total order

== English == === Noun === total order (plural total orders) (set theory, order theory) A partial order, ≤, (a binary relation that is reflexive, antisymmetric, and transitive) on some set S, such that any two elements of S are comparable (for any x, y ∈ S, either x ≤ y or y ≤ x). 2013, Nick Huggett, Tiziana Vistarini, Christian Wüthrich, 15: Time in Quantum Gravity, Adrian Bardon, Heather Dyke (editors), A Companion to the Philosophy of Time, Wiley, 2016, Paperback, page 245, A binary relation R defines a total order on a set X just in case for all x, y, z ∈ X, the following four conditions obtain: (1) Rxx (reflexivity), (2) Rxy & Ryz → Rxz (transitivity), (3) Rxy & Ryx → x = y (weak antisymmetry), and (4) Rxy ∨ Ryx (comparability). Bearing in mind that the relata of the total order are not events in E {\displaystyle {\mathcal {E}}} , but entire equivalence classes E / S {\displaystyle {\mathcal {E}}/S} of simultaneous events, it is straightforward to ask ≤ to be a total order of E / S {\displaystyle {\mathcal {E}}/S} . ==== Synonyms ==== (partial order which applies an order to any two elements): linear order, linear ordering, total ordering, total ordering relation (rare) ==== Hypernyms ==== (partial order that applies an order to any two elements): partial order preorder ==== Hyponyms ==== (partial order that applies an order to any two elements): well-order ==== Related terms ==== totally ordered totally ordered set ==== Translations ==== === See also === chain connex property connex relation trichotomy === Further reading === Comparability on Wikipedia.Wikipedia Lexicographical order on Wikipedia.Wikipedia Prefix order on Wikipedia.Wikipedia Suslin's problem on Wikipedia.Wikipedia Well-order on Wikipedia.Wikipedia