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

top
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

top

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

top
  1. [1] J. Duskin, Simplicial Methods and the Interpretation of 'Triple' Cohomology, Mem. Amer. Math. Soc., No. 163, 1975. Zbl0376.18011
  2. [2] A.W. Goldie, On direct decompositions. I, and II, Proc. Cambridge Philos. Soc. 48 (1952), 1-34. Zbl0046.02901
  3. [3] M. Gran and M.C. Pedicchio, n-Permutable locally finite presentable categories, Theory Appl. Categ. 8 (2001), 1-15. 
  4. [4] H.-P. Gumm, Geometrical Methods in Congruence Modular Varieties, Mem. Amer. Math. Soc, No. 286. doi: 1983 
  5. [5] J. Hagemann and C. Herrmann, A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity, Arch. Math. (Basel) 32 (1979), 234-245. Zbl0419.08001
  6. [6] J. Hagemann and A. Mitscke, On n-permutable congruences, Algebra Universalis 3 (1973), 8-12. 
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.