[…]we conclude §2.1 by showing how, given a triangulation Δ (i.e., simplicial decomposition) of a closed oriented 3-manifilld M, and a total order '<' on the set of vertices of Δ, as well as a choice of a system ℬ of orthonormal bases for various Hilbert spaces that get specified in the process, we may obtain a complex number ⟨(M,Δ,<,ℬ)⟩.

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 ℰ, but entire equivalence classes ℰ/S of simultaneous events, it is straightforward to ask ≤ to be a total order of ℰ/S.

Total order

Meaning

Noun

Synonyms