Definitions for heyting algebra
heyt·ing al·ge·bra

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1. Heyting algebranoun

   A bounded lattice equipped with a binary operation ab of implication such that (ab)a b, and moreover ab is the greatest such in the sense that if ca b then c ab.

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1. Heyting algebra

   In mathematics, a Heyting algebra (also known as pseudo-Boolean 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. 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 ¬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 c ∧ a ≤ 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. 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 intuitive content of ¬a is the proposition that to assume a would lead to a contradiction. The definition implies that a ∧ ¬a = 0. It can further be shown that a ≤ ¬¬a, although the converse, ¬¬a ≤ a, is not true in general, that is, double negation elimination does not hold in general in a Heyting algebra. Heyting algebras generalize Boolean algebras in the sense that Boolean algebras are precisely the Heyting algebras satisfying a ∨ ¬a = 1 (excluded middle), equivalently ¬¬a = a. Those elements of a Heyting algebra H 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. 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. The open sets of any topological space form a complete Heyting algebra. Complete Heyting algebras thus become a central object of study in pointless topology. 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 subdirectly irreducible by adjoining a new greatest element. It follows that even among the finite Heyting algebras there exist infinitely many that are subdirectly irreducible, no two of which have the same equational theory. Hence no finite set of finite Heyting algebras can supply all the counterexamples to non-laws of Heyting algebra. This is in sharp contrast to Boolean algebras, whose only subdirectly irreducible one is the two-element one, which on its own therefore suffices for all counterexamples to non-laws of Boolean algebra, the basis for the simple truth table decision method. Nevertheless, it is decidable whether an equation holds of all Heyting algebras.Heyting algebras are less often called pseudo-Boolean algebras, or even Brouwer lattices, although the latter term may denote the dual definition, or have a slightly more general meaning.

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1. Heyting algebra

   In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then 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. As lattices, Heyting algebras can be shown to be distributive. Every Boolean algebra is a Heyting algebra when a→b is defined as usual as ¬a∨b, as is every complete distributive lattice 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 a complete distributive lattice and hence a Heyting algebra. In the finite case every nonempty distributive lattice, in particular every nonempty finite chain, is automatically bounded and complete and hence a Heyting algebra.

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Numerology

1. Chaldean Numerology

   The numerical value of heyting algebra in Chaldean Numerology is: 5

2. Pythagorean Numerology

   The numerical value of heyting algebra in Pythagorean Numerology is: 8

References

1. ^ Wiktionary
   https://en.wiktionary.org/wiki/Heyting_Algebra
2. ^ Wikipedia
   https://en.wikipedia.org/wiki/Heyting_Algebra
3. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=heyting algebra