Home > Lattice (order)
:See lattice for other mathematical as well as non-mathematical meanings of the term.In mathematics, a lattice is a partially ordered set in which all nonempty finite subsets have both a supremum (join) and an infimum (meet). On the other hand, lattices can also be characterized as algebraic structures that satisfy certain identities. Since both views can be used interchangeably, lattice theory can draw upon applications and methods both from order theory and from universal algebra. Lattices constitute one of the most prominent representatives of a series of "lattice-like" structures which admit order-theoretic as well as algebraic descriptions, such as semilattices, Heyting algebras, or Boolean algebras. The term "lattice" derives from the shape of the Hasse diagrams that result from depicting these orders.
This article treats the most basic definitions of lattice theory, including the case of bounded lattices, i.e lattices that have top and bottom elements.
1 Formal definition
As mentioned above, lattices can be characterized both as posets and as algebraic structures. Both approaches and their relationship are explained below.
1.1 Lattices as posets
Consider a partially ordered set (L, ≤). L is a lattice if
- for all elements x and y of L, the set {x, y} has both a least upper bound (join) and a greatest lower bound (meet).
In this situation, the join and meet of x and y are denoted by x'y and x'y, respectively. Clearly, this defines binary operations and on lattices. Also note that the above definition is equivalent to requiring L to be both a meet- and a join-semilattice.
It will be stated explicitly whenever a lattice is required to have a least or greatest elementIn mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S''. The term least element is defined dually. Formally, given a partially. If both of these special elements do exist, then the poset is a bounded lattice.
Using an easy inductionMathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers, or otherwise is true of all members of an infinite sequence. A somewhat more general form of argument used in mathe argument, one can also conclude the existence of all suprema and infima of non-empty finite subsets of any lattice. Further conclusions may be possible in the presence of other properties. See the article on completeness in order theoryIn the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set. In a more special usage of the term one also talks about complete partial orders or complete lattices. for more discussion on this subject. This article also discusses how one may rephrase the above definition in terms of the existence of suitable Galois connectionIn mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets ("posets"). Galois connections generalize the correspondence between subgroups and subfields investigated in Galois theory.s between related posets -- an approach that is of special interest for category theoreticCategory theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "generalized abstract nonsense". See list of category theory topics for a breakdown of relevan investigations of the concept.
