total order
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English[edit]
Noun[edit]
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,
Synonyms[edit]
 (partial order which applies an order to any two elements): linear order, linear ordering, total ordering, total ordering relation (rare)
Hypernyms[edit]
 (partial order that applies an order to any two elements):
Hyponyms[edit]
 (partial order that applies an order to any two elements):
Related terms[edit]
Translations[edit]
partial order that applies an order to any two elements


See also[edit]
Further reading[edit]
 Comparability on Wikipedia.Wikipedia
 Lexicographical order on Wikipedia.Wikipedia
 Prefix order on Wikipedia.Wikipedia
 Suslin's problem on Wikipedia.Wikipedia
 Wellorder on Wikipedia.Wikipedia