c + (1 t) (a b)+e e ,k). − · − i − j fore it can be represented as a subquandle of an affine quandle. 1 In particular, L(0,i)L(0−,z)((c, k)) = (c + ei,k). Now we prove by induction that n Theorem 3.5. Let X be a set. The free medial quandle over X is isomorphic 1 L n − 1 L(0,i)L(0,z) ((c, j)) = (c + t ei,j) for each i X−, j X, and n Z. to a subquandle of the affine quandle M = Aff( x X Z[t, t− ],t). · ∈ ∈ ∈ ∈ − 508 P. Jedlicˇka,P. Jedliˇcka, A. Pilitowska, A. Pilitowska, and A. andZamojska A. Zamojska AlgebraAlgebra Univers. univers.

1 Proof. Let Λ: [t, t− ] be the group homomorphism x X− Z x X− Z ∈ → ∈ 1 induced by the evaluation homomorphism Z[t, t− ] Z, where t 1. Denote → → Q = a M Λ(a)=e , for some i X . Then Q is a subquandle of M { ∈ | i ∈ } since

Λ(L (b)) = Λ((1 t) a + t b)=0 Λ(a)+1 Λ(b)=Λ(b), a − · · · · 1 and analogously, Λ(La− (b)) = Λ(b). We shall prove that Q is a free quandle over e x X . { x | ∈ } Take the quandle F from Theorem 3.3. Note that the orbit F (0,z) is isomorphic to M through the bijection (a, z) a. Now consider the embedding → R : F F (0,z). Clearly, R ((0,i)) = (e , 0). Therefore, the subquan- (0,z) → (0,z) i dle of M generated by Y = e x X is free. { x | ∈ } The only thing left to prove is to show that Q = Y . Clearly, Y Q and ⊂ Q is a subquandle of M, hence Q Y . ⊇ On the other hand, for Q Y , we notice that ⊆ Q = a M a e (mod (1 t)), for some i X . { ∈ | ≡ i − ∈ } 1 Ln n Moreover, we have (Lex Le−z ) : Q Q; u u + (1 t)t ex, with an → →1 f(L) − · analogous proof as in Theorem 3.3. Then (Lex Le−z ) (u)=u+(1 t)f ex for 1 − · each f Z[t, t− ]. Now, for each element a Q, a = ei+(1 t)g for some i X ∈ ∈ − 1 gx(L)∈ and g =(gx)x X− M. Hence, we have that a = x X Lex Le−z (ei). ∈ ∈ − Therefore, a Y . ∈ ∈ Example 3.6. We describe now the free medial quandle on three generators. Let X = 0, 1, 2 ; let e1 = (1, 0), e2 = (0, 1), and e0 = (0, 0). Now 1{ } 1 1 1 2 M = Aﬀ(Z[t, t− ] Z[t, t− ],t) and Λ: Z[t, t− ] Z[t, t− ] Z is the group ho- × × → 1 momorphism induced by the evaluation homomorphism Z[t, t− ] Z, where → t 1. Denote Q = a M Λ(a) (0, 0), (1, 0), (0, 1) . By Theo- → { ∈ | ∈{ }} rem 3.5, Q is a subquandle of M and it is the free quandle generated by the set (0, 0), (1, 0), (0, 1) . For example, the element a = (1 t, 1+t t2) lies { } − − in Q since Λ(a) = (0, 1). Now a can be represented as

(0, 1)+(1 t)(1,t) = (0, 1)+(1 t) (1, 0)+(1 t)t (0, 1) − − · − · 1 1 L 1 1 L − − =(L(1,0)L(0,0))(L(0,1)L(0,0)) (0, 1)=(Le1 Le−0 )(Le2 Le−0 ) (e2).

4. Free quandles in subvarieties

In this section, we study free n-symmetric and free m-reductive medial quandles. Both types of varieties have a similar property: they can be char- acterized by an identity on Dis(Q).

