• In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total ordering on S with the property that every non-empty subset...
    12 KB (1,882 words) - 17:26, 22 September 2024
  • the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set X is well-ordered by a strict total order if...
    7 KB (989 words) - 17:35, 14 March 2024
  • In mathematics, the well-ordering principle states that every non-empty subset of nonnegative integers contains a least element. In other words, the set...
    7 KB (1,144 words) - 09:34, 17 August 2024
  • In mathematics, specifically order theory, a well-quasi-ordering or wqo on a set X {\displaystyle X} is a quasi-ordering of X {\displaystyle X} for which...
    18 KB (3,049 words) - 11:59, 22 May 2024
  • In order theory, a partial order is called well-founded if the corresponding strict order is a well-founded relation. If the order is a total order then...
    10 KB (1,382 words) - 15:55, 4 November 2024
  • lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences...
    23 KB (3,352 words) - 07:24, 16 November 2023
  • mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation ≤...
    21 KB (3,093 words) - 12:07, 5 October 2024
  • ∗ {\displaystyle \sigma ^{*}} . The order type of a well-ordered set X is sometimes expressed as ord(X). The order type of the integers and rationals is...
    5 KB (695 words) - 12:50, 4 September 2024
  • Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing...
    31 KB (4,508 words) - 03:55, 24 August 2024
  • Thumbnail for Monotonic function
    or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus,...
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  • In the mathematical field of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism...
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    linear order such that every non-empty subset has a least element is called a well-order. The axiom of choice implies that every set can be well-ordered...
    48 KB (6,712 words) - 03:10, 2 November 2024
  • Thumbnail for Transfinite induction
    transfinite induction: First, well-order the real numbers (this is where the axiom of choice enters via the well-ordering theorem), giving a sequence ⟨...
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  • In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion...
    13 KB (1,766 words) - 09:56, 30 January 2024
  • New World Order is a non-fiction book written by H.G. Wells and was published by Secker & Warburg in January 1940. In The New World Order, Wells proposed...
    32 KB (4,109 words) - 13:27, 14 November 2024
  • In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (poset)...
    13 KB (1,924 words) - 19:19, 15 October 2024
  • is called orderable or linearly orderable if there exists a total order on its elements such that the order topology induced by that order and the given...
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  • Thumbnail for Partially ordered set
    In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other....
    40 KB (5,378 words) - 08:40, 19 November 2024
  • Thumbnail for Weak ordering
    In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose...
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  • Thumbnail for Georg Cantor
    natural numbers. It begins by defining well-ordered sets. Ordinal numbers are then introduced as the order types of well-ordered sets. Cantor then defines...
    84 KB (10,026 words) - 20:52, 9 November 2024
  • order relations on the set of monomials that are not well-orders. In the case of finitely many variables, well-ordering of a monomial order is equivalent...
    14 KB (2,070 words) - 18:51, 25 January 2024
  • this preorder is even a partial order (called the specialization order). On the other hand, for T1 spaces the order becomes trivial and is of little...
    10 KB (1,414 words) - 18:14, 11 November 2024
  • {\displaystyle L} . It is well known that the axiom of choice is equivalent to the ability to well-order every set. Being able to well-order the proper class V...
    32 KB (6,092 words) - 05:10, 28 August 2024
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    Preorder (redirect from Quasi-order)
    In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. The name preorder is meant...
    23 KB (3,351 words) - 13:50, 21 July 2024
  • and Boolean algebras. These lattice-like structures all admit order-theoretic as well as algebraic descriptions. The sub-field of abstract algebra that...
    40 KB (5,665 words) - 11:17, 1 November 2024
  • Thumbnail for Order of the British Empire
    The Most Excellent Order of the British Empire is a British order of chivalry, rewarding contributions to the arts and sciences, work with charitable and...
    58 KB (5,975 words) - 07:04, 22 November 2024
  • Thumbnail for Product order
    B} , respectively, the product order (also called the coordinatewise order or componentwise order) is a partial ordering ≤ {\displaystyle \leq } on the...
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  • Thumbnail for Order of Australia
    The Order of Australia is an Australian honour that recognises Australian citizens and other persons for outstanding achievement and service. It was established...
    57 KB (4,715 words) - 05:21, 9 November 2024
  • Thumbnail for Cyclic order
    mathematics, a cyclic order is a way to arrange a set of objects in a circle.[nb] Unlike most structures in order theory, a cyclic order is not modeled as...
    53 KB (6,391 words) - 13:03, 8 November 2024
  • incomparabilities) Well-orders, total orders in which every non-empty subset has a least element Well-quasi-orderings, a class of preorders generalizing the well-orders...
    2 KB (198 words) - 21:00, 15 December 2022