Abstract
A category is universal if it contains a full subcategory isomorphic to the category Γ of all directed graphs without loops and isolated points. LetV be a universal semigroup variety,S a semigroup inV, andV S = {T eV;S is a homomorphic image ofT} the full subcategory ofV of all coextensions ofS withinV. We establish the universality ofV S in two cases:(a) ifV is the varietyS of all semigroups andS has an idempotent, and(b) ifV is an arbitrary universal semigroup variety andS has a kernel.
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