The structure of medial quandles

Abstract

Medial quandles are represented using a heterogeneous affine structure. As a consequence, we obtain numerous structural properties, including enumeration of isomorphism classes of medial quandles up to 13 elements.

Free medial quandles

“…Mediality is another name for entropicity in the binary case. For a more detailed history of medial quandles, we refer to [9]. Here we generalize the result of Joyce but not directly.…”

“…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 medial quandles, these groups are free Z[t, t −1 ]-modules and we can construct the free medial quandles based on these modules.…”

“…This section recalls some important notions from [9] where the structure of medial quandles was described. Key ingredients are two permutation groups acting on quandles.…”

“…In [9], 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…”

Multivariate Alexander quandles, II. The involutory medial quandle of a link

“…The proof of Theorem 6 requires some ideas from the general theory of quandles. We summarize these ideas in this section, and refer to the work of Jedlička, Pilitowska, Stanovský and Zamojska-Dzienio [4,5] for a more thorough discussion. Proofs are included for the reader's convenience.…”