Characterization Of Monoids By Condition(IP)

The S-acts theory of semigroups is an important branch of semigroups theory.Using the flatness properties of S-acts to characterize the internal structure of monoids is a major part.Let S be a monoid.We characterize monoids by using Condition(IP)of S-acts.Firstly,we define the Condition(IP)and give the fundamental properties of Condition(IP).We study the implication relation among the Condition(IP)and other flatness properties.In particular,we characterize Condition(IP)by pullback diagrams.Secondly,we investigate the homological classification problems of monoids by using Condition(IP).Based on the characterization that cyclic acts and monocyclic acts satisfy Condition(IP),we study the homological classification problem of Condition(IP)in cyclic acts.In particular,flat(weakly flat)cyclic S-act satisfies the Condition(IP)if and only if the idempotent that is not identity of S is a right zero element.In addition,we call an ideal K with |E(K)|≥ 2 of a monoid S I ideal if for all x ∈ S,t ∈ E(S),xt ∈ K imply that x∈K,E(K)(?)St and give the homological classification of Condition(IP)on general Rees factor acts and special Rees factor acts.Finally,we study Condition(IP)covers and give the necessary condition that any cyclic S-acts have Condition(IP)cover.We define the monoid with I decomposition and give the sufficient and necessary condition for all cyclic S-acts have Condition(IP)cover by this kind of monoids.