Lattice

Summary

A lattice is a partially ordered set in which every two elements have a join and a meet. These are a least upper bound and a greatest lower bound. Equivalently, it is a set with two binary operations $\vee$ and $\wedge$ that are commutative and associative and satisfy the absorption laws. Lattices occur all over mathematics. Subsets of a set, subgroups of a group, subspaces of a vector space and open sets of a space all form lattices. Special classes such as distributive lattices and Boolean algebras are central in algebra and logic.