totally ordered

== English == === 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). 1996, Scientific Books staff (translators), Vasiliǐ M. Kopytov, Nikolaǐ Ya. Medvedev, Right-Ordered Groups, Scientific Books, page 98, We introduce the following notation: G {\displaystyle G} is a transitive group of order automorphisms of a totally ordered set X {\displaystyle X} , θ {\displaystyle \theta } is a convex G {\displaystyle G} -congruence on the totally ordered set X {\displaystyle X} , ξ {\displaystyle \xi } is the order type of the totally ordered set X {\displaystyle X} , η ¯ {\displaystyle {\overline {\eta }}} is the order type of some class Y = x η {\displaystyle Y=x\eta } of the congruence η {\displaystyle \eta } , ζ {\displaystyle \zeta } is the order type of the totally ordered quotient set X / θ {\displaystyle X/\theta } of the totally ordered set X {\displaystyle X} by the congruence θ {\displaystyle \theta } . ==== Synonyms ==== (equipped with a total order): linearly ordered ==== Derived terms ==== totally ordered set ==== Translations ==== === See also === chain === Further reading === totally ordered on Wikipedia.Wikipedia