A partial order on a set S is a relation ≤ that is reflexive (a ≤ a), transitive (a ≤ b and b ≤ c imply a ≤ c), and antisymmetric (a ≤ b and b ≤ a imply a = b). A set equipped with a partial order is a partially ordered set, or poset.
“Partial” means that not every pair of elements need be comparable: it is possible that neither a ≤ b nor b ≤ a holds. A total order (or linear order) is a partial order where every pair is comparable.
A poset viewed as a category is a thin category: objects are elements, and there is at most one morphism from a to b (existing when a ≤ b). This perspective connects order theory to category theory and makes posets the simplest nontrivial categories.
A poset that has meets and joins for every pair of elements is a lattice. A complete lattice has meets and joins for all subsets. A complete lattice with the Heyting implication is a complete Heyting algebra — the algebraic structure underlying the semiotic universe. The Heyting algebra H is a partial order whose elements are semantic values and whose ordering expresses entailment: a ≤ b means “a entails b.”