Lattice (order)

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Lattice (order)

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a lattice is a partially ordered set

Partially ordered set

In mathematics, especially order theory, a partially ordered set formalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set ....
(also called a poset) in which subsets of any two elements have a unique supremum

Supremum

In mathematics, given a subset S of a partially ordered set T, the supremum of S, if it exists, is the greatest element of T that is greater than or equal to each element of S....
(the elements' least upper bound; called their join
Join (mathematics)

In mathematical order theory, join is a binary operation on a partially ordered set that gives the supremum of its arguments, provided the least upper bound exists....
) and an infimum

Infimum

In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is less than or equal to all elements of the subset....
(greatest lower bound; called their meet
Meet (mathematics)

In mathematics, a meet on a set is defined either as the unique infimum with respect to a partial order on the set, provided an infimum exists, or as a commutative and associative binary operation satisfying an idempotency law....
).

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Encyclopedia

In mathematics

Mathematics

Partially ordered set

Supremum

Infimum

Axiom

In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evidence, or subject to necessary decision....
identities

Identity (mathematics)

In mathematics, the term identity has several different important meanings:*An identity is an equality that remains true regardless of the values of any variables that appear within it, to distinguish it from an Equality which is true under more particular conditions....
. Since the two definitions are equivalent, lattice theory draws on both order theory

Order theory

Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another....
and universal algebra

Universal algebra

Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures.For instance, rather than take particular groups as the object of study, in universal algebra one takes "the theory of groups" as an object of study....
. Semilattice

Semilattice

Heyting algebra

In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebra s, named after Arend Heyting....
and Boolean algebras. These "lattice-like" structures all admit order-theoretic as well as algebraic descriptions.

Lattices as posets

A poset

Partially ordered set

In mathematics, especially order theory, a partially ordered set formalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set ....
(L, =) is a lattice if it satisfies the following two axioms.

Existence of binary joins:

For any two elements a and b of L, the set has a join
Join (mathematics)

In mathematical order theory, join is a binary operation on a partially ordered set that gives the supremum of its arguments, provided the least upper bound exists....
(also known as least upper bound or supremum).

Existence of binary meets:

For any two elements a and b of L, the set has a meet
Meet (mathematics)

In mathematics, a meet on a set is defined either as the unique infimum with respect to a partial order on the set, provided an infimum exists, or as a commutative and associative binary operation satisfying an idempotency law....
(also known as greatest lower bound or infimum).

The join and meet of a and b are denoted by and , respectively. This definition makes and binary operation

Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator....
s. The first axiom says that L is a join-semilattice; the second says that L is a meet-semilattice. Both operations are monotone with respect to the order: a₁ = a₂ and b₁ = b₂ implies that a₁ b₁ = a₂ b₂ and a₁b₁ = a₂b₂.

It follows by an induction

Mathematical induction

Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then...
argument that every non-empty finite subset of a lattice has a join (supremum) and a meet (infimum). With additional assumptions, further conclusions may be possible; see Completeness (order theory)

Completeness (order theory)

In the mathematics area of order theory, completeness properties assert the existence of certain infimum or supremum of a given partially ordered set ....
for more discussion of this subject. That article also discusses how one may rephrase the above definition in terms of the existence of suitable Galois connection

Galois connection

In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets . Galois connections generalize the correspondence between subgroups and field investigated in Galois theory....
s between related posets – an approach of special interest for the category theoretic

Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
approach to lattices.

A bounded lattice has a greatest

Greatest 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....
and least element, denoted 1 and 0 by convention (also called top and bottom). Any lattice can be converted into a bounded lattice by adding a greatest and least element, and every finite lattice is bounded, by taking the join (resp., meet) of all elements, denoted by (resp.) where .

A poset is a bounded lattice if and only if every finite set of elements (including the empty set) has a join and a meet. Here, the join of an empty set of elements is defined to be the least element , and the meet of the empty set is defined to be the greatest element . This convention is consistent with the associativity and commutativity of meet and join: the join of a union of finite sets is equal to the join of the joins of the sets, and dually, the meet of a union of finite sets is equal to the meet of the meets of the sets, i.e., for finite subsets A and B of a poset L,

and

hold. Taking B to be the empty set,

and

which is consistent with the fact that .

