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Discussiones Mathematicae General Algebra and Applications 23(2) (2003) 149-161
DOI: https://doi.org/10.7151/dmgaa.1070

ON THE CHARACTERISATION OF MAL'TSEV AND JÓNSSON-TARSKI ALGEBRAS

Jonathan D.H. Smith

Department of Mathematics
Iowa State University
Ames, IA 50011, USA
e-mail: jdhsmith@math.iastate.edu

Abstract

There are very strong parallels between the properties of Mal'tsev and Jónsson-Tarski algebras, for example in the good behaviour of centrality and in the factorization of direct products. Moreover, the two classes between them include the majority of algebras that actually arise ``in nature''. As a contribution to the research programme building a unified theory capable of covering the two classes, along with other instances of good centrality and factorization, the paper presents a common framework for the characterisation of Mal'tsev and Jónsson-Tarski algebras. Mal'tsev algebras are characterized by simplicial identities in the product complex of an algebra. In the dual of a pointed variety, a simplicial object known as the pointed complex is then constructed. The basic simplicial Mal'tsev identity in the pointed complex characterises Jónsson-Tarski algebras. Higher-dimensional simplicial Mal'tsev identities in the pointed complex are characteristic of a class of algebras lying properly between Goldie and Jónsson-Tarski algebras.

Keywords: Mal'tsev variety, Mal'tsev algebra, Jónsson-Tarski variety, Jónsson-Tarski algebra, Goldie variety, Goldie algebra, congruence permutability, simplicial object.

2000 Mathematics Subject Classification: 08B05, 08C05, 18G30.

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Received 16 May 2003