On Slender Generalized Groups
Mohammad Reza Ahmadi Zand, Hamid Torabi Ardakani
TL;DR
The paper addresses extending slenderness from abelian groups to generalized groups (completely simple semigroups) and analyzes how slenderness behaves under homomorphisms and substructures. Slender generalized groups are defined via homomorphisms from the generalized product $\prod_{\mathbb{N}}^{g}\mathbb{Z}$, and the authors show that in the abelian case this generalized notion coincides with the classical slenderness. Key results include preservation of slenderness under surjective generalized group homomorphisms, closure under direct sums, that $G_{e(a)}$ is slender when $G$ is, and a characterization of slender abelian groups via $\prod_{\mathbb{N}}^{g}\mathbb{Z}$. The work also provides examples and non-examples (notably the non-slender generalized product $\prod_{\mathbb{N}}^{g}\mathbb{Z}$) and suggests exploring topological slenderness in generalized groups.
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.