Lattices as algebraic structures

An algebraic structure

Algebraic structure

In algebra, a branch of pure mathematics, an algebraic structure consists of one or more Set Closure under one or more Operation , satisfying some axiom....
(L, ), consisting of a set L and two binary operations

Operation (mathematics)

In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values....
), and ), on L is a lattice if the following axiomatic identities hold for all elements a, b, c of L.

The following identity can be derived from the axioms. Idempotent laws , .

These axioms assert that both (L,) and (L,) are semilattice

Semilattice

A semilattice is a mathematical concept with two definitions, one as a type of ordered set, the other as an algebraic structure. In order theory, a semilattice is a partially ordered set closed under one of two binary operations, either supremum or infimum ....
s. The absorption laws, the only axioms above in which both meet and join appear, distinguish a lattice from a random pair of semilattices and assure that the two semilattices interact appropriately. In particular, each semilattice is the dual

Duality (order theory)

In the mathematics area of order theory, every partially ordered set P gives rise to a dual partially ordered set which is often denoted by Pop or Pd....
of the other.

A bounded lattice is an algebraic structure of the form (L, , 1, 0) such that (L, ) is a lattice, 0 (the lattice's bottom) is the identity element

Identity element

In mathematics, an identity element is a special type of element of a Set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them....
for the join operation , and 1 (the lattice's top) is the identity element for the meet operation . See semilattice

Semilattice

A semilattice is a mathematical concept with two definitions, one as a type of ordered set, the other as an algebraic structure. In order theory, a semilattice is a partially ordered set closed under one of two binary operations, either supremum or infimum ....
for further details.

Lattices have some connections to the family of group-like algebraic structures

Magma (algebra)

In abstract algebra, a magma is a basic kind of algebraic structure. Specifically, a magma consists of a Set M equipped with a single binary operation M × M ? M....
. Because meet and join both commute and associate, a lattice can be viewed as consisting of two commutative semigroups having the same domain. For a bounded lattice, these semigroups are in fact commutative monoid

Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element....
s. The absorption law

Absorption law

In algebra, the absorption law is an identity linking a pair of binary operations.Any two binary operations, say $ and %, are subject to the absorption law if:...
is the only defining identity that is peculiar to lattice theory.

By commutativity and associativity one can think of join and meet as operations that are defined on non-empty finite sets, rather than pairs, of elements. In a bounded lattice the empty join and the empty meet can also be defined (as 0 and 1, respectively). This makes bounded lattices somewhat more natural than general lattices, and many authors require all lattices to be bounded.

The algebraic interpretation of lattices plays an essential role in universal algebra

Universal algebra

Connection between the two definitions

An order-theoretic lattice gives rise to the two binary operations and . Since the commutative, associative and absorption laws can easily be verified for these operations, they make (L, , ) into a lattice in the algebraic sense. The ordering can be recovered from the algebraic structure because a = b holds if and only if

If and only if

If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements....
a = a?b.

The converse is also true. Given an algebraically defined lattice (L, , ), one can define a partial order = on L by setting

a = b if and only if a = a'b, or

a = b if and only if b = a'b,

for all elements a and b from L. The laws of absorption ensure that both definitions are equivalent. One can now check that the relation = introduced in this way defines a partial ordering within which binary meets and joins are given through the original operations and .

Since the two definitions of a lattice are equivalent, one may freely invoke aspects of either definition in any way that suits the purpose at hand.

Examples

For any set A, the collection of all subsets of A (called the power set
Power set

In mathematics, given a Set S, the power set of S, written , P, ℘ or Power set#Representing subsets as functions, is the set of all subsets of S....
of A) can be ordered via subset inclusion
Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
to obtain a lattice bounded by A itself and the null set. Set intersection
Intersection (set theory)

In mathematics, the intersection of two Set A and B is the set that contains all elements of A that also belong to B , but no other elements....
and union
Union (set theory)

In set theory, the term Union refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets....
interpret meet and join, respectively.
For any set A, the collection of all finite subsets of A, ordered by inclusion, is also a lattice, and will be bounded if and only if A is finite.
The natural number
Natural number

In mathematics, a natural number can mean either an element of the Set = *n = = ? = ? ...
s (including 0) in their usual order form a lattice, under the operations of "min" and "max". 0 is bottom; there is no top.
The Cartesian square of the natural numbers, ordered by = so that (a,b) = (c,d) ? (a = c) & (b = d). (0,0) is bottom; there is no top.
The natural numbers also form a lattice under the operations of taking the greatest common divisor
Greatest common divisor

