International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 30673, 26 pages

On semilattices of groups whose arrows are epimorphisms

M. El-Ghali M. Abdallah, L. N. Gab-Alla, and Sayed K. M. Elagan

Department of Mathematics, Faculty of Science, Menoufiya University, Shebin El-Kom 32511, Egypt

Received 6 April 2006; Accepted 6 April 2006

Copyright © 2006 M. El-Ghali M. Abdallah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 eE(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 SP(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.