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Discussiones Mathematicae General Algebra and Applications 21(1) (2001) 105-114
DOI: https://doi.org/10.7151/dmgaa.1031

BALANCED CONGRUENCES

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

Let V be a variety with two distinct nullary operations 0 and 1. An algebra \frakA ∈ V is called balanced if for each φ,ψ ∈ Con (\frakA), we have [0]φ = [0]Ψ if and only if [1]φ = [1]Ψ. The variety V is called balanced if every \frakA ∈ V is balanced. In this paper, balanced varieties are characterized by a Mal'cev condition (Theorem 3). Furthermore, some special results are given for varieties of bounded lattices.

Keywords: balanced congruence, balanced algebra, balanced variety, Mal'cev condition.

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

References

[1]	I. Chajda, Locally regular varieties, Acta Sci. Math. (Szeged) 64 (1998),431-435.
[2]	I. Chajda and G. Eigenthaler, A remark on congruence kernels in complemented lattices and pseudocomplemented semilattices, Contributions to General Algebra 11 (1999), 55-58.
[3]	G. Grätzer and E.T. Schmidt, Ideals and congruence relations in lattices, Acta Math. Sci. Hungar. 9 (1958), 137-175.
[4]	A.I. Mal'cev, On the general theory of algebraic systems (Russian), Mat. Sb. 35 (1954), 3-20.

Received 27 March 2000
Revised 2 August 2000