See well-order in All languages combined, or Wiktionary
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"text": "1986, G. Richter, Noetherian semigroup rings with several objects, G. Karpilovsky (editor), Group and Semigroup Rings, Elsevier (North-Holland), page 237,\n̲X is well-order enriched iff every morphism set ̲X(X,Y) carries a well-order ≤_(XY) such that\nf≨_(XY)g⇒h•f≨_(XY)h•g\nfor every h:Y→Z."
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"text": "2001, Robert L. Vaught, Set Theory: An Introduction, Springer (Birkhäuser), 2nd Edition, Softcover, page 71,\nSome 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."
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"text": "2014, Abhijit Dasgupta, Set Theory: With an Introduction to Real Point Sets, Springer (Birkhäuser), page 378,\nDefinition 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).\nClearly the examples listed by von Neumann above, namely\nempty , empty, empty ,empty, empty ,empty,empty ,empty, …\nare 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."
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},
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},
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Download raw JSONL data for well-order meaning in English (7.3kB)
This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-11-06 from the enwiktionary dump dated 2024-10-02 using wiktextract (fbeafe8 and 7f03c9b). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
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