Summary: 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 \(x\neq y\), \(t(x,x,z)=z)\), while \(\bigvee({\mathcal K}^ t)\) 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 \(\bigvee({\mathcal K}^ t)\) 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 \(\bigvee({\mathcal K}^ t)\) 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 \(\bigvee({\mathcal K}^ t)\) is decidable.