Dominions in varieties generated by simple groups

Let S be a nite nonabelian simple group, and let H be a subgroup of S. In this work, the dominion (in the sense of Isbell) of H in S in Var(S) is determined, generalizing an example of B.H. Neumann. A necessary and suucient condition for H to be epimorphically embedded in S is obtained. These results are then extended to a variety generated by a family of nite nonabelian simple groups. An epimorphism in a given category C is deened to be a right cancellable function. That is, given C , a map f: A ! B in C is an epimorphisms if and only if for every object C 2 C , and every pair of maps g; h: B ! C , if g f = h f then g = h. In many familiar categories, such as Group, being an epimorphism is equivalent to being a surjective map (for a proof of this, see 9]). On the other hand, this is not the case in other familiar categories. For example, in the category of rings, the embedding Z, ! Q is an epimorphisms, even though it is not surjective. Isbell 4] has introduced the concept of dominion to study epimorphisms in categories of algebras (in the sense of Universal Algebra). Recall that given a full subcategory C of the category of all algebras of a given type, and A 2 C with a subalgebra B of A, we deene the dominion of B in A in the category C to be the intersection of all equalizer subalgebras of A containing B. Explicity, dom C A (B) = n a 2 A 8C 2 C; 8f; g: A ! C; if fj B = gj B then f(a) = g(a) o : It is clear that B is epimorphically embedded in A (in the category C) if and only if dom C A (B) = A. If dom C A (B) = B, we will say that the dominion of B in A is trivial (meaning it is as small as possible), and we will say it is nontrivial otherwise. As Isbell notes, an arbitrary morphism f: A 0 ! A of algebras may be factored as a surjection onto f(A 0) followed by the embedding of f(A 0) into A. The surjection A 0 7 ! f(A 0) is well understood in terms of congruences, and …