# Heyting algebra

A Heyting algebra is a bounded lattice (a set with an order, plus greatest and least elements) equipped with an implication operation \( a \to b \) characterized by the adjunction \( a \wedge x \le b \) if and only if \( x \le a \to b \) for all elements \( a, b, x \). It is the algebraic counterpart of intuitionistic logic in the same way that a [Boolean algebra](https://www.edgepedia.org/boolean-algebra) is the algebraic counterpart of classical logic.<sup>[1](https://mathworld.wolfram.com/HeytingAlgebra.html)</sup><sup> • </sup><sup>[2](https://plato.stanford.edu/entries/logic-algebraic-propositional/)</sup>

| Key fact | Detail |
|---|---|
| Definition | A bounded lattice \( H \) with a binary operation \( \to \) such that \( a \wedge x \le b \) iff \( x \le a \to b \), for all \( a, b, x \in H \)<sup>[1](https://mathworld.wolfram.com/HeytingAlgebra.html)</sup> |
| Negation | Defined as \( \neg a = a \to 0 \), the pseudocomplement of \( a \); in general \( \neg a \vee a \ne 1 \), so excluded middle can fail<sup>[1](https://mathworld.wolfram.com/HeytingAlgebra.html)</sup><sup> • </sup><sup>[3](https://planetmath.org/heytingalgebra)</sup> |
| Distributivity | Every Heyting algebra is a distributive lattice<sup>[1](https://mathworld.wolfram.com/HeytingAlgebra.html)</sup> |
| Boolean case | Every Boolean algebra is Heyting with \( a \to b = \neg a \vee b \); non-Boolean Heyting algebras show that excluded middle is not valid in all of them, not that it must fail in each one |
| Logical role | The class HA of Heyting algebras is the algebraic semantics for intuitionistic logic, as the class BA of Boolean algebras is for classical logic<sup>[2](https://plato.stanford.edu/entries/logic-algebraic-propositional/)</sup> |
| Canonical example | The open sets of any topological space \( X \), ordered by inclusion, form a Heyting algebra<sup>[1](https://mathworld.wolfram.com/HeytingAlgebra.html)</sup> |
| Origin | Introduced by Arend Heyting in 1930 to formalize intuitionistic logic; also called pseudo-Boolean algebras<sup>[4](https://marcobenini.me/wp-content/uploads/2024/12/slides.pdf)</sup><sup> • </sup><sup>[3](https://planetmath.org/heytingalgebra)</sup> |

## Definition and basic properties

The defining condition says that \( a \to b \) is the largest element \( x \) whose meet with \( a \) stays below \( b \). This operation is called the relative pseudocomplement of \( a \) with respect to \( b \). A lattice in which this relative pseudocomplement exists for every pair of elements, together with a bottom element \( 0 \), is a Heyting algebra; the synonym Brouwerian lattice is used when the bottom element is not assumed.<sup>[1](https://mathworld.wolfram.com/HeytingAlgebra.html)</sup><sup> • </sup><sup>[3](https://planetmath.org/heytingalgebra)</sup><sup> • </sup><sup>[5](https://encyclopediaofmath.org/wiki/Pseudo-Boolean_algebra)</sup>

Negation is not primitive. It is defined by \( \neg a = a \to 0 \), and this operation is the pseudocomplement: the largest element disjoint from \( a \). Several consequences follow directly from the adjunction. The top element satisfies \( 1^{*} = 0 \), an element satisfies \( a^{*} = 1 \) only when \( a = 0 \), and every element obeys \( a \le a^{**} \), where \( a^{*} \) denotes the pseudocomplement of \( a \). The reverse inequality \( a^{**} \le a \) does not hold in general; a Heyting algebra in which \( a^{**} \le a \) for all \( a \) is exactly a Boolean algebra, and there the pseudocomplement coincides with the ordinary complement and \( a \to b \) reduces to the material implication \( \neg a \vee b \).<sup>[3](https://planetmath.org/heytingalgebra)</sup>

Every Heyting algebra is a distributive lattice, a consequence of the adjunction.<sup>[1](https://mathworld.wolfram.com/HeytingAlgebra.html)</sup> A Heyting algebra homomorphism is a lattice homomorphism that preserves \( 0 \), \( 1 \), and \( \to \); such a map automatically preserves pseudocomplementation as well.<sup>[3](https://planetmath.org/heytingalgebra)</sup>

