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See also: total-order



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total order (plural total orders)

  1. (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, yS, either xy or yx).
    • 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 3-manifilld  , 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 RT.
    • 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, zX, the following four conditions obtain: (1) Rxx (reflexivity), (2) Rxy & RyzRxz (transitivity), (3) Rxy & Ryxx = y (weak antisymmetry), and (4) RxyRyx (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  .




  • (partial order that applies an order to any two elements):

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