Sublattices of the lattices of strong P-congruences on P-inversive semigroups

In this paper, we describe strong P-congruences and sublattice-structure of the strong P-congruence lattice C-P(S) of a P-inversive semigroup S(P). It is proved that the set of all strong P-congruences C-P(S) on S(P) is a complete lattice. A close link is discovered between the class of P-inversive semigroups and the well-known class of regular *-semigroups. Further, we introduce concepts of strong normal partition/equivalence, C-trace/kernel and discuss some sublattices of C-P(S). It is proved that the set of strong P-congruences, which have C-traces (C-kernels) equal to a given strong normal equivalence of P (C-kernel), is a complete sublattice of C-P(S). It is also proved that the sublattices determined by C-trace-equaling relation theta and C-kernel-equaling relation k, respectively, are complete sublattices of C-P(S) and the greatest elements of these sublattices are given.