A Structural and Categorical Study of Quasi-Ideal Transversals in Abundant Semigroups

Abstract

Quasi-ideal transversals serve as central tools for analyzing the structure of abundant semigroups. While generalized Green’s relations—L ∗ , R∗ , and H∗—extend classical decomposition techniques, a full characterization of quasi-ideal transversals in terms of these relations has remained incomplete. Existing studies are mostly restricted to adequate or special U-abundant semigroups, lacking a unified structural and categorical framework. This paper develops a comprehensive theory of quasi-ideal transversals in abundant semigroups using generalized Green’s relations. We provide necessary and sufficient conditions for their existence formulated purely through L ∗ - and R∗ -classes, and prove canonical factorization theorems expressing every element as a “sandwich” of transversal representatives and related elements. Building on these factorizations, we establish categorical reconstruction results, showing that an abundant semigroup can be recovered up to isomorphism from transversal data and generalized Green’s classes. Compatibility with morphisms and closure under sandwich operations is also demonstrated, integrating and extending previous results on adequate and quasi-adequate transversals. Our findings offer a unified structural and categorical treatment of quasi-ideal transversals, consolidating disparate results in the literature and providing a foundation for further algebraic and categorical investigations in non-regular semigroup theory.