On Slender Generalized Groups

TL;DR

This paper extends the concept of slender abelian groups to generalized groups, establishing foundational properties, structural results, and illustrating the theory with examples.

Contribution

It introduces the notion of slender generalized groups, explores their properties, and shows how classical slenderness concepts extend to this broader setting.

Findings

Slenderness is preserved under homomorphisms and subgroups.

In abelian cases, classical and generalized slenderness coincide.

Group components of slender generalized groups are classically slender.

Abstract

This paper introduces the concept of slender generalized groups, extending the classical notion of slender abelian groups to the setting of generalized groups (completely simple semigroups). We establish fundamental properties of slender generalized groups and prove that, in the abelian case, the classical and generalized definitions of slenderness coincide. Several structural results are provided, including the behavior of slenderness under homomorphisms and subgroups. We also present examples and non-examples to illustrate the theory. Our results demonstrate that slenderness is preserved under taking generalized subgroups and that the group components $G_{e(a)}$ of a slender generalized group are slender in the classical sense.