Minimal conditions on Clifford semigroup congruences. (Q885606)

A partial group \(S\) is a semigroup with some additional properties, thus it may be viewed as a strong semilattice of groups \(S=[E(S);S_e,\varphi_{e,f}]\) where \(S_e\) is the maximal subgroup of \(S\) with identity \(e\) (\(e\in E(S)\)) and for \(e\geq f\) in \(E(S)\), \(\varphi_{e,f}\) is the homomorphism of groups \(S_e\to S_f\), \(x\mapsto xf\); here \(E(S)\) is the semilattice (\(e\geq f\) if and only if \(ef=f\)) of idempotents (partial identities) in \(S\). A partial group is precisely a Clifford semigroup. In the paper, a known result in groups concerning the inheritance of minimal conditions on normal subgroups by subgroups with finite indexes is extended to semilattices of groups \([E(S);S_e,\varphi_{e,f}]\) with identities in which all \(\varphi_{e,f}\) are epimorphisms (called partial groups). A formulation of this result in terms of \(q\) congruences is also obtained.