Decidable discriminator varieties from unary classes
journal contributionposted on 01.01.1990, 00:00 authored by Ross Willard
Abstract: "Let K be a class of (universal) algebras of fixed type. K[superscript t] denotes the class obtained by augmenting each member of K by the ternary discriminator function (t(x,y,z) = x if x [does not equal] y, t(x,x,z) = z), while V(K[superscript t]) is the closure of K[superscript t] under the formulation of subalgebras, homomorphic images, and arbitrary Cartesian products. For example, the class of Boolean algebras is definitionally equivalent to V(K[superscript 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 V(K[superscript t]) is known as a discriminator variety. Building on recent work of S. Burris, R. McKenzie and M. Valeriote, we characterize these locally finite universal classes K of unary algebras of finite type for which the first-order theory of V(K[superscript t]) is decideable."