Free medial quandles

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

Here we generalize the result of Joyce but not directly. We choose the path started in [8] instead and we study certain permutation groups acting on quandles, called displacement groups. It turns out that, in the case of free medial quandles, these groups are free Z[t, t −1 ]-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 n−times * y) · · · ) ≈ y.
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 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 each finite medial quandle embeds into a product of a reductive quandle and a quasigroup [9]. The paper contents four Sections. In Section 2 we recall and present some facts about general medial quandles. Section 3 contains the main results. Theorem 3.3 gives a description of the free medial quandles and Theorem 3.5 a construction of affine quandles into which the free quandles embed. Section 4 is devoted to n-symmetric and m-reductive free medial quandles. In both cases, the displacement group of the free algebra turns out to be a free Z[t]/(f )-module, for a suitable polynomial f . The description of the free quandles in these varieties is analogous as for general medial quandles.
Note that when studying left quasigroups, important tools are the mappings L e : x → e * x, called the left translations. We use also the right translations R e : x → x * e. The idempotency and the mediality imply that both L e and R e are endomorphisms. The left quasigroup property means that L e is an automorphism.

Preliminaries
This section recalls some important notions from [8] where the structure of medial quandles was described. Key ingredients are two permutation groups acting on quandles.
Definition 2.1. Let Q be a quandle. The left multiplication group of Q is the group The displacement group is the group It was proved in [7, Proposition 2.1] that the actions of both groups on Q have the same orbits. We use, in the sequel, the word orbit plainly without explicitly mentioning the acting groups. The orbit of Q containing x is denoted by Qx and the stabilizer subgroup of x is denoted by Dis(Q) x . For two permutations α, β, we write α β = βαβ −1 . The commutator is defined by [β, α] = α β α −1 . The identity permutation is denoted by 1.
It is also useful to understand the structure of the displacement group.
. From this lemma, we can clearly see that Dis(Q) is a normal subgroup of LMlt(Q). Moreover, in our context, the group is commutative. Since Dis(Q) is abelian, conjugations by elements from the same coset of Dis(Q) yield the same results.
x L y αL −1 y = α Ly due to the abelianess of Dis(Q).
From now on, by writing α L , we mean α Lx , for an arbitrary x ∈ Q, since the conjugation does not depend on the element x.
It is easy to see that, for α ∈ Aut(Q) and x ∈ Q, L α(x) = L α x . In particular, for α = L y , we obtain L y * x = L Ly x . This implies that LMlt(Q) has only few generators. On the other hand, DisQ, 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 {L x ; x ∈ X}; • the group Dis(Q) is generated by {(L x L −1 z ) L k ; x ∈ X, k ∈ Z}. Proof. The generating set for LMlt(Q) is obtained by the induction using L x * y = L x L y L −1 x and L x\y = L −1 x L y L x . Suppose now α ∈ Dis(Q). By Lemma 2.2 and the previous observation we can suppose α = We prove the claim by an induction on n. For n = 2 the claim is true.
Let the induction hypothesis holds for all words of length at most n − 2. If ε 1 = ε n then clearly w = w 1 w 2 with w i ∈ Dis(Q) and we use the induction hypothesis. Let now ε 1 = 1 and ε n = −1. Then w = L x 1 w ′ L −1 xn and w ′ is, by the induction hypothesis, a product of elements from The argument is similar for ε 1 = −1 and ε n = 1.
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 Z[t, t −1 ]-module, it suffices to pick an automorphism of Dis(Q). A natural choice is the inner automorphism α → α L . Hence, from now on, the group Dis(Q) is treated, depending on the situation, either as a permutation group acting on Q or as an It was proved in [8, 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 Dis(Q)/Dis(Q) x by defining the group operation on Qx as: The group so defined is denoted by Orb Q (x) and called the orbit group for Qx. Moreover, ]-module and we can call Orb Q (x) the orbit module for Qx.

Free medial quandle
In this section we present the free medial quandles. Regarding the generating set, we see that, in any quandle Q, for all x, y ∈ Q, y * x ∈ Qx as well as y\x ∈ Qx. Hence each orbit has to contain at least one generator.
The following proposition characterizes the free medial quandles. Formally, it is pronounced as a sufficient condition only but we can see in Theorem 3.3 that such an object exists, making the condition necessary too.
Proposition 3.2. Let F be a medial quandle generated by a set X ⊂ F . Choose z ∈ X arbitrarily. Then F is free over X if the following conditions are satisfied: (1) each element of X lies in a different orbit; Proof. Observe first that, for any y ∈ F , there exists exactly one x ∈ X and exactly one α ∈ Dis(F ) such that y = α(x). Indeed, the existence of x, comes from Lemma 3.1, and its uniqueness from (1). The uniqueness of α is due to (3).
Let Q be a medial quandle and let Y ⊂ Q. Let ψ be a mapping X → Y . We prove that ψ can be extended to a homomorphism Ψ : F → Q. We define first a Z[t, t −1 ]-module homomorphism Φ : Dis(F ) → Dis(Q) on the basis of Dis(F ) by setting Φ(L for all α ∈ Dis(F ) and x ∈ X.
Mapping Ψ is well defined since every element of F has a unique representation by α and x.
In the sequel, we use the following notation: let X be a set. We choose z ∈ X arbitrarily and we denote by X − the set X {z}. We often do not specify the element z since we actually rarely need it explicitly. Let now 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 ; i ∈ X − }, and by defining e z = 0 ∈ M , we have defined e i as an element of M , for each i ∈ X. Proof. The idempotency is evident. The mediality is proved by the observation that ((a, i) * (b, j)) * ((c, k) * (d, n)) = ((1−t) 2 ·a+(t−t 2 )·(b+c)+t 2 ·d+(1−t)·e i +t·(e j +e k )−(1+t)·e n , n).
The left-quasigroup operation is given by the formula 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 ).
The case n = 0 was already proved. Now suppose n > 0.
Let (f i ) i∈X − , j be in F . We now prove that this element lies in the subquandle generated by {(0, i); i ∈ X}. But it is not difficult to see that The product is finite since only finitely many f i are non-zero. Hence {(0, i); i ∈ X} = F . Moreover, we see that different generators lie in different orbits.
Since the set L (0,i) L −1 (0,z) L n ; i ∈ X, n ∈ Z generates Dis(F ), due to Proposition 2.5, we see that Dis(F ) acts freely on every orbit of F . That means also that Dis(F ) is isomorphic to M and {L (0,i) L (0,z) ; i ∈ X − } is clearly its free basis. According to Proposition 3.2, F is free over {(0, i); i ∈ X}.
In [8], the structure of medial quandles was represented using a heterogeneous 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 now constructed is the sum of the affine mesh Recall that subquandles of affine quandles are quasi-affine. Every (both sided) cancellative medial quandle is quasi-affine -to see this we can either use a result by Kearnes [11] for idempotent cancellative algebras having a central binary operation or a result by Romanowska and Smith (see e.g. [14]) for cancellative modes. Nevertheless, a direct proof is simple.

