Geodesic
Congruences and Boolean filters of quasi-modular p-algebras
Discussiones Mathematicae. General Algebra and Applications, Tome 34 (2014) no. 1, pp. 109-123

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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 
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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/