Strong P-Congruences On P-Inversive Semigroups And The Lattice

Abstract: Let S(P) be a P-inversive semigroup. In this paper we study the strong P-congruences on S(P) and the lattice. In Chapter 1, we give the definitions of P-inversive semigroups and so-called strong P-congruences, and it is given that the least element of the lattice of strong P-congruences Cp(S). Some examples are given to illustrate that P-inversive semigroups are a special subclass of the class of E-inversive semigroups and there exist P-inversive semigroups which are neither P-regular semigroups nor E-inversive E-semigroups. A powerful link is discovered between the relatively large class of P-inversive semigroups and the well-known class of regular *-semigroups. A sufficient and necessary condition is given for a congruence on S(P) to be a strong P-congruence. In Chapter 2, we shall characterize some sublattices of strong P-congruence lattice Cp(S). We first consider a strong normal partition of P, it is proved that the set of strong P-congruences, whose characteristic traces equal to a strong normal equivalence induced by a given strong normal partition of P, is a complete sublattice of Cp(S). Moreover, we give the definitions of characteristic traces and characteristic kernels. It is proved that the sublattices determined by the characteristic trace relation θ and characteristic kernel relation κ, respectively, are complete sublattices of Cp(S). In Chapter 3, we describe the strong P-congruences on S(P) in terms of their strong P-kernel normal systems. It is proved that there is a bijection between the strong P-congruences and the strong P-kernel normal systems. Further, a characterization of strong P-kernel normal systems is given for the join and meet of two strong P-congruences which are in the same θ-class. Finally, it is also proved that the strong P-congruence lattice and the lattice of strong P-kernel normal systems on S(P) are complete lattice isomorphism. In Chapter 4, we discuss the constructions of strong P-congruence pairs on a strong P-semilattice of P-inversive semigroups. The strong P-congruences on a strong P-semilattice of P-inversive semigroups are characterized using by each of P-inversive semigroups.