Order theory

Order theory studies relations that express comparisons such as “less than,” including partial orders, lattices, and structures defined by them.

Partially ordered set: A set equipped with a reflexive, antisymmetric, transitive relation. Partially ordered sets are the central objects order theory studies.
Lattice: A partially ordered set in which every pair has a meet and a join. Lattices add two fundamental operations to partial order.
Domain theory: A mathematical theory of ordered structures used to model computation and recursive definitions. Its domains make limits of increasingly informative computations precise.
Category theory: A mathematical framework built from objects, morphisms, and their composition. A preorder can be viewed as a category with at most one arrow between objects.
Preorder: A reflexive, transitive relation that need not be antisymmetric. Preorders weaken partial orders by allowing distinct elements to compare both ways.
Partial order: A reflexive, antisymmetric, transitive relation on a set. This relation allows some elements to remain incomparable.
Meet and join: The greatest lower bound and least upper bound of elements in an ordered set. These operations define the basic algebra of a lattice.
Boolean algebra: A complemented distributive lattice with least and greatest elements. Boolean algebras use lattice order to organize logical operations.
Universal algebra: The study of algebraic structures through operations and equations. Lattices are algebraic structures whose operations arise from order.
Strict partial order: An irreflexive, transitive relation that represents strict comparison. It expresses order without the reflexive convention used for partial orders.

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