Projective and indecomposable S-acts

Let S be a semigroup and E the set of all idempotents in S. Let S-Act be the category of all S-acts. Let (?) be a full subcategory of S-Act which contains _sS and is closed under coproducts and summands. It is proved that, in (?), an S-act P is projective and unitary if and only if P (?) Ц_i∈1 Se_i, e_i∈E. In particular, P is a projective, indecomposable and unitary object if and only if P (?) Se for some e∈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.