Global determinism of semigroups having regular globals
The global, or power semigroup, of a semigroup \(S\) is the semigroup \(P(S)\) of all nonempty subsets of \(S\), equipped with set-wise multiplication. A class \({\mathcal C}\) of semigroups is \textit{globally determined} if any two members of \(S\) having isomorphic globals are themselves isomorphic, and the class is said to \textit{satisfy the strong isomorphism property} if, further, any isomorphism from one member of \({\mathcal C}\) to another maps the singleton subsets of the first to those of the second (thereby inducing a `canonical' isomorphism of the semigroups). By results of \textit{J. Pelikán} [Period. Math. Hung. 4, 103--106 (1973; Zbl 0277.20083)] and [\textit{B. Pondelicek}, in: Algebraic theory of semigroups, Proc. Conf. Szeged 1976, Colloq. Math. Soc. Janos Bolyai, 20, 453--461 (1979; Zbl 0408.20044)], if \(P(S)\) is regular, then it satisfies \(x^3 = x\) (and so is completely regular). It is shown here that (1) the class of semigroups for which the global satisfies \(x^2 = x\) (that is, it is a band) is globally determined and (2) the class for which the global is regular, but not a band, satisfies the strong isomorphism property. In combination, it follows that the class for which the global is regular is globally determined. The authors do not address the question of whether or not the entire class actually satisfies the strong isomorphism property.
scientific article; zbMATH DE number 6699247
scientific article