In mathematics, the greatest common divisor , sometimes known as the greatest common factor or highest common factor , of two non-zero integers, is the largest positive integer that divisor both numbers without remainder....
and least common multiple
Least common multiple

In arithmetic and number theory, the least common multiple or lowest common multiple or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple both of a and of b....
, with divisibility as the order relation: a ≤ b if a divides b. 1 is bottom; 0 is top.
Any complete lattice
Complete lattice

In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science....
(also see below) is a (rather specific) bounded lattice. This class gives rise to a broad range of practical examples.
The set of compact element
Compact element

In the mathematics area of order theory, the compact or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not already contain members above the compact element....
s of an arithmetic complete lattice is a lattice with a least element, where the lattice operations are given by restricting the respective operations of the arithmetic lattice. This is the specific property which distinguishes arithmetic lattices from algebraic lattices, for which the compacts do only form a join-semilattice
Semilattice

A semilattice is a mathematical concept with two definitions, one as a type of ordered set, the other as an algebraic structure. In order theory, a semilattice is a partially ordered set closed under one of two binary operations, either supremum or infimum ....
. Both of these classes of complete lattices are studied in domain theory
Domain theory

Domain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. Consequently, domain theory can be considered as a branch of order theory....
.

Further examples are given for each of the additional properties discussed below.

Morphisms of lattices

The appropriate notion of a morphism

Morphism

In mathematics, a morphism is an Abstraction derived from structure-preserving map between two mathematical structures.The study of morphisms and of the structures over which they are defined, is central to category theory....
between two lattices flows easily from the above algebraic definition. Given two lattices (L, ?_L, ?_L) and (M, ?_M, ?_M), a homomorphism of lattices or lattice homomorphism is a function f : L ? M such that

f(a?_Lb) = f(a) ?_M f(b), and

f(a?_Lb) = f(a) ?_M f(b).

Thus f is a homomorphism

Homomorphism

In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ???? meaning "same" and ???f? meaning "shape"....
of the two underlying semilattice

Semilattice

A semilattice is a mathematical concept with two definitions, one as a type of ordered set, the other as an algebraic structure. In order theory, a semilattice is a partially ordered set closed under one of two binary operations, either supremum or infimum ....
s. When lattices with more structure are considered, the morphisms should 'respect' the extra structure, too. Thus, a morphism f between two bounded lattices L and M should also have the following property:

f(0_L) = 0_M , and

f(1_L) = 1_M .

In the order-theoretic formulation, these conditions just state that a homomorphism of lattices is a function preserving binary meets and joins. For bounded lattices, preservation of least and greatest elements is just preservation of join and meet of the empty set.

Any homomorphism of lattices is necessarily monotone with respect to the associated ordering relation; see preservation of limits. The converse is of course not true: monotonicity by no means implies the required preservation of meets and joins, although an order-preserving

Monotonic function

In mathematics, a monotonic function is a function which preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....
bijection

Bijection

In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that f = y....
is a homomorphism if its inverse

Inverse function

In mathematics, if ƒ is a function from A to B then an inverse function for ƒ is a function in the opposite direction, from B to A, with the property that a round trip from A to B to A returns each element of the initial set to itself....
is also order-preserving.

Given the standard definition of isomorphism

Isomorphism

In abstract algebra, an isomorphism is a bijection map f such that both f and its inverse function f −1 are homomorphisms, i.e., structure-preserving mappings....
s as invertible morphisms, a lattice isomorphism is just a bijective lattice homomorphism. Similarly, a lattice endomorphism is a lattice homomorphism from a lattice to itself, and a lattice automorphism is a bijective lattice endomorphism. Lattices and their homomorphisms form a category

Category theory

Properties of lattices

We now introduce a number of important properties that lead to interesting special classes of lattices. One, boundedness, has already been discussed.

Completeness

A poset is called a complete lattice if all its subsets have both a join and a meet. In particular, every complete lattice is a bounded lattice. While bounded lattice homomorphisms in general preserve only finite joins and meets, complete lattice homomorphisms are required to preserve arbitrary joins and meets.

Every poset that is a complete semilattice is also a complete lattice. Related to this result is the interesting phenomenon that there are various competing notions of homomorphisms for this class of posets, depending on whether they are seen as complete lattices, complete join-semilattices, complete meet-semilattices, or as join-complete or meet-complete lattices.

