KnowraTotal orderTotal orderA binary relation that is reflexive, antisymmetric, and transitive, and in which every pair of elements is comparable.BriefConnectBinary relation: A relation between elements of two sets, represented as a set of ordered pairs. A total order is a binary relation on a set.Partial order: A reflexive, antisymmetric, and transitive relation that may leave some pairs incomparable. A partial order lacks the comparability requirement that distinguishes total orders.Lexicographic order: An order on sequences that compares entries from left to right until they differ. It builds total orders on strings and tuples from orders on their entries.Real number: An element of the complete ordered field used to represent points on a continuous number line. The usual less-than-or-equal relation totally orders the real numbers.Reflexive relation: A relation in which every element is related to itself. Reflexivity is one of the properties required of a total order.Strict total order: An irreflexive, transitive relation in which every pair of distinct elements is related in exactly one direction. It expresses the same linear arrangement using strict comparisons.Topological sorting: A linear ordering of a directed acyclic graph's vertices that respects every directed edge. It produces a total order extending the graph's partial precedence constraints.Natural number: A counting number, with conventions differing on whether zero is included. The usual numerical order gives a familiar total order on natural numbers.Antisymmetric relation: A relation where two elements related in both directions must be equal. Antisymmetry prevents distinct elements from being equivalent in both directions.Preorder: A reflexive and transitive relation that need not be antisymmetric. Preorders retain two order properties but may identify distinct elements in both directions.Show all 20Linked from 12 pagesMerge sortRelated: Merge sort requires a consistent ordering to decide which item comes next.Well-orderNarrower topic: Every well-order is total, but totality alone does not guarantee least elements for all subsets.Partially ordered setCompared with: It adds comparability for every pair, which a partial order does not require.Chain (order theory)Related: The order induced on a chain is total.Well-founded relationCompared with: Totality is independent of well-foundedness: a well-founded relation can leave pairs incomparable.Greatest element and least elementRelated: In a total order, greatest and least elements coincide with the familiar largest and smallest values.Lattice (order theory)Compared with: Every total order is a lattice, but lattices can contain incomparable elements.Strict partial orderCompared with: It combines reflexive ordering with the comparability absent from partial orders generally.Show all 12