Decidable discriminator varieties from unary classes

Let K be a class of (universal) algebras of fixed type. K t ' denotes the class obtained by augmenting each member of K by the ternary discriminator function (t(x, y, z) = x if x ¬= y, t(x, x, z) = z), while ∨(K t ) is the closure of 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 ) where 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 ) 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 of unary algebras of finite type for which the first-order theory of ∨(K t ) is decidable