totally ordered

English

Adjective

totally ordered (not comparable)

1. (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 $\theta$.

Synonyms

• (equipped with a total order): linearly ordered

Derived terms

• totally ordered set

Translations

that is equipped with a total order

• German: total geordnet

See also

• chain

Further reading

• totally ordered on Wikipedia.Wikipedia