1 Deﬁnition 4.1. Let I Z[t, t− ]. We say that a medial quandle Q is an ⊂ I-quandle if αf = 1 for each α Dis(Q) and f I. ∈ ∈ 8 P. Jedliˇcka, A. Pilitowska, and A. Zamojska Algebra univers. Vol. 00, XX Free medial quandles 519

1 1 1 Proof. Let Λ: [t, t− ] be the group homomorphism If I is an ideal of [t, t− ] and a [t, t− ]-module M satisfies the identity x X− Z x X− Z Z Z ∈ → ∈ 1 induced by the evaluation homomorphism Z[t, t− ] Z, where t 1. Denote f a = 0 for each a M and f I, then M can be viewed as a module over → → · 1 ∈ ∈ Q = a M Λ(a)=ei, for some i X . Then Q is a subquandle of M Z[t, t− ]/I. { ∈ | ∈ } since In our context, the set I shall usually be a principal ideal, that means I = 1 (f) for some f Z[t, t− ]. We then write that Q is an f-quandle, rather Λ(La(b)) = Λ((1 t) a + t b)=0 Λ(a)+1 Λ(b)=Λ(b), ∈ n r − · · · · than f -quandle or (f)-quandle. If, moreover, f = r=0 crt and the coeffi- { } 1 1 n r 1 and analogously, Λ(L 1(b)) = Λ(b). We shall prove that Q is a free quandle cient c0 is invertible, then Z[t, t− ]/f ∼= Z[t]/f since t− r=1 crt − a− 1 ≡− · c− (mod f). We use these remarks since working with the ring Z[t]/f is often over ex x X . 0 { | ∈ } 1 Take the quandle F from Theorem 3.3. Note that the orbit F (0,z) is iso- easier than working with the ring Z[t, t− ]. morphic to M through the bijection (a, z) a. Now consider the embedding We prepared the framework of I-quandles to work with symmetric and → R : F F (0,z). Clearly, R ((0,i)) = (e , 0). Therefore, the subquan- reductive medial quandles at once. First note that if Q is an I-quandle, then (0,z) → (0,z) i I f dle of M generated by Y = e x X is free. clearly OrbQ(x) = α (x) α Dis(Q),f I = x . On the other hand, { x | ∈ } f { | ∈ ∈ } { } The only thing left to prove is to show that Q = Y . Clearly, Y Q and let α (x)=x for arbitrary α Dis(Q) with f I and each x Q. Hence, ⊂ f ∈ f∈ ∈ Q is a subquandle of M, hence Q Y . the action of α on Q is trivial and this means α = 1 since Dis(Q) is faithful. ⊇ On the other hand, for Q Y , we notice that This immediately gives the following lemma. ⊆ 1 Lemma 4.2. Let Q be a medial quandle and let I Z[t, t− ]. Then Q is an Q = a M a ei (mod (1 t)), for some i X . I ⊂ { ∈ | ≡ − ∈ } I-quandle if and only if OrbQ(x) = x for each x Q. 1 Ln n { } ∈ Moreover, we have (Lex Le−z ) : Q Q; u u + (1 t)t ex, with an n → →1 f(L) − · Recall that a quandle Q is n-symmetric if Lx = 1 for each x Q. analogous proof as in Theorem 3.3. Then (Lex Le−z ) (u)=u+(1 t)f ex for ∈ 1 − · each f Z[t, t− ]. Now, for each element a Q, a = ei+(1 t)g for some i X Proposition 4.3. A