 FACTOID # 14: North Carolina has a larger Native American population than North Dakota, South Dakota and Montana combined.

 Home Encyclopedia Statistics States A-Z Flags Maps FAQ About

 WHAT'S NEW

SEARCH ALL

FACTS & STATISTICS    Advanced view

Search encyclopedia, statistics and forums:

(* = Graphable)

Encyclopedia > Lattice (order)

In mathematics, a lattice is a partially ordered set (or poset) in which every pair of elements has a unique supremum (the elements' least upper bound; called their join) and an infimum (greatest lower bound; called their meet). Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These "lattice-like" structures all admit order-theoretic as well as algebraic descriptions. Image File history File links Lattice_of_partitions_of_an_order_4_set. ... Image File history File links Lattice_of_partitions_of_an_order_4_set. ... In the mathematical discipline known as order theory, a Hasse diagram (pronounced HAHS uh, named after Helmut Hasse (1898â€“1979)) is a simple picture of a finite partially ordered set, forming a drawing of the transitive reduction of the partial order. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, especially order theory, a partially ordered set (or poset) is a set equipped with a partial order relation. ... In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ... In mathematics, a join on a set is defined either as unique suprema (least upper bounds) with respect to a partial order on the set, provided such suprema exist, or (abstractly) as a commutative and associative binary operation satisfying an idempotency law. ... 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 other elements of the subset. ... In mathematics, a meet on a set is defined either as the unique infimum (greatest lower bound) with respect to a partial order on the set, provided an infimum exists, or (abstractly) as a commutative and associative binary operation satisfying an idempotency law. ... In abstract algebra, an algebraic structure consists of a set together with a collection of operations or relations defined on it which satisfy certain axioms. ... An axiom is a sentence or proposition that is not proved or demonstrated and is considered as obvious or as an initial necessary consensus for a theory building or acceptation. ... In mathematics, the term identity has several important uses: identity can refer to 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. ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... Universal algebra (sometimes called General algebra) is the field of mathematics that studies the ideas common to all algebraic structures. ... A semilattice is a mathematical concept with two definitions, one as a type of ordered set, the other as an algebraic structure. ... In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras. ... In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ...

## Lattices as posets GA_googleFillSlot("encyclopedia_square");

Consider a poset (L, ≤). L is a lattice if In mathematics, especially order theory, a partially ordered set (or poset) is a set equipped with a partial order relation. ...

For all elements x and y of L, the set {x, y} has both a least upper bound in L (join, or supremum) and a greatest lower bound in L (meet, or infimum).

The join and meet of x and y are denoted by and , respectively. Because joins and meets are assumed to exist in a lattice, and are binary operations. Hence this definition is equivalent to requiring L to be both a join- and a meet-semilattice. In mathematics, a join on a set is defined either as unique suprema (least upper bounds) with respect to a partial order on the set, provided such suprema exist, or (abstractly) as a commutative and associative binary operation satisfying an idempotency law. ... In mathematics, a meet on a set is defined either as the unique infimum (greatest lower bound) with respect to a partial order on the set, provided an infimum exists, or (abstractly) as a commutative and associative binary operation satisfying an idempotency law. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... In mathematical order theory, a semilattice is a partially ordered set (poset) within which all binary sets have a supremum (join) or infimum (meet), respectively. ... In mathematical order theory, a semilattice is a partially ordered set (poset) within which all binary sets have a supremum (join) or infimum (meet), respectively. ...

A bounded lattice has a greatest 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. 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. ... 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. ...

Using an easy induction argument, one can deduce the existence of suprema (joins) and infima (meets) of all non-empty finite subsets of any lattice. With additional assumptions, further conclusions may be possible; see Completeness (order theory) 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 connections between related posets — an approach of special interest for the category theoretic approach to lattices. Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ... In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set. ... In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets (posets). Galois connections generalize the correspondence between subgroups and subfields investigated in Galois theory. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...

## Lattices as algebraic structures

Let L be a set with two binary operations, and . A lattice is an algebraic structure of type , such that the following axiomatic identities hold for all members a, b, and c of L: In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In logic and mathematics, an operation Ï‰ is a function of the form Ï‰ : X1 Ã— â€¦ Ã— Xk â†’ Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k is called the arity of the operation. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ...

 Commutative laws: Associative laws: Absorption laws:

The following important identity follows from the above: Mathematical meaning A map or binary operation is said to be commutative when, for any x in A and any y in B . ... In mathematics, associativity is a property that a binary operation can have. ... In algebra, the absorption law is an identity linking a pair of binary operations. ...