## Relation to Boolean algebras

The two structures differ precisely in the behavior of negation. In a Boolean algebra every element has a complement, so \( \neg a \vee a = 1 \) always. In a general Heyting algebra the pseudocomplement \( \neg a \) is only the largest element disjoint from \( a \), and the join \( \neg a \vee a \) can fall short of the top. The two-element chain, which is the smallest Boolean algebra, is the algebra a classical truth table computes in; among finite Heyting algebras, the Boolean ones are exactly those whose underlying poset has no nontrivial order relations, and every other finite Heyting algebra refutes excluded middle.<sup>[6](https://boole.stanford.edu/cs353/handouts/book3.pdf)</sup><sup> • </sup><sup>[7](https://www.matpic.com/essays/not-one-step-but-a-continuum/)</sup>

Any chain with a greatest element \( 1 \) forms a Heyting algebra: when \( p \le q \) one must set \( p \to q = 1 \), and otherwise \( p \to q = q \).<sup>[6](https://boole.stanford.edu/cs353/handouts/book3.pdf)</sup> These linearly ordered examples, studied by [Alfred Horn](https://www.edgepedia.org/alfred-horn) in 1969, give the simplest non-Boolean models.<sup>[8](https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/decidability-problem-for-finite-heyting-algebras/AA714467711BA071CD9C6B321E95E42F)</sup>

## Algebraic semantics for intuitionistic logic

Heyting algebras provide the standard algebraic semantics for intuitionistic propositional logic. A valuation assigns to each formula an element of a Heyting algebra, with the greatest element \( 1^{\bA} \) interpreting the constant \( \top \). The soundness and completeness statement takes the form: \( \Gamma \vdash_{\mathrm{L}} \phi \) if and only if, for every algebra \( \bA \) in the class and every valuation \( v \) on \( \bA \), if \( v(\psi) = 1^{\bA} \) for all \( \psi \in \Gamma \), then \( v(\phi) = 1^{\bA} \).<sup>[2](https://plato.stanford.edu/entries/logic-algebraic-propositional/)</sup>

The correspondence extends to a dictionary between logics and equation classes. Each formula \( \varphi \) of intuitionistic propositional calculus corresponds to the equation \( \varphi \approx 1 \) in the theory of Heyting algebras, and this yields a one-to-one correspondence between superintuitionistic logics, the logics between intuitionistic and classical propositional calculus, and equational theories of Heyting algebras, that is, varieties of Heyting algebras.<sup>[9](https://staff.fnwi.uva.nl/n.bezhanishvili//MSL/MSL2017/Int-logic-and-HA.pdf)</sup> Classical logic sits above them: \( \mathrm{CPC} = \mathrm{IPC} + (p \vee \neg p) \), and the law of excluded middle is not derivable in intuitionistic logic.<sup>[9](https://staff.fnwi.uva.nl/n.bezhanishvili//MSL/MSL2017/Int-logic-and-HA.pdf)</sup> By Glivenko's theorem, the class of Heyting algebras also serves, with the equation \( \neg \neg p \approx 1 \), as an algebraic semantics for classical logic, while it is the standard \( p \approx 1 \)-semantics for intuitionistic logic.<sup>[2](https://plato.stanford.edu/entries/logic-algebraic-propositional/)</sup>

## Examples from topology and Kripke models

The canonical non-Boolean example comes from topology. For any topological space \( X \), the set \( \mathrm{Open}(X) \) of open sets, ordered by inclusion, is a lattice under union and intersection, and it carries a Heyting implication: \( U \to V \) is the largest open set \( W \) with \( W \cap U \subseteq V \), namely the interior of the complement of \( U \) union \( V \).<sup>[1](https://mathworld.wolfram.com/HeytingAlgebra.html)</sup><sup> • </sup><sup>[10](https://pages.physics.ua.edu/faculty/fabi/CT/Mac%20Lane%2C%20Ieke%20Moerdijk%20-%20Sheaves%20in%20Geometry%20and%20Logic_%20A%20First%20Introduction%20to%20Topos%20Theory-.pdf)</sup> The negation of an open set \( U \) is the interior of its complement, not the complement itself, which in general fails to be open.<sup>[11](https://projects.lsv.ens-paris-saclay.fr/topology/?page_id=1224)</sup> For any topological space the poset of open sets is in fact a complete Heyting algebra, also called a frame.<sup>[12](https://awswan.github.io/files/intuitionisticlogicnotes.pdf)</sup><sup> • </sup><sup>[11](https://projects.lsv.ens-paris-saclay.fr/topology/?page_id=1224)</sup>

[Kripke semantics](https://www.edgepedia.org/kripke-semantics) connects to the same algebra.

