Contents

Definition

This is equivalent to the following definition.

The implication $a\Rightarrow b$ is the exponential object $b^a$.

The definition of Heyting algebra may be recast into purely equational form, and so we can speak of an internal Heyting algebra in any category with products.

Relation to other concepts

To logic

In logic, Heyting algebras are precisely algebraic models for intuitionistic propositional calculus, just as Boolean algebras model classical propositional calculus. As one might guess from this description, the “law of the excluded middle” does not generally hold in a Heyting algebra; see the discussion below.

In a Heyting category, every subobject poset $Sub(A)$ is a Heyting algebra. In particular, this holds for every topos. Furthermore, in a topos, the power object $\mathcal{P}(A)$ is an internal Heyting algebra that corresponds to the external Heyting algebra $Sub(A)$. In a boolean topos, the internal Heyting algebras are all internal boolean algebras. In general, however, the internal logic of a topos (or other Heyting category) is intuitionistic.

To topology

One of the chief sources of Heyting algebras is given by topologies. As a poset, the topology of a topological space $X$ is a complete lattice (it has arbitrary joins and meets, although the infinitary meets are not in general given by intersection), and the implication operator is given by

where $U, V$ are open sets, $U^c$ is the set-theoretic complement of $U$, and $int(S)$ denotes the interior of a subset $S \subseteq X$.

Somewhat more generally, a frame (a sup-lattice in which finite meets distribute over arbitrary sups) also carries a Heyting algebra structure. In a frame, we may define

and the distributivity property guarantees that the universal property for implication holds. (The detailed proof is a “baby” application of an adjoint functor theorem.)

Thus frames are extensionally the same thing as complete Heyting algebras. However, intensionally they are quite different; that is, a morphism of frames is not usually a morphism of complete Heyting algebras: they do not preserve the implication operator.

A locale is the same thing as a frame, but again the morphisms are different; they are reversed.

Topologies that are Boolean algebras are the exception rather than the rule; basic examples include topologies of Stone spaces; see Stone duality. Another example is the topology of a discrete space $X$.

To Boolean algebras

In any Heyting algebra $L$, we may define a negation operator

by $\neg x = (x \Rightarrow 0)$, where $0$ is the bottom element of the lattice. A Heyting algebra is Boolean if the double negation

coincides with the identity map; this gives one of many ways of defining a Boolean algebra.

In any Heyting algebra $L$, we have for all $a, b \in L$ the inequality

and another characterization of Boolean algebra is a Heyting algebra in which this is an equality for all $a, b$.

There are several ways of passing back and forth between Boolean algebras and Heyting algebras, having to do with the double negation operator. A useful lemma in this regard is

The proof can be made purely equational, and is therefore internally valid in any category with products. Applied to the internal Heyting algebra $L = \Omega$ of a topos, that is the subobject classifier, this lemma says exactly that the double negation operator $\neg \neg\colon \Omega \to \Omega$ defines a Lawvere–Tierney topology. Similarly, we get the double negation sublocale of any locale.

Now let $L_{\neg\neg}$ denote the poset of regular elements of $L$, that is, those elements $x$ such that $\neg\neg x = x$. (When $L$ is the topology of a space, an open set $U$ is regular if and only if it is the interior of its closure, that is if it is a regular element of the Heyting algebra of open sets described above.) With the help of the lemma above, we may prove

Thus $\neg\neg\colon L \to L_{\neg\neg}$ preserves finite joins and finite meets and implication. In the other direction, we have an inclusion $i\colon L_{\neg\neg} \to L$, and this preserves meets but not joins. It also preserves negations; more generally and perhaps surprisingly, it preserves implications as well.

Regular elements are not to be confused with complemented element?s, i.e., elements $x$ in a Heyting algebra such that $x \vee \neg x = 1$, although it is true that every complemented element is regular. An example of a regular element which is not complemented is given by the unit interval $(0, 1)$ as an element of the topology of $\mathbb{R}$; a complemented element in a Heyting algebra given by a topology is the same as a clopen subset.

Complemented elements furnish another universal relation between Boolean algebras and Heyting algebras: the set of complemented elements in a Heyting algebra $H$ is a Boolean algebra $Comp(H)$, and the inclusion $Comp(H) \to H$ is a Heyting algebra map which is universal among Heyting algebra maps $B \to H$ out of Boolean algebras $B$. In other words, we have the following result.

Proofs

We prove the lemma and theorems of the preceding section.

To toposes

An elementary topos is a vertical categorification of a Heyting algebra: the notion of Heyting algebra is essentially equivalent to that of (0,1)-topos. Note that a Grothendieck $(0,1)$-topos is a frame or locale.

Examples

In particular for $X = \Omega$ the subobject classifier, $Sub(\Omega)$ is a Heyting algebra.

In $\mathcal{T} =$ Set for every set $S$ we have that $Sub(S)$ is the Boolean algebra of subset of $S$.

More details and examples are spelled out at internal logic.

References

Named after Arend Heyting.