well-order

plural well-orders · alternative well order

1. mathematicsA 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, ̲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.

   2 more examples

   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∈X, we have x=y∈X|y<x (that is x is equal to the set Pred(x) consisting of its predecessors). Clearly the examples listed by von Neumann above, namely empty , empty, empty ,empty, empty ,empty,empty ,empty, … are all von Neumann well-orders if ordered by the membership relation "∈," 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: well-ordering

present singular third person well-orders · participle present well-ordering · participle past well-ordered · past well-ordered · alternative well order

1. transitivemathematicsTo impose a well-order on (a set).

   The set of positive integers is well-ordered by the relation ≤.

   3 more examples

   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.

   Then #92;le#95;C is a well defined order on C, and (C,#92;le#95;C) belongs to #92;mathcal#123;X#125; (that is, #92;le#95;C well orders C) and is an upper bound for #92;mathcal#123;C#125;.

   2006, Charalambos D. Aliprantis, Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, 3rd edition, Springer, page 18