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., S P R,R Q S , , ) with and surjective. For a factorisable semigroup S, we denote S = {(s 1, s 2) SS|ss 1 = ss 2, sS}, S' = S/ S and US-FAct = { S MS Act |SM = M and SHom S (S, M) M}. We show that, for factorisable semigroups S and M, 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 semigroups to be strongly Morita equivalent to a sandwich semigroup, local units semigroup, monoid and group separately are also given. Moreover, we show that a semigroup S is completely simple if and only if S is strongly Morita equivalent to a group and for any index set I, SSHom S (S, iI S) iI S, st (st) is an S-isomorphism.