On S-acts containing no maximal S-subacts

Let S be a semigroup and let A be an S-act. Some necessary and sufficient conditions that S-subacts of A are maximal S-subacts are given. A relation B which is similar to the Green relation in semigroups is defined. By the relation B, it is proved that a non-empty set L of A is a maximal S-subact if and only if A／L is a （maximal） B-class. Finally, the concept of a C-subact is defined, some properties of C-subacts are discussed, and it is proved that A contains no maximal S-subacts if and only if every cyclic S-subact of A is a C-subact. Consequently, the results obtained by Imrich Fabrici that semigroups contain no maximal （left） ideals are the corollary of this paper.

On Monoids over which All Strongly Flat Right S-Acts Are Regular

This paper investigates the characterizations of monoids over which all strongly flat right S-acts are regular. It is shown that all strongly flat right S-acts are regular if and only if S is a right PSF monoid and every left collapsible submonoid of S contains a left zero. This result gives a new answer to the problem in Kilp and Knauer.

Projective and indecomposable S-acts

LetS be a semigroup andE the set of all idempotents inS. LetS-Act be the category of allS-acts. LetC be a full subcategory ofS-Act which containss S and is closed under coproducts and summands. It is proved that, inC, anS-actP is projective and unitary if and only ifP? ? j? I Se i ,e i ?E. In particular,P is a projective, indecomposable and unitary object if and only ifP ?Se for somee ∈E. These generalize some results obtained by Knauer and Talwar.

Morita Equivalence for Factorisable Semigroups

Recall that the semigroups S and R are said to be strongly Morita equivalent if there exists a unitary Morita context （S,R,SPR,RQS,） with and surjective.For a factorisable semigroup S,we denote ζS={（s1,s2）∈S×S|ss1=ss2,S∈S},S′=S/ζS and US-FAct={SM∈ S-Act|SM=M and SHomS（S,M）≌M}.We show that,for factorisable semigroups S and R,the categories US-FAct and UR-FAct are equivalent if and only if the semigroups S′ and R′ are strongly Morita equivalent.Some conditions for a factorisable semigroup to be strongly Morita equivalent to a sandwich semigroup,local units semigroup,monoid and group separately are also given.Moreover,we show that a seinigroup S is completely simple if and only if S is strongly Morita equivalent to a group and for any index set I,SSHomS(S,i∈IS)→i∈IS,st·f（st）f is an S-isomorphism.