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

Jonathan D.H. Smith

Discussiones Mathematicae - General Algebra and Applications (2003)

  • Volume: 23, Issue: 2, page 149-161
  • ISSN: 1509-9415

Abstract

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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.

How to cite

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Jonathan D.H. Smith. "On the characterisation of Mal'tsev and Jónsson-Tarski algebras." Discussiones Mathematicae - General Algebra and Applications 23.2 (2003): 149-161. <http://eudml.org/doc/287730>.

@article{JonathanD2003,
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.},
author = {Jonathan D.H. Smith},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {Mal'tsev variety; Mal'tsev algebra; Jónsson-Tarski variety; Jónsson-Tarski algebra; Goldie variety; Goldie algebra; congruence permutability; simplicial object},
language = {eng},
number = {2},
pages = {149-161},
title = {On the characterisation of Mal'tsev and Jónsson-Tarski algebras},
url = {http://eudml.org/doc/287730},
volume = {23},
year = {2003},
}

TY - JOUR
AU - Jonathan D.H. Smith
TI - On the characterisation of Mal'tsev and Jónsson-Tarski algebras
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2003
VL - 23
IS - 2
SP - 149
EP - 161
AB - 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.
LA - eng
KW - Mal'tsev variety; Mal'tsev algebra; Jónsson-Tarski variety; Jónsson-Tarski algebra; Goldie variety; Goldie algebra; congruence permutability; simplicial object
UR - http://eudml.org/doc/287730
ER -

References

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  7. [7] B. Jónsson and A. Tarski, Direct Decompositions of Finite Algebraic Systems, Notre Dame Mathematical Lectures, Notre Dame, IN, 1947. Zbl0041.34501
  8. [8] S. Mac Lane, Categories for the Working Mathematician, Springer-Verlag, New York, NY, 1971. 
  9. [9] A.I. Mal'tsev, K obshche teorii algebraicheskikh sistem, Mat. Sb. (N.S.) 35 (77) (1954), 3-20. 
  10. [10] A.I. Mal'cev, On the general theory of algebraic systems, (translation of [] by H. Alderson), Transl. Amer. Math. Soc. 27 (1963), 125-140. 
  11. [11] J.D.H. Smith, Mal'cev Varieties, Springer-Verlag, Berlin 1976. Zbl0344.08002
  12. [12] J.D.H. Smith, Centrality, Abstr. Amer. Math. Soc. 1 (1980), 774-A21. 
  13. [13] J.D.H. Smith and A.B. Romanowska, Post-Modern Algebra, Wiley, New York, NY, 1999. 
  14. [14] S.T. Tschantz, More conditions equivalent to congruence modularity, 'Universal Algebra and Lattice Theory,' Springer-Verlag, Berlin 1985, 270-282. Zbl0586.08006

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