On connected component decompositions of quandles

Abstract

We give a formula of the connected component decomposition of the Alexander quandle: $\mathbb{Z}[t^{\pm 1}]/(f_1(t),\dots ,f_k(t))={⨆}_{i=0}^{a-1}\mathrm{Orb}(i)$, where $a=\gcd (f_1(1),\dots ,f_k(1))$. We show that the connected component $\mathrm{Orb}(i)$ is isomorphic to $\mathbb{Z}[t^{\pm 1}]/J$ with an explicit ideal $J$. By using this, we see how a quandle is decomposed into connected components for some Alexander quandles. We introduce a decomposition of a quandle into the disjoint union of maximal connected subquandles. In some cases, this decomposition is obtained by iterating a connected component decomposition. We also discuss the maximal connected sub-multiple conjugation quandle decomposition.