Dominions in product varieties

Dominions, in the sense of Isbell, are investigatedin the contextof decomposable varieties of groups. An upper and lower bound for dominions in such a variety is given in terms of the two varietal factors, and the internal structure of the group being analyzed. Finally, the following result is established: If a variety N has instances of nontrivial dominions, then for any proper subvariety Q of Group, NQ also has instances of nontrivial dominions. Section 1. Introduction Suppose that a group G, a subgroup H of G, and a class C of groups containing G is given. Are there any elements g 2 G n H such that any two morphisms between G and a C-group which agree on H must also agree on g? To put this question in a more general context, let C be a full subcategory of the category of all algebras (in the sense of Universal Algebra) of a xed type which is closed under passing to subalgebras. Let A 2 C, and let B be a subalgebra of A. Recall that, in this situation, Isbell 2] deenes the dominion of B in A (in the category C) to be the intersection of all equalizer subalgebras of A containing B. Explicitly, 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 : Therefore, the question with which we opened this discussion may be rephrased in terms of the dominion of H in G in the category of context. If H = dom C G (H) we say that the dominion of H in G (in the category C) is trivial, and say it is nontrivial otherwise. In this work we will study dominions when the category C is a product of two proper nontrivial varieties of Group. In Section 2 we will recall the basic properties of varieties that we will need; we refer the reader to Hanna Neumann's excellent book 8] for more information on varieties of groups. Since wreath products are closely related to products of varieties, and are used in the proofs of our results, we will also recall some of their properties. In Section 3 we will state and prove the main results of this work, which give an upper and lower bounds for the dominion of a subgroup in a product variety. Finally, in Section …