totally ordered

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Total order

Wikipedia

Adjective

totally ordered (not comparable)

(set theory, order theory) That is equipped with a total order, that is a subset of (the ground set of) a partially ordered set whose partial order is a total order with respect to said subset.
Hypernym: partially ordered

Hyponym: well-ordered
- 1976, K. D. Stroyan, W. A. J. Luxemburg, Introduction to the Theory of Infinitesimals, Harcourt Brace Jovanovich (Academic Press), page 67,
  (A.2.5) THEOREM If A is a totally ordered ring and if I is a proper order ideal, then A/I is a totally ordered ring (with the operations and order given above).
- 1982, A. G. Hamilton, Numbers, Sets and Axioms: The Apparatus of Mathematics, Cambridge University Press, page 91:
  The set {1, 2, 3, 4, 6, 8, 12, 24}, ordered by 'divides' is not totally ordered, since (for example) 6 and 8 are not related. However, the set {1, 2, 4, 12, 24} is totally ordered by the relation 'divides'.
- 1996, Scientific Books staff (translators), Vasiliǐ M. Kopytov, Nikolaǐ Ya. Medvedev, Right-Ordered Groups, Scientific Books, page 98,
  We introduce the following notation:
  
  $G$ is a transitive group of order automorphisms of a totally ordered set $X$ ,
  
  $\theta$ is a convex $G$ -congruence on the totally ordered set $X$ ,
  
  $\xi$ is the order type of the totally ordered set $X$ ,
  
  ${\overline {\eta }}$ is the order type of some class $Y=x\eta$ of the congruence $\eta$ ,
  
  $\zeta$ is the order type of the totally ordered quotient set $X/\theta$ of the totally ordered set $X$ by the congruence $\theta$ .