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Discussiones Mathematicae General Algebra and Applications 21(1) (2001) 57-66
DOI: https://doi.org/10.7151/dmgaa.1027

SOME PROPERTIES OF CONGRUENCE RELATIONS ON ORTHOMODULAR LATTICES

Gerhard Dorfer

Institute of Algebra and Computational Mathematics
Vienna University of Technology
Wiedner Hauptstr. 8-10/118, A-1040 Vienna, Austria
e-mail: g.dorfer@tuwien.ac.at

Abstract

In this paper congruences on orthomodular lattices are studied with particular regard to analogies in Boolean algebras. For this reason the lattice of p-ideals (corresponding to the congruence lattice) and the interplay between congruence classes is investigated. From the results adduced there, congruence regularity, uniformity and permutability for orthomodular lattices can be derived easily.

Keywords: orthomodular lattice, congruence relation, congruence-regular, congruence-uniform, congruence-permutable.

2000 Mathematics Subject Classification: 06C15, 08A30, 03G12, 81P10.

References

[1]	L. Beran, Orthomodular Lattices , D. Reidel Publishing Company, Dordrecht-Boston 1985.
[2]	R.P. Dilworth, The structure of relatively complemented lattices , Ann. of Math. 51 (1950), 348-359.
[3]	P.D. Finch, Congruence relations on orthomodular lattices , J. Austral. Math. Soc. 6 (1966), 46-54.
[4]	J. Hashimoto, Congruence relations and congruence classes in lattices , Osaka Math. J. 15 (1963), 71-86.
[5]	G. Kalmbach, Orthomodular Lattices , Academic Press, London 1983.
[6]	T. Katrinák, Die Kennzeichnung der distributiven pseudokomplementären Halbverbände , J. Reine Angew. Math. 241 (1970), 160-179.
[7]	P. Pták and S. Pulmannová, Orthomodular Structures as Quantum Logics , Kluwer Academic Publishers, Dordrecht 1991.

Received 30 August 1999
Revised 19 April 2001