Abstract

A group is positively discriminating if any finite subset of positive equations u = v, which are not laws in G, can be simultaneously falsified in G. All known groups which are not positively discriminating satisfy positive laws. The question whether every group without positive laws must be positively discriminating is open. We give an affirmative answer to this question for the class of locally graded groups.

AMS Classification: 20E10 (primary), 20M07 (secondary).

An equation in a group is an expression of the form u = v, where u = u(x1,…,xn), v = v(x1, …, xn) are different words (v may be the empty word 1) in the free group F, freely generated by x1, x2, …. If n = 2, the equation is called binary and we use x, y instead of x1, x2. The equation is called positive if u and v are written without the inverses of the xi's. A positive equation is called balanced if the exponent sum of xi is the same in u and v for each fixed i. A balanced equation u = v is of degree n if the x-length of u and v is equal to n. We say that the n-tuple of elements g1, …, gn in G satisfies the equation u = v, if under the substitution xi → gi we get the equality u(g1, …, gn) = v(g1, …, gn).