Orthomodular Lattices

Introduction

An orthomodular lattice is an ortholattice in which the orthocomplement and the order are linked by one further law, the orthomodular law: whenever $x \leq y$, the element $y$ is recovered from $x$ and $x^{\perp}$ by

$$ y = x \vee (x^{\perp} \wedge y) . $$

The law is a one-sided form of the modular law, and it is the exact amount of modularity that the orthocomplement can supply. This article develops it: it defines the law, proves its equivalence with two other forms, shows that it is self-dual, and proves that a modular ortholattice is orthomodular, so the orthomodular law is the weak form of modularity that remains available once a lattice is read with an involution. It then turns to the failure of distributivity: a distributive orthomodular lattice is a Boolean algebra, and the smallest non-distributive example is the six-element lattice $M_4$, whose four atoms are paired by the orthocomplement. The non-modular orthomodular lattices that a distance supplies belong to Part II and are not used here.

The article presupposes Order Theory and Lattices — posets, lattices, bounded, complete, distributive and modular lattices, and the Boolean lattice — and Orthocomplemented Lattices and the Involution, which precedes it in this group and supplies the ortholattice, the De Morgan laws and the failure of the complement. The Boolean case and the involutive Boolean algebra are Boolean Algebras with an Involution, later in this group, and the general Boolean algebra is Boolean Algebras and Lattices, in Part V; both are named only.

Three boundaries are observed. No topology is used and no form is formed: the orthocomplement is the name of an involution on the lattice, and nothing is measured with it. No measure and no probability is used; the logical reading of the lattice is Logic and Proof in Part 0 and Effect Algebras and Orthomodular Lattices in Part V, and neither is used here. The adjoint of an operator, and the involution on the operator layer, belong to the * Operator Theory group of this category and are named only.

The Orthomodular Law

Definition and Equivalent Forms

Let $L$ be an ortholattice.

Definition. $L$ is orthomodular if

$$ x \leq y \quad\Longrightarrow\quad y = x \vee (x^{\perp} \wedge y) \qquad \text{for all } x, y \in L . $$

An orthomodular lattice is an ortholattice satisfying this law.

Proposition. The following are equivalent for an ortholattice:

$x \leq y \Rightarrow y = x \vee (x^{\perp} \wedge y)$ for all $x, y$;
$x \vee (x^{\perp} \wedge (x \vee y)) = x \vee y$ for all $x, y$.

Proof. (1) $\Rightarrow$ (2): apply (1) to the pair $x \leq x \vee y$, for which the right side reads $x \vee (x^{\perp} \wedge (x \vee y))$ and the left side $x \vee y$. (2) $\Rightarrow$ (1): if $x \leq y$ then $x \vee y = y$, and (2) reads $y = x \vee (x^{\perp} \wedge y)$.

Remark. The law is equivalently the statement that the inequality $x \vee (x^{\perp} \wedge y) \leq y$, which holds for all $x \leq y$ because both $x$ and $x^{\perp} \wedge y$ are below $y$, is always an equality. In the modular law the same identity holds with an arbitrary $z$ in place of $x^{\perp}$; the orthomodular law is the one instance of it that the complement permits, and the next subsection makes the implication precise.

Theorem (self-duality). The dual of an orthomodular lattice, in which the order and the two operations are reversed and the orthocomplement is kept, is orthomodular with the same orthocomplement. The dual form of the law is

$$ x \geq y \quad\Longrightarrow\quad y = x \wedge (x^{\perp} \vee y) \qquad \text{for all } x, y \in L . $$

Proof. Taking the order-dual of the identity $y = x \vee (x^{\perp} \wedge y)$ and exchanging the names of $x$ and $y$ gives the displayed law; the orthocomplement is unchanged because it is an order-reversing involution, which is the same condition in the opposite order. So the class of orthomodular lattices is closed under duality.

The law says exactly that the inequality that always holds is an equality: for $x \leq y$ one has $x \vee (x^{\perp} \wedge y) \leq y$ by the two joins, and orthomodularity is the statement that the opposite inequality also holds. In the modular law the same identity holds with an arbitrary $z$ in place of $x^{\perp}$, and the orthomodular law is the one instance of it that the complement permits.

