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

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