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Abstract
In this thesis a q partial group is defined to be a partial group, i.e. a strong semilattice of groups S=[E(S); Se, ?e,ƒ] such that S has an identity 1 and ?1,e is 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 extend to idempotent separating congruences on S, it is proved that Q (Pk) =PQ(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 iff 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. A known result in groups concerning the inheritance of minimal conditions on normal subgroups by subgroups with finite indexes is extended to q partial groups. Formulation of this result in terms of so called q congruences is also obtained