A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the power set of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection. Another example is given by the natural numbers, partially ordered by divisibility, for which the supremum is the least common multiple and the infimum is the greatest common divisor.
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. The class of lattices can be generalized to semilattices, and some notable subclasses of lattices are Heyting algebras, Boolean algebras, distributive lattices, and geometric lattices (matroids). These lattice-like structures all admit order-theoretic as well as algebraic descriptions.
The sub-field that studies lattices is called lattice theory.
Definition A lattice can be defined either order-theoretically as a partially ordered set, or as an algebraic structure.
As partially ordered set A partially ordered set (poset) is called a lattice if it is both a join- and a meet-semilattice, i.e. each two-element subset has a join (i.e. least upper bound, denoted by ) and dually a meet (i.e. greatest lower bound, denoted by ). This definition makes and binary operations. Both operations are monotone with respect to the given order: and implies that and
It follows by an induction argument that every non-empty finite subset of a lattice has a least upper bound and a greatest lower bound. 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 partially ordered sets—an approach of special interest for the category theoretic approach to lattices, and for formal concept analysis.
Given a subset of a lattice, meet and join restrict to partial functions – they are undefined if their value is not in the subset The resulting structure on is called a . In addition to this extrinsic definition as a subset of some other algebraic structure (a lattice), a partial lattice can also be intrinsically defined as a set with two partial binary operations satisfying certain axioms.
As algebraic structure A lattice is an algebraic structure , consisting of a set and two binary, commutative and associative operations and on satisfying the following axiomatic identities (sometimes called ) for all elements :
The following two identities are also usually regarded as axioms, even though they follow from the two absorption laws taken together.. "since and dually". Birkhoff attributes this to These are called .
These axioms assert that both and are semilattices. The absorption laws, the only axioms above in which both meet and join appear, distinguish a lattice from an arbitrary pair of semilattice structures and assure that the two semilattices interact appropriately. In particular, each semilattice is the dual of the other. The absorption laws can be viewed as a requirement that the meet and join semilattices define the same partial order.
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 into a lattice in the algebraic sense.
The converse is also true. Given an algebraically defined lattice one can define a partial order on by setting
for all elements The laws of absorption ensure that both definitions are equivalent:
and dually for the other direction.
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.
Bounded lattice
A bounded lattice is a lattice that additionally has a (also called , or element, and denoted by or and a (also called , or , denoted by or which satisfy
A bounded lattice may also be defined as an algebraic structure of the form such that is a lattice, (the lattice's bottom) is the identity element for the join operation and (the lattice's top) is the identity element for the meet operation
A partially ordered set is a bounded lattice if and only if every finite set of elements (including the empty set) has a join and a meet. Every finite lattice is bounded.
Every lattice can be embedded into a bounded lattice by adding a greatest and a least element. Furthermore, every non-empty finite lattice is bounded, by taking the join (respectively, meet) of all elements, denoted by (respectively ) where is the set of all elements.
Connection to other algebraic structures Lattices have some connections to the family of group-like algebraic structures. 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 monoids. The absorption law is the only defining identity that is peculiar to lattice theory. A bounded lattice can also be thought of as a commutative without the distributive axiom.
By commutativity, associativity and idempotence one can think of join and meet as operations on non-empty finite sets, rather than on pairs of elements. In a bounded lattice the join and meet of the empty set can also be defined (as and 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.
Examples
For any set the collection of all subsets of (called the power set of ) can be ordered via subset inclusion to obtain a lattice bounded by itself and the empty set. In this lattice, the supremum is provided by set union and the infimum is provided by set intersection (see Pic. 1). For any set the collection of all finite subsets of ordered by inclusion, is also a lattice, and will be bounded if and only if is finite. For any set the collection of all partitions of ordered by refinement, is a lattice (see Pic. 3). The positive integers in their usual order form an unbounded lattice, under the operations of "min" and "max". 1 is bottom; there is no top (see Pic. 4). The Cartesian square of the natural numbers, ordered so that if The pair is the bottom element; there is no top (see Pic. 5). The natural numbers also form a lattice under the operations of taking the greatest common divisor and least common multiple, with divisibility as the order relation: if divides is bottom; is top. Pic. 2 shows a finite sublattice. Every 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 that distinguishes arithmetic lattices from algebraic lattices, for which the compacts only form a join-semilattice. Both of these classes of complete lattices are studied in domain theory.
Further examples of lattices are given for each of the additional properties discussed below.
Examples of non-lattices
Most partially ordered sets are not lattices, including the following. A discrete poset, meaning a poset such that implies is a lattice if and only if it has at most one element. In particular the two-element discrete poset is not a lattice. Although the set partially ordered by divisibility is a lattice, the set so ordered is not a lattice because the pair 2, 3 lacks a join; similarly, 2, 3 lacks a meet in The set partially ordered by divisibility is not a lattice. Every pair of elements has an upper bound and a lower bound, but the pair 2, 3 has three upper bounds, namely 12, 18, and 36, none of which is the least of those three under divisibility (12 and 18 do not divide each other). Likewise the pair 12, 18 has three lower bounds, namely 1, 2, and 3, none of which is the greatest of those three under divisibility (2 and 3 do not divide each other).
Morphisms of lattices
The appropriate notion of a morphism between two lattices flows easily from the above algebraic definition. Given two lattices and a lattice homomorphism from L to M is a function such that for all
Thus is a homomorphism of the two underlying semilattices. When lattices with more structure are considered, the morphisms should "respect" the extra structure, too. In particular, a bounded-lattice homomorphism (usually called just "lattice homomorphism") between two bounded lattices and should also have the following property:
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 Limit preserving function. The converse is not true: monotonicity by no means implies the required preservation of meets and joins (see Pic. 9), 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 is just a bijective lattice homomorphism. Similarly, a is a lattice homomorphism from a lattice to itself, and a is a bijective lattice endomorphism. Lattices and their homomorphisms form a category.
Let and be two lattices with 0 and 1. A homomorphism from to is called 0,1-separating if and only if ( separates 0) and ( separates 1).
Sublattices A of a lattice is a subset of that is a lattice with the same meet and join operations as That is, if is a lattice and is a subset of such that for every pair of elements both and are in then is a sublattice of Burris, Stanley N., and Sankappanavar, H. P., 1981. A Course in Universal Algebra. Springer-Verlag. .
A sublattice of a lattice is a of if and implies that belongs to for all elements
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 if 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 homomorphism 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.
"Partial lattice" is not the opposite of "complete lattice" – rather, "partial lattice", "lattice", and "complete lattice" are increasingly restrictive definitions.
Conditional completeness