Congruence lattices of distributive \(p\)-algebras (Q1239177)

An algebra \(\langle L;\cup,\cap,*,0,1\rangle\) is said to be a distributive \(p\)-algebra if \(\langle L;\cup,\cap,0,1\rangle\) is a bounded distributive lattice in which every element \(a\in L\) has a pseudocomplement \(a^*\), i.e. \(a\cap x=0\) if and only if \(x\leq a^*\). In the paper an abstract characterization of the congruence lattice of a distributive \(p\)-algebra is given. More precisely, an algebraic lattice \(A\) is a congruence lattice of a distributive \(p\)-algebra if and only if the join-semilattice of all compact elements of \(A\) forms a dual Heyting algebra of order 3 (Theorem 3). The proof of this theorem bases upon the fact that any distributive \(p\)-algebra \(L\) can be embedded in a Heyting algebra \(H\) of order 3 such that every congruence relation on \(L\) has only one extension to \(H\) (Theorem 2). We recall that a Heyting algebra \(H\) has order 3 if \(D(H) = \{x\in H: x^*=0\}\) is relatively complemented.