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1987•Czechoslovak Mathematical JournalOpen access

The subalgebra lattice of a Heyting algebra

L. Vrancken-Mawet, Georges Hansoul

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Abstract

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

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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

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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

Key concepts: Subalgebra, Heyting algebra, Mathematics, Lattice (music), Algebra over a field, Pure mathematics, Physics, Acoustics

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