The theory of actions of semigroups on sets is applied in many branches of mathematical sciences, such as algebra, dynamical systems, computer science, automata, and computational mathematics. The book [37] is a good reference for most of we know about acts from an abstract point of view. General ordered algebraic structures play a role in a wide range of areas, including analysis, logic, theoretical computer science, and physics. Combining the notions of a poset and an act, one important of these structures, the category of S-posets, which are the representations of a posemigroup or a pomonoid S by order preserving maps of partially ordered sets, is of interest to some mathematicians, and many algebraic and categorical properties of the category of actions of a posemigroup or a pomonoid on a poset have been studied in [12{15,28,67].