 Idempotent laws:

These axioms assert that (L,) and (L,) are each semilattices. The absorption laws, the only axioms above in which both meet and join appear, distinguish a lattice from a pair of semilattices and assure that the two semilattices interact appropriately. In particular, each semilattice is the dual of the other. A bounded lattice requires that meet and join each have a neutral element, called 1 and 0 by convention. See the entry semilattice. In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... A semilattice is a mathematical concept with two definitions, one as a type of ordered set, the other as an algebraic structure. ... In mathematics, duality has numerous meanings. ... A semilattice is a mathematical concept with two definitions, one as a type of ordered set, the other as an algebraic structure. ...

Lattices have some connections to the groupoid family. Because meet and join both commute and associate, a lattice can be viewed as consisting of two commutative semigroups having the same carrier. If a lattice is bounded, these semigroups are also commutative monoids. The absorption law is the only defining identity that is peculiar to lattice theory. In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure. ... A closed system S(#) with an associative operation (#) is called a semigroup , where: . . . for all a, b, c in S holds (a#b)#c = a#(b#c) Normal string concatenation is associative, and it is the standard for notation, hence : (ab)c = a(bc) = abc (unique, independent of bracketing, also... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In algebra, the absorption law is an identity linking a pair of binary operations. ...

The closure of L under both meet and join implies, by induction, the existence of the meet and join of any finite subset of L, with one exception: the meet and join of the empty set are the greatest and least elements, respectively. Therefore a lattice contains all finite (including empty) meets and joins only if it is bounded. For this reason, some authors define a lattice so as to require that 0 and 1 be members of L. While defining a lattice in this manner entails no loss of generality, because any lattice can be embedded in a bounded lattice, this definition will not be adopted here. In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ... The empty set is the set containing no elements. ...

The algebraic interpretation of lattices plays an essential role in universal algebra. Universal algebra (sometimes called General algebra) is the field of mathematics that studies the ideas common to all algebraic structures. ...

## Connection between the two definitions

The algebraic definition of a lattice implies the order theoretic one, and vice versa.

Obviously, an order-theoretic lattice gives rise to two binary operations and . It is easy to see that these operations make (L, , ) into a lattice in the algebraic sense. The converse is true also: Consider an algebraically defined lattice (M, , ). Now define a partial order ≤ on M by setting

xy if and only if x = xy

or, equivalently, This article does not cite any references or sources. ...

xy if and only if y = xy

for elements x and y in M. The laws of absorption ensure that both definitions are indeed 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 . Conversely, the order induced by the algebraically defined lattice (L, , ) that was derived from the order theoretic formulation above coincides with the original ordering of L.

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 of A) can be ordered via subset inclusion to obtain a lattice bounded by A itself and the null set. Set intersection and union 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 numbers (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) ↔ (ac) & (bd). (0,0) is bottom; there is no top.
• The positive integers also form a lattice under the operations of taking the greatest common divisor and least common multiple, with divisibility as the order relation: ab if a divides b. Bottom is 1; there is no top.
• Any complete lattice (also see below) is a (rather specific) bounded lattice. This class gives rise to a broad range of practical examples.
• The set of compact elements 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. Both of these classes of complete lattices are studied in domain theory.

Further examples are given for each of the additional properties discussed below. In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, the Cartesian product (or direct product) of two sets X and Y, denoted X Ã— Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. X Ã— Y = { (x, y) | x âˆˆ X and y... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ... In arithmetic and number theory, the least common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both a and b. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). ... In the mathematical 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 directed set that does not already contain members above the compact element. ... An arithmetic lattice is one derived from a division algebra such as for example quaternions. ... In the mathematical 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 directed set that does not already contain members above the compact element. ... A semilattice is a mathematical concept with two definitions, one as a type of ordered set, the other as an algebraic structure. ... Domain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. ...

## Morphisms of lattices

The appropriate notion of a morphism between two lattices flows easily from the above algebraic definition. Given two lattices (L, , ) and (M, , ), a homomorphism of lattices or lattice homomorphism is a function f : LM such that In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...

f(xy) = f(x) f(y), and
f(xy) = f(x) f(y).

Thus f is a homomorphism of the two underlying semilattices. 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: In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). ... A semilattice is a mathematical concept with two definitions, one as a type of ordered set, the other as an algebraic structure. ...

f(0L) = 0M , and
f(1L) = 1M .

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. In the mathematical area of order theory, one often speaks about functions that preserve certain limits, i. ...

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 properties. In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... In the mathematical area of order theory, one often speaks about functions that preserve certain limits, i. ...

Given the standard definition of isomorphisms 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. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...

## 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 highly relevant class of lattices are the complete lattices. A lattice is complete if all of its subsets have both a join and a meet, which should be contrasted to the above definition of a lattice where one only requires the existence of all (non-empty) finite joins and meets. Details can be found within the respective article. In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). ...

