Congruences and Boolean filters of quasi-modular p-algebras
Journal Title: Discussiones Mathematicae - General Algebra and Applications - Year 2014, Vol 34, Issue 1
Abstract
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. Keywords: p-algebras, quasi-modular p-algebras, Boolean filters, direct products, congruences. 2010 Mathematics Subject Classification: 06A06, 06A20, 06A30, 06D15.
Authors and Affiliations
Abd El-Mohsen Badawy, Kar-Ping Shum
Keywords
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