# well-order
## English
### Alternative forms

* well order

### Noun

well-order (plural well-orders)
* (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
       $\emptyset ,\quad \{\emptyset \},\quad \{\emptyset ,\{\emptyset \}\},\quad \{\emptyset ,\{\emptyset \},\{\emptyset ,\{\emptyset \}\}\},\quad \dots$
    are all von Neumann well-orders if ordered by the membership relation "$\in$," 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
### Verb

well-order (third-person singular simple present well-orders, present participle well-ordering, simple past and past participle well-ordered)
* (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, 3rd edition, Springer, page 18:Then $\leq _{C}$ is a well defined order on $C$, and $(C,\leq _{C})$ belongs to ${\mathcal {X}}$ (that is, $\leq _{C}$ well orders $C$) and is an upper bound for ${\mathcal {C}}$.

#### Translations
### 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