Decidable discriminator varieties from unary classes

Abstract

Let K \mathcal {K} be a class of (universal) algebras of fixed type. K t {\mathcal {K}^t} denotes the class obtained by augmenting each member of K \mathcal {K} by the ternary discriminator function ( t ( x , y , z ) = x (t(x,y,z) = x if x ≠ y , t ( x , x , z ) = z ) x \ne y,t(x,x,z) = z) , while ∨ ( K t ) \vee ({\mathcal {K}^t}) is the closure of K t {\mathcal {K}^t} under the formation of subalgebras, homomorphic images, and arbitrary Cartesian products. For example, the class of Boolean algebras is definitionally equivalent to ∨ ( K t ) \vee ({\mathcal {K}^t}) where K \mathcal {K} consists of a two-element algebra whose only operations are the two constants. Any equationally defined class (that is, variety) of algebras which is equivalent to some ∨ ( K t ) \vee ({\mathcal {K}^t}) is known as a discriminator variety. Building on recent work of S. Burris, R. McKenzie, and M. Valeriote, we characterize those locally finite universal classes K \mathcal {K} of unary algebras of finite type for which the first-order theory of ∨ ( K t ) \vee ({\mathcal {K}^t}) is decidable.