| Let S be a monoid.In[2]it is proved that every right S-act has a SF-cover and CP-cover over a finite generated monoid with a finite geometric type,The first part of this paper proves that every right S-act has a SPF-cover over a finite generated monoid with a finite geometric type;Furthermore,GP flatness,condi-tion(P’),condition(E’),condition(EP)are the new flatness properties in S-act theory.This paper proves that every class of S-acts having the above flatness prop-erty is closed under directed colimits.Secondly,the notion of factorization systems for S-acts is introduced and give the equivalent characterizations of factorization systems for S-acts.We obtain the necessary and sufficient condition of every S-act having χ-covers with the unique mapping property.Furthermore,Let C be a class of S-maps,denote by 丄C the class of all S-maps that have the unique left lifting property with respect to each S-map in C,this paper proves that ⊥C is saturated.Let PWF be the class of principally weakly flat S-acts,denote by PWF-mono the class of S-monomorphisms f:X → Y such that the Rees quotient Y/X∈ PWF,it is obtained that PWF-mono is also saturated. |
