Lattice (order theory)

A partially ordered set in which every pair of elements has a least upper bound and a greatest lower bound. These operations are called join and meet.

Partially ordered set: A set equipped with a reflexive, antisymmetric, and transitive relation. A lattice is a partially ordered set with joins and meets for every pair.
Join (order theory): The least upper bound of two elements in a partially ordered set, when it exists. It is one of the two operations guaranteed for every pair in a lattice.
Power set: The set of all subsets of a given set, ordered by inclusion. Union and intersection give joins and meets for every pair of subsets.
Semilattice: A partially ordered set in which every pair has either a join or a meet, depending on the chosen type. A semilattice requires only one of the two operations that a lattice requires.
Upper bound: An element greater than or equal to every element in a specified subset of a partially ordered set. The least upper bound in a lattice is the join of the subset's pair.
Meet (order theory): The greatest lower bound of two elements in a partially ordered set, when it exists. It is the other operation guaranteed for every pair in a lattice.
Divisibility lattice: The positive integers ordered by divisibility, with greatest common divisor and least common multiple as meet and join. It realizes lattice operations as familiar arithmetic constructions.
Distributive lattice: A lattice in which meet distributes over join and join distributes over meet. Distributivity is an additional law, not part of the definition of every lattice.
Lower bound: An element less than or equal to every element in a specified subset of a partially ordered set. The greatest lower bound in a lattice is the meet of the subset's pair.
Lattice homomorphism: A map between lattices that preserves joins and meets. It preserves the operations that define a lattice's structure.

Linked from 15 pages