Lattice-Ordered Set

A lattice-ordered set is a poset (L,<=) in which each two-element subset {a,b} has an infimum, denoted inf{a,b}, and a supremum, denoted sup{a,b}. There is a natural relationship between lattice-ordered sets and lattices. In fact, a lattice (L, ^ , v ) is obtained from a lattice-ordered poset (L,<=) by defining a ^ b=inf{a,b} and a v b=sup{a,b} for any a,b in L. Also, from a lattice (L, ^ , v ), one may obtain a lattice-ordered set (L,<=) by setting a<=b in L if and only if a=a ^ b. One obtains the same lattice-ordered set (L,<=) from the given lattice by setting a<=b in L if and only if a v b=b. (In other words, one may prove that for any lattice, (L, ^ , v ), and for any two members a and b of L, a ^ b=b if and only if a=a v b.)

Lattice-ordered sets abound in mathematics and its applications, and many authors do not distinguish between them and lattices. From a universal algebraist's point of view, however, a lattice is different from a lattice-ordered set because lattices are algebraic structures that form an equational class or variety, but lattice-ordered sets are not algebraic structures, and therefore do not form a variety.

A lattice-ordered set is bounded provided that it is a bounded poset, i.e., if it has an upper bound and a lower bound. For a bounded lattice-ordered set, the upper bound is frequently denoted 1 and the lower bound is frequently denoted 0. Given an element x of a bounded lattice-ordered set (L,<=), we say that x is complemented in (X,<=) if there exists an element y in X such that infx,y=0 and sup(x,y)=1.

See also

Lattice, Partially Ordered Set

This entry contributed by Matt Insall (author's link)

Explore with Wolfram|Alpha

Cite this as:

Insall, Matt. "Lattice-Ordered Set." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications