Dominions in decomposable varieties (Q1866811)

In the paper dominions [see \textit{J. R. Isbell}, Proc. Conf. Categor. Algebra, La Jolla 1965, 232-246 (1966; Zbl 0194.01601)] are investigated in the context of decomposable varieties of groups. For \(\mathcal C\), a full subcategory of the category of all algebras of a fixed type which is closed under passing to subalgebras, \(A\in{\mathcal C}\) and \(B\) a subalgebra of \(A\), the dominion of \(B\) in \(A\) (in the category \(\mathcal C\)) is the intersection of all equalizer subalgebras of \(A\) containing \(B\). Here dominions are studied when the category \(\mathcal C\) is a product of two proper nontrivial varieties of groups. An upper and lower bound for dominions in such a case is given in terms of the two varietal factors, and the internal structure of the group being analyzed. The following result is established: If a variety \(\mathcal N\) has instances of nontrivial dominions, then for any proper subvariety \(\mathcal Q\) of \({\mathcal G}roup\), \(\mathcal{NQ}\) has also instances of nontrivial dominions.