Distributivity

Since lattices come with two binary operations, it is natural to ask whether one of them distributes

Distributivity

In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra....
over the other, i.e. whether one or the other of the following dual

Duality (order theory)

In the mathematics area of order theory, every partially ordered set P gives rise to a dual partially ordered set which is often denoted by Pop or Pd....
laws holds for any three elements a, b, c of L:

Distributivity of ? over ?

a?(b?c) = (a?b) ? (a?c).

Distributivity of ? over ?

a?(b?c) = (a?b) ? (a?c).

A lattice that satisfies the first or, equivalently (as it turns out), the second axiom, is called a distributive lattice.

For an overview of stronger notions of distributivity which are appropriate for complete lattices and which are used to define more special classes of lattices such as frames

Complete Heyting algebra

In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra which is completeness as a lattice . Complete Heyting algebras are the object of three different category ; the category CHey, the category Loc of locales, and its opposite , the category Frm of frames....
and completely distributive lattice

Completely distributive lattice

In the mathematical area of order theory, a completely distributive lattice is a complete lattice in which arbitrary joins distributivity over arbitrary meets....
s, see distributivity in order theory

Distributivity (order theory)

In the mathematics area of order theory, there are various notions of the common concept of distributivity, applied to the formation of supremum and infimum....
.

Modularity

For some applications the distributivity condition is too strong, and the following weaker property is often useful. A lattice (L, ?, ?) is modular if, for all elements a, b, c of L, the following identity holds. Modular identity: (a ? c) ? (b ? c) = [(a ? c) ? b] ? c. This condition is equivalent to the following axiom. Modular law: a = c implies a ? (b ? c) = (a ? b) ? c. Besides distributive lattices, examples of modular lattices are the lattice of submodules of a module

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalar to lie in a field , the "scalars" may lie in an arbitrary ring....
, and the lattice of normal subgroup

Normal subgroup

In mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup. Normal subgroups are important because they can be used to construct quotient groups from a given group ....
s of a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
.

Semimodularity

A finite lattice is modular if and only if it is both upper and lower semimodular

Semimodular lattice

In the branch of mathematics known as order theory, a semimodular lattice, is a lattice that satisfies the following condition:Semimodular law: a ? b <: a implies b <: a ? b....
. For a graded

Graded poset

In mathematics, a graded poset, sometimes called a ranked poset , is a partially ordered set P equipped with a rank function ρ from P to Z compatible with the ordering such that whenever y covering relation x, then ....
lattice, (upper) semimodularity is equivalent to the following condition on the rank function r: . Another equivalent (for graded lattices) condition is Birkhoff's condition:

for each x and y in L, if x and y both cover , then covers both x and y.

A lattice is called lower semimodular if its dual is semimodular. For finite lattices this means that the previous conditions hold with and exchanged, "covers" exchanged with "is covered by", and inequalities reversed.

Continuity and algebraicity

In domain theory

Domain theory

Domain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. Consequently, domain theory can be considered as a branch of order theory....
, it is natural to seek to approximate the elements in a partial order by "much simpler" elements. This leads to the class of continuous posets, consisting of posets where any element can be obtained as the supremum of a directed set

Directed set

In mathematics, a directed set is a nonempty Set A together with a reflexive relation and transitive relation binary relation = , with the additional property that every pair of elements has an upper bound....
of elements that are way-below the element. If one can additionally restrict these to the compact element

Compact element

In the mathematics area of order theory, the compact or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not already contain members above the compact element....
s of a poset for obtaining these directed sets, then the poset is even algebraic. Both concepts can be applied to lattices as follows:

A continuous lattice is a complete lattice that is continuous as a poset.
An algebraic lattice is a complete lattice that is algebraic as a poset.

Both of these classes have interesting properties. For example, continuous lattices can be characterized as algebraic structures (with infinitary operations) satisfying certain identities. While such a characterization is not known for algebraic lattices, they can be described "syntactically" via Scott information system

Scott information system

In domain theory, a branch of mathematics and computer science, a Scott information system is a primitive kind of logical deductive system often used as an alternative way of presenting Scott domains....
s.

