Regular *- Congruences On Quasi Regular *- Semigroups

As a generalization of regular *-semigroups, the notation of quasi regular*-semigroup is introduced. A unitary semigroup S is called a quasi regular *-semigroup if it satisfies exactly the identities {x^***=x^*, (xy)^*=y^*x^*,x^*x^**x^*=x^*}, where x^** and x^*** denote (x^*)^* and ((x^*)^*)^*, respectively. In this paper, someproperties of quasi regular *-semigroup are discussed and an example shows thata quasi regular *-semigroup need not be a regular *-semigroup.The congruence plays an important role in studying semigroups. Kernel nor-mal system and congruence pair are the primary methods of describing the congru-ences on semigroups. To describe the congruences on quasi regular *-semigroups,we define the concepts of projection kernel normal system and regular *- congru-ence pair for quasi regular *-semigroups, and prove that each regular *-congruenceon a quasi regular *-semigroup is uniquely determined by its projection kernel nor-mal system and regular *- congruence pair. In addition, we find that for arbitraryρ,σ∈C_R^*(S),ρ=σif and only ifρ|s^*=σ|s^*, where C_R^*(S) denotes the setof the regular *-congruences on a quasi regular *-semigroup S. In chapter 4, wedefine a relationθon C^*(S), the complete lattice of *-congruences on a quasiregular *-semigroup S, and prove thatθis a congruence on C^*(S) and eachθ-class is a complete sublattice of C^*(S), and characterize the largest *-congruenceof eachθ-class and the largest projection separating *-congruence on S. In theend, we discuss the projection kernel normal system of the meet and union of the*-congruencesρandσ, whereρ,σ∈C_R^*(S), (ρ,σ)∈θ.