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Lattice

A lattice is a universal algebra <L; ^ , v > such that L is a nonempty set and ^ and v are binary operations on L that are commutative and associative. Every element is idempotent under both operations, and the two operations 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). A point lattice can be identified with Z^n and then ordered componentwise. This ordering depends on the chosen identification.

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 a universal 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 universal algebra L^a=<L; ^ , v > is a lattice.

2. Let the universal 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 universal 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 inequality.

The associated lattice-ordered set (L,<=) can equivalently be defined by setting a<=b iff a v b=b. In particular, for any two elements a,b in L, the identity a ^ b=b holds iff a=a v b.

See also

Congruence Lattice, Cubic Lattice, Distributive Lattice, Finite Lattice Representation Problem, Integration Lattice, Laminated Lattice, Lattice-Ordered Set, Lattice Theory, Modular Lattice, Point Lattice, Sublattice, Toric Variety

Portions 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

Lattice

Cite this as:

Weisstein, Eric W., with contributions by Matt Insall. "Lattice." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Lattice.html

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