Some Studies On The Flatness Of S-acts (S-posets)

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?atness 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 de?nitions and facts that are exactly what we need in this thesis.In Chapter 2, we study S-acts satisfying Condition(P F). First, we introduce Condition(P F) in the category of S-acts, and discuss relations between this property and weak pullback ?atness(resp., Condition(P)). Moreover. we prove that Condition(P F) coincides exactly with the conjunction of Conditions(P) and(E). Second,we give some characterizations of monoids by Condition(P F) of(cyclic, Rees factor)acts. Particularly, we characterize monoids under which Condition(P F) coincides with Condition(P)(resp., weak pullback ?atness, strong ?atness) for Rees factor acts. Finally, we investigate Conditions(P),(E) and(P F) using the purity of epimorphisms.In Chapter 3, we study S-acts satisfying Condition GP-(P). Corresponding to GP-?atness in the category of S-acts, we ?rst de?ne Condition GP-(P), which is a generalization of Condition(P W P). Moreover, we study the homological classi?cation 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(P W P) 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(P W P)).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 ?atness properties of S-posets, such as, strongly ?atness, Conditions(E),(E),(P),(Pw),(W P),(W P)w,(P W P) and(P W P)w. Thereby, we give other equivalent conditions in the Stenstr¨om-Govorov-Lazard Theorem for S-posets. Second, we prove that these new epimorphisms are closed under directed colimits. In particular, we provide a new approach to show that most of ?atness properties of S-posets can be transferred to their directed colimit.In Chapter 5, we study GP-po-?atness of S-posets. First, we give some basic properties about GP-po-?at S-posets. Second, we investigate the homological classi-?cation problems of pomonoids by using this new property. Finally, we consider direct products of GP-po-?at S-posets. As an application, characterizations of pomonoids over which direct products of nonempty families of principally weakly po-?at S-posets are principally weakly po-?at are obtained, and some results of Khosravi in 2004 are generalized.In Chapter 6, we study weak torsion freeness of S-posets. First, by giving counterexamples, we show that the implication: “principal weak ?atness  torsion freeness” that Bulman-Fleming et al. have provided in 2006, and the equivalent descriptions of ?atness, weak ?atness and principal weak ?atness that Golchin et al. have given in 2009, are incorrect. In view of the above, we present correct descriptions of these properties, and study weak torsion freeness of S-posets. Moreover, we characterize almost regular pomonoids and po-cancellable pomonoids by using this property.