# Separation properties in congruence lattices of lattices

Colloquium Mathematicae (2000)

- Volume: 83, Issue: 1, page 71-84
- ISSN: 0010-1354

## Access Full Article

top## Abstract

top## How to cite

topPloščica, Miroslav. "Separation properties in congruence lattices of lattices." Colloquium Mathematicae 83.1 (2000): 71-84. <http://eudml.org/doc/210775>.

@article{Ploščica2000,

abstract = {We investigate the congruence lattices of lattices in the varieties $ℳ _n$. Our approach is to represent congruences by open sets of suitable topological spaces. We introduce some special separation properties and show that for different n the lattices in $ℳ _n$ have different congruence lattices.},

author = {Ploščica, Miroslav},

journal = {Colloquium Mathematicae},

keywords = {algebraic lattice; topological representation; uniform separation; congruence lattices; separation properties},

language = {eng},

number = {1},

pages = {71-84},

title = {Separation properties in congruence lattices of lattices},

url = {http://eudml.org/doc/210775},

volume = {83},

year = {2000},

}

TY - JOUR

AU - Ploščica, Miroslav

TI - Separation properties in congruence lattices of lattices

JO - Colloquium Mathematicae

PY - 2000

VL - 83

IS - 1

SP - 71

EP - 84

AB - We investigate the congruence lattices of lattices in the varieties $ℳ _n$. Our approach is to represent congruences by open sets of suitable topological spaces. We introduce some special separation properties and show that for different n the lattices in $ℳ _n$ have different congruence lattices.

LA - eng

KW - algebraic lattice; topological representation; uniform separation; congruence lattices; separation properties

UR - http://eudml.org/doc/210775

ER -

## References

top- [1] D. Clark and B. A. Davey, Dualities for the Working Algebraist, Cambridge Univ. Press, 1998. Zbl0910.08001
- [2] B. A. Davey and H. Werner, Dualities and equivalences for varieties of algebras, in: Contributions to Lattice Theory, Colloq. Math. Soc. János Bolyai 33, North-Holland, 1983, 101-276.
- [3] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove and D. S. Scott, A Compendium of Continuous Lattices, Springer, 1980. Zbl0452.06001
- [4] G. Grätzer, General Lattice Theory, 2nd ed., Birkhäuser, 1998. Zbl0909.06002
- [5] A. Hajnal and A. Máté, Set mappings, partitions and chromatic numbers, in: Logic Colloquium '73, Stud. Logic Found. Math. 80, North-Holland, 1975, 347-379.
- [6] J. L. Kelley, General Topology, Van Nostrand, 1955.
- [7] K. Kuratowski, Sur une caractérisation des alephs, Fund. Math. 38 (1951), 14-17. Zbl0044.27302
- [8] R. McKenzie, R. McNulty and W. Taylor, Algebras, Lattices, Varieties I, Wadsworth & Brooks/Cole, 1987.
- [9] M. Ploščica and J. Tůma, Uniform refinements in distributive semilattices, in: Contributions to General Algebra 10 (Klagenfurt '97), Verlag Johannes Heyn, 1998. Zbl0907.06002
- [10] M. Ploščica, J. Tůma and F. Wehrung, Congruence lattices of free lattices in nondistributive varieties, Colloq. Math. 76 (1998), 269-278. Zbl0904.06005
- [11] H. A. Priestley, Representation of distributive lattices by means of ordered Stone spaces, Bull. London Math. Soc. 2 (1970), 186-190. Zbl0201.01802
- [12] M. H. Stone, Topological representation of distributive lattices and Brouwerian logics, Čas. Pěst. Mat. Fyz. 67 (1937), 1-25. Zbl0018.00303
- [13] F. Wehrung, Non-measurability properties of interpolation vector spaces, Israel J. Math. 103 (1998), 177-206. Zbl0916.06018
- [14] F. Wehrung, A uniform refinement property of certain congruence lattices, Proc. Amer. Math. Soc. 127 (1999), 363-370. Zbl0902.06006

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.