Complemented lattice
Encyclopedia
In the mathematical
discipline of order theory
, a complemented lattice is a bounded lattice
in which every element a has a complement, i.e. an element b satisfying a ∨ b = 1 and a ∧ b = 0.
A relatively complemented lattice is a lattice such that every interval
[c, d] is complemented. Complements need not be unique.
An orthocomplementation on a complemented lattice is an involution which is order-reversing and maps each element to a complement. An orthocomplemented lattice satisfying a weak form of the modular law
is called an orthomodular lattice.
In distributive lattices, complements are unique. Every complemented distributive lattice has a unique orthocomplementation and is in fact a Boolean algebra.
1), in which every element a has a complement, i.e. an element b such that
In general an element may have more than one complement. However, in a bounded distributive lattice
every element will have at most one complement. A lattice in which every element has exactly one complement is called a uniquely complemented lattice.
A lattice with the property that every interval is complemented is called a relatively complemented lattice. In other words, a relatively complemented lattice is characterized by the property that for every element a in an interval [c, d] there is an element b such that
Such an element b is called a complement of a relative to the interval. A distributive lattice is complemented if and only if it is bounded and relatively complemented.
Complement law: a⊥ ∨ a = 1 and a⊥ ∧ a = 0.
Involution law: a⊥⊥ = a.
Order-reversing: if a ≤ b then b⊥ ≤ a⊥.
An orthocomplemented lattice or ortholattice is a bounded lattice which is equipped with an orthocomplementation. The lattices of subspaces of inner product space
s, and the orthogonal complement operation in these lattices, provide examples of orthocomplemented lattices that are not, in general, distributive.
Boolean algebras are a special case of orthocomplemented lattices, which in turn are a special case of complemented lattices (with extra structure). These structures are most often used in quantum logic
, where the closed
subspaces
of a separable Hilbert space
represent quantum propositions and behave as an orthocomplemented lattice.
Orthocomplemented lattices, like Boolean algebras, satisfy de Morgan's laws:
if for all elements a, b and c the implication
holds. This is weaker than distributivity. A natural further weakening of this condition for orthocomplemented lattices, necessary for applications in quantum logic, is to require it only in the special case b = a⊥. An orthomodular lattice is therefore defined as an orthocomplemented lattice such that for any two elements the implication
holds.
Lattices of this form are of crucial importance for the study of quantum logic
, since they are part of the axiomisation of the Hilbert space
formulation
of quantum mechanics
. Garrett Birkhoff
and John von Neumann
observed that the propositional calculus in quantum logic is "formally indistinguishable from the calculus of linear subspaces [of a Hilbert space] with respect to set products, linear sums and orthogonal complements" corresponding to the roles of and, or and not in Boolean lattices. This remark has spurred interest in the closed subspaces of a Hilbert space, which form an orthomodular lattice.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
discipline of order theory
Order theory
Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
, a complemented lattice is a bounded lattice
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
in which every element a has a complement, i.e. an element b satisfying a ∨ b = 1 and a ∧ b = 0.
A relatively complemented lattice is a lattice such that every interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
[c, d] is complemented. Complements need not be unique.
An orthocomplementation on a complemented lattice is an involution which is order-reversing and maps each element to a complement. An orthocomplemented lattice satisfying a weak form of the modular law
Modular lattice
In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition:Modular law: x ≤ b implies x ∨ = ∧ b,where ≤ is the partial order, and ∨ and ∧ are...
is called an orthomodular lattice.
In distributive lattices, complements are unique. Every complemented distributive lattice has a unique orthocomplementation and is in fact a Boolean algebra.
Definition and basic properties
A complemented lattice is a bounded lattice (with least element 0 and greatest elementGreatest element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually...
1), in which every element a has a complement, i.e. an element b such that
-
- a ∨ b = 1 and a ∧ b = 0.
In general an element may have more than one complement. However, in a bounded distributive lattice
Distributive lattice
In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection...
every element will have at most one complement. A lattice in which every element has exactly one complement is called a uniquely complemented lattice.
A lattice with the property that every interval is complemented is called a relatively complemented lattice. In other words, a relatively complemented lattice is characterized by the property that for every element a in an interval [c, d] there is an element b such that
-
- a ∨ b = d and a ∧ b = c.
Such an element b is called a complement of a relative to the interval. A distributive lattice is complemented if and only if it is bounded and relatively complemented.
Orthocomplementation
An orthocomplementation on a bounded lattice is a function that maps each element a to an "orthocomplement" a⊥ in such a way that the following axioms are satisfied:Complement law: a⊥ ∨ a = 1 and a⊥ ∧ a = 0.
Involution law: a⊥⊥ = a.
Order-reversing: if a ≤ b then b⊥ ≤ a⊥.
An orthocomplemented lattice or ortholattice is a bounded lattice which is equipped with an orthocomplementation. The lattices of subspaces of inner product space
Inner product space
In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...
s, and the orthogonal complement operation in these lattices, provide examples of orthocomplemented lattices that are not, in general, distributive.
Boolean algebras are a special case of orthocomplemented lattices, which in turn are a special case of complemented lattices (with extra structure). These structures are most often used in quantum logic
Quantum logic
In quantum mechanics, quantum logic is a set of rules for reasoning about propositions which takes the principles of quantum theory into account...
, where the closed
Closed
Closed may refer to:Math* Closure * Closed manifold* Closed orbits* Closed set* Closed differential form* Closed map, a function that is closed.Other* Cloister, a closed walkway* Closed-circuit television...
subspaces
Linear subspace
The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
of a separable Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
represent quantum propositions and behave as an orthocomplemented lattice.
Orthocomplemented lattices, like Boolean algebras, satisfy de Morgan's laws:
- (a ∨ b)⊥ = a⊥ ∧ b⊥
- (a ∧ b)⊥ = a⊥ ∨ b⊥.
Orthomodular lattices
A lattice is called modularModular lattice
In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition:Modular law: x ≤ b implies x ∨ = ∧ b,where ≤ is the partial order, and ∨ and ∧ are...
if for all elements a, b and c the implication
-
- if a ≤ c, then a ∨ (b ∧ c) = (a ∨ b) ∧ c
holds. This is weaker than distributivity. A natural further weakening of this condition for orthocomplemented lattices, necessary for applications in quantum logic, is to require it only in the special case b = a⊥. An orthomodular lattice is therefore defined as an orthocomplemented lattice such that for any two elements the implication
-
- if a ≤ c, then a ∨ (a⊥ ∧ c) = c
holds.
Lattices of this form are of crucial importance for the study of quantum logic
Quantum logic
In quantum mechanics, quantum logic is a set of rules for reasoning about propositions which takes the principles of quantum theory into account...
, since they are part of the axiomisation of the Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
formulation
Mathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as...
of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
. Garrett Birkhoff
Garrett Birkhoff
Garrett Birkhoff was an American mathematician. He is best known for his work in lattice theory.The mathematician George Birkhoff was his father....
and John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
observed that the propositional calculus in quantum logic is "formally indistinguishable from the calculus of linear subspaces [of a Hilbert space] with respect to set products, linear sums and orthogonal complements" corresponding to the roles of and, or and not in Boolean lattices. This remark has spurred interest in the closed subspaces of a Hilbert space, which form an orthomodular lattice.