Morita Equivalence for Factorisable Semigroups

Abstract

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 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, S⊗SHomS(S, ∐i∈IS) →∐i∈IS, s⊗t·ƒ↦ (st)ƒ is an S-isomorphism.