The concept of Boolean filters in p-algebras is introduced. Some properties of Boolean filters are studied. It is proved that the class of all Boolean filters BF(L) of a quasi-modular p-algebra L is a bounded distributive lattice. The Glivenko congruence Φ on a p-algebra L is defined by (x,y) ∈ Φ iff x** = y**. Boolean filters [Fₐ), a ∈ B(L) , generated by the Glivenko congruence classes Fₐ (where Fₐ is the congruence class [a]Φ) are described in a quasi-modular p-algebra L. We observe that the set is a Boolean algebra on its own. A one-one correspondence between the Boolean filters of a quasi-modular p-algebra L and the congruences in [Φ,∇] is established. Also some properties of congruences induced by the Boolean filters [Fₐ), a ∈ B(L) are derived. Finally, we consider some properties of congruences with respect to the direct products of Boolean filters.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmal_1212,
author = {Abd El-Mohsen Badawy and K.P. Shum},
title = {Congruences and Boolean filters of quasi-modular p-algebras},
journal = {Discussiones Mathematicae - General Algebra and Applications},
volume = {34},
year = {2014},
pages = {109-123},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmal_1212}
}
Abd El-Mohsen Badawy; K.P. Shum. Congruences and Boolean filters of quasi-modular p-algebras. Discussiones Mathematicae - General Algebra and Applications, Tome 34 (2014) pp. 109-123. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmal_1212/
[000] [1] R. Balbes and A. Horn, Stone lattices, Duke Math. J. 37 (1970) 537-543. doi: 10.1215/S0012-7094-70-03768-3 | Zbl 0207.02802
[001] [2] R. Balbes and Ph. Dwinger, Distributive Lattices (Univ. Miss. Press, 1975).
[002] [3] G. Birkhoff, Lattice theory, Amer. Math. Soc., Colloquium Publications, 25, New York, 1967. | Zbl 0153.02501
[003] [4] G. Grätzer, A generalization on Stone's representations theorem for Boolean algebras, Duke Math. J. 30 (1963) 469-474. doi: 10.1215/S0012-7094-63-03051-5 | Zbl 0121.26702
[004] [5] G. Grätzer, Lattice Theory, First Concepts and Distributive Lattice (W.H. Freeman and Co., San-Francisco, 1971). | Zbl 0232.06001
[005] [6] G. Grätzer, General Lattice Theory (Birkhäuser Verlag, Basel and Stuttgart, 1978). | Zbl 0436.06001
[006] [7] O. Frink, Pseudo-complments in semi-lattices, Duke Math. J. 29 (1962) 505-514. doi: 10.1215/S0012-7094-62-02951-4 | Zbl 0114.01602
[007] [8] T. Katriŭák and P. Mederly, Construction of p-algebras, Algebra Universalis 4 (1983) 288-316. | Zbl 0536.06004
[008] [9] M. Sambasiva Rao and K.P. Shum, Boolean filters of distributive lattices, Int. J. Math. and Soft Comp. 3 (2013) 41-48.
[009] [10] P.V. Venkatanarasimhan, Ideals in semi-lattices, J. Indian. Soc. (N.S.) 30 (1966) 47-53. | Zbl 0158.01604