English
well-order
Noun

well-order

  1. (set theory, order theory) A total order of some set such that every nonempty subset contains a least element.
    • 1986, G. Richter, Noetherian semigroup rings with several objects, G. Karpilovsky (editor), Group and Semigroup Rings, Elsevier (North-Holland), page 237 ↗,
      \underline{X} is well-order enriched iff every morphism set \underline{X}(X,Y) carries a well-order \leq_{XY} such that
      f\lneqq_{XY} g \Rightarrow h\bullet f\lneqq_{XY} h\bullet g
      for every h : Y \rightarrow Z.
    • 2001, Robert L. Vaught, Set Theory: An Introduction, Springer (Birkhäuser), 2nd Edition, Softcover, page 71 ↗,
      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 limit element - i.e., x has a predecessor but no immediate predecessor. The structure (∅, ∅) is a well-order.
    • 2014, Abhijit Dasgupta, Set Theory: With an Introduction to Real Point Sets, Springer (Birkhäuser), page 378 ↗,
      Definition 1226 (Von Neumann Well-Orders). A well-order X is said to be a von Neumann well-order if for every x\in X, we have x=\{y\in X\vert y< x\} (that is x is equal to the set \mathrm{Pred}(x) consisting of its predecessors).
      Clearly the examples listed by von Neumann above, namely
      \empty,\quad \{\empty\},\quad \{\empty, \{\empty\}\},\quad \{\empty, \{\empty\}, \{\empty, \{\empty\}\}\},\quad\dots
      are all von Neumann well-orders if ordered by the membership relation "\in," and the process can be iterated through the transfinite. Our immediate goal is to show that these and only these are the von Neumann well-orders, with exactly one von Neumann well-order for each ordinal (order type of a well-order). This is called the existence and uniqueness result for the von Neumann well-orders.
Synonyms Verb

well-order (well-orders, present participle well-ordering; past and past participle well-ordered)

  1. (set theory, order theory, transitive) To impose a well-order on (a set).
    The set of positive integers is well-ordered by the relation ≤.
    • 1950, Frederick Bagemihl (translator), Erich Kamke, Theory of Sets, 2006, Dover (Dover Phoenix), page 111 ↗,
      Starting from these special well-ordered subsets, it is then possible to well-order the entire set.
    • 1975 [The Williams & Wilkins Company], Dennis Sentilles, A Bridge to Advanced Mathematics, Dover, 2011, page 182 ↗,
      To carry the analogy a bit further, the axiom of choice implies the ability to well order any set.
    • 2006, Charalambos D. Aliprantis, Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, Springer, 3rd Edition, page 18 ↗,
      Then \le_C is a well defined order on C, and (C,\le_C) belongs to \mathcal{X} (that is, \le_C well orders C) and is an upper bound for \mathcal{C}.


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