On the Multiplicative Quasi-Adequate Transversals of an Abundant Semigroup

In this article, we explore the multiplicative quasi-adequate transversals of an abundant semigroup. Let S be an abundant semigroup with multiplicative quasi-adequate transversals. Then the product of any two multiplicative quasi-adequate transversals is also a multiplicative quasi-adequate transversal. Moreover, all multiplicative quasi-adequate transversals of S form a rectangular band. Let S°and S ^? be multiplicative quasi-adequate transversals of S, and let δ°(δ^?) be the δ-relation on S°(S ^?). Then there exists a bijection ? from S°/δ°onto S ^?/δ^?. In particular, if δ°and δ^? are congruences, then the bijection ? is an isomorphism.