A simple example of a lattice in R^{n} is the subgroup Z^{n}. A more complicated example is the [Leech lattice]?, which is a subgroup of R^{24}.
See also Minkowski's theorem.
A lattice can also be algebraically defined as a set L, together with two binary operations ^ and v (pronounced meet and join, respectively), such that for any a, b, c in L,
a v a = a | a ^ a = a | (idempotency laws) |
a v b = b v a | a ^ b = b ^ a | (commutativity laws) |
a v (b v c) = (a v b) v c | a ^ (b ^ c) = (a ^ b) ^ c | (associativity laws) |
a v (a ^ b) = a | a ^ (a v b) = a | (absorption laws) |
If the two operations satisfy these algebraic rules then they define a partial order <= on L by the following rule:
Conversely, if a lattice (L, <=) is given, and we write a v b for the least upper bound of {a, b} and a ^ b for the greatest lower bound of {a, b}, then (L, v, ^) satisfies all the axioms of an algebraically defined lattice.
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 v 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. Complete lattices are always bounded. Many of the most important lattices are complete. Examples include:
The class of all lattices forms a category if we define a homomorphism between two lattices (L, ^, v) and (N, ^, v) to be a function f : L -> N such that
Two important types of lattices are totally ordered sets and Boolean algebras. Lattices are also used to formulate pointless topology.