Total order

Definition

In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation on some set , which satisfies the following for all and in : (reflexive). If and then (transitive). If and then (antisymmetric). or .

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Affinely extended real number system Alexandrov topology Alphabetical order Antichain Antisymmetric relation Artinian ring Ascending chain condition Asymmetric relation Banach lattice Better-quasi-ordering Betweenness relation Bijection Binary relation Boolean algebra (structure)Boolean prime ideal theorem Bounded lattice Cantor's isomorphism theorem Cantor–Bernstein theorem Cartesian product Category (mathematics)Chain-complete partial order Cofinal (mathematics)Cofinality Commutative ring Compact space Comparability Comparability graph Complemented lattice Complete lattice Completeness (order theory)Composition of relations Connected relation Connectedness Converse relation Countable set Counting Countryman line Covering relation Cyclic order Decidability (logic)Dedekind-complete Dense order Dense set Descending chain condition Dilworth's theorem Dimension of a vector space Dimension theory Direct product Directed set Distributive lattice Duality (order theory)Dushnik–Miller theorem Empty set Encyclopedia of Mathematics Equivalence relation Eulerian poset European Mathematical Society Filter (mathematics)Finite set First-order logic Foundational relation Fréchet lattice Garrett Birkhoff Glossary of order theory Graded poset Graph (discrete mathematics)Group (mathematics)Gunther Schmidt Hamel bases Hasse diagram Hausdorff maximal principle Heyting algebra Hilary Priestley Homeomorphism Homogeneous relation Ideal (order theory)Ideal (ring theory)Idempotent relation If and only if Index set Initial object Initial segment Injective function Integers Interval (mathematics)Introduction to Lattices and Order Irreflexive relation Isomorphism Join-semilattice Join and meet Knaster–Tarski theorem Krull dimension Kruskal's tree theorem Lattice (order)Laver's theorem Least upper bound Lexicographic order Lexicographical order Linear extension Linear order (linguistics)Linear subspace List of Boolean algebra topics List of order structures in mathematics List of order theory topics Locally convex vector lattice Lower bound Map (mathematics)Mathematical structure Mathematics Maximal ideal Meet-semilattice Mirsky's theorem Monadic second-order logic Monotone sequence Monotonic function Morphism Natural numbers Net (mathematics)Noetherian ring Normal cone (functional analysis)Normal space Normed lattice Order embedding Order isomorphic Order isomorphism Order theory Order topology Order topology (functional analysis)Order type Ordered field Ordered pair Ordered topological vector space Ordered vector space Ordinal number Partial equivalence relation Partial order Partially ordered group Partially ordered set Partially ordered space Paul R. Halmos Permutation Point-pair separation Positive cone of a partially ordered group Positive cone of an ordered field Positive cone of an ordered vector space Prefix order Preorder Prewellordering Prime ideal Product order Ranking Rational number Rational numbers Real number Real numbers Reflexive closure Reflexive reduction Reflexive relation Regular chain Riesz space Ring (mathematics)Roland Fraïssé S2S (mathematics)Semilattice Semiorder Series-parallel partial order Set (mathematics)Singleton set Specialization (pre)order Star product Strict partial order Strict total order Strict weak ordering Subcategory Subnet (mathematics)Subset Supremum Suslin's problem Symmetric closure Symmetric relation Szpilrajn extension theorem Tolerance relation Topological vector lattice Topology Total preorder Total relation Totally ordered group Transitive closure Transitive relation Undergraduate Texts in Mathematics Unit interval Upper bound Upper set Vector space Walk (graph theory)Weak ordering Well-founded order Well-founded relation Well-order Well-quasi-ordering Well order Yiannis N. Moschovakis Young's lattice Zorn's lemma

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