Complete Characterization of Quasi-Ideal Transversals in Abundant Semigroups Using Generalized Green’s Relations

2025-12-29T06:16:00+00:00

This paper develops a precise and comprehensive characterization of quasi-ideal adequate transversals of abundant semigroups by means of generalized Green relations. Building on and refining results of Fountain, El-Qallali, Saito, and Al-Bar & Renshaw, we establish necessary and sufficient conditions under which a subsemigroupT of an abundant semigroup S serves as an adequate transversal with the quasi-ideal property. In particular, we prove that if S is H_e-abundant, then T is a quasi-ideal adequate transversal of S if and only if T meets each H_e-class of S in exactly one element and satisfies STS ⊆T. The result subsumes and extends earlier structure theorems for adequate, quasi-adequate, and inverse transversals. We also describe the canonical factorization s = etffor all s ∈S with t ∈T and e,f∈E(S), and show that the semilattice E(T) induces a congruence decomposition on S. Furthermore, we explore categorical interpretations by relating quasi-ideal transversals to wide subcategories in the Ehresmann category associated with S. Concrete examples and spined-product constructions are provided to illustrate the theory. This resolves all open assertions outlined in the abstract and clarifies the structure of quasi-ideal transversals in the general abundant setting.

IJO - International Journal of Mathematics (ISSN: 2992-4421 )

Complete Characterization of Quasi-Ideal Transversals in Abundant Semigroups Using Generalized Green’s Relations