Congruences and Boolean filters of quasi-modular p-algebras

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 ${\displaystyle F_B\left(L\right)=\{[Fₐ):a\in B\left(L\right)\}}$ 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.