Boolean Algebras with an Involution

Introduction

Two different involutions act on the elements of a Boolean algebra. The complement $\neg$ is an order-reversing involution, of order two, that exchanges the meet and the join; it is the canonical involution of the structure, and it is an anti-automorphism, not an automorphism. An involution of a Boolean algebra, in the sense of this article, is on the contrary an order-preserving map $\sigma$ of order two that preserves the Boolean operations, that is, a Boolean automorphism with $\sigma^2 = \mathrm{id}$. A **Boolean algebra with an involution** is a Boolean algebra $B$ together with such a $\sigma$; its fixed part $B^{\sigma}$, the elements with $\sigma x = x$, is a Boolean subalgebra, and the pair $(B, \sigma)$ has a representation by a power set with an involution induced by an involutive permutation of the underlying set. This article separates the two involutions, treats the morphisms of the involutive structure and the fixed part, and gives the representation.

The article presupposes Order Theory and Lattices — distributive, complemented and Boolean lattices — and Boolean Algebras and Lattices, earlier in this Part, where the complement and the Boolean homomorphisms are fixed; it uses Orthocomplemented Lattices and the Involution, earlier in this group, for the order-reversing involution, and Involutive Set Theory and the Symmetric Difference, which follows, for the power-set representation. The Stone representation of a general Boolean algebra is Stone Duality and Boolean Algebras, in Part V, and is named only; the article proves the representation for the finite power-set case and states the general case as a forward reference. The complement and the involution of a Boolean ring are Involutive Algebras and the - Algebras group; the star of an algebra with involution is not used here.

Three boundaries are observed. No topology is used; the Stone space is named and deferred to Part V. No form is used, and the measure theory of a Boolean algebra is not touched. The reading of the Boolean operations as connectives is Logic and Proof, in Part 0, and is named only.

The Complement as an Involution

Definition. Let $B$ be a Boolean algebra. The complement is the map $\neg : B \to B$ with $x \wedge \neg x = 0$ and $x \vee \neg x = 1$; it is the orthocomplement of Orthocomplemented Lattices and the Involution, the unique complement of each element.

Theorem. The complement is an order-reversing involution and satisfies the De Morgan laws and the two unit laws:

$$ \neg\neg x = x, \qquad x \leq y \Rightarrow \neg y \leq \neg x, \qquad \neg(x \wedge y) = \neg x \vee \neg y, \qquad \neg(x \vee y) = \neg x \wedge \neg y, \qquad \neg 0 = 1, \quad \neg 1 = 0 . $$

It is an anti-automorphism of the Boolean algebra: it reverses the order and exchanges the two operations, and it is not an automorphism unless the algebra is trivial.

Proof. The involution, the order reversal and the De Morgan laws are those of an orthocomplement in a Boolean algebra, proved in Orthocomplemented Lattices and the Involution. The last assertion is the definition of an anti-automorphism: an automorphism is order-preserving, while the complement reverses the order, and it preserves the meet, while the complement exchanges the meet and the join.

So the complement is the order-reversing involution of the structure, and it is not an automorphism. An involution of a Boolean algebra in the sense used below is a second, order-preserving map; the two commute, since an automorphism of a Boolean algebra preserves the complement.

Proposition. If $B$ is a Boolean algebra with an involution $\sigma$, then $\sigma \circ \neg = \neg \circ \sigma$, and $\sigma$ is determined by its restriction to the atoms in the finite case.

Proof. An automorphism of a Boolean algebra preserves the complement, which is defined by the two Boolean identities, so $\sigma(\neg x) = \neg \sigma(x)$; for the finite case, every element is the join of the atoms below it and an automorphism permutes the atoms.

Involutive Boolean Algebras and their Homomorphisms

Definition. An involution of a Boolean algebra $B$ is a map $\sigma : B \to B$ with $\sigma^2 = \mathrm{id}$, $\sigma(x \wedge y) = \sigma x \wedge \sigma y$, $\sigma(x \vee y) = \sigma x \vee \sigma y$, $\sigma 0 = 0$ and $\sigma 1 = 1$. A **Boolean algebra with an involution** is a pair $(B, \sigma)$ with $\sigma$ an involution.

Proposition. An involution of a Boolean algebra is a Boolean automorphism of order two; conversely every Boolean automorphism of order two is an involution. It preserves the complement, the order, and every Boolean expression in the elements.

Proof. The preservation of the meet and the join together with $\sigma 0 = 0$ and $\sigma 1 = 1$ makes $\sigma$ a Boolean homomorphism; the involution makes it bijective with inverse itself, hence an automorphism of order two. The preservation of the complement follows from the identities and the uniqueness of the complement.

Definition. A morphism of involutive Boolean algebras $(B, \sigma) \to (B', \sigma')$ is a Boolean homomorphism $h : B \to B'$ with $h \circ \sigma = \sigma' \circ h$. The involutive Boolean algebras and their morphisms form a category, and the forgetful functor to Boolean algebras is faithful but not full.

Proof. The composition of two such homomorphisms commutes with the involutions, so the class is a category; the forgetful functor is injective on hom-sets but its image omits the homomorphisms that do not commute with the involutions, so it is not full.

