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

Abstract

Green’s relations have played a central role in semigroup theory since their introduction by Green [3], providing a powerful means of analyzing the internal structure of semigroups through equivalence classes determined by principal ideals. In the regular and inverse settings, these relations lead to precise structural decompositions and representation results, many of which are now classical. The theory of semigroups has been extensively studied through the lens of Green’s relations [3, 1, 2], which allow a detailed understanding of internal structure and class decomposition. In non-regular settings, generalized Green’s relations, including L^∗, R^∗, and H^∗, have been used to extend classical structural insights to abundant and U-abundant semigroups [4, 14, 16]. A comprehensive characterization of quasi-ideal adequate transversals in abundant semigroups via generalized Green’s relations has been recently established, providing canonical decompositions and categorical interpretations for these structures [10].

Transversals, and particularly quasi-ideal transversals, provide a powerful mechanism for decomposing semigroups into canonical components [5, 19, 13]. These tools have been successfully applied to structural analysis, canonical factorization, and categorical reconstruction [15, 12].

Recent work has also highlighted intrinsic limitations of classical structural characterizations based on independence and generating systems in semigroups, demonstrating that such notions do not always capture the internal organization of non-regular and non-commutative semigroups adequately [11]. This reinforces the need for alternative structural frameworks—such as generalized Green’s relations and transversal-based decompositions—capable of encoding deeper algebraic behavior.

In non-regular contexts, however, the classical Green’s relations are often too coarse to capture essential structural features. This limitation motivated the introduction of generalized Green’s relations, notably the relations L^∗, R^∗, and H^∗, which arise naturally in the study of abundant semigroups following the work of Fountain [4]. These relations refine the classical framework while retaining sufficient flexibility to apply beyond regular semigroups.

Parallel to these developments, the concept of a transversal has emerged as a key tool for decomposing semigroups into more manageable components. In particular, adequate and quasi-ideal transversals have been used to isolate canonical representatives from Green-type classes and to facilitate reconstruction of the ambient semigroup. Foundational work by El-Qallali and Fountain [5] established the importance of transversals in abundant semigroups, while later contributions refined the theory in special settings.

Despite these advances, existing results concerning quasi-ideal transversals remain fragmented. Studies such as those of Al-Bar and Renshaw [6] and Kong, Wang, and Tang [7] focus primarily on specific subclasses of abundant semigroups or emphasize structural decomposition without establishing complete equivalence results formulated purely in terms of generalized Green’s relations. Moreover, the categorical implications of quasi-ideal transversals—particularly their role in reconstructing semigroups up to isomorphism—have not been fully developed.

The purpose of the present paper is to provide a comprehensive and unified theory of quasi-ideal transversals in abundant semigroups using generalized Green’s relations. We establish necessary and sufficient conditions for the existence of such transversals, prove canonical factorization theorems, and develop categorical reconstruction results that extend and consolidate earlier work.

The paper is organized as follows. Section 2 recalls the necessary preliminaries on semigroups, abundant structures, and generalized Green’s relations. In Section 3 we examine structural properties of abundant semigroups relevant to transversal theory. Section 4 introduces quasi-ideal transversals and studies their basic properties. The main equivalence theorems are proved in Section 5, followed by canonical factorization results in Section 6. Section 7 develops categorical reconstruction results, while Section 8 investigates closure properties under standard operations. Examples and applications are presented in Section 9, and the paper concludes with a summary and directions for future research.