Existence theorems for weakly symmetric operations

Abstract

A k-ary weak near-unanimity operation (or k-WNU) on A is an operation that satisfies the equations w(x, . . . x) ≈ x and w(y, x, . . . , x) ≈ w(x, y, . . . , x) ≈ · · · ≈ w(x, x, . . . , x, y) . If an algebra A has a k-NU (or a k-WNU) term operation, we say that A satisfies NU(k) (or WNU(k), respectively). Likewise, a variety is said to satisfy NU(k) (or WNU(k), respectively), it it has a k-variable term satisfying these equations. It has been conjectured that a finite idempotent algebra A has finite relational width if and only if V(A) (the variety generated by A) has meet semi-distributive congruence lattices. The concept of “finite relational width” arises in the theory of complexity of algorithms, in the algebraic study of constraint-satisfaction problems. Actually, there are several different definitions of this concept and it is not known if they are equivalent. One version of the concept and the conjecture mentioned above are due to B. Larose and L. Zadori [10]. The important family of varieties with meet semi-distributive congruence lattices has various known characterizations. There is a characterization by a certain Maltsev condition; also, it is known that a locally finite variety has this property iff it omits congruence covers of types 1 and 2 (defined in the tame congruence theory of D. Hobby, R. McKenzie [6]). E. Kiss showed that a finite idempotent algebra of relational width k must have an m-WNU term operation for every m ≥ k. E. Kiss and M. Valeriote then observed that a finite algebra with a k-WNU term operation, k > 1, must omit congruence covers of type 1. These observations led M. Valeriote to make two conjectures: any locally finite variety omits congruence covers of type 1 iff it satisfies WNU(k) for some k > 1; any locally finite variety has meet semi-distributive congruence lattices if and only if for some k, it satisfies WNU(m) for all m ≥ k. In this paper, we prove both of these conjectures of M. Valeriote. The family of locally finite varieties omitting type 1 is the largest family of locally finite varieties defined by a nontrivial idempotent Maltsev condition. For this