Residual finiteness in completely regular semigroup varieties
Let \({\mathcal V}\) be a group variety and \({\mathcal C}{\mathcal R}({\mathcal V})\) [\({\mathcal C}{\mathcal S}({\mathcal V})]\) the completely regular semigroup variety consisting of all completely regular semigroups [completely simple semigroups] whose maximal subgroups are in \({\mathcal V}\). For X a countably infinite set, denote by F\({\mathcal C}{\mathcal R}({\mathcal V})_ X\) [F\({\mathcal V}_ X\), F\({\mathcal C}{\mathcal S}({\mathcal V})_ X]\) the free object in \({\mathcal C}{\mathcal R}({\mathcal V})\) [\({\mathcal V}\), \({\mathcal C}{\mathcal S}({\mathcal V})]\) on X. It is shown that the word problem for F\({\mathcal V}_ X\) is solvable if and only if the word problem is solvable for F\({\mathcal C}{\mathcal R}({\mathcal V})_ X\). This result is used to show that F\({\mathcal C}{\mathcal R}({\mathcal V})_ X\) is residually finite (i.e. can be written as a subdirect product of finite completely regular semigroups) if and only if the same is true for F\({\mathcal V}_ X\) or F\({\mathcal C}{\mathcal S}({\mathcal V})_ X\). It follows in particular that the variety of completely regular semigroups is generated by its finite members. Let \({\mathcal W}\) be a variety of completely regular semigroups and \({\mathcal G}\) the variety of groups. If F\({\mathcal W}_ X\) is residually finite, then F(\({\mathcal W}\cap {\mathcal G})_ X\) is residually finite but in general the converse is false.
variety of completely regular semigroups
scientific article; zbMATH DE number 4019333
scientific article