Lattice (order)

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 two elements have 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 natural numbers, partially ordered by divisibility, for which the unique supremum is the least common multiple and the unique 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. 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 partially ordered sets

If (L, ≤) is a partially ordered set (poset), and S ⊆ L is an arbitrary subset, then an element u ∈ L is said to be an upper bound of S if s ≤ u for each s ∈ S. A set may have many upper bounds, or none at all. An upper bound u of S is said to be its least upper bound, or join, or supremum, if u ≤ x for each upper bound x of S. A set need not have a least upper bound, but it cannot have more than one. Dually, l ∈ L is said to be a lower bound of S if l ≤ s for each s ∈ S. A lower bound l of S is said to be its greatest lower bound, or meet, or infimum, if x ≤ l for each lower bound x of S. A set may have many lower bounds, or none at all, but can have at most one greatest lower bound.

A partially ordered set (L, ≤) is called a join-semilattice and a meet-semilattice if each two-element subset {a, b} ⊆ L has a join (i.e. least upper bound) and a meet (i.e. greatest lower bound), denoted by a ∨ b and a ∧ b, respectively. (L, ≤) is called a lattice if it is both a join- and a meet-semilattice. This definition makes ∨ and ∧ binary operations. Both operations are monotone with respect to the order: a1 ≤ a2 and b1 ≤ b2 implies that a1 ∨ b1 ≤ a2 ∨ b2 and a1 ∧ b1 ≤ a2 ∧ b2.

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.

A bounded lattice is a lattice that additionally has a greatest element 1 and a least element 0, which satisfy

0 ≤ x ≤ 1 for every x in L.

The greatest and least element is also called the maximum and minimum, or the top and bottom element, and denoted by ⊤ and ⊥, respectively. Every lattice can be converted into a bounded lattice by adding an artificial greatest and least element, and every non-empty finite lattice is bounded, by taking the join (resp., meet) of all elements, denoted by (resp.) where .

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. For every element x of a poset it is trivially true (it is a vacuous truth) that and , and therefore every element of a poset is both an upper bound and a lower bound of the empty set. This implies that the join of an empty set is the least element , and the meet of the empty set is the greatest element . This 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 .

A lattice element y is said to cover another element x, if y > x, but there does not exist a z such that y > z > x. Here, y > x means x ≤ y and x ≠ y.

A lattice (L, ≤) is called graded, sometimes ranked (but see Ranked poset for an alternative meaning), if it can be equipped with a rank function r from L to ℕ, sometimes to ℤ, compatible with the ordering (so r(x) < r(y) whenever x < y) such that whenever y covers x, then r(y) = r(x) + 1. The value of the rank function for a lattice element is called its rank.

Given a subset of a lattice, H ⊂ L, meet and join restrict to partial functions – they are undefined if their value is not in the subset H. The resulting structure on H is called a partial lattice. 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.[1]

Lattices as algebraic structures

General lattice

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

Commutative laws
a ∨ b = b ∨ a,
a ∧ b = b ∧ a.
     
Associative laws
a ∨ (b ∨ c) = (a ∨ b) ∨ c,
a ∧ (b ∧ c) = (a ∧ b) ∧ c.
     
Absorption laws
a ∨ (a ∧ b) = a,
a ∧ (a ∨ b) = a.

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

Idempotent laws
a ∨ a = a,
a ∧ a = a.

These axioms assert that both (L, ∨) and (L, ∧) are semilattices. The absorption laws, the only axioms above in which both meet and join appear, distinguish a lattice from an arbitrary 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, ∨, ∧, 0, 1) such that (L, ∨, ∧) is a lattice, 0 (the lattice's bottom) is the identity element for the join operation ∨, and 1 (the lattice's top) is the identity element for the meet operation ∧.

Identity laws
a ∨ 0 = a,
a ∧ 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 ∨ 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 converse is also true. Given an algebraically defined lattice (L, ∨, ∧), one can define a partial order ≤ on L by setting

a ≤ b if a = a ∧ b, or
a ≤ b if b = a ∨ b,

for all elements a and b from L. The laws of absorption ensure that both definitions are equivalent:

a = a ∧ b implies b = b ∨ (b ∧ a) = (a ∧ b) ∨ b = a ∨ b

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.

Examples

Pic.5: Lattice of nonnegative integer pairs, ordered componentwise.
Pic.4: Lattice of positive integers, ordered by <.
Pic.3: Lattice of partitions of {1, 2, 3, 4}, ordered by "refines".
Pic.2: Lattice of integer divisors of 60, ordered by "divides".
Pic.1: Lattice of subsets of {x, y, z}, ordered by "is subset of". The name "lattice" is suggested by the form of the Hasse diagram depicting it.

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

Counter-examples

Pic.8: Non-lattice poset: a and b have common lower bounds 0, d, g, h, and i, but none of them is the greatest lower bound.
Pic.7: Non-lattice poset: b and c have common upper bounds d, e, and f, but none of them is the least upper bound.
Pic.6: Non-lattice poset: c and d have no common upper bound.

