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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: a <= b if and only if a v b = b, or, equivalently, a ^ b = a. L, together with the partial order <= so defined, will then be a lattice in the above ordertheoretic sense.
Conversely, if an ordertheoretic 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 KnasterTarski 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 v (y ^ (x v z)) = (x v y) ^ (x v 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 then for all y we have the identity x v (y ^ z) = (x v y) ^ z.
A lattice is called distributive if v distributes over ^, that is, x v (y ^ z) = (x v y) ^ (x v z). Equivalently, ^ distributes over v. 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. Distributive lattices are used to formulate pointless topology.
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
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