Complements and pseudo-complements

Let L be a bounded lattice with greatest element 1 and least element 0. Two elements x and y of L are complements of each other if and only if:

and

In this case, we write ¬x = y and equivalently, ¬y = x. A bounded lattice for which every element has a complement is called a complemented lattice

Complemented lattice

In the mathematics discipline of order theory, and in particular, in lattice , a complemented lattice is a lattice , in which every element a has a complement, i.e....
. The corresponding unary operation

Operation (mathematics)

In its simplest meaning in mathematics and logic, an operation is an action or procedure which produces a new value from one or more input values....
over L, called complementation, introduces an analogue of logical negation

Negation

In logic and mathematics, negation or not is an operation on logical values, for example, the logical value of a proposition, that sends true to false and false to true....
into lattice theory. The complement is not necessarily unique, nor does it have a special status among all possible unary operations over L. A complemented lattice that is also distributive is a Boolean algebra. For a distributive lattice, the complement of x, when it exists, is unique.

Heyting algebra

Heyting algebra

In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebra s, named after Arend Heyting....
s are an example of distributive lattices having at least some members lacking complements. Every element x of a Heyting algebra has, on the other hand, a pseudo-complement, also denoted ¬x. The pseudo-complement is the greatest element y such that x'y = 0. If the pseudo-complement of every element of a Heyting algebra is in fact a complement, then the Heyting algebra is in fact a Boolean algebra.

Sublattices
A sublattice of a lattice L is a nonempty subset of L which is a lattice with the same meet and join operations as L. That is, if L is a lattice and M is a subset of L such that for every pair of elements a, b in M both a'b and a'b are in M, then M is a sublattice of L.

A sublattice M of a lattice L is a convex sublattice of L, if x ≤ z ≤ y and x, y in M implies that z belongs to M, for all elements x, y, z in L.

Free lattices
Any set X may be used to generate the free semilattice FX. The free semilattice is defined to consist of all of the finite subsets of X, with the semilattice operation given by ordinary set union. The free semilattice has the universal property
Universal property

In various branches of mathematics, a useful construction is often viewed as the ?most efficient solution? to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise....
.

Important lattice-theoretic notions
In the following, let L be a lattice. We define some order-theoretic notions that are of particular importance in lattice theory.

An element x of L is called join irreducible if and only if
x = a v b implies x = a or x = b for any a, b in L,
if L has a 0, x is sometimes required to be different from 0.
When the first condition is generalized to arbitrary joins Va_i, x is called completely join irreducible. The dual notion is called meet irreducibility. Sometimes one also uses the terms v-irreducible and ^-irreducible, respectively.

An element x of L is called join prime if and only if
x = a v b implies x = a or x = b,
if L has a 0, x is sometimes required to be different from 0.
Again, this can be generalized to obtain the notion completely join prime and dualized to yield meet prime. Any join-prime element is also join irreducible, and any meet-prime element is also meet irreducible. If the lattice is distributive the converse is also true.

An element x of L is an atom, if L has a 0, 0 < x, and there exists no element y of L such that 0 < y < x. L is atomic

Atomic (order theory)

In the mathematical field of order theory, given two elements a and b of a partially ordered set, one says that b covers a, and writes a a, if a < b and there is no element c such that a < c < b....
, if for every nonzero element x of L there exists an atom a of L such that . L is atomistic

Atomic (order theory)

Supremum

In mathematics, given a subset S of a partially ordered set T, the supremum of S, if it exists, is the greatest element of T that is greater than or equal to each element of S....
of atoms, that is, for all a, b in L such that , there exists an atom x of L such that and .

Other important notions in lattice theory are ideal

Ideal (order theory)

In mathematics order theory, an ideal is a special subset of a partially ordered set . Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion....
and its dual notion filter

Filter (mathematics)

In mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the ordered set under consideration is just the power set of some set, ordered by set inclusion....
. Both terms describe special subsets of a lattice (or of any partially ordered set in general). Details can be found in the respective articles.

External links

Eric W. Weisstein et al. "" From MathWorld--A Wolfram Web Resource.

J.B. Nation, , unpublished course notes available as two PDF files.

Ralph Freese, "".

Lattice (order)

Lattices as posets

Lattices as algebraic structures

Connection between the two definitions

Examples

Morphisms of lattices

Properties of lattices

Completeness

Distributivity

Modularity

Semimodularity

Continuity and algebraicity

Complements and pseudo-complements

Sublattices

Free lattices

Important lattice-theoretic notions

See also

External links