12.8 (1992)

Let $\mathfrak{V}$ be a non-trivial variety of groups and let $a_1, \dots, a_r$ freely generate a free group $F_r(\mathfrak{V})$ in $\mathfrak{V}$. We say $F_r(\mathfrak{V})$ strongly discriminates $\mathfrak{V}$ just in case every finite system of inequalities $w_i(a_1, \dots, a_r, x_1, \dots, x_k) \neq 1$ for $1 \leqslant i \leqslant n$ having a solution in some $F_s(\mathfrak{V})$ containing $F_r(\mathfrak{V})$ as a varietally free factor in the sense of $\mathfrak{V}$, already has a solution in $F_r(\mathfrak{V})$. Does there exist $\mathfrak{V}$ such that for some integer $r > 0$, $F_r(\mathfrak{V})$ discriminates but does not strongly discriminate $\mathfrak{V}$? What about the analogous question for general algebras in the context of universal algebra?