medial quandle Q is n-symmetric if and only if Q is a ∈ ∈ − gx(L)∈ n 1 r 1 ( − t )-quandle. and g =(gx)x X− M. Hence, we have that a = x X Lex Le−z (ei). r=0 ∈ ∈ ∈ − Therefore, a Y . ∈ Proof. According to [9, Proposition 7.2], a medial quandle Q is n-symmetric n 1 r Example 3.6. We describe now the free medial quandle on three genera- if and only if for each x Q, − (1 ϕ) α(x)=x, where ϕ: Qx Qx is ∈ r=0 − → tors. Let X = 0, 1, 2 ; let e = (1, 0), e = (0, 1), and e = (0, 0). Now defined by α(x) [α, L](x). 1 2 0 1{ } 1 1 1 2 → L t M = Aff(Z[t, t− ] Z[t, t− ],t) and Λ: Z[t, t− ] Z[t, t− ] Z is the group ho- Then (1 ϕ)(α(x)) = α(x) [α, L](x)=α[L, α](x)=α (x)= α (x), and −n 1 r − × × → 1 − t momorphism induced by the evaluation homomorphism Z[t, t− ] Z, where therefore α r=0 (x)=x for each x Q. Hence, according to Lemma 4.2, Q → ∈ n 1 r t 1. Denote Q = a M Λ(a) (0, 0), (1, 0), (0, 1) . By Theo- is n-symmetric if and only if it is a ( − t )-quandle. → { ∈ | ∈{ }} r=0 rem 3.5, Q is a subquandle of M and it is the free quandle generated by the Recall that a quandle Q is m-reductive if Rm(y)=x for all x, y Q. set (0, 0), (1, 0), (0, 1) . For example, the element a = (1 t, 1+t t2) lies x ∈ { } − − in Q since Λ(a) = (0, 1). Now a can be represented as Proposition 4.4. A medial quandle Q is m-reductive if and only if it is a m 1 (1 t) − -quandle. (0, 1)+(1 t)(1,t) = (0, 1)+(1 t) (1, 0)+(1 t)t (0, 1) − − − · − · 1 1 L 1 1 L Proof. According to [9, Proposition 6.2], a medial quandle Q is m-reductive if =(L(1,0)L− )(L(0,1)L− ) (0, 1)=(Le1 Le− )(Le2 Le− ) (e2). (0,0) (0,0) 0 0 m 1 and only if ϕ − α(x)=x for each x Q, where ϕ: Qx Qx is defined by ∈ → α(x) [α, L](x). m 1 → 1 L (1 t) (1 t) − 4. Free quandles in subvarieties Then ϕ(α(x))=[α, L](x)=α(α− ) (x)=α − (x), so α − (x)=x for each x. Hence, according to Lemma 4.2, Q is m-reductive if and only if it m 1 In this section, we study free n-symmetric and free m-reductive medial is a (1 t) − -quandle. − quandles. Both types of varieties have a similar property: they can be char- n 1 r m 1 Both mentioned polynomials, i.e., − t and (1 t) − , have the property acterized by an identity on Dis(Q). r=0 − that the leading as well as the absolute coefficients are invertible. 1 Definition 4.1. Let I Z[t, t− ]. We say that a medial quandle Q is an In this case, not only can Dis(Q) be treated as a Z[t]/f-module, but it has ⊂ I-quandle if αf = 1 for each α Dis(Q) and f I. fewer generators even as a group. ∈ ∈ 5210 P. Jedlicˇka,P. Jedliˇcka, A. Pilitowska, A. Pilitowska, and A. andZamojska A. Zamojska AlgebraAlgebra Univers. univers.

