On semilattices of groups whose arrows are epimorphisms

A q partial group is defined to be a partial group, that is, a strong semilattice of groups S= [E(S);Se,φe, f ] such that S has an identity 1 and φ1,e is an epimorphism for all e ∈ E(S). Every partial group S with identity contains a unique maximal q partial group Q(S) such that (Q(S))1 = S1. This Q operation is proved to commute with Cartesian products and preserve normality. With Q extended to idempotent separating congruences on S, it is proved that Q(ρK ) = ρQ(K) for every normal K in S. Proper q partial groups are defined in such a way that associated to any group G, there is a proper q partial group P(G) with (P(G))1 = G. It is proved that a q partial group S is proper if and only if S ∼= P(S1) and hence that if S is any partial group, there exists a group M such that S is embedded in P(M). P epimorphisms of proper q partial groups are defined with which the category of proper q partial groups is proved to be equivalent to the category of groups and epimorphisms of groups.