Green Relations And Congruences On Semigroup

First, we defined a particular regular semigroup which we called weaken U-semigroup. Then the Green relations on the regular semigroup are investigated.Second, a structure of the smallest semilattice congruence on general semigroup is given. We redefine C-subsemigroup of S, if T is a C-subsemigroup of S then each semilattice congruence of T can be distended to a semilattice on SThen, we proved that the set of all semilattice congrunces on S forms a com-plete sublattice of the congruence lattice of S. Furthermore, we characterized the relationship between normal trace and semilattice congruences. By using these, a structure theorem for an arbitrary semilattice congruence on S is given. It is also proved that there exists a bijection between the semilattice congruence and the set of all normal trace of the regular semigroiups.Finally, we proved that the set of all rectangular band congruences on S forms a complete sublattice of the congruence lattice of S. Furthermore, we characterized the relationship between normal trace and rectangular band congruences. By using these, a structure theorem for an arbitrary rectangular band congruence on S is given. It is also proved that there exists a bijection between the rectangular band congruence and the set of all normal trace of the eventually regular semigroup.