Congruences on distributive pseudocomplemented lattices

Abstract

Let B0 ⊂ B1 ⊂ Bn ⊂ … ⊂: Bw be all the non-trivial varieties of distributive pseudocomplemented lattices (L; ∨, ∧, *, 0, 1) considered as algebras of type (2, 2, 1, 0, 0). A subset J of such an algebra L is a congruence-kernel if and only if it is a lattice-ideal and x** ∈ J for each x ∈ J. The smallest congruence having J as its kernel is Θ(J), where a ≡ b (Θ(J)), (a, b ∈ L) if and only if a ∧ c* = b ∧ c* for some c ∈ J. For given 0 ≤ n ≤ w, let Σn (J) be the smallest congruence having J as its kernel and such that the associated quotient algebra is in Bn. Of course, Σw (J) = Θ(J) and the main result of this paper shows that for 1 ≤ n < w, Σn(J) = ∩{Θ(P1 ∩ P2 ∩ … ∩ Pn): J ⊆ P1, P2, …, Pn ⊂ L are minimal prime ideals}.