Implicative Identities in Groups

Recently W. McCune and D. A. Sands have considered, in a note Amer Math. Monthly 103 (1996), 888{892], implicative axioms in groups of the form if u = v, then w = t, where u, v, w, t are group words in some variables. Their interest is in such identical implications that deene the variety of groups, or the variety of abelian groups. What I want to present is, slightly more generally, identical implications of the form if u1 = v1 and : : : and un = vn, then w = t. where again the ui, vi, w, t are words in the algebraic system I consider. This is a group enriched with a further binary relation that imitates in many respects the commutator x; y] := x ?1 y ?1 x y ; and is designed to investigate the dependence on, or independence from, certain commutator laws from others. I. D. Macdonald and I have introduced such enriched groups, called kappa-groups, in three axiomatic studies of the interdependence of commutator identical relations; and I have used a similar device in a corresponding study in rings to be published ]. The identical implications to be considered here are the analogues of if x; z] = y; z] = e, then x y; z] = e.