On connected component decompositions of quandles

Original · EN

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