On semilattices of groups whose arrows are epimorphisms. (Q885612)

Let \(S\) be a monoid and a strong semilattice \((Y;S_\alpha,\varphi_{\alpha,\beta})\) of groups such that every \(\varphi_{\alpha,\beta}\) is an epimorphism (the authors use the confusing name `a partial group'). Then for every Clifford semigroup \(T\) which is a subsemigroup of \(S\) containing the identity of \(S\) there exists a unique maximal subsemigroup \(\mathbf Q(T)\) of \(T\) such that \(\mathbf Q(T)\) is a monoid and a strong semilattice \((Y',T_\alpha,\psi_{\alpha,\beta})\) of groups such that every \(\psi_{\alpha,\beta}\) is an epimorphism. The operator \(\mathbf Q\) commutes with categorical products and preserves normality of subgroups. The congruences of \(S\) separating idempotent elements of \(S\) form a modular lattice. It is proved that the category of monoids \(S\) which are strong semilattices \((Y,S_\alpha,\varphi_{\alpha,\beta})\) of groups such that every \(\varphi_{\alpha,\beta}\) is an epimorphism and all epimorphisms between them and the category of all groups and all epimorphims between them are equivalent.