We give a formula of the connected component decomposition of the Alexander quandle: Z[t ?]/(f1(t), . , fk(t)) = a-1 i=0 Orb(i), where a = gcd(f1(1), . , fk(1)). We show that the connected component Orb(i) is isomorphic to Z[t ?]/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.
- Alexander quandle
- connected component decomposition
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