With Q-inverse Transversals Regular Semigroup Congruence Grid

In this paper ,we first give a survey of the research on the congruence of regular semigroups with inverse transversals .Then we study the congruence and congruence lattices of regular semigroups with Q-inverse transversals on the basis of the structure related the build bricks R and L.For a regular semigroup S, ainverse subsemigroup S° of S is an inverse transversal if |/V(x) ∩ S°| = 1 for each x ∈ S, where V(x) denotes the set of inverses of x. In this case ,the unique element of V{x) ∩ S° is denoted by x° and x~∞ denotes (x°)° E° denotes the semilattice of idempotents of S°. Some notations are defined as follows ,The structure of regular with Q- inverse transversals 5° was given by Mcalister and Mcfadden in 1982. Congruence lattices of regular semigroups with inverse transversals were first studied by Wang Limin. Wang Limin and Tang Xilin extended the method used above to investigate the congruence lattice to the general case .Saito gave the structure theorem of regular semigroups with Q-inverse transversals on the basis of the structure related the" build bricks" R and L .In the second chapter ,we study regular semigroups with Q-inverse transversals on the basis of the structure related the build bricks R and L. We will obtain the so called congruence pair based on Saito structure theorem, so that it produces a congruence on S abstractly. We consider the lattice of congruence pair ordered by componentwise inclusion and establish an isomorphism between the congruence lattice and the lattice of congruence pair .We also offer some special congruences on regular semigroups with Q-inversal transversal.Two complete congruence on the congruence lattice are analysed . The classes of each complete congruence which are intervals are discussed and their least and greast elements presented clearly. The extremal values of classes of complete congruence for the universal relation are exhibited and some of their relation are explored.Finally ,we arrive at a subdirect decompositions of the congruence lattice.