Nilpotent groups, solvable groups and factorizable inverse monoids

TL;DR

This paper introduces new concepts of idempotent series in inverse semigroups, establishes their properties, and extends group-theoretic results like the Jordan-Hölder theorem to semigroup theory, providing new insights into nilpotent and solvable groups.

Contribution

It develops the theory of subcentral and central idempotent series in inverse semigroups and extends classical group results to semigroup context, including characterizations of nilpotent and solvable groups.

Findings

Isomorphism of composition subcentral and central idempotent series in factorizable inverse monoids

Characterizations of coset monoids of nilpotent and solvable groups

Extension of Jordan-Hölder theorem analogue to inverse semigroups

Abstract

In this paper subcentral (resp., central) idempotent series and composition subcentral (resp., central) idempotent series in an inverse semigroup are introduced and investigated. It is shown that if $S=EG$ is a factorizable inverse monoids with semilattice $E$ of idempotents and the group $G$ of units such that the natural connection $\theta$ is a dual isomorphism from $E$ to a sublattice of $L(G)$, then any two composition subcentral (resp., central) idempotent series in $S$ are isomorphic. It may be considered as an appropriate analogue in semigroup theory of Jordan-H\"{o}lder Theorem in group theory. Based on this,$G$-nilpotent and $G$-solvable inverse monoids are also introduced and studies. Some characterizations of the coset monoid of nilpotent groups and solvable groups are given. This extends the main result in Semigroup Forum 20: 255-267, 1980 and also provides another effective approach for the study of nilpotent and solvable groups. Finally, some open problems related to nilpotent and solvable groups are translated to semigroup theory, which may be helpful for us to solve these open problems.