1. In some mathematical usage, a lattice is a discrete subgroup of Rn or Cn. A simple example is the subgroup Zn. A more complicated example is the [Leech lattice]?, which is a subgroup of R24. See also [Minkowski's Theorem]? |
1. In some mathematical usage, a lattice is a discrete subgroup of Rn or Cn. Every lattice can be generated from a basis for the underlying vector space by considering all linear combinations with integral coefficients. A simple example of a lattice in Rn is the subgroup Zn. A more complicated example is the [Leech lattice]?, which is a subgroup of R24. See also Minkowski's theorem. |
2. In other mathematical usage, a lattice is a partially ordered set in which all nonempty finite subsets have a least upper bound and a greatest lower bound (also called supremum and infimum, respectively). If L is the set and <= is the partial order then the least upper bound and greatest lower bound of a subset V of L are defined as follows: |
2. In other mathematical usage, a lattice is a partially ordered set in which all nonempty finite subsets have a least upper bound and a greatest lower bound (also called supremum and infimum, respectively). If L is the set and <= is the partial order then the least upper bound and greatest lower bound of a subset V of L are defined as follows: |
It can easily be shown the least upper bound and greatest lower bound of a finite set are always unique. For example, assume that x and y are both a least upper bound of V then it follows that x <= y and y <= x, and since <= is antisymmetric it follows that x = y. |
It can easily be shown that the least upper bound and greatest lower bound of any set are always unique: if x and y are both a least upper bound of V then it follows that x <= y and y <= x, and since <= is antisymmetric it follows that x = y. |
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 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 b = b if, and only if, a <= b. Note that a v b = b is equivalent with a ^ b = a, so that can also be used in the rule. |
: a <= b if, and only if a v b = b. Note that a v b = b is equivalent with a ^ b = a, so that the latter can also be used as the definition of the partial order. L, together with the partial order <= so defined, will then be a lattice in the above order theoretic sense. |
It is easy to see that the two definitions are equivalent if we assume that a v b is the least upper bound of {a, b} and a ^ b is the greatest lower bound of {a, b}. |
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 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 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. |
* The lattice of all topologies on a set. |
* The lattice of all subsets of a given set. |
* The lattice of all convex subsets of a real or complex vector space. |
* The lattice of all ideals of a ring. |
* The lattice of all ideals of a ring. * The lattice of all topologies on a set. |
* The lattice of all convex subsets of a real or complex vector space. |
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 :f(a ^ b) = f(a) ^ f(b) :f(a v b) = f(a) v f(b) for all a, b in L. A bijective homomorphism whose inverse is also a homomorphism is called an isomorphism of lattices, and two involved lattices are called isomorphic. |
Lattices are also used to formulate pointless topology. |
A simple example of a lattice in Rn is the subgroup Zn. A more complicated example is the [Leech lattice]?, which is a subgroup of R24.
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.