total order

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English

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total order

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).
- 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 $\Delta$ (i.e., simplicial decomposition) of a closed oriented 3-manifilld $M$ , and a total order ' $\leq$ ' on the set of vertices of $\Delta$ , as well as a choice of a system ${\mathcal {B}}$ of orthonormal bases for various Hilbert spaces that get specified in the process, we may obtain a complex number $\langle (M,\Delta ,\leq ,{\mathcal {B}})\rangle$ .
- 2006, Daniel J. Velleman, How to Prove It: A Structured Approach, 2nd edition, Cambridge University Press, 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 ${\mathcal {E}}$ , but entire equivalence classes ${\mathcal {E}}/S$ of simultaneous events, it is straightforward to ask ≤ to be a total order of ${\mathcal {E}}/S$ .

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

Translations

partial order that applies an order to any two elements

Czech: lineární uspořádání n, úplné uspořádání n
Esperanto: tuteca ordo
Finnish: täydellinen järjestys
German: Totalordnung f, totale Ordnung f
Spanish: orden total m

See also

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

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