Strong Morita equivalence of semigroups with local units.
This paper is based on Morita contexts to develop Morita equivalence for semigroups. Let \(S\) and \(T\) be semigroups, \(E(S)\) be the set of idempotent elements in \(S\). A semigroup with local units is a semigroup \(S\) such that for any \(s\in S\) there exist \(e,f\in E(S)\) such that \(s=esf\). A left \(S\)-act \(_SA\) (a biact \(_SA_T\)) is said to be unitary if \(SA=A\) (\(SA=AT=A\)). A semigroup \(S\) is factorisable if the \(S\)-act \(_SS\) is unitary. A unitary Morita context is a six-tuple \((S,T,{_SP_T},{_TQ_S},\theta,\varphi)\), where \(\theta\colon{_S(P\otimes_TQ)_S}\to{_SS_S}\) and \(\varphi\colon{_T(Q\otimes_SP)_T}\to{_TT_T}\) are biact morphisms such that for any \(p,p'\in P\) and \(q,q'\in Q\), \(\theta(p\otimes q)p'=p\varphi(q\otimes p')\) and \(q\theta(p\otimes q')=\varphi(q\otimes p)p'\). Semigroups \(S\) and \(T\) are called strongly Morita equivalent if there exists a unitary Morita context such that the mappings \(\varphi\) and \(\theta\) are surjective. For a factorisable semigroup \(S\), it is proved that a Rees matrix semigroup \(\mathcal M=\mathcal M(S,U,V,M)\) is strongly Morita equivalent to \(S\) if and only if \(S\text{im}(M)S=S\), where \(\text{im}(M)\subseteq S\) is the set of the entries of the sandwich matrix \(M\). It is proved that if \(S\) and \(T\) are strongly Morita equivalent semigroups then there exists a Rees matrix semigroup \(\mathcal M=\mathcal M(S,U,V,M)\) and an epimorphism \(\tau\colon\mathcal M\to T\) such that the restriction of \(\tau\) to every subsemigroup \(a\mathcal Mb\) of \(\mathcal M\) is injective, where \(a\in\mathcal Ma\) and \(b\in b\mathcal M\) (such an epimorphism is called a strict local isomorphism), and the idempotents and regular elements lift along \(\tau\), that means for any idempotent \(f\in T\) (resp. any regular element \(t\in T\)) there exists an idempotent \(e\in\mathcal M\) (resp. a regular element \(s\in\mathcal M\)) such that \(\tau(e)=f\) (resp. \(\tau(t)=s\)). Some elegant characterizations of Morita equivalence for semigroups with local units are given. Let \(S\) and \(T\) be semigroups with local units. Then the following are equivalent: (1) \(S\) and \(T\) are (strongly) Morita equivalent; (2) there exists a Rees matrix semigroup \(\mathcal M=\mathcal M(S,U,V,M)\) over \(S\) with \(S\text{im}(M)S=S\) and a strict local isomorphism \(\tau\colon\mathcal M\to T\) along which idempotents lift; (3) there is a surjective defined unitary Morita semigroup \(Q\otimes_SP\) and a strict local isomorphism \(\tau\colon Q\otimes{_SP}\to T\) along which idempotents lift.
scientific article; zbMATH DE number 5941012