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)
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(2)
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(3)
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(4)
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(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 .)