Congruences on pseudocomplemented semilattices

Zuzana Heleyová

Discussiones Mathematicae - General Algebra and Applications (2000)

  • Volume: 20, Issue: 2, page 219-231
  • ISSN: 1509-9415

Abstract

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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.

How to cite

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Zuzana Heleyová. "Congruences on pseudocomplemented semilattices." Discussiones Mathematicae - General Algebra and Applications 20.2 (2000): 219-231. <http://eudml.org/doc/287639>.

@article{ZuzanaHeleyová2000,
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.},
author = {Zuzana Heleyová},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {pseudocomplemented semilattice; congruence lattice; p-algebra; Stone algebra; (relative) (Lₙ)-lattice; pseudocomplemented semilattices; congruence lattices},
language = {eng},
number = {2},
pages = {219-231},
title = {Congruences on pseudocomplemented semilattices},
url = {http://eudml.org/doc/287639},
volume = {20},
year = {2000},
}

TY - JOUR
AU - Zuzana Heleyová
TI - Congruences on pseudocomplemented semilattices
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2000
VL - 20
IS - 2
SP - 219
EP - 231
AB - 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.
LA - eng
KW - pseudocomplemented semilattice; congruence lattice; p-algebra; Stone algebra; (relative) (Lₙ)-lattice; pseudocomplemented semilattices; congruence lattices
UR - http://eudml.org/doc/287639
ER -

References

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  1. [1] G. Grätzer, General Lattice Theory, Birkhäuser-Verlag, Basel 1978. Zbl0436.06001
  2. [2] M. Haviar and T. Katrinák, Semi-discrete lattices with (Ln)-congruence lattices, Contribution to General Algebra 7 (1991), 189-195. Zbl0758.06004
  3. [3] K.B. Lee, Equational classes of distributive pseudo-complemented lattices, Canad. J. Math. 22 (1970), 881-891. Zbl0244.06009
  4. [4] H.P. Sankappanavar, Congruence lattices of pseudocomplemented semilattices, Algebra Universalis 9 (1979), 304-316. Zbl0424.06001
  5. [5] H.P. Sankappanavar, On pseudocomplemented semilattices with Stone congruence lattices, Math. Slovaca 29 (1979), 381-395. Zbl0416.06007
  6. [6] H.P. Sankappanavar, On pseudocomplemented semilattices whose congruence lattices are distributive, (preprint). Zbl0416.06007

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