totally ordered

adjective

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.

Similar
linearly ordered
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Meanings

adjectivetotally ordered (not comparable)
  1. not comparablemathematicsThat 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.

    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).

    2 more examples

    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

    G is a transitive group of order automorphisms of a totally ordered set X, θ is a convex G-congruence on the totally ordered set X, ξ is the order type of the totally ordered set X, ◌̅η is the order type of some class Y=xη of the congruence η,

    1996, Scientific Books staff (translators), Vasiliǐ M. Kopytov, Nikolaǐ Ya. Medvedev, Right-Ordered Groups, Scientific Books, page 98, We introduce the following notation

    Synonyms: linearly ordered

Synonyms & antonyms

adjectiveThat 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.

Similar
linearly ordered

Translations

German total geordnet
  • total geordnetthat is equipped with a total order
Polish liniowy
  • liniowythat is equipped with a total order
Cite this entry
APA

totally ordered. (n.d.). In WatkinsX Dictionary. Retrieved November 3, 2026, from https://dictionary.watkinsx.com/word/totally%20ordered

MLA

“totally ordered.” WatkinsX Dictionary, https://dictionary.watkinsx.com/word/totally%20ordered. Accessed November 3, 2026.

Chicago

WatkinsX Dictionary, s.v. “totally ordered,” accessed November 3, 2026, https://dictionary.watkinsx.com/word/totally%20ordered.

Source: totally ordered on Wiktionary, CC BY-SA 4.0.