Proposition 3.4.
Let Q be a cancellative medial quandle. Then Q embeds into any of its orbits.
Proof. R x is an endomorphism of Q, for each x ∈ Q. The right cancellativity ensures that R x is injective. Hence, for each x ∈ Q, R x embeds Q into Qx.
The free medial quandle, we have constructed, is cancellative and therefore it can be represented as a subquandle of an affine quandle.
Theorem 3.5. Let X be a set. The free medial quandle over X is isomorphic to a subquandle of the affine quandle M = Aff( x∈X − Z[t, t −1 ], t).
and analogously Λ(L −1 a (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 (0,z) : F → F (0, z). Clearly R (0,z) ((0, i)) = (e i , 0). Therefore the subquandle of M generated by Y = {e x ; x ∈ X} is free.
The only thing left to prove is to On the other hand, for Q ⊆ Y , we notice that Q = {a ∈ M ; a ≡ e i (mod (1 − t)), for some i ∈ X}. Moreover, we have (L ex L −1 ez ) L n : Q → Q; u → u + (1 − t)t n · e x , with an analogous proof as in Theorem 3.3.
. Now, for each element a ∈ Q, we have a = e i + (1 − t)g, for some i ∈ X and g = (g x ) x∈X − ∈ M . Hence, we have Therefore a ∈ Y .

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 characterized by an identity on Dis(Q).
. We say that a medial quandle Q is an I-quandle, if α f = 1, for each α ∈ Dis(Q) and f ∈ I.
If I is an ideal of Z[t, t −1 ] and a Z[t, t −1 ]-module M satisfies the identity f · a = 0, for each a ∈ M and f ∈ I, then M can be viewed as a module over Z[t, t −1 ]/I. In our context, the set I shall usually be a principal ideal, that means I = (f ), for some f ∈ Z[t, t −1 ]. We then write that Q is an f -quandle, rather than {f }-quandle or (f )-quandle.
If, moreover, f = n r=0 c r t r and the coefficient c 0 is invertible then We use these remarks since working with the ring Z[t]/f is often easier than working with the ring Z[t, t −1 ].
We prepared the framework of I-quandles to work with symmetric and reductive medial quandles at once. First note that if Q is an I-quandle then clearly Orb Q (x) I = {α f (x) | α ∈ Dis(Q), f ∈ I} = {x}. On the other hand, let α f (x) = x for arbitrary α ∈ Dis(Q), f ∈ I and each x ∈ Q. Hence the action of α f on Q is trivial and this means α f = 1 since Dis(Q) is faithful. This immediately gives the following lemma. Recall that a quandle Q is n-symmetric, if L n x = 1, for each x ∈ Q. Proposition 4.3. A medial quandle Q is n-symmetric if and only if Q is a ( n−1 r=0 t r )-quandle. Proof. According to [8,Proposition 7.2], a medial quandle Q is n-symmetric if and only if, for each and therefore it means that α n−1 r=0 t r (x) = x, for each x ∈ Q. Hence, according to Lemma 4.2, Q is n-symmetric if and only if it is a ( n−1 r=0 t r )-quandle. Recall that a quandle Q is m-reductive, if R m x (y) = x, for all x, y ∈ Q. In this case, not only Dis(Q) can be treated as a Z[t]/f -module but it has less generators even as a group.
Proposition 4.5. Let f = s r=0 c r t r be a polynomial with c 0 and c k 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 {(L x L −1 z ) L r ; x ∈ X {z} and 0 ≤ r < s}. Proof. According to Proposition 2.5, the group Dis(Q) is generated by (L x L −1 z ) L k , for x ∈ X {z} and k ∈ Z. But now Similarly for all (L x L −1 z ) L r , where r < −1 or r > s. The structure of free medial f -quandles can be described exactly in the same way as the structure of general free medial quandles.
Proof. It is well known [17] that Z[ζ k ] is a Dedekind domain, for each k. Free n-symmetric quandles embed into products of Aff( x∈X − Z[ζ k ], ζ k ). Each of these affine quandles embeds into any of its orbits, i.e. into a module over Z[ζ k ].
Note that applying our idea of I-quandles one obtains the description of free n-symmetric m-reductive medial quandles if we consider I = { n−1 r=0 t r , (1 − t) m−1 }. In particular, the free 2-reductive n-symmetric medial quandle over X is isomorphic to x∈X − Z n × X with the operation (a, i) * (b, j) = (b + e i − e j , j) [13,Proposition 2.4].