Lattice-Ordered Set
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31-12-2021
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Lattice-Ordered Set
A lattice-ordered set is a poset 
in which each two-element subset ![<span style=]()
{a,b}" src="https://mathworld.wolfram.com/images/equations/Lattice-OrderedSet/Inline2.gif" style="height:15px; width:32px" /> has an infimum, denoted ![inf<span style=]()
{a,b}" src="https://mathworld.wolfram.com/images/equations/Lattice-OrderedSet/Inline3.gif" style="height:15px; width:52px" />, and a supremum, denoted ![sup<span style=]()
{a,b}" src="https://mathworld.wolfram.com/images/equations/Lattice-OrderedSet/Inline4.gif" style="height:15px; width:54px" />. There is a natural relationship between lattice-ordered sets and lattices. In fact, a lattice 
is obtained from a lattice-ordered poset 
by defining ![a ^ b=inf<span style=]()
{a,b}" src="https://mathworld.wolfram.com/images/equations/Lattice-OrderedSet/Inline7.gif" style="height:15px; width:96px" /> and ![a v b=sup<span style=]()
{a,b}" src="https://mathworld.wolfram.com/images/equations/Lattice-OrderedSet/Inline8.gif" style="height:15px; width:98px" /> 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 
.
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