Lattice (order)

Lattices as posets

A poset (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 {a, b} has a join: $a \lor b$ (also known as the least upper bound, or the supremum).
Existence of binary meets: For any two elements a and b of L, the set {a, b} has a meet: $a \land b$ (also known as the greatest lower bound, or the infimum).

The join and meet of a and b are denoted by $a \lor b$ and $a \land b$ , respectively. This definition makes $\lor$ and $\land$ binary operations. 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 $\lor$ b₁ ≤ a₂ $\lor$ b₂ and a₁ $\land$ b₁ ≤ a₂ $\land$ b₂.

It follows by an induction 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) 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.

A bounded lattice has a greatest (or maximum) and least (or minimum) 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 non-empty finite lattice is bounded, by taking the join (resp., meet) of all elements, denoted by $\bigvee A=a_1\lor\cdots\lor a_n$ (resp. $\bigwedge A=a_1\land\cdots\land a_n$ ) where $A=\{a_1,\ldots,a_n\}$ .

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 $\bigvee\varnothing=0$ , and the meet of the empty set is defined to be the greatest element $\bigwedge\varnothing=1$ . 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,

$\bigvee \left( A \cup B \right)= \left( \bigvee A \right) \vee \left( \bigvee B \right)$

and

$\bigwedge \left( A \cup B \right)= \left(\bigwedge A \right) \wedge \left( \bigwedge B \right)$

hold. Taking B to be the empty set,

$\bigvee \left( A \cup \emptyset \right) = \left( \bigvee A \right) \vee \left( \bigvee \emptyset \right) = \left( \bigvee A \right) \vee 0 = \bigvee A$

and

$\bigwedge \left( A \cup \emptyset \right) = \left( \bigwedge A \right) \wedge \left( \bigwedge \emptyset \right) = \left( \bigwedge A \right) \wedge 1 = \bigwedge A$

which is consistent with the fact that $A \cup \emptyset = A$ .

Lattices as algebraic structures

General lattice

An algebraic structure (L, $\lor, \land$ ), consisting of a set L and two binary operations $\lor$ , and $\land$ , on L is a lattice if the following axiomatic identities hold for all elements a, b, c of L.

Commutative laws: $a \lor b = b \lor a$ ,; $a \land b = b\land a$ .

Associative laws: $a \lor(b \lor c) = (a \lor b)\lor c$ ,; $a \land(b \land c) = (a \land b)\land c$ .

Absorption laws: $a \lor(a \land b) = a$ ,; $a \land (a \lor b) = a$ .

The following two identities are also usually regarded as axioms, even though they follow from the two absorption laws taken together.^[1]

Idempotent laws: $a \lor a = a$ ,; $a \land a = a$ .

These axioms assert that both (L, $\lor$ ) and (L, $\land$ ) are semilattices. 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 of the other.

Bounded lattice

A bounded lattice is an algebraic structure of the form (L, $\lor, \land$ , 1, 0) such that (L, $\lor, \land$ ) is a lattice, 0 (the lattice's bottom) is the identity element for the join operation $\lor$ , and 1 (the lattice's top) is the identity element for the meet operation $\land$ .

Identity laws: $a \lor 0 = a$ ,; $a \land 1 = a$ .

See semilattice for further details.

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.

By commutativity and associativity one can think of join and meet as binary operations that are defined on non-empty finite sets, rather than on 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.

Connection between the two definitions

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

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

a ≤ b if and only if a = a $\land$ b, or

a ≤ b if and only if b = a $\lor$ 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 $\lor$ and $\land$ .

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.
For any set A, the collection of all partitions of A, ordered by refinement, is a lattice.
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 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: a ≤ b if a divides b. 1 is bottom; 0 is 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.

Most posets are not lattices, including the following.

A discrete poset, meaning a poset such that x ≤ y implies x = y, 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 {1,2,3,6} partially ordered by divisibility is a lattice, the set {1,2,3} so ordered is not a lattice because the pair 2,3 lacks a join, and it lacks a meet in {2,3,6}.
The set {1,2,3,12,18,36} 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).

