Lattice (order)

The name "lattice" is suggested by the form of the Hasse diagram depicting it.

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.

Lattices as posets

Consider a poset (L, ≤). L is a lattice if

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

The join and meet of x and y are denoted by Failed to parse (Missing texvc executable; please see math/README to configure.): x \vee y

and Failed to parse (Missing texvc executable; please see math/README to configure.): x \wedge y

, respectively. Because joins and meets are assumed to exist in a lattice, Failed to parse (Missing texvc executable; please see math/README to configure.): \vee

and Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge
are binary operations. Hence this definition is equivalent to requiring L to be both a join- and a meet-semilattice.

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, and every finite lattice is bounded, by taking the join (resp., meet) of all elements.

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.

Lattices as algebraic structures

Let L be a set with two binary operations, Failed to parse (Missing texvc executable; please see math/README to configure.): \vee

and Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge

. A lattice is an algebraic structure Failed to parse (Missing texvc executable; please see math/README to configure.): \langle L,\vee,\wedge\rangle

of type Failed to parse (Missing texvc executable; please see math/README to configure.): \langle2,2\rangle

, such that the following axiomatic identities hold for all members a, b, and c of L:

The following important identity follows from the above:

These axioms assert that (L,Failed to parse (Missing texvc executable; please see math/README to configure.): \vee ) and (L,Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge ) 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.

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.

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.

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

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 Failed to parse (Missing texvc executable; please see math/README to configure.): \vee

and Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge

. It is easy to see that these operations make (L, Failed to parse (Missing texvc executable; please see math/README to configure.): \vee , Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge ) into a lattice in the algebraic sense. The converse is true also: Consider an algebraically defined lattice (M, Failed to parse (Missing texvc executable; please see math/README to configure.): \vee , Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge ). Now define a partial order ≤ on M by setting

x ≤ y if and only if x = xFailed to parse (Missing texvc executable; please see math/README to configure.): \wedge

y

or, equivalently,

x ≤ y if and only if y = xFailed to parse (Missing texvc executable; please see math/README to configure.): \vee

y

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 Failed to parse (Missing texvc executable; please see math/README to configure.): \vee

and Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge

. Conversely, the order induced by the algebraically defined lattice (L, Failed to parse (Missing texvc executable; please see math/README to configure.): \vee , Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge ) 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) ↔ (a ≤ c) & (b ≤ d). (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: a ≤ b 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.

Morphisms of lattices

The appropriate notion of a morphism between two lattices flows easily from the above algebraic definition. Given two lattices (L, Failed to parse (Missing texvc executable; please see math/README to configure.): \vee , Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge ) and (M, Failed to parse (Missing texvc executable; please see math/README to configure.): \cup , Failed to parse (Missing texvc executable; please see math/README to configure.): \cap ), a homomorphism of lattices or lattice homomorphism is a function f : L → M such that

f(xFailed to parse (Missing texvc executable; please see math/README to configure.): \vee

y) = f(x) Failed to parse (Missing texvc executable; please see math/README to configure.): \cup

f(y), and

f(xFailed to parse (Missing texvc executable; please see math/README to configure.): \wedge

y) = f(x) Failed to parse (Missing texvc executable; please see math/README to configure.): \cap

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:

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 bijection is a homomorphism if its inverse is also order-preserving.

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.

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.

Distributivity

Main article: Distributive lattice

Since any lattice comes with two binary operations, it is natural to consider whether one distributes over the other. A lattice (L, Failed to parse (Missing texvc executable; please see math/README to configure.): \vee , Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge ) is distributive, if the following condition is satisfied for every three elements x, y and z of L:

Failed to parse (Missing texvc executable; please see math/README to configure.): x \vee (y \wedge z) = (x \vee y) \wedge (x \vee z).

This condition is equivalent to the dual statement:

Failed to parse (Missing texvc executable; please see math/README to configure.): x \wedge (y \vee z) = (x \wedge y) \vee (x \wedge z).

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.

