Supernilpotent Semigroups

TL;DR

This paper investigates the structure of supernilpotent semigroups, showing that certain structural decompositions hold only in orthodox semigroups and providing counterexamples for non-orthodox cases.

Contribution

It extends the understanding of semigroup structure by identifying conditions under which supernilpotent semigroups decompose similarly to abelian semigroups.

Findings

Supernilpotent semigroups decompose similarly to abelian semigroups only if they are orthodox.

Counterexamples demonstrate that such decompositions do not hold in non-orthodox regular semigroups.

The paper clarifies the structural limitations for supernilpotent semigroups.

Abstract

Regular abelian semigroups are isomorphic to a direct product of an abelian group and a rectangular band (Warne, 1994). Seeking for a similar result for nilpotency, solvability and supernilpotency of regular semigroups, we obtain that analogous statement is true only in orthodox semigroups. We provide an example that shows that the same does not have to be true in regular semigroups that are not orthodox.