total order

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, 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 ${\displaystyle \Delta }$ (i.e., simplicial decomposition) of a closed oriented 3-manifilld ${\displaystyle M}$ , and a total order '${\displaystyle \leq }$ ' on the set of vertices of ${\displaystyle \Delta }$ , as well as a choice of a system ${\displaystyle {\mathcal {B}}}$ of orthonormal bases for various Hilbert spaces that get specified in the process, we may obtain a complex number ${\displaystyle \langle (M,\Delta ,\leq ,{\mathcal {B}})\rangle }$ .

   • 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 ${\displaystyle {\mathcal {E}}}$ , but entire equivalence classes ${\displaystyle {\mathcal {E}}/S}$ of simultaneous events, it is straightforward to ask ≤ to be a total order of ${\displaystyle {\mathcal {E}}/S}$ .

• (partial order which applies an order to any two elements): linear order, linear ordering, total ordering, total ordering relation (rare)

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

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

• totally ordered
• totally ordered set

• chain
• connex property
• connex relation
• trichotomy

• 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