The subalgebra lattice of a Heyting algebra
L. Vrancken-Mawet, Georges Hansoul · Czechoslovak Mathematical Journal · 1987
In [7], L. Vrancken-Mawet investigates the subalgebra lattice of a finite Hey ting algebra.In this paper, we consider infinite Heyting algebras.Minimal (non trivial) and maximal (proper) subalgebras of a Heyting algebra L are determined.This enables to prove that the subalgebra lattice of L is always upper semimodular and that it is atomistic if and only if L is a Stone algebra.Also we characterize those Heyting algebras whose subalgebra lattice is Boolean.In § 1, we briefly recall Priestley's duality ([5]), adapting it for Heyting algebras.The problems are solved in the dual category and reinterpreted in terms of Heyting algebras in § 3 (Theorem 2.13).We use standard set theoretic symbols.Note that cz denotes strict inclusion anddenotes complement (in some given universe). PRIESTLEY'S DUALITY