A Study on P-algebras and Its Underlying Lattices

Abstract

In this thesis we study p-algebras and its underlying lattices. An algebra L = ⟨L, ∧, ∨, ∗ , 0, 1⟩ is called a p-algebra if (i) ⟨L, ∧, ∨, 0, 1⟩ is a bounded lattice, and (ii) for all x, y ∈ L, x ∧ x ∗ = 0 and x ∧ y = 0 implies y 6 x ∗ . The lattice ⟨L, ∧, ∨⟩ is called the underlying lattice of L. There are many research works on p-algebras where the underlying lattices are distributive. In this thesis we study p-algebras in general. The underlying lattice of a p-algebra is 0-distributive but not every un- derlying lattice of a p-algebra is 0-modular or 1-distributive. We study some classes of ideals of 0-distributive lattice for study of p-algebras. We charac- terize some subclasses of p-algebras. We introduce a notion of DM-algebras which is a nice subclass of p-algebras. P-ideals and p-filters of a p-algebra play an important role to study p- algebras. I ∗ (L), the set of all p-ideals of a p-algebra L, itself form a com- plete distributive p-algebra which is isomorphic to the complete distributive p-algebra formed by the set of all p-filters of L. We study congruences, particularly, kernel ideals of a p-algebra. We show that an ideal of a p-algebra is a kernel ideal if and only if it is a p-ideal. We also characterize co-kernel filters of a p-algebra. Finally, we show that if f : L → M is an epimorphism of p-algebras, then there is an epimorphism f ∗ : I ∗ (L) → I ∗ (M) if and only if ker f is a principal ideal. We also show that for any p-algebra L we have I ∗ (L) ∼ = I(S(L)).