Lattice
An algebra 
is called a lattice if 
is a nonempty
set, 
and 
are binary
operations on 
, both 
and 
are idempotent,
commutative, and associative,
and they satisfy the absorption law. The study
of lattices is called lattice theory.
Note that this type of lattice is distinct from the regular array of points known as a point lattice (or informally as a mesh or grid). While every point lattice is a lattice under the ordering inherited from the plane, many lattices are not point lattices.
Lattices offer a natural way to formalize and study the ordering of objects using a general concept known as the partially ordered set. A lattice as an algebra is equivalent to a lattice as a partially ordered set (Grätzer 1971, p. 6) since
1. Let the partially ordered set 
be a lattice. Set 
and 
.
Then the algebra 
is a lattice.
2. Let the algebra 
be a lattice. Set
iff 
. Then 
is a partially
ordered set, and the partially ordered set 
is a lattice.
3. Let the partially ordered set 
be a lattice. Then 
.
4. Let the algebra 
be a lattice. Then
.
The following inequalities hold for any lattice:
 |
(1)
|
 |
(2)
|
 |
(3)
|
 |
(4)
|
(Grätzer 1971, p. 35). The first three are the distributive inequalities, and the last is the modular identity.
A lattice 
can be 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 
.)
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