Some Studies On Abundant Semigroups With Quasi-adequate Transversals

This thesis is mainly devoted to study an abundant semigroup with a quasi-adequate transversal. Details are as follows:In chapter 2, we introduced the definition of multiplicative quasi-adequate transversa then got a structure theorem for abundant semigroups with a multiplicative quasi-adequate transversal. At last, we apply the structure theorem to some special cases, which shows that our structure covers all regular semigroups with a multiplicative inverse transversal, all regular semigroups with a multiplicative orthodox semigroups as well as all abundant semigroups with a multiplicative adequate transversal.In chapter 3, it is devoted to study the relationship between multiplicative quasi-adequate transversals of an abundant semigroup. We proved that the multiplication of two multiplicative quasi-adequate transversals is also a multiplicative quasi-adequate transversal. Furthermore, the set of all multiplicative quasi-adequate transversal for S is a rectangular band. It is investigated that for any multiplicative quasi-adequate transversals S°and S~b of an abundant semigroup S, ifδ°andδ°areδ-relation on S°and S~b respectively, then there exists a bijection from S°/δ°to S~b/δ~b. In particular, if theδ-relation is a congruence either on S°or S~b then S°/δ°(?)S~b/δ~b.In chapter 4, it is devoted to study an abundant semigroup with a quasi-ideal quasi-adequate transversal and characterize the structure of a class of such abundant semigroups. Furthermore, we also apply the structure theorem to a special case, which shows that our structure covers all regular semigroups with a quasi-adequate transversal.In chapter 5, we mainly study the good homomorphic image of an abundant semigroup with a multiplicative normal quasi-adequate transversal and the good congruence on a class of such abundant semigroups. Then by using the good congruence triple (η,θ,τ) which is the restriction of a good congruence on the blocks I, S°and A of an abundant semigroup S with a multiplicative normal quasi-adequate transversal, we characterized the representation theorem for good congruences on a class of such abundant semigroups.