{'fraction_non_alphanumeric': 0.06694560669456066, 'fraction_numerical': 0.0, 'mean_word_length': 4.258241758241758, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
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 FB(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.