Abstract

This article provides an in-depth examination of (m, n, γ)-regular le-Γ- semigroups, focusing on the characterization and properties of various types of ideal el- ements within these structures. Specifically, the discussion encompasses (m, n, γ)-ideal elements, (m, 0, γ)-ideal elements, and (0, n, γ)-ideal elements, highlighting their sig- nificance and interrelationships. Furthermore, the article investigates the (m, n, α, β)- regularity associated with the set denoted as I(m,n,α,β), which consists of all (m, n, α, β)- ideal elements. In conjunction, the study explores the set Q(m,n,α,β), which comprises all (m, n, α, β)-quasi-ideal elements of le-Γ-semigroups, detailing the implications of these classifications on the structure and behavior of the semigroups. Additionally, the research delves into the concept of 0-minimality, particularly concerning (0, m, γ)-ideal elements in both poe-Γ-semigroups and le-Γ-semigroups. This aspect of the study aims to clarify the foundational properties of ideal elements and their roles in the broader context of semi- group theory. The findings presented in this article contribute to a deeper understanding of the algebraic properties of le-Γ-semigroups and their ideal elements, paving the way for future research in this area.