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June 2019 On Connected Component Decompositions of Quandles
Yusuke IIJIMA, Tomo MURAO
Tokyo J. Math. 42(1): 63-82 (June 2019). DOI: 10.3836/tjm/1502179252

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.

Citation

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Yusuke IIJIMA. Tomo MURAO. "On Connected Component Decompositions of Quandles." Tokyo J. Math. 42 (1) 63 - 82, June 2019. https://doi.org/10.3836/tjm/1502179252

Information

Published: June 2019
First available in Project Euclid: 18 July 2019

zbMATH: 07114901
MathSciNet: MR3982050
Digital Object Identifier: 10.3836/tjm/1502179252

Subjects:
Primary: 57M25
Secondary: 57M27

Rights: Copyright © 2019 Publication Committee for the Tokyo Journal of Mathematics

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Vol.42 • No. 1 • June 2019
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