Lattice-Ordered Set

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

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  of a bounded lattice-ordered set , we say that  is complemented in  if there exists an element  such that  and .