Some modifications of congruence permutability and dually congruence regular varietie

Ivan Chajda; Günther Eigenthaler

Discussiones Mathematicae - General Algebra and Applications (2001)

  • Volume: 21, Issue: 2, page 165-174
  • ISSN: 1509-9415

Abstract

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It is well known that every congruence regular variety is n-permutable (in the sense of [9]) for some n ≥ 2. For the explicit proof see e.g. [2]. The connections between this n and Mal'cev type characterizations of congruence regularity were studied by G.D. Barbour and J.G. Raftery [1]. The concept of local congruence regularity was introduced in [3]. A common generalization of congruence regularity and local congruence regularity was given in [6] under the name "dual congruence regularity with respect to a unary term g". The natural problem arises what modification of n-permutability is satisfied by dually congruence regular varieties. The aim of this paper is to find out such a modification, to characterize varieties satisfying it by a Mal'cev type condition and to show connections with normally presented varieties (see e.g. [5], [8], [11]). The latter concept was introduced already by J. P≥onka under a different term; the names "normal identity" and "normal variety" were firstly used by E. Graczyńska in [8].

How to cite

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Ivan Chajda, and Günther Eigenthaler. "Some modifications of congruence permutability and dually congruence regular varietie." Discussiones Mathematicae - General Algebra and Applications 21.2 (2001): 165-174. <http://eudml.org/doc/287616>.

@article{IvanChajda2001,
abstract = {It is well known that every congruence regular variety is n-permutable (in the sense of [9]) for some n ≥ 2. For the explicit proof see e.g. [2]. The connections between this n and Mal'cev type characterizations of congruence regularity were studied by G.D. Barbour and J.G. Raftery [1]. The concept of local congruence regularity was introduced in [3]. A common generalization of congruence regularity and local congruence regularity was given in [6] under the name "dual congruence regularity with respect to a unary term g". The natural problem arises what modification of n-permutability is satisfied by dually congruence regular varieties. The aim of this paper is to find out such a modification, to characterize varieties satisfying it by a Mal'cev type condition and to show connections with normally presented varieties (see e.g. [5], [8], [11]). The latter concept was introduced already by J. P≥onka under a different term; the names "normal identity" and "normal variety" were firstly used by E. Graczyńska in [8].},
author = {Ivan Chajda, Günther Eigenthaler},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {congruence regularity; local congruence regularity; dual congruence regularity; local n-permutability; dually congruence regular varieties; locally n-permutable at a unary term; Mal'tsev type characterization; normal variety; normal closure of a variety; dually congruence regular algebras},
language = {eng},
number = {2},
pages = {165-174},
title = {Some modifications of congruence permutability and dually congruence regular varietie},
url = {http://eudml.org/doc/287616},
volume = {21},
year = {2001},
}

TY - JOUR
AU - Ivan Chajda
AU - Günther Eigenthaler
TI - Some modifications of congruence permutability and dually congruence regular varietie
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2001
VL - 21
IS - 2
SP - 165
EP - 174
AB - It is well known that every congruence regular variety is n-permutable (in the sense of [9]) for some n ≥ 2. For the explicit proof see e.g. [2]. The connections between this n and Mal'cev type characterizations of congruence regularity were studied by G.D. Barbour and J.G. Raftery [1]. The concept of local congruence regularity was introduced in [3]. A common generalization of congruence regularity and local congruence regularity was given in [6] under the name "dual congruence regularity with respect to a unary term g". The natural problem arises what modification of n-permutability is satisfied by dually congruence regular varieties. The aim of this paper is to find out such a modification, to characterize varieties satisfying it by a Mal'cev type condition and to show connections with normally presented varieties (see e.g. [5], [8], [11]). The latter concept was introduced already by J. P≥onka under a different term; the names "normal identity" and "normal variety" were firstly used by E. Graczyńska in [8].
LA - eng
KW - congruence regularity; local congruence regularity; dual congruence regularity; local n-permutability; dually congruence regular varieties; locally n-permutable at a unary term; Mal'tsev type characterization; normal variety; normal closure of a variety; dually congruence regular algebras
UR - http://eudml.org/doc/287616
ER -

References

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  1. [1] G.D. Barbour and J.G. Raftery, On the degrees of permutability of subregular varieties, Czechoslovak Math. J. 47 (1997), 317-325. Zbl0927.08001
  2. [2] R. Belohlávek and I. Chajda, Congruence classes in regular varieties, Acta Math. Univ. Com. (Bratislava) 68 (1999), 71-76. Zbl1014.08002
  3. [3] I. Chajda, Locally regular varieties, Acta Sci. Math. (Szeged) 64 (1998), 431-435. Zbl0913.08006
  4. [4] I. Chajda, Semi-implication algebras, Tatra Mt. Math. Publ. 5 (1995), 13-24. Zbl0856.08004
  5. [5] I. Chajda, Normally presented varieties, Algebra Universalis 34 (1995), 327-335. 
  6. [6] I. Chajda and G. Eigenthaler, Dually regular varieties, Contributions to General Algebra 12 (2000), 121-128. Zbl0965.08006
  7. [7] I. Chajda and H. Länger, Ring-like operations in pseudocomplemented semilattices, Discuss. Math. Gen. Algebra Appl. 20 (2000), 87-95. Zbl0968.06004
  8. [8] E. Graczyńska, On normal and regular identities, Algebra Universalis 27 (1990), 387-397. Zbl0713.08007
  9. [9] J. Hagemann and A. Mitschke, On n-permutable congruences, Algebra Universalis 3 (1973), 8-12. 
  10. [10] A.I. Mal'cev, On the general theory of algebraic systems (Russian), Mat. Sbornik 35 (1954), 8-20. 
  11. [11] I.I. Melnik, Nilpotent shifts of varieties (Russian), Mat. Zametki 14 (1973), 703-712. 

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