Further examples of lattices 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, ∨_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 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 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

Main article: Complete lattice

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 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.

Conditional completeness

A conditionally complete lattice is a poset in which every nonempty subset that has an upper bound has a join (i.e., a least upper bound). Such lattices provide the most direct generalization of the completeness axiom of the real numbers. A conditionally complete lattice is either a complete lattice, or a complete lattice without its maximum element 1, its minimum element 0, or both.

Distributivity

Main article: Distributive lattice

Since lattices come with two binary operations, it is natural to ask whether one of them distributes over the other, i.e. whether one or the other of the following dual 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 and completely distributive lattices, see distributivity in order theory.

Modularity

Main article: Modular lattice

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, and the lattice of normal subgroups of a group.

Semimodularity

Main article: Semimodular lattice

A finite lattice is modular if and only if it is both upper and lower semimodular. For a graded lattice, (upper) semimodularity is equivalent to the following condition on the rank function r:

$r(x)%2Br(y) \ge r(x \wedge y) %2B r(x \vee y).$

Another equivalent (for graded lattices) condition is Birkhoff's condition:

for each x and y in L, if x and y both cover $x \wedge y$ , then $x \vee y$ 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 $\vee$ and $\wedge$ exchanged, "covers" exchanged with "is covered by", and inequalities reversed.^[2]

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:

$x \vee y = 1$ and $x \wedge y = 0.$

In the case the complement is unique, 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 where some members might be 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 $\wedge$ 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 M $\not=\varnothing$ is a subset of L such that for every pair of elements a, b in M both a $\wedge$ b and a $\vee$ b are in M, then M is a sublattice of L.^[3]

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

We now define some order-theoretic notions of importance to lattice theory. In the following, let x be an element of some lattice L. If L has a 0, x≠0 is sometimes required. x is:

Join irreducible iff x = a∨b implies x = a or x = b for any a,b in L. When the first condition is generalized to arbitrary joins $\lor a_i$ , x is called completely join irreducible (or ∨-irreducible). The dual notion is meet irreducibility (∧-irreducible);
Join prime iff x ≤ a ∨ b implies x ≤ a or x ≤ b. This too can be generalized to obtain the notion completely join prime. The dual notion is meet prime. Any join-prime element is also join irreducible, and any meet-prime element is also meet irreducible. The converse holds if L is distributive.

Let L have a 0. An element x of L is an atom if 0 < x and there exists no element y of L such that 0 < y < x. We then say that L is:

Atomic if for every nonzero element x of L, there exists an atom a of L such that a ≤ x ;
Atomistic if every element of L is a supremum of atoms. That is, for all a, b in L such that $a\nleq b,$ there exists an atom x of L such that $x\leq a$ and $x\nleq b.$

The dual notions of ideals and filters refer to particular kinds of subsets of any partially ordered set, and are therefore important for lattice theory. Details can be found in the respective entries.

Notes

^ $a \lor a=a\lor(a\land(a\lor a))=a$ , and dually for the other idempotent law. Dedekind, Richard (1897), "Ueber Zerlegungen von Zahlen durch ihre grössten gemeinsamen Teiler", Braunschweiger Festschrift: 1–40 .
^ Stanley, Richard P, Enumerative Combinatorics (vol. 1), Cambridge University Press, pp. 103–104, ISBN 0521663512
^ Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. A Course in Universal Algebra. Springer-Verlag. ISBN 3-540-90578-2.

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.
Nation, J. B., Notes on Lattice Theory. Chapters 1-6. Chapters 7-12; Appendices 1-3.

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.; Priestley, H. A. (2002), Introduction to Lattices and Order, Cambridge University Press, ISBN 978-0-521-78451-1

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.

External links

Weisstein, Eric W., "Lattice" from MathWorld.
J.B. Nation, Notes on Lattice Theory, unpublished course notes available as two PDF files.
Ralph Freese, "Lattice Theory Homepage".