Local Malcev Conditions
Alden F. Pixley · Canadian Mathematical Bulletin · 1972
Abstract 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 of for each ∊L( ) if and only if for each finite subset F of and 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.