The Category Of S-dcpos And Congruences On S-lattices

The act of an ordered semigroup on a poset is widely used in semigroup theory.In this paper, some theoretical properties of Category Pos-S will be extended to thecategory of S-dcpos. We also discuss the congruences on S-lattices.In the first part, we brie?y introduce the history and the current research situationon S-posets, S-lattices, directed complete posets, congruences and the categories whichare cartesian closed in the Domain theory, and give a clean reins of them. Throughciting and analyzing numerous works in this field, we also simply introduce our research.For understanding this paper, we give some preliminary knowledge.In the second part, based on theories of S-posets, the definition of directed com-plete partially ordered monoid (or simply, a dcpomonoid) is introduced. Accordingto this definition, we give the concept of S-dcpo when a dcpomonoid acts on a dcpo.Then we study some theoretical properties of Category Dcpo-S and prove that thiscategory is cartesian closed.In the third part, we study the congruences on S-lattices by the concept of pseudo-congruences. Some homomorphism theorems of S-lattices are given, which are similarto the ones in ordered semigroups. Finally, like the S-posets congruences theorems,the construction of S-lattices congruencesν(H) induced on the distributive S-latticeA by any binary relation H in A is given.