PDF

Discussiones Mathematicae General Algebra and Applications 21(2) (2001) 165-174
DOI: https://doi.org/10.7151/dmgaa.1035

SOME MODIFICATIONS OF CONGRUENCE PERMUTABILITY AND DUALLY CONGRUENCE REGULAR VARIETIE

Ivan Chajda Department of Algebra and Geometry Palacký University of Olomouc Tomkova 40, CZ-77900 Olomouc, Czech Republic e-mail: chajda@risc.upol.cz

Günther Eigenthaler Institut für Algebra und Computermathematik Technische Universität Wien Wiedner Hauptstraß e 8-10, A-1040 Wien, Austria e-mail: g.eigenthaler@tuwien.ac.at

Abstract

It is well known that every congruence regular variety is n-permutable (in the sense of [9]) for some n ł 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. Płonka under a different term; the names "normal identity" and "normal variety" were firstly used by E. Graczyńska in [8].

Keywords: congruence regularity, local congruence regularity, dual congruence regularity, local n-permutability.

2000 Mathematics Subject Classification: Primary 08A30, Secondary 08B05.

References

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

Received 18 December 2000
Revised 6 June 2001