, Volume 17, Issue 3, pp 437-454

Morita Equivalence for Factorisable Semigroups

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Abstract

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 ζ S = {(s 1, s 2) ∈S×S|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, ∐ i∈I S) →∐ i∈I S, st·ƒ↦ (st)ƒ is an S-isomorphism.