On Morita equivalence of partially ordered semigroups with local units (Q2906187)

In this paper, Pos denotes the category of posets and monotone maps, Pos\(_S\) the Pos-category of right \(S\)-posets and right \(S\)-poset morphisms, FPos\(_S\) the full Pos-subcategory of Pos\(_S\) generated by closed \(S\)-posets. NEWLINENEWLINENEWLINE These are the main results of the paper: Let \(S\) and \(T\) be ordered semigroups (po-semigroups). If \(S\) and \(T\) have common local units and their Cauchy completions \(C(S)\) and \(C(T)\) are Pos-equivalent, then \(S\) and \(T\) have a joint enlargement \(R\) which also has common local units and the biposets \(SRT\) and \(TRS\) are Pos-unitary. If the Pos-categories Pos\(_S\) and Pos\(_T\) are Pos-equivalent, then \(S\) and \(T\) are isomorphic. If \(S\) and \(T\) have common local units and \(C(S)\), \(C(T)\) are Pos-equivalent, then so are FPos\(_S\) and FPos\(_T\). If \(S\) and \(T\) have ordered local units and FPos\(_S\), FPos\(_T\) are Pos-equivalent, then so are \(C(S)\) and \(C(T)\). NEWLINENEWLINENEWLINE Recall that if a po-semigroup \(S\) has common local units, then it has ordered local units, and if \(S\) has ordered local units, then it has local units.