Inverse semigroups with isomorphic bundles of \(i\)-correspondences
Let \(S\) be a semigroup and \(C(S)\) the set of all correspondences of \(S\) (i.e. subsemigroups of \(S\times S\) including the empty subset). The bundle (it is denoted \(C(S)\), too) of all correspondences of \(S\), i.e. the algebraic system \((C(S),\circ,{}^{-1},\subset)\) is considered. If \(S\) is an inverse semigroup, then the set of all \(i\)-correspondences (inverse subsemigroups of \(S\times S\)) forms a subbundle \(C_i(S)\) of the bundle \(C(S)\). If \(\Phi:C(S)\to C(T)\) (resp. \(C_i(S)\to C_i(T))\) is an isomorphism of bundles, it is called a \(C\) (\(C_i\))-isomorphism \(S\) onto \(T\) and these semigroups are called \(C\)-(\(C_i\))-isomorphic. Some consequences of the fact that the inverse semigroups \(S\) and \(T\) are \(C_i\)-isomorphic are given. These results lead to the main theorem, which states that every \(C_i\)-isomorphism \(\Phi\) of a fundamental inverse semigroup \(S\) onto an inverse semigroup \(T\) is induced by a unique isomorphism of \(S\) onto \(T\). Moreover, that isomorphism and its composition with the natural involution on \(T\) are the only bijections of \(S\) onto \(T\) inducing \(\Phi\). Comparing \(C\)- and \(C_i\)-isomorphisms, the author obtains that if \(\Psi\) is a \(C\)-isomorphism of an inverse semigroup \(S\) with \(\text{ind }S>1\) onto a semigroup \(T\) then \(T\) is an inverse semigroup and \(\Psi|C_i(S)\) is a \(C_i\)-isomorphism of \(S\) onto \(T\). As a corollary, the main result of the author's paper [J. Algebra 125, No. 2, 474-488 (1989; Zbl 0677.20050)] is obtained.
scientific article; zbMATH DE number 934247
scientific article