Lattice
An algebra 👁 <L; ^ , v >
is called a lattice if 👁 L
is a nonempty set, 👁 ^
and 👁 v
are binary operations on 👁 L
, both 👁 ^
and 👁 v
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 👁 L=<L;<=>
be a lattice. Set 👁 a ^ b=inf{a,b}
and 👁 a v b=sup{a,b}
. Then the algebra 👁 L^a=<L; ^ , v >
is a lattice.
2. Let the algebra 👁 L=<L; ^ , v >
be a lattice. Set 👁 a<=b
iff 👁 a ^ b=a
.
Then 👁 L^p=<L;<=>
is a partially
ordered set, and the partially ordered set 👁 L^p
is a lattice.
3. Let the partially ordered set 👁 L=<L;<=>
be a lattice. Then 👁 (L^a)^p=L
.
4. Let the algebra 👁 L=<L; ^ , v >
be a lattice. Then 👁 (L^p)^a=L
.
The following inequalities hold for any lattice:
| 👁 (x ^ y) v (x ^ z)<=x ^ (y v z) |
(1)
|
| 👁 x v (y ^ z)<=(x v y) ^ (x v z) |
(2)
|
| 👁 (x ^ y) v (y ^ z) v (z ^ x)<=(x v y) ^ (y v z) ^ (z v x) |
(3)
|
| 👁 (x ^ y) v (x ^ z)<=x ^ (y v (x ^ z)) |
(4)
|
(Grätzer 1971, p. 35). The first three are the distributive inequalities, and the last is the modular identity.
A lattice 👁 (L, ^ , v )
can be 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
.)
See also
Cubic Lattice, Distributive Lattice, Integration Lattice, Laminated Lattice, Lattice-Ordered Set, Lattice Theory, Modular Lattice, Point Lattice, Toric VarietyPortions of this entry contributed by Matt Insall (author's link)
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References
Grätzer, G. Lattice Theory: First Concepts and Distributive Lattices. San Francisco, CA: W. H. Freeman, 1971.Referenced on Wolfram|Alpha
LatticeCite this as:
Insall, Matt and Weisstein, Eric W. "Lattice." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Lattice.html