well-order

well-order

English

Alternative forms

• well order

Noun

well-order (plural well-orders)

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,

     ${\\displaystyle {\\underline {X}}}$ is well-order enriched iff every morphism set ${\\displaystyle {\\underline {X}}(X,Y)}$ carries a well-order ${\\displaystyle \\leq _{XY}}$ such that

     ${\\displaystyle f\\lneqq _{XY}g\\Rightarrow h\\bullet f\\lneqq _{XY}h\\bullet g}$

     for every ${\\displaystyle h:Y\\rightarrow Z}$.

   • 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 ${\\displaystyle X}$ is said to be a von Neumann well-order if for every ${\\displaystyle x\\in X}$, we have ${\\displaystyle x=\\{y\\in X\\vert y<x\\}}$ (that is ${\\displaystyle x}$ is equal to the set ${\\displaystyle \\mathrm {Pred} (x)}$ consisting of its predecessors).

     Clearly the examples listed by von Neumann above, namely

     ${\\displaystyle \\emptyset ,\\quad \\{\\emptyset \\},\\quad \\{\\emptyset ,\\{\\emptyset \\}\\},\\quad \\{\\emptyset ,\\{\\emptyset \\},\\{\\emptyset ,\\{\\emptyset \\}\\}\\},\\quad \\dots }$

     are all von Neumann well-orders if ordered by the membership relation "${\\displaystyle \\in }$," 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

• (type of total order): well-ordering

Hypernyms

• (type of total order):
  • total order
    • partial order
      • preorder

Translations

a type of total order

Danish: velordning c Dutch: welordening f Finnish: hyvinjärjestys German: Wohlordnung f Norwegian: Bokmål: velordning m or f Nynorsk: velordning f Polish: porządek uporządkowany m Swedish: välordning c

Verb

well-order (third-person singular simple present well-orders, present participle well-ordering, simple 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 ${\\displaystyle \\leq _{C}}$ is a well defined order on ${\\displaystyle C}$, and ${\\displaystyle (C,\\leq _{C})}$ belongs to ${\\displaystyle {\\mathcal {X}}}$ (that is, ${\\displaystyle \\leq _{C}}$ well orders ${\\displaystyle C}$) and is an upper bound for ${\\displaystyle {\\mathcal {C}}}$.

Translations

Translations

Finnish: hyvinjärjestää German: wohlordnen Polish: uporządkowywać impf, uporządkować (pl) pf

See also

• ordinal number
• tree

Further reading

• Ordinal number on Wikipedia.Wikipedia
• Well-ordering theorem on Wikipedia.Wikipedia
• Well-ordering principle on Wikipedia.Wikipedia
• Tree (set theory) on Wikipedia.Wikipedia
• Well-founded relation on Wikipedia.Wikipedia