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 .)

REFERENCES:

Grätzer, G. Lattice Theory: First Concepts and Distributive Lattices. San Francisco, CA: W. H. Freeman, 1971.