## Heyting algebras in topos theory

In a topos, a category that behaves like a universe of sets, the subobjects of any object form a lattice, and in an elementary topos these subobject lattices are Heyting algebras. The [subobject classifier](https://www.edgepedia.org/subobject-classifier) \( \Omega \) itself carries the structure of an internal Heyting algebra, and the Heyting algebra structure of the subobject lattices is induced by this internal structure via the Yoneda Lemma.<sup>[13](https://www.oliviacaramello.com/Teaching/CambridgeToposTheoryCourseLectures17to20.pdf)</sup> This is what makes the internal logic of a topos intuitionistic: the truth values of the topos form a Heyting algebra rather than a two-element Boolean algebra.<sup>[14](https://ncatlab.org/nlab/show/Heyting%2Balgebra)</sup> The double negation operator \( \neg \neg \colon \Omega \to \Omega \) on this internal Heyting algebra defines a Lawvere–Tierney topology.<sup>[14](https://ncatlab.org/nlab/show/Heyting%2Balgebra)</sup>

## History

The algebras are named for [Arend Heyting](https://www.edgepedia.org/arend-heyting) (1898–1980), a student of L. E. J. Brouwer at the [University of Amsterdam](https://www.edgepedia.org/university-of-amsterdam). In 1927 the Dutch Mathematical Association announced a prize question asking for the formalization of intuitionistic logic and mathematics; Heyting wrote the single submission, under the motto "Stones for bread", and the jury crowned his work in 1928.<sup>[15](https://plato.stanford.edu/entries/intuitionistic-logic-development/)</sup> The revised papers appeared in 1930 as *Die formalen Regeln der intuitionistischen Logik* and *Die formalen Regeln der intuitionistischen Mathematik II, III*, the first partially successful attempt to present a formalized version of Brouwer's theories.<sup>[16](https://archive.illc.uva.nl/KNAW/Heyting/en/About_Heyting/index.html)</sup> Heyting's formalization covered propositional and predicate logic, arithmetic, and analysis in one system, and, together with his 1956 book *Intuitionism: An Introduction*, it remains among the most influential intuitionistic publications.<sup>[15](https://plato.stanford.edu/entries/intuitionistic-logic-development/)</sup> His clear expositions are credited with keeping intuitionism a live research subject, and intuitionistic logic later came to play an important role in the development of type theory and automated proof-checking.<sup>[16](https://archive.illc.uva.nl/KNAW/Heyting/en/About_Heyting/index.html)</sup>

## References

1. [Heyting Algebra, Wolfram MathWorld](https://mathworld.wolfram.com/HeytingAlgebra.html)
2. [Algebraic Propositional Logic, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/logic-algebraic-propositional/)
3. [Heyting algebra, PlanetMath](https://planetmath.org/heytingalgebra)
4. [Mathematical Logic lecture notes (Marco Benini)](https://marcobenini.me/wp-content/uploads/2024/12/slides.pdf)
5. [Pseudo-Boolean algebra, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Pseudo-Boolean_algebra)
6. [Boolean and Heyting Algebras for Logic, Stanford CS353 course notes](https://boole.stanford.edu/cs353/handouts/book3.pdf)
7. [Not one step but a continuum, Matpic](https://www.matpic.com/essays/not-one-step-but-a-continuum/)
8. [Decidability problem for finite Heyting algebras, Journal of Symbolic Logic](https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/decidability-problem-for-finite-heyting-algebras/AA714467711BA071CD9C6B321E95E42F)
9. [Varieties of Heyting algebras and superintuitionistic logics (N. Bezhanishvili, MSL 2017)](https://staff.fnwi.uva.nl/n.bezhanishvili//MSL/MSL2017/Int-logic-and-HA.pdf)
10. [Sheaves in Geometry and Logic (Mac Lane and Moerdijk)](https://pages.physics.ua.edu/faculty/fabi/CT/Mac%20Lane%2C%20Ieke%20Moerdijk%20-%20Sheaves%20in%20Geometry%20and%20Logic_%20A%20First%20Introduction%20to%20Topos%20Theory-.pdf)
11. [The O functor does not preserve binary products, Non-Hausdorff Topology and Domain Theory](https://projects.lsv.ens-paris-saclay.fr/topology/?page_id=1224)
12. [Intuitionistic logic notes (A. Swan)](https://awswan.github.io/files/intuitionisticlogicnotes.pdf)
13. [Topos Theory, Lectures 17–20: The interpretation of logic in categories (O. Caramello)](https://www.oliviacaramello.com/Teaching/CambridgeToposTheoryCourseLectures17to20.pdf)
14. [Heyting algebra, nLab](https://ncatlab.org/nlab/show/Heyting%2Balgebra)
15. [The Development of Intuitionistic Logic, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/intuitionistic-logic-development/)
16. [About Heyting, Arend Heyting Stichting](https://archive.illc.uva.nl/KNAW/Heyting/en/About_Heyting/index.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Boolean and logic-related algebras › Heyting algebras*

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