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In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written β¨ and β§ and with least element 0 and greatest element 1) equipped with a binary operation a β b called implication such that (c β§ a) β€ b is equivalent to c β€ (a β b). In a Heyting algebra a β€ b can be found to be equivalent to 1 β€ a β b; i.e. if a β€ b then a proves b. From a logical standpoint, A β B is by this definition the weakest proposition for which modus ponens, the inference rule A β B, A β’ B, is sound. Like Boolean algebras, Heyting algebras form a variety axiomatizable with finitely many equations. Heyting algebras were introduced in 1930 by Arend Heyting to formalize intuitionistic logic.
Heyting algebras are distributive lattices. Every Boolean algebra is a Heyting algebra when a β b is defined as Β¬a β¨ b, as is every complete distributive lattice satisfying a one-sided infinite distributive law when a β b is taken to be the supremum of the set of all c for which c β§ a β€ b. In the finite case, every nonempty distributive lattice, in particular every nonempty finite chain, is automatically complete and completely distributive, and hence a Heyting algebra.