Word.ws English Dictionary

well-order

well-order noun: a total order of some set such that every nonempty subset contains a least element.

Noun

well-order (plural well-orders)

  1. A total order of some set such that every nonempty subset contains a least element.
    • ̲X is well-order enriched iff every morphism set ̲X(X,Y) carries a well-order ≤_(XY) such that
      f≨_(XY)g⇒h•f≨_(XY)h•g
      for every h:Y→Z.1986, G. Richter, Noetherian semigroup rings with several objects, G. Karpilovsky (editor), Group and Semigroup Rings, Elsevier (North-Holland), page 237
    • Some simple facts and terminology about well-orders were already given in and just before 1.8.4. Here are some more: In a well-order A, every element x is clearly of just one of these three kinds: x is the first element; x is a successor element - i.e., x has an immediate predecessor; or x is a...2001, Robert L. Vaught, Set Theory: An Introduction, Springer (Birkhäuser), 2nd Edition, Softcover, page 71
    Synonyms:
    well-ordering

Verb

well-order (third-person singular simple present well-orders, present participle well-ordering, simple past and past participle well-ordered)

  1. (transitive) To impose a well-order on (a set).
    • The set of positive integers is well-ordered by the relation ≤.
    • Starting from these special well-ordered subsets, it is then possible to well-order the entire set.1950, Frederick Bagemihl (translator), Erich Kamke, Theory of Sets, 2006, Dover (Dover Phoenix), page 111

Synonyms of well-order

Forms of well-order

Noun
pluralwell-orders
alternative formwell order
Verb
third-person singular presentwell-orders
present participlewell-ordering
past participlewell-ordered
pastwell-ordered

Questions about well-order

What does well-order mean?
Well-order means a total order of some set such that every nonempty subset contains a least element.
What part of speech is well-order?
Well-order can be a noun or a verb.
What is the plural of well-order?
The plural of well-order is well-orders.

Words near well-order in the dictionary

  1. well-off
  2. well-offness
  3. well-oiled
  4. well-oiled machine
  5. well-oiled machines
  6. well-order
  7. well-orderable
  8. well-ordered
  9. well-ordering
  10. well-orderings
  11. well-orders

Source and license

Adapted from Wiktionary: well-order (page history and authors). Text available under the Creative Commons Attribution-ShareAlike 4.0 license. Word.ws reformatted the entry and added links. Wiktionary data of 2026-09-02.

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