Let p and q be polynomial symbols of a type of algebras having operations ∨, ∧, and; (interpreted as the join, meet, and product of congruence relations). If is an algebra, L(), the local variety of , is the class of all algebras such that for each finite subset G of there is a finite subset F of such that every identity of F is also an identity of G.
THEOREM. There is an algorithm which, for each inequality
p≤q,
and pair of integers n, k≥2, determines a set Un, k of (Malcev) equations with the property:
For each algebra, p≤q is true in the congruence lattice offor each∊L() if and only if for each finite subset F ofand integer n≥2 there is a k=k(n, F) such that Un, kare identities of F.
This generalizes a corresponding result for varieties due to Wille (Kongruenzklassengeometrien, Lect. Notes in Math. Springer- Verlag, Berlin-Heidelberg, New York, 1970) and at the same time provides a more direct proof.