See totally ordered on Wiktionary
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"text": "1976, K. D. Stroyan, W. A. J. Luxemburg, Introduction to the Theory of Infinitesimals, Harcourt Brace Jovanovich (Academic Press), page 67,\n(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)."
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"ref": "1982, A. G. Hamilton, Numbers, Sets and Axioms: The Apparatus of Mathematics, Cambridge University Press, page 91:",
"text": "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'.",
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"ref": "1996, Scientific Books staff (translators), Vasiliǐ M. Kopytov, Nikolaǐ Ya. Medvedev, Right-Ordered Groups, Scientific Books, page 98, We introduce the following notation",
"roman": "ζ is the order type of the totally ordered quotient set X/θ of the totally ordered set X by the congruence θ.",
"text": "G is a transitive group of order automorphisms of a totally ordered set X,\nθ is a convex G-congruence on the totally ordered set X,\nξ is the order type of the totally ordered set X,\n◌̅η is the order type of some class Y=xη of the congruence η,"
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"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."
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"(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."
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"text": "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'.",
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Download raw JSONL data for totally ordered meaning in All languages combined (2.8kB)
This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-12-21 from the enwiktionary dump dated 2024-12-04 using wiktextract (d8cb2f3 and 4e554ae). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
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