Modularity

Main article: Modular lattice

Distributivity is too strong a condition for certain applications. A strictly weaker property is modularity: a lattice (L, Failed to parse (Missing texvc executable; please see math/README to configure.): \vee , Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge ) is modular if, for all elements x, y, and z of L, we have

Failed to parse (Missing texvc executable; please see math/README to configure.): x \vee (y \wedge (x \vee z)) = (x \vee y) \wedge (x \vee z)

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

Failed to parse (Missing texvc executable; please see math/README to configure.): x \vee (y \wedge z) = (x \vee y) \wedge z

Equivalently,^[1] it is possible to define an upper semimodular, or simply semimodular, lattice L as one in which the following equivalent conditions holds:

L is graded, and the rank function r of L satisfies, for each x and y in L, Failed to parse (Missing texvc executable; please see math/README to configure.): r(x)+r(y) \ge r(x \wedge y) + r(x \vee y)

or

for each x and y in L, if x and y both cover Failed to parse (Missing texvc executable; please see math/README to configure.): x \wedge y

, then Failed to parse (Missing texvc executable; please see math/README to configure.): x \vee y

covers both x and y.

A finite lattice is said to be lower semimodular if its dual is upper semimodular (that is, the previous conditions hold with Failed to parse (Missing texvc executable; please see math/README to configure.): \vee

and Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge
exchanged, "covers" exchanged with "is covered by", and inequalities reversed). With these definitions, a finite lattice is modular if and only if it is both upper and lower semimodular.

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.

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:

• 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.

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:

Failed to parse (Missing texvc executable; please see math/README to configure.): x \vee y = 1

and Failed to parse (Missing texvc executable; please see math/README to configure.): x \wedge y = 0.

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 unique.

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 xFailed to parse (Missing texvc executable; please see math/README to configure.): \wedge 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 MFailed to parse (Missing texvc executable; please see math/README to configure.): \not=\varnothing

is a subset of L such that for every pair of elements a, b in M both aFailed to parse (Missing texvc executable; please see math/README to configure.): \wedge

b and aFailed to parse (Missing texvc executable; please see math/README to configure.): \vee b are in M, then M is a sublattice of L.^[2]

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.

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. We say that L is atomic (or a point lattice), if every nonzero element of L is a join of atoms. 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 Failed to parse (Missing texvc executable; please see math/README to configure.): a\nleq b , there exists an atom x of L such that Failed to parse (Missing texvc executable; please see math/README to configure.): x\leq a

and Failed to parse (Missing texvc executable; please see math/README to configure.): x\nleq b

.

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.

See also

• Map of lattices
• Ontology (computer science)
• Orthocomplemented lattice

References

Monographs available free online:

• Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. A Course in Universal Algebra. Springer-Verlag. ISBN 3-540-90578-2.

• Jipsen, Peter, and Henry Rose, Varieties of Lattices, Lecture Notes in Mathematics 1533, Springer Verlag, 1992. ISBN 0-387-56314-8.

Elementary texts recommended for those with limited mathematical maturity:

• Donnellan, Thomas, 1968. Lattice Theory. Pergamon.
• Grätzer, G., 1971. Lattice Theory: First concepts and distributive lattices. W. H. Freeman.

The standard contemporary introductory text, somewhat harder than the above:

• Davey, B.A., and H. A. Priestley, 2002. Introduction to Lattices and Order. Cambridge University Press.

Advanced monographs:

• Garrett Birkhoff, 1967. Lattice Theory, 3rd ed. Vol. 25 of AMS Colloquium Publications. American Mathematical Society.
• Robert P. Dilworth and Crawley, Peter, 1973. Algebraic Theory of Lattices. Prentice-Hall. ISBN 9780130222695.

On free lattices:

• R. Freese, J. Jezek, and J. B. Nation, 1985. "Free Lattices". Mathematical Surveys and Monographs Vol. 42. Mathematical Association of America.
• Johnstone, P.T., 1982. Stone spaces. Cambridge Studies in Advanced Mathematics 3. Cambridge University Press.

Notes

1. ^ Stanley, Richard P. Enumerative Combinatorics (vol. 1). Cambridge University Press, 103-104. ISBN 0521663512. 
2. ^ Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. A Course in Universal Algebra. Springer-Verlag. ISBN 3-540-90578-2.

External links

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

• J.B. Nation, Notes on Lattice Theory, unpublished course notes available as two PDF files.

• Ralph Freese, "Lattice Theory Homepage".cs:Svaz (matematika)

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