s r Proposition 4.5. Let f = r=0 crt be a polynomial with c0 and ck invertible and let Q be an f-quandle generated by X Q. Let z X be an arbitrary ⊂ ∈ element. Then Dis(Q) is generated by 1 Lr (LxL− ) x X z and 0 r~~s. x z − The structure of free medial f-quandles can be described exactly in the same way as the structure of general free medial quandles.~~

Proposition 4.6. Let f Z[t] be a polynomial with the leading and the abso- ∈ lute coefficients invertible. Let F be an f-quandle generated by a set X F . ⊂ Choose z X arbitrarily. Then F is a free f-quandle over X if the following ∈ conditions are satisfied: (1) each two elements of X lie in different orbits; 1 (2) Dis(F ) is a free Z[t]/f-module with LxL− x X z as a free basis; { z | ∈ { }} (3) the action of Dis(F ) on F is free. Proof. The proof is nearly the same as the proof of Proposition 3.2. The only difference is that displacement groups appearing here are Z[t]/f-modules. Theorem 4.7. Let f Z[t] be a polynomial with the leading and the absolute ∈ coefficients invertible. Let X be a set. Let M = [t]/f. Let us denote s X− Z by F the set M X equipped with the operation ∈ × (a, i) (b, j) = ((1 t) a + t b + e e ,j). ∗ − · · i − j Then (F, ) is a free f-quandle over (0,i) i X . ∗ { | ∈ } Proof. The proof is nearly the same as of Theorem 3.3, with the usage of Proposition 4.6. The only thing to show is that αf = 1 for all α Dis(F ). ∈ But this follows from Dis(F ) ∼= M. Non-trivial reductive medial quandles are never right-cancellative since the multiplication by (1 t) is not injective. On the other hand, free n-symmetric − quandles are cancellative and we can embed them in their orbits. Moreover, the polynomial tr is a product of cyclotomic polynomials. 10 P. Jedliˇcka, A. Pilitowska, and A. Zamojska Algebra univers. Vol. 00, XX Free medial quandles 5311 s r n 1 r Proposition 4.5. Let f = r=0 crt be a polynomial with c0 and ck invertible Theorem 4.8. Let X be a set. Let n N and let r=0− t = j fj, ∈ ∈J and let Q be an f-quandle generated by X Q. Let z X be an arbitrary where fj are irreducible in Z[t] and Mj = Aff( Z[t]/fj,t) for each ⊂ ∈ x X − element. Then Dis(Q) is generated by j . Each free n-symmetric medial quandle is isomorphic∈ to the subquandle r ∈J 1 L of Mj generated by (ei)j i X . (LxL− ) x X z and 0 rs. modulo 2. This confirms the result of Joyce [11, Theorem 10.5]. x z − The structure of free medial f-quandles can be described exactly in the Theorem 4.8 can be reformulated as follows: let ζk be a primitive k-th root n 1 same way as the structure of general free medial quandles. of unity in C. It is well known that Z[t]/(1 + t + + t − ) = Z[ζk]. ··· ∼ k n,k>1 Hence, the free X -generated n-symmetric quandle is the subquandle| of Proposition 4.6. Let f Z[t] be a polynomial with the leading and the abso- | | ∈ lute coefficients invertible. Let F be an f-quandle generated by a set X F . Aff( Z[ζk],ζk) ⊂ k n,k>1 x X Choose z X arbitrarily. Then F is a free f-quandle over X if the following | ∈ − ∈ conditions are satisfied: generated by (e ,e ,...,e ) i X . { i i i | ∈ } (1) each two elements of X lie in different orbits; Example 4.10. For X = 2, the free 2-generated n-symmetric medial quan- 1 | | (2) Dis(F ) is a free Z[t]/f-module with LxLz− x X z as a free basis; dle F is the subquandle of Aff( [ζ ],ζ ) generated by (0,...,0) and { | ∈ { }} k n,k>1 Z k k (3) the action of Dis(F ) on F is free. | (1,...,1), and F consists of the tuples (ak)k n,k>1 where ak 0 (mod (1 ζk)) | ≡ − for all k n with k>1, or a 1 (mod (1 ζ )) for all k n with k>1. Proof. The proof is nearly the same as the proof of Proposition 3.2. The only | k ≡ − k | difference is that displacement groups appearing here are Z[t]/f-modules. Every finite quandle is n-symmetric for some n. For studying finite medial 1 quandles, it is nice to know that we need not always consider Z[t, t− ]-modules Theorem 4.7. Let f Z[t] be a polynomial with the leading and the absolute ∈ but we can sometimes focus on nicer rings, or even domains. coefficients invertible. Let X be a set. Let M = s X Z[t]/f. Let us denote ∈ − by F the set M X equipped with the operation Corollary 4.11. Let n N. The variety of n-symmetric medial quandles × ∈ is generated by quandles that are polynomially equivalent to modules over (a, i) (b, j) = ((1 t) a + t b + ei ej,j). ∗ − · · − Dedekind domains. Then (F, ) is a free f-quandle over (0,i) i X . ∗ { | ∈ } Proof. From [19], Z[ζk] is a Dedekind domain for each k. Free n-symmetric Proof. The proof is nearly the same as of Theorem 3.3, with the usage of quandles embed into products of Aff( x X Z[ζk],ζk). Each of these affine ∈ − Proposition 4.6. The only thing to show is that αf = 1 for all α Dis(F ). quandles embeds into any of its orbits, i.e., into a module over [ζ ]. ∈ Z k But this follows from Dis(F ) = M. ∼ Note that by applying our idea of I-quandles, one obtains the descrip- Non-trivial reductive medial quandles are never right-cancellative since the tion of free n-symmetric m-reductive medial quandles if we consider I = n 1 r m 1 multiplication by (1 t) is not injective. On the other hand, free n-symmetric − t , (1 t) − . In particular, the free 2-reductive n-symmetric me- − { r=0 − } quandles are cancellative and we can embed them in their orbits. Moreover, dial quandle over X is isomorphic to Zn X with the operation x X− × the polynomial tr is a product of cyclotomic polynomials. (a, i) (b, j)=(b + e e ,j), [15, Proposition∈ 2.4]. ∗ i − j 5412 P. Jedlicˇka,P. Jedliˇcka, A. Pilitowska, A. Pilitowska, and A. and Zamojska A. Zamojska AlgebraAlgebra Univers. univers.

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~~Premyslˇ Jedlickaˇ Department of Mathematics, Faculty of Engineering, Czech University of Life Sciences, Kam´yck´a129, 16521 Praha 6, Czech Republic e-mail: [email protected]~~

~~Agata Pilitowska Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland e-mail: [email protected]~~

~~Anna Zamojska-Dzienio Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland e-mail: [email protected]~~

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