Decidable Discriminator Varieties from Unary Classes

Abstract

Let $\mathcal {K}$ be a class of (universal) algebras of fixed type. ${\mathcal {K}^t}$ denotes the class obtained by augmenting each member of $\mathcal {K}$ by the ternary discriminator function $(t(x,y,z) = x$ if $x \ne y,t(x,x,z) = z)$, while $\vee ({\mathcal {K}^t})$ is the closure of ${\mathcal {K}^t}$ under the formation of subalgebras, homomorphic images, and arbitrary Cartesian products. For example, the class of Boolean algebras is definitionally equivalent to $\vee ({\mathcal {K}^t})$ where $\mathcal {K}$ 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 $\vee ({\mathcal {K}^t})$ 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 $\mathcal {K}$ of unary algebras of finite type for which the first-order theory of $\vee ({\mathcal {K}^t})$ is decidable.