Some properties of filters on a lattice L are studied with respect to a congruence on L. The notion of a θ-filter of L is introduced and these filters are then characterized in terms of classes of θ. For distributive L, an isomorphism between the lattice of θ-filters of L and the lattice of filters of is obtained.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmal_1223,
author = {M. Sambasiva Rao and Abd El-Mohsen Badawy},
title = {Filters of lattices with respect to a congruence},
journal = {Discussiones Mathematicae - General Algebra and Applications},
volume = {34},
year = {2014},
pages = {213-219},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmal_1223}
}
M. Sambasiva Rao; Abd El-Mohsen Badawy. Filters of lattices with respect to a congruence. Discussiones Mathematicae - General Algebra and Applications, Tome 34 (2014) pp. 213-219. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmal_1223/
[000] [1] R. Balbes and P. Dwinger, Distributive Lattices (University of Missouri Press, Columbia, Mo., 1974).
[001] [2] G. Birkhoff, Lattice Theory (Amer. Math. Soc. Colloq. XXV, Providence, USA, 1967).
[002] [3] T.S. Blyth, Ideals and filters of pseudo-complemeneted semi-lattices, Proc. Edinburgh Math. Soc. 23 (1980) 301-316. doi: 10.1017/S0013091500003850 | Zbl 0484.06004
[003] [4] B.A. Davey and H.A. Priestly, Introduction to Lattice and Order (Cambridge University Press, USA, 2002). doi: 10.1017/CBO9780511809088
[004] [5] G. Grätzer and E.T. Schmidt, Ideals and congruence relations in lattices, Acta Math. Acad. Sci. Hungary 9 (1958) 137-175. doi: 10.1007/BF02023870 | Zbl 0085.02002
[005] [6] G. Grätzer and E.T. Schmidt, On congruence relations of lattices, Acta Math. Acad. Sci. Hungary 13 (1962) 179-185. doi: 10.1007/BF02033636 | Zbl 0101.02103
[006] [7] T.P. Speed, Two congruences on distributive lattices, Bulletin de la Societe Royale des Sciences de Liege, 38e (3-4) (1969) 86-95.