Geodesic
Congruences and Boolean filters of quasi-modular p-algebras
Discussiones Mathematicae. General Algebra and Applications, Tome 34 (2014) no. 1, pp. 109-123 Cet article a éte moissonné depuis la source Library of Science

Voir la notice de l'article

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 F_B(L) = [Fₐ): a ∈ B(L) 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.
Keywords: p-algebras, quasi-modular p-algebras, Boolean filters, direct products, congruences 
@article{DMGAA_2014_34_1_a7,
     author = {El-Mohsen Badawy, Abd and Shum, K.},
     title = {Congruences and {Boolean} filters of quasi-modular p-algebras},
     journal = {Discussiones Mathematicae. General Algebra and Applications},
     pages = {109--123},
     year = {2014},
     volume = {34},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGAA_2014_34_1_a7/}
}
TY  - JOUR
AU  - El-Mohsen Badawy, Abd
AU  - Shum, K.
TI  - Congruences and Boolean filters of quasi-modular p-algebras
JO  - Discussiones Mathematicae. General Algebra and Applications
PY  - 2014
SP  - 109
EP  - 123
VL  - 34
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/DMGAA_2014_34_1_a7/
LA  - en
ID  - DMGAA_2014_34_1_a7
ER  - 
%0 Journal Article
%A El-Mohsen Badawy, Abd
%A Shum, K.
%T Congruences and Boolean filters of quasi-modular p-algebras
%J Discussiones Mathematicae. General Algebra and Applications
%D 2014
%P 109-123
%V 34
%N 1
%U http://geodesic.mathdoc.fr/item/DMGAA_2014_34_1_a7/
%G en
%F DMGAA_2014_34_1_a7
El-Mohsen Badawy, Abd; Shum, K. Congruences and Boolean filters of quasi-modular p-algebras. Discussiones Mathematicae. General Algebra and Applications, Tome 34 (2014) no. 1, pp. 109-123. http://geodesic.mathdoc.fr/item/DMGAA_2014_34_1_a7/

[1] R. Balbes and A. Horn, Stone lattices, Duke Math. J. 37 (1970) 537-543. doi: 10.1215/S0012-7094-70-03768-3

[2] R. Balbes and Ph. Dwinger, Distributive Lattices (Univ. Miss. Press, 1975).

[3] G. Birkhoff, Lattice theory, Amer. Math. Soc., Colloquium Publications, 25, New York, 1967.

[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

[5] G. Grätzer, Lattice Theory, First Concepts and Distributive Lattice (W.H. Freeman and Co., San-Francisco, 1971).

[6] G. Grätzer, General Lattice Theory (Birkhäuser Verlag, Basel and Stuttgart, 1978).

[7] O. Frink, Pseudo-complments in semi-lattices, Duke Math. J. 29 (1962) 505-514. doi: 10.1215/S0012-7094-62-02951-4

[8] T. Katriŭák and P. Mederly, Construction of p-algebras, Algebra Universalis 4 (1983) 288-316.

[9] M. Sambasiva Rao and K.P. Shum, Boolean filters of distributive lattices, Int. J. Math. and Soft Comp. 3 (2013) 41-48.

[10] P.V. Venkatanarasimhan, Ideals in semi-lattices, J. Indian. Soc. (N.S.) 30 (1966) 47-53.