Lattice (order)

See lattice for other meanings of the term, both within and without mathematics.

In mathematics, a lattice is a partially ordered set in which all nonempty finite subsets have a least upper bound (also called supremum or join) and a greatest lower bound (also called infimum or meet). The term "lattice" comes from the shape of the Hasse diagrams of such orders.

Algebraic definition

A lattice can also be algebraically defined as a set L, together with two binary operations ∧ and ∨ (pronounced meet and join, respectively), such that for any a, b, c in L,

a ∨ a = a	a ∧ a = a	idempotency laws
a ∨ b = b ∨ a	a ∧ b = b ∧ a	commutativity laws
a ∨ (b ∨ c) = (a ∨ b) ∨ c	a ∧ (b ∧ c) = (a ∧ b) ∧ c	associativity laws
a ∨ (a ∧ b) = a	a ∧ (a ∨ b) = a	absorption laws

(The idempotency laws can be deduced from the absorption laws and therefore don't have to be stated separately.)

If the two operations satisfy these algebraic rules then they define a partial order ≤ on L by the following rule: a ≤ b if and only if a ∨ b = b, or, equivalently, a ∧ b = a. L, together with the partial order ≤ so defined, will then be a lattice in the above order-theoretic sense.

Conversely, if an order-theoretic lattice (L, ≤) is given, and we write a ∨ b for the least upper bound of {a, b} and a ∧ b for the greatest lower bound of {a, b}, then (L, ∨, ∧) satisfies all the axioms of an algebraically defined lattice.

Homomorphisms

The class of all lattices forms a category if we define a homomorphism between two lattices (L, ∧, ∨) and (N, ∩, ∪) to be a function f : L → N such that

f(a ∧ b) = f(a) ∩ f(b)

f(a ∨ b) = f(a) ∪ f(b)

Every homomorphism is a monotone map between the two lattices, but not every monotone map yields a lattice homomorphism: in addition we need the compatibility with finite suprema and infima.

Properties of lattices, examples

A lattice is said to be bounded if it has a greatest element and a least element. The greatest element is often denoted by 1 and the least element by 0. If x is an element of a bounded lattice then any element y of the lattice satisfying x ∧ y = 0 and x ∨ y = 1 is called a complement of x. A bounded lattice in which every element has a (not necessarily unique) complement is called a complemented lattice.

A lattice in which every subset (including infinite ones) has a supremum and an infimum is called a complete lattice. (It is enough to require that every subset have a supremum; the existence of all infima then follows.) Complete lattices are always bounded. Many of the most important lattices are complete. Examples include:

• The subsets of a given set, ordered by inclusion. The supremum is given by the union and the infimum by the intersection of subsets.
• The unit interval [0,1] and the extended real number line, with the familiar total order and the ordinary suprema and infima.
• The non-negative integers, ordered by divisibility. The supremum is given by the least common multiple and the infimum by the greatest common divisor.
• The subgroups of a group, ordered by inclusion. The supremum is given by the subgroup generated by the union of the groups and the infimum is given by the intersection.
• The submodules of a module, ordered by inclusion. The supremum is given by the sum of submodules and the infimum by the intersection.
• The ideals of a ring, ordered by inclusion. The supremum is given by the sum of ideals and the infimum by the intersection.
• The open sets of a topological space, ordered by inclusion. The supremum is given by the union of open sets and the infimum by the interior of the intersection.
• The convex subsets of a real or complex vector space, ordered by inclusion. The infimum is given by the intersection of convex sets and the supremum by the convex hull of the union.
• The topologies on a set, ordered by inclusion. The infimum is given by the intersection of topologies, and the supremum by the topology generated by the union of topologies.
• The lattice of all transitive binary relations on a set.
• The lattice of all sub-multisets of a multiset.
• The lattice of all equivalence relations on a set; the equivalence relation ~ is considered to be smaller (or "finer") than ≈ if x~y always implies x≈y.

The Knaster-Tarski theorem states that the set of fixed points of a monotone function on a complete lattice is again a complete lattice.

The lattice of submodules of a module and the lattice of normal subgroups of a group have the special property that x ∨ (y ∧ (x ∨ z)) = (x ∨ y) ∧ (x ∨ z) for all x, y and z in the lattice. A lattice with this property is called a modular lattice. The condition of modularity can also be stated as follows: If x ≤ z then for all y we have the identity x ∨ (y ∧ z) = (x ∨ y) ∧ z.

A lattice is called distributive if ∨ distributes over ∧, that is, x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z). Equivalently, ∧ distributes over ∨. All distributive lattices are modular. Two important types of distributive lattices are totally ordered sets and Boolean algebras (like the lattice of all subsets of a given set). The lattice of natural numbers, ordered by divisibility, is also distributive. A lattice is said to be completely distributive if the above distributivity law hold for arbitrary (infinite) meets and joins. Distributive lattices are used to formulate pointless topology.

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 iff

• x = a ∨ 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.

An element x of L is called join-prime iff

• x ≤ a ∨ b implies x ≤ a or x ≤ b,
• if L has a 0, x is sometimes required to be different from 0.

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.

Literature

• B. A. Davey, H. A. Priestley: Introduction to Lattices and Order. Cambridge University Press, 2002. (ISBN 0521784514)

Referenced By