totally ordered

Adjective

totally ordered (not comparable)

1. 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.
   • (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).1976, K. D. Stroyan, W. A. J. Luxemburg, Introduction to the Theory of Infinitesimals, Harcourt Brace Jovanovich (Academic Press), page 67
   • 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'.1982, A. G. Hamilton, Numbers, Sets and Axioms: The Apparatus of Mathematics, Cambridge University Press, page 91

   Synonyms:

   linearly ordered