A Class of Homomorphism Theories for Groupoids

Abstract

In a well-behaved homomorphism theory for a class of algebraic systems certain “closed objects” relative to a given G ∈ are distinguished which act as kernels of homomorphisms. For example, if is the class of groups then the closed objects relative to a given group G are the normal subgroups of G; if is the class of semigroups with zero element then one can devise a homomorphism theory in which the closed objects relative to a given S ∈ are the ideals of S[cf. Rees (3)]; in the class of groupoids one may define the closed objects relative to a given groupoid G to be the congruence relations on G, that is, subsets π⊆G×G which are equivalence relations having the property that (x1y1, x2y2) ∈ π whenever (x1, x2), (y1, y2) ∈ π. Given such a closed object N relative to G there exists a “factor” system G/N and a (canonical) homomorphism η: G→G/N characterised by the property: If σ G→H is a homomorphism with kernel N then there is a unique homomorphism : G/N→H such that . η = σ and the kernel of is trivial in the sense that the kernel of is the unique smallest closed object relative to G/N.