|Other Abstract||The S-act (S-poset) theory of (ordered) semigroups as an important branch of the (ordered) semigroup theory, it plays an important role not only in studying properties of (ordered) semigroups but also in other mathematical areas, such as graph theory and algebraic automata theory. In this thesis, we characterize (po-)monoids by using flatness properties of S-acts (S-posets) over (po-)monoids. This thesis contains six chapters.In Chapter 1, we introduce the research background about the theory of S-acts (S-posets), and then list the main results of this thesis. Finally, we introduce some basic definitions and facts that are exactly what we need in this thesis.In Chapter 2, we study S-acts satisfying Condition (PF'). First, we introduce Condition (PF') in the category of S-acts, and discuss relations between this property and weak pullback flatness (resp., Condition (P')). Moreover. we prove that Condition (PF') coincides exactly with the conjunction of Conditions (P') and (E'). Second, we give some characterizations of monoids by Condition (P'') of (cyclic, Rees factor) acts. Particularly, we characterize monoids under which Condition (PF') coincides with Condition (P') (resp., weak pullback flatness, strong flatness) for Rees factor acts. Finally, we investigate Conditions (P'), (E') and (PF') using the purity of epimorphisms.
In Chapter 3, we study S-acts satisfying Condition GP-(P). Corresponding to GP-flatness in the category of S-acts, we first define Condition GP-(P), which is a generalization of Condition (PWP). Moreover, we study the homological classification problems of monoids by using Condition GP-(P) of their (cyclic, Rees factor) acts. Especially, the classes of some important monoids are characterized, such as right cancellative monoids, generally regular monoids, groups and so on. And then the correlative conclusions about Condition (PWP) are generalized. Second, we present some equivalent characterizations of monoids (such as cancellative monoids, left groups, etc.) by investigating their diagonal acts satisfying Condition GP-(P). Finally, using quasi G-2-pure epimorphisms and quasi-2-pure epimorphisms, we obtain some new equivalent descriptions of Condition GP-(P) (resp., Condition (PWP)).In Chapter 4, we research on purity conditions of epimorphisms in the category of S-posets. First, we introduce some new types of epimorphisms with certain purity conditions, and obtain equivalent descriptions of various flatness propert...|