totally ordered

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Total order

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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:

     ${\displaystyle G}$ is a transitive group of order automorphisms of a totally ordered set ${\displaystyle X}$,

     ${\displaystyle \theta }$ is a convex ${\displaystyle G}$-congruence on the totally ordered set ${\displaystyle X}$,

     ${\displaystyle \xi }$ is the order type of the totally ordered set ${\displaystyle X}$,

     ${\displaystyle {\overline {\eta }}}$ is the order type of some class ${\displaystyle Y=x\eta }$ of the congruence ${\displaystyle \eta }$,

     ${\displaystyle \zeta }$ is the order type of the totally ordered quotient set ${\displaystyle X/\theta }$ of the totally ordered set ${\displaystyle X}$ by the congruence ${\displaystyle \theta }$.

• (equipped with a total order): linearly ordered

• totally ordered set

• chain

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