Implicit definition of the quaternary discriminator

Abstract

Let A be an algebra. A function f: A n → A is implicitly definable by a system of term equations Λ t i(x i,...,x n,z)if f is the only n-ary operation on A making the identities t i(x,f(x))≈ s i(x,f(x)) hold in A. Let K be a class of non-trivial algebras. We prove that the quaternary discriminator is implicitly definable on every member of K (via the same system) iff K is contained in the class of relatively simple members of some relatively semisimple quasivariety with equationally definable relative principal congruences. As an application, we obtain a characterization of the relatively permutable members of such type of quasivarieties. Furthermore, we prove that every algebra in such a quasivariety has a unique relatively permutable extension. © 2012 Springer Basel AG.