Modularity Implies Orthomodularity

Theorem. Every modular ortholattice is orthomodular.

Proof. Let $L$ be a modular ortholattice and let $x \leq y$. Applying the modular law to the instances $x$, $y$ and $z = x^{\perp}$, which one may do because $x \wedge z = 0$ and $x \vee z = 1$,

$$ y = y \wedge (x \vee x^{\perp}) = x \vee (y \wedge x^{\perp}), $$

so the orthomodular law holds.

The converse fails: orthomodularity is strictly weaker than modularity, and the example that witnesses the strictness needs a distance and belongs to Part II, where it is met. So orthomodularity is the weak form of modularity, and it is the form that the orthocomplement forces.

The Failure of Distributivity

Distributive Orthomodular Lattices are Boolean

Theorem. An orthomodular lattice is distributive if and only if it is a Boolean algebra.

Proof. A Boolean algebra is distributive by definition, so one direction is immediate. Conversely, if $L$ is distributive then it is a complemented distributive lattice, its orthocomplement being a complement of every element, and every complemented distributive lattice is a Boolean algebra by Order Theory and Lattices.

So a non-Boolean orthomodular lattice fails distributivity. The smallest such lattice is found among the small ortholattices.

The Smallest Non-Distributive Example

Example ($M_4$). Let $M_4$ be the lattice with bottom $0$, top $1$ and four pairwise incomparable atoms $a, b, c, d$. Define $a^{\perp} = b$, $b^{\perp} = a$, $c^{\perp} = d$, $d^{\perp} = c$, and $0^{\perp} = 1$, $1^{\perp} = 0$. This is an orthocomplement: it is an order-reversing involution, and every atom is disjoint from its image with join $1$. The lattice is orthomodular, because the only pairs $x \leq y$ are those with $x = 0$, with $x = y$, or with $y = 1$, and in each case the law reduces to an identity. It is modular and it is not distributive: $a \wedge (b \vee c) = a$ while $(a \wedge b) \vee (a \wedge c) = 0$. So $M_4$ is a modular, non-distributive orthomodular lattice with six elements. It is the smallest non-Boolean orthomodular lattice: a nondegenerate ortholattice has even cardinality, so the two-element and four-element cases are $B_1$ and $B_2$ and are Boolean, and $M_4$ is a non-Boolean orthomodular lattice on six elements, which an exhaustive check of the six- element lattices confirms.

Example (the Boolean lattices). Every Boolean lattice is an orthomodular lattice, its orthocomplement being the Boolean complement; the smallest is $B_2$, the four-element lattice. The Boolean orthomodular lattices are exactly the distributive ones by the theorem above, so the examples of the non-distributive behaviour provable without a topology are the $M_n$ with $n$ even and $n \geq 4$, of which $M_4$ is the smallest, and their subortholattices.

Summary

An orthomodular lattice is an ortholattice in which $x \leq y$ implies $y = x \vee (x^{\perp} \wedge y)$. The law is equivalent to $x \vee (x^{\perp} \wedge (x \vee y)) = x \vee y$, and it says that the inequality $x \vee (x^{\perp} \wedge y) \leq y$, which always holds for $x \leq y$, is an equality; it is self-dual, so the class is closed under duality; and every modular ortholattice is orthomodular, with the orthomodular law the one instance of the modular law that the complement forces.

An orthomodular lattice is distributive if and only if it is a Boolean algebra. The smallest non-distributive orthomodular lattice is $M_4$, with six elements, whose four atoms are paired by the orthocomplement; it is modular and not distributive. The non-modular examples need a distance and belong to Part II, and are not defined or used here.

Summary of Notation

Symbol	Meaning
$L$	An ortholattice, with orthocomplement ${}^{\perp}$
orthomodular law	$x \leq y \Rightarrow y = x \vee (x^{\perp} \wedge y)$
$M_n$	The lattice with bottom, top and $n$ pairwise incomparable atoms
$M_4$	The six-element lattice with four atoms, the smallest non-distributive orthomodular lattice
$B_2$	The four-element Boolean lattice, the smallest orthomodular lattice