Implicit definition of the quaternary discriminator

CAMPERCHOLI, MIGUEL; VAGGIONE, DIEGO

ALGEBRA UNIVERSALIS

BIRKHAUSER VERLAG AG

Año: 2012 vol. 68 p. 1 - 16

0002-5240

Let A be an algebra. A function f:Aⁿ→A is implicitly definable by a system of term equations &t_{i}(x₁,...,x_{n},z)=s_{i}(x₁,...,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.