Most partial ordered sets are not lattices, including the following.

Morphisms of lattices

Pic.9: Monotonic map f between lattices that preserves neither joins nor meets, since f(u) ∨ f(v) = u′ ∨ u′ = u′ ≠ 1′ = f(1) = f(u ∨ v) and f(u) ∧ f(v) = u′ ∧ u′ = u′ ≠ 0′ = f(0) = f(u ∧ v).

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 lattice homomorphism from L to M is a function f : L → M such that for all a, b ∈ L:

f(a ∨L b) = f(a) ∨M f(b), and
f(a ∧L b) = 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. In particular, a bounded-lattice homomorphism (usually called just "lattice homomorphism") f between two bounded lattices L and M should also have the following property:

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.

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 (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 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.

Sublattices

A sublattice of a lattice L is a nonempty subset of L that 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.[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.

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 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.

Note that "partial lattice" is not the opposite of "complete lattice" – rather, "partial lattice", "lattice", and "complete lattice" are increasingly restrictive definitions.

Conditional completeness

A conditionally complete lattice is a lattice 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

Pic.11: Smallest non-modular (and hence non-distributive) lattice N5.
The labelled elements violate the distributivity equation c ∧ (a ∨ b) = (c ∧ a) ∨ (c ∧ b), but satisfy its dual c ∨ (a ∧ b) = (c ∨ a) ∧ (c ∨ b).
Pic.10: Smallest non-distributive (but modular) lattice M3.

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 every 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. The only non-distributive lattices with fewer than 6 elements are called M3 and N5;[3] they are shown in picture 10 and 11, respectively. A lattice is distributive if and only if it doesn't have a sublattice isomorphic to M3 or N5.[4] Each distributive lattice is isomorphic to a lattice of sets (with union and intersection as join and meet, respectively).[5]

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

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.

A lattice is modular if and only if it doesn't have a sublattice isomorphic to N5 (shown in pic.11).[4] Besides distributive lattices, examples of modular lattices are the lattice of two-sided ideals of a ring, the lattice of submodules of a module, and the lattice of normal subgroups of a group. The set of first-order terms with the ordering "is more specific than" is a non-modular lattice used in automated reasoning.

Semimodularity

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) + r(y) ≥ r(x ∧ y) + r(x ∨ 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 ∧ y, then x ∨ 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 ∨ and ∧ exchanged, "covers" exchanged with "is covered by", and inequalities reversed.[6]

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 every 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:

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 ∨ y = 1 and x ∧ y = 0.

In general, some elements of a bounded lattice might not have a complement, and others might have more than one complement. For example, the set {0, ½, 1} with its usual ordering is a bounded lattice, and ½ does not have a complement. In the bounded lattice N5, the element a has two complements, viz. b and c (see Pic.11).

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 ∧ 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.

Jordan–Dedekind chain condition

A chain from x0 to xn is a set , where . The length of this chain is n, or one less than its number of elements. A chain is maximal if xi covers xi−1 for all 1 ≤ i ≤ n.

If for any pair, x and y, where x < y, all maximal chains from x to y have the same length, then the lattice is said to satisfy the Jordan–Dedekind chain condition.

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. For the free lattice over a set X, Whitman gave a construction based on polynomials over X's members.[7][8]

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 bottom element 0, x ≠ 0 is sometimes required. x is called:

Let L have a bottom element 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. Then L is called:

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

See also

Applications that use lattice theory

Note that in many applications the sets are only partial lattices: not every pair of elements has a meet or join.

Notes

  1. ↑ a ∨ a = a ∨ (a ∧ (a ∨ a)) = a, and dually for the other idempotent law. Dedekind, Richard (1897), "Über Zerlegungen von Zahlen durch ihre grössten gemeinsamen Teiler", Braunschweiger Festschrift: 1–40.

References

  1. ↑ Grätzer 1996, p. 52.
  2. ↑ Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. A Course in Universal Algebra. Springer-Verlag. ISBN 3-540-90578-2.
  3. ↑ Davey & Priestley (2002), Exercise 4.1, p. 104.
  4. 1 2 Davey & Priestley (2002), Theorem 4.10, p. 89.
  5. ↑ Davey & Priestley (2002), Theorem 10.21, pp. 238–239.
  6. ↑ Stanley, Richard P, Enumerative Combinatorics (vol. 1), Cambridge University Press, pp. 103–104, ISBN 0-521-66351-2
  7. ↑ Philip Whitman (1941). "Free Lattices I". Annals of Mathematics. 42: 325–329. doi:10.2307/1969001.
  8. ↑ Philip Whitman (1942). "Free Lattices II". Annals of Mathematics. 43: 104–115. doi:10.2307/1968883.

Monographs available free online:

Elementary texts recommended for those with limited mathematical maturity:

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

Advanced monographs:

On free lattices:

On the history of lattice theory:

On applications of lattice theory:

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