A q partial group is defined to be a partial group, that is, a strong semilattice of groups S = [E(S);S e ,ϕ e, f ] such that S has an identity 1 and ϕ 1,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 = S 1. This Q operation is proved to commute with Cartesian… (More)
The reaction of 5-amino-3-(arylamino)-1H-pyrazole-4-carboxamides 1a,b with acetylacetone 2 and arylidenemalononitriles 5a-c yielded the pyrazolo[1,5-a]-pyrimidine derivatives 4a,b and 7a-f respectively. On the other hand, Schiff bases 9a,b and 12a-j were obtained upon treatment of carboxamides 1a,b with isatin 8 and some selected aldehydes 11a-e. The newly… (More)
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 , ϕ e, f ] with identities in which all ϕ e, f are epimorphisms (called q partial groups). Formulation of this result in terms of q congruences is also obtained.