Divisibility in categories of a class which includes the category of binary relations

Abstract

Green's relations are essential for “co-ordinatizing” semigroups. Jacqueline Klasa, in applying cognate ideas to categories [4, 5], has shown that divisibility in suitably-behaved categories may be described in terms of subobjects and quotients.Here it is shown that adjoint functors which are onto objects preserve divisibility (in a certain sense). The inclusion functor of the category of sets into the category R of binary relations is such a functor. A slight modification of its right adjoint allows the representation of R as a full subcategory in a category CSL of complete semilattice morphisms.