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There are 7 letters in HEETING ( E1G2H4I1N1T1 )
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In mathematics, a Heyting algebra is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation a → b of implication such that c ∧ a ≤ b is equivalent to c ≤ a → b. From a logical standpoint, A → B is by this definition the weakest proposition for which modus ponens, the inference rule A → B, A ⊢ B, is sound. Equivalently a Heyting algebra is a residuated lattice whose monoid operation a⋅b is a ∧ b; yet another definition is as a posetal cartesian closed category with all finite sums. Like Boolean algebras, Heyting algebras form a variety axiomatizable with finitely many equations. Heyting algebras were introduced by Arend Heyting (1930) to formalize intuitionistic logic. * As lattices, Heyting algebras are distributive. Every Boolean algebra is a Heyting algebra when a → b is defined as usual as ¬a ∨ b, as is every complete distributive lattice satisfying a one-sided infinite distributive law when a → b is taken to be the supremum of the set of all c for which a ∧ c ≤ b. The open sets of a topological space form such a lattice, and therefore a (complete) Heyting algebra. 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. * It follows from the definition that 1 ≤ 0 → a, corresponding to the intuition that any proposition a is implied by a contradiction 0. Although the negation operation ¬a is not part of the definition, it is definable as a → 0. The definition implies that a ∧ ¬a = 0, making the intuitive content of ¬a the proposition that to assume a would lead to a contradiction, from which any other proposition would then follow. It can further be shown that a ≤ ¬¬a, although the converse, ¬¬a ≤ a, is not true in general, that is, double negation does not hold in general in a Heyting algebra. * Heyting algebras generalize Boolean algebras in the sense that a Heyting algebra satisfying a ∨ ¬a = 1 (excluded middle), equivalently ¬¬a = a (double negation), is a Boolean algebra. Those elements of a Heyting algebra of the form ¬a comprise a Boolean lattice, but in general this is not a subalgebra of H (see below). * Heyting algebras serve as the algebraic models of propositional intuitionistic logic in the same way Boolean algebras model propositional classical logic. Complete Heyting algebras are a central object of study in pointless topology. The internal logic of an elementary topos is based on the Heyting algebra of subobjects of the terminal object 1 ordered by inclusion, equivalently the morphisms from 1 to the subobject classifier Ω. * Every Heyting algebra whose set of non-greatest elements has a greatest element (and forms another Heyting algebra) is subdirectly irreducible, whence every Heyting algebra can be made an SI by adjoining a new greatest elem... |