DM-GAA

ISSN 1509-9415 (print version)

ISSN 2084-0373 (electronic version)

https://doi.org/10.7151/dmgaa

Discussiones Mathematicae - General Algebra and Applications

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Discussiones Mathematicae - General Algebra and Applications

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Discussiones Mathematicae General Algebra and Applications 20(2) (2000) 219-231
DOI: https://doi.org/10.7151/dmgaa.1019

CONGRUENCES ON PSEUDOCOMPLEMENTED SEMILATTICES

Zuzana Heleyová

College of Business and Management, Technical University
Technicka 2, 616 69 Brno, Czech Republic
e-mail:zheleyova@iol.cz

Abstract

It is known that congruence lattices of pseudocomplemented semilattices are pseudocomplemented [4]. Many interesting properties of congruences on pseudocomplemented semilattices were described by Sankappanavar in [4], [5], [6]. Except for other results he described congruence distributive pseudocomplemented semilattices [6] and he characterized pseudocomplemented semilattices whose congruence lattices are Stone, i.e. belong to the variety B1 [5].

In this paper we give a partial solution to a more general question: Under what condition on a pseudocomplemented semilattice its congruence lattice is element of the variety Bn (n > 2)?

In the last section we widen the Sankappanavar's result to obtain the description of pseudocomplemented semilattices with relative Stone congruence lattices. A partial solution of the description of pseudocomplemented semilattices with relative (Ln)-congruence lattices (n > 2) is also given.

Keywords: pseudocomplemented semilattice, congruence lattice, p-algebra, Stone algebra, (relative) (Ln)-lattice.

1991 Mathematics Subject Classification: Primary 06D15, 06A99; Secondary 08A30

References

[1] G. Grätzer, General Lattice Theory, Birkhäuser-Verlag, Basel 1978.
[2] M. Haviar and T. Katrinák, Semi-discrete lattices with (Ln)-congruence lattices, Contribution to General Algebra 7 (1991), 189-195.
[3] K.B. Lee, Equational classes of distributive pseudo-complemented lattices, Canad. J. Math. 22 (1970), 881-891.
[4] H.P. Sankappanavar, Congruence lattices of pseudocomplemented semilattices, Algebra Universalis 9 (1979), 304-316.
[5] H.P. Sankappanavar, On pseudocomplemented semilattices with Stone congruence lattices, Math. Slovaca 29 (1979), 381-395.
[6] H.P. Sankappanavar, On pseudocomplemented semilattices whose congruence lattices are distributive, (preprint).

Received 27 October 1998
Revised 1 October 1999

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