Context equivalence of semigroups. (Q532117)

Let \(S\) and \(T\) be arbitrary semigroups. A Morita context is a six-tuple \((S,T,{_SP_T},{_TQ_S},\theta,\varphi)\), where \(_SP_T\) is an \((S,T)\)-bi-act, \(_TQ_S\) is a \((T,S)\)-bi-act, and \(\theta\colon{_S(P\otimes_TQ)_S}\to{_SS_S}\), \(\varphi\colon{_T(Q\otimes_SP)_T}\to{_TT_T}\) are bi-act morphisms such that, for every \(p,p'\in P\) and \(q,q'\in Q\), \(\theta(p\otimes q)p'=p\varphi(q\otimes p')\), \(q\theta(p\otimes q')=\varphi(q\otimes p)q'\). Semigroups \(S\) and \(T\) are called strongly Morita equivalent if there exists a Morita context such that \(_SP_T\) and \(_TQ_S\) are unitary bi-acts and the mappings \(\theta\) and \(\varphi\) are surjective. Semigroups \(S\) and \(T\) are called context equivalent if there exists a Morita context such that the mappings \(\theta\) and \(\varphi\) are surjective. It is proved that context equivalence is an equivalence relation on the class of factorizable semigroups and that Cauchy completions of context equivalent semigroups are equivalent categories. Among other results, the following is proved: a semigroup \(S\) is strongly Morita equivalent (or, equivalently, factorizable and context equivalent) 1) to a semigroup of local units if and only if \(S\) is a sandwich semigroup; 2) to a monoid if and only if \(S=SeS\) for some \(e\in E\); 3) to a group if and only if \(S\) is completely simple; 4) to a one-element semigroup if and only if \(S\) is a rectangular band.