well-order

Example Sentences

• "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."
• "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
• "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."