total order
See also: totalorder
Contents
EnglishEdit
NounEdit
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).
 2001, Vijay Kodiyalam, V. S. Sunder, Topological Quantum Field Theories from Subfactors, CRC Press (Chapman & Hall), page 2,
 […] we conclude §2.1 by showing how, given a triangulation (i.e., simplicial decomposition) of a closed oriented 3manifilld , and a total order ' ' on the set of vertices of , as well as a choice of a system of orthonormal bases for various Hilbert spaces that get specified in the process, we may obtain a complex number .
 2006, Daniel J. Velleman, How to Prove It: A Structured Approach, Cambridge University Press, 2nd Edition, page 269,
 Example 6.2.2. Suppose A is a finite set and R is a partial order on A. Prove that R can be extended to a total order on A. In other words, prove that there is a total order T on A such that R ⊆ T.
 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 , but entire equivalence classes of simultaneous events, it is straightforward to ask ≤ to be a total order of .
 2001, Vijay Kodiyalam, V. S. Sunder, Topological Quantum Field Theories from Subfactors, CRC Press (Chapman & Hall), page 2,
SynonymsEdit
 (partial order which applies an order to any two elements): linear order, linear ordering, total ordering, total ordering relation (rare)
HypernymsEdit
 (partial order that applies an order to any two elements):
HyponymsEdit
 (partial order that applies an order to any two elements):
Related termsEdit
TranslationsEdit
partial order that applies an order to any two elements


See alsoEdit
Further readingEdit
 Comparability on Wikipedia.Wikipedia
 Lexicographical order on Wikipedia.Wikipedia
 Prefix order on Wikipedia.Wikipedia
 Suslin's problem on Wikipedia.Wikipedia
 Wellorder on Wikipedia.Wikipedia