Meaning of total order | Babel Free

Definitions

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).

Equivalents

Examples

“[…]we conclude⟳ §2.1 by showing how, given a triangulation #92;Delta (i.e., simplicial decomposition) of a closed oriented 3-manifilld M, and a total order⟳ '#92;le' on the set⟳ of vertices of #92;Delta, as well as a choice of a system #92;mathcal#123;B#125; of orthonormal bases for various Hilbert spaces that get⟳ specified in the process⟳, we may obtain⟳ a complex number⟳ #92;langle(M,#92;Delta,#92;le,#92;mathcal#123;B#125;)#92;rangle.”

“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.”

CEFR level