Example (the power set). Let $X$ be a set and let $s : X \to X$ be an involution of the set, $s^2 = \mathrm{id}_X$. The map $\sigma_s$ on the power set $\mathcal{P}(X)$ defined by $\sigma_s(A) = s(A)$ is an involution of the Boolean algebra $\mathcal{P}(X)$; it is the involution induced by $s$. Every involution of a finite power set is of this form, with $s$ the restriction of $\sigma$ to the atoms, that is, the singletons.

The Fixed Part

Definition. The fixed part of an involutive Boolean algebra $(B, \sigma)$ is $B^{\sigma} = {x \in B : \sigma x = x}$.

Theorem. The fixed part is a Boolean subalgebra of $B$ containing $0$ and $1$. Its atoms, when $B$ is finite, are the orbits of $\sigma$ on the atoms of $B$; in particular every orbit of atoms a singleton $\{a\}$ with $\sigma a = a$ contributes one atom, and every two-element orbit ${a, \sigma a}$ contributes the element $a \vee \sigma a$, which is fixed and is an atom of the fixed part.

Proof. Fixed elements are closed under the Boolean operations because $\sigma$ preserves them, and $0, 1$ are fixed. An element of $B$ is the join of the atoms below it, and it is fixed if and only if the set of atoms below it is stable under $\sigma$; so the fixed elements are exactly the unions of orbits, and the minimal such unions are the orbits, which are therefore the atoms of the fixed part.

Corollary. For a finite Boolean algebra with $n$ atoms, the fixed part has one atom per orbit of $\sigma$ on the atoms, so its cardinality is $2^{f + (n - f)/2}$, where $f$ is the number of atoms fixed by $\sigma$; in particular the number of fixed atoms and $n$ have the same parity.

Proof. The orbits are the $f$ fixed atoms and the $(n-f)/2$ two-element orbits; the fixed part is the power set of its atoms by the theorem.

Example. On $\mathcal{P}(X)$ with the involution $s$, the fixed part is the power set of the set of orbits of $s$ on $X$: a subset is fixed exactly when it is a union of orbits, so $A$ is determined by which orbits it meets, and the fixed part is $\mathcal{P}(X/{\sim})$ with ${\sim}$ the orbit relation. The map $A \mapsto$ its set of orbits is a Boolean isomorphism onto the power set of $X/{\sim}$.

Representation

Theorem (finite representation). Every involution $\sigma$ of a finite Boolean algebra $B$ is induced by an involution $s$ of the set of atoms of $B$, and $B \cong \mathcal{P}(\mathrm{At}(B))$ as Boolean algebras, with $\sigma$ corresponding to $\sigma_s$. Hence every finite Boolean algebra with an involution is isomorphic to a power set with an involution, and the fixed part is the power set of the orbits.

Proof. The atoms are permuted by $\sigma$ because $\sigma$ is an automorphism and the atoms are the elements covering $0$; the restriction $s$ is an involution, and the induced map $\sigma_s$ agrees with $\sigma$ on the atoms, hence on all joins of atoms, which is all of $B$.

Theorem (general representation, forward reference). Every Boolean algebra with an involution $(B, \sigma)$ is isomorphic to the Boolean algebra of clopen subsets of a compact Hausdorff space $S$ carrying a homeomorphism $s$ with $s^2 = \mathrm{id}$, in such a way that $\sigma$ corresponds to the map $U \mapsto s^{-1}(U)$ on clopen sets, and the fixed part $B^{\sigma}$ corresponds to the clopen subsets fixed by that map. The space is the Stone space of $B$ and the theorem is the Stone representation, which is Stone Duality and Boolean Algebras, in Part V; this article states it and does not prove it.

The finite theorem is the case in which the space is discrete and the algebra is a power set, and it is the case used in the examples of this article.

Summary

A Boolean algebra carries the complement, an order-reversing involution that exchanges the meet and the join and is an anti-automorphism, and it may carry an involution $\sigma$, an order-preserving automorphism of order two; the two commute and are different. An involutive Boolean algebra is a pair $(B, \sigma)$; its morphisms are the Boolean homomorphisms commuting with the involutions.

The fixed part $B^{\sigma}$ is a Boolean subalgebra; over a finite algebra its atoms are the orbits of $\sigma$ on the atoms of $B$, so its cardinality is $2^{f + (n-f)/2}$ for $n$ atoms with $f$ of them fixed. Every finite Boolean algebra with an involution is a power set with the involution induced by an involution of the underlying set, and the general case is the Stone representation, deferred to Part V; the fixed part is the power set of the orbits.

Summary of Notation

Symbol	Meaning
$\neg$	The complement, an order-reversing involution and an anti-automorphism
$(B, \sigma)$	A Boolean algebra with an involution $\sigma$
$B^{\sigma}$	The fixed part, $\{x : \sigma x = x\}$
$\sigma_s$	The involution of $\mathcal{P}(X)$ induced by an involution $s$ of $X$
orbit	A set $\{a, \sigma a\}$ of atoms; the atoms of the fixed part
$\mathrm{At}(B)$	The set of atoms of $B$