Algebra universalis

, Volume 68, Issue 1–2, pp 1–16 | Cite as

Implicit definition of the quaternary discriminator

Article

Abstract

Let A be an algebra. A function f: A n A is implicitly definable by a system of term equations \({\bigwedge t_{i}(x_{1}, . . . , x_{n}, z) = s_{i}(x_{1}, . . . ,x_{n}, z)}\) if f is the only n-ary operation on A making the identities \({t_{i}(\overrightarrow{x}, f(\overrightarrow{x})) \approx s_{i}(\overrightarrow{x}, f(\overrightarrow{x}))}\) hold in A. Let \({\mathcal{K}}\) be a class of non-trivial algebras. We prove that the quaternary discriminator is implicitly definable on every member of \({\mathcal{K}}\) (via the same system) iff \({\mathcal{K}}\) 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.

2010 Mathematics Subject Classification

Primary: 08C15 

Key words and phrases

quaternary discriminator implicit equational definition equationally definable principal congruences 

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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Facultad de Matemática, Astronomía y FísicaUniversidad Nacional de CórdobaCórdobaArgentina

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