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 
.