### Distributivity

Since any lattice comes with two binary operations, it is natural to consider whether one distributes over the other. A lattice (L, , ) is distributive, if the following condition is satisfied for every three elements x, y and z of L: In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...

This condition is equivalent to the dual statement: In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop. ...

Other characterizations exist, and can be found in the article on distributive lattices. For complete lattices one can formulate various stronger properties, giving rise to the classes of frames and completely distributive lattices. For an overview of these different notions, see distributivity in order theory. In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. ... In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra which is complete as a lattice. ... In the mathematical area of order theory, a completely distributive lattice is a complete lattice in which arbitrary joins distribute over arbitrary meets. ... In the mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of suprema and infima. ...

### Modularity

Distributivity is too strong a condition for certain applications. A strictly weaker property is modularity: a lattice (L, , ) is modular if, for all elements x, y, and z of L, we have See lattice for other mathematical as well as non-mathematical meanings of the term. ...

Another equivalent statement of this condition is as follows: if xz then for all y one has

For example, the lattice of submodules of a module, and the lattice of normal subgroups of a group, all have this special property. Moreover, every distributive lattice is modular. In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element gâˆ’1ng is still in N. The statement N is a normal subgroup of G is written: . There are...

### Continuity and algebraicity

In domain 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 of elements that are way-below the element. If one can additionally restrict these to the compact elements of a poset for obtaining these directed sets, then the poset is even algebraic. Both concepts can be applied to lattices as follows: Domain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. ... In mathematics, a directed set is a set A together with a binary relation &#8804; having the following properties: a &#8804; a for all a in A (reflexivity) if a &#8804; b and b &#8804; c, then a &#8804; c (transitivity) for any two a and b in A, there... Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. ... In the mathematical 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 directed set that does not already contain members above the compact element. ... In the mathematical 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. ...

• 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 systems. In the mathematical 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 directed set that does not already contain members above the compact element. ... In Domain Theory, a Scott Information System is a primitive kind of logical deductive system often used as an alternative way of presenting Scott domains. ...

### 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. The corresponding unary operation over L, called complementation, introduces an analogue of logical negation 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 provably unique. In the mathematical discipline of order theory, and in particular, in lattice theory, a complemented lattice is a bounded lattice in which each element x has a complement, defined as a unique element ~ x such that and A Boolean algebra may be defined as a complemented distributive lattice. ... In logic and mathematics, an operation Ï‰ is a function of the form Ï‰ : X1 Ã— â€¦ Ã— Xk â†’ Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k is called the arity of the operation. ... A complementation test is used in genetics to decide if two recessive mutant phenotypes are determined by mutations in the same gene or two different genes. ... Negation (i. ... In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ...

Heyting algebras 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 xy = 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. In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras. ...

## 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 ab and ab 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

Main article: Free lattice

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. In mathematics, in the area of order theory, a free lattice is the free object corresponding to a lattice. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

## 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 Vai, 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 xa or xb,
• 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. We say that L is atomic, if every nonzero element of L lies above some atom of L. We say that L is atomistic, if every element of L is a supremum of atoms, that is, for all a, b in L such that , there exists an atom x of L such that and . In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ...

Other important notions in lattice theory are ideal and its dual notion filter. Both terms describe special subsets of a lattice (or of any partially ordered set in general). Details can be found in the respective articles. In mathematical order theory, an ideal is a special subset of a partially ordered set. ... In mathematics, a filter is a special subset of a partially ordered set. ...

In both computer science and information science, an ontology is a data model that represents a domain and is used to reason about the objects in that domain and the relations between them. ... An orthocomplemented lattice is an algebraic structure consisting of a bounded lattice along with a unary function called an orthocomplementation. ... Results from FactBites:

 Lattice (order) - Wikipedia, the free encyclopedia (2421 words) The set of compact elements 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. For example, the lattice of submodules of a module, and the lattice of normal subgroups of a group, all have this special property. These conditions basically amount to saying that there is a functor from the category of sets and functions to the category of lattices and lattice homomorphisms which is left adjoint to the forgetful functor from lattices to their underlying sets.
 Lattice (913 words) Every lattice can be generated from a basis for the underlying vector space by considering all linear combinations with integral coefficients. In another mathematical usage, a lattice is a partially ordered set in which all nonempty finite subsets have a least upper bound and a greatest lower bound (also called supremum and infimum, respectively). The lattice of submodules of a module and the lattice of normal subgroups of a group have the special property that x v (y ^ (x v z)) = (x v y) ^ (x v z) for all x, y and z in the lattice.
More results at FactBites »

Share your thoughts, questions and commentary here
Press Releases | Feeds | Contact