Some Studies For Po-semigroups

The study of po-semigroups has taken an important part in the theory of semigroups. The semilattice decomposition is the main method in the study of semigroups so that the complete semilattice congruence plays an important role in the structure of po-semigroups. Since there is a close connection between principal filters and the smallest complete semi-lattice congruences on a po-semigroups, the study of the structure of filters has attracted a number of authors. For example, Kehayopulu, Xie X.Y, Cao Y.L and Gao Z.L all did some works on principal filters. Although principal filters pay a crucial role in the structure of po-semigroups, there is so far no explicit description for the principal filters.The structure of principal filters for any po-semigroup is investigated in this thesis. It is proved that every principal filter of any po-semigroup S can be uniquely determined by its N-classes where N is the smallest complete semilattice congruence on 5. In Particular, we prove that N is the equality relation on S if and only if S is a semilattice and N is the universal relation on S if and only if 5 is the only principal filter. Also, the complete semilattice congruence classes on S is considered, and it is proved that every complete semilattice congruence class of S is the union of some N-classes of S. The relation between principal filters and complete semilattice congruence classes of a po-semigroup S is also investigated.By using the left and right translations of po-semigroups, we can construct chains formed by some po-semigroups. Furthermore, a method of constructing the chains ofpo-semigroups is also given.