On the characterisation of Mal'tsev and Jónsson-Tarski algebras

• Autorzy: Jonathan D.H. Smith
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Mal'tsev variety; Mal'tsev algebra; Jónsson-Tarski variety; Jónsson-Tarski algebra; Goldie variety; Goldie algebra; congruence permutability; simplicial object

Kategorie tematyczne

• 08B05: Equational logic, Mal\cprime cev (Mal\cprime tsev) conditions
• 08C05: Categories of algebras
• 18G30: Simplicial sets, simplicial objects (in a category)

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae - General Algebra and Applications
• Rocznik: 2003
• Tom: 23
• Numer: 2
• Strony: 149-161

Daty

wydano

2003

otrzymano

2003-05-16

Twórcy

autor

Jonathan D.H. Smith

• Department of Mathematics, Iowa State University, Ames, IA 50011, USA

Bibliografia

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• [9] A.I. Mal'tsev, K obshche teorii algebraicheskikh sistem, Mat. Sb. (N.S.) 35 (77) (1954), 3-20.
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• [12] J.D.H. Smith, Centrality, Abstr. Amer. Math. Soc. 1 (1980), 774-A21.
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• [14] S.T. Tschantz, More conditions equivalent to congruence modularity, 'Universal Algebra and Lattice Theory,' Springer-Verlag, Berlin 1985, 270-282.

Identyfikatory

DOI

10.7151/dmgaa.1070