A NOTE ON CONGRUENCE LATTICES OF DISTRIBUTIVE p–ALGEBRAS

A (distributive) p-algebra is an algebra 〈L;∨,∧, ∗, 0, 1〉 whose reduct 〈L;∨,∧, 0, 1〉 is a bounded (distributive) lattice and whose unary operation ∗ is characterized by x ≤ a if and only if a ∧ x = 0. If L is a p-algebra, B(L) = { x ∈ L : x = x } and D(L) = { x ∈ L : x = 1 } then 〈B(L);∪,∧, 0, 1〉 is a Boolean algebra when a ∪ b is defined to be (a∗ ∧ b∗)∗, for any a, b ∈ B(L), D∗(L) = { x ∨ x∗ : x ∈ L } and is a filter of L. A (distributive) dual p-algebra is an algebra 〈L;∨,∧,+, 0, 1〉 whose reduct 〈L;∨,∧, 0, 1〉 is a bounded (distributive) lattice and whose unary operation + is characterized by x ≥ a if and only if a ∨ x = 1. In such an algebra, D(L) = {x ∈ L : x = 0} is an ideal of L. A distributive p-algebra (dual p-algebra) L is said to be of order 3 if and only if D∗(L) (D(L)) is relatively complemented. By a congruence relation of a p-algebra L we mean a lattice congruence θ of L preserving ∗ and, for a ∈ L, we denote {x ∈ L : x ≡ a(θ)} by [a]θ. The relation φ defined on L by (a, b) ∈ φ if and only if a∗ = b∗ is a congruence called the Glivenko congruence of L, L/φ ∼= B(L) and [1]φ = D(L). θ(a, b)(θlat(a, b)) will denote the principal congruence of L (of the lattice reduct of L) collapsing a pair a, b ∈ L and, for any filter F of L, Θ(F ) (Θlat(F )) will denote the smallest congruence of L (of the lattice reduct of L) collapsing F . The congruence lattice of L will be denoted Con (L): it is distributive and algebraic and its join subsemilattice of compact elements will be denoted Comp (Con (L)).