# On connected component decompositions of quandles

## TL;DR

This paper provides explicit formulas and methods for decomposing Alexander quandles into connected components and maximal subquandles, enhancing understanding of their structure.

## Contribution

It introduces a formula for connected component decomposition of Alexander quandles and describes how to decompose quandles into maximal connected subquandles.

## Key findings

- Explicit decomposition formula for Alexander quandles.
- Isomorphism characterization of connected components.
- Method to decompose quandles into maximal connected subquandles.

## Abstract

We give a formula of the connected component decomposition of the Alexander quandle: $\mathbb{Z}[t^{\pm1}]/(f_1(t),\ldots, f_k(t))=\bigsqcup^{a-1}_{i=0}\mathrm{Orb}(i)$, where $a=\gcd (f_1(1),\ldots, f_k(1))$. We show that the connected component $\mathrm{Orb}(i)$ is isomorphic to $\mathbb{Z}[t^{\pm1}]/J$ 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.