Yu Qun CHEN(1),K. P. SHUM(2)
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 surjec-tive. For a factorisable semigroup S, we denote ζS = {(s1, s2)∈ S× S| ss1 = ss2, s ∈ S}, S' = S/ζs and U S-FAct= {SM ∈ S-Act | SM = M and SHoms(S, M)≌ M}. We show that, for factorisable semigroups S and R, the categories U S-FAct and U R-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 respectively 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, S SHoms(S, i∈I S)→ i∈I S, s t·f → (st)f is an S-isomorphism.