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...
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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...
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In mathematics, the well-ordering principle states that every non-empty subset of nonnegative integers contains a least element. In other words, the set...
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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...
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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...
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lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences...
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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 ≤...
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∗ {\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...
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Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing...
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Monotonic function (redirect from Order-preserving)
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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Ordinal number (section Well-ordered sets)
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...
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Transfinite induction (redirect from Well-ordered 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...
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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...
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In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (poset)...
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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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Partially ordered set (redirect from Partial order)
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....
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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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Georg Cantor (redirect from Absolute infinite, well-ordering theorem, and paradoxes)
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...
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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...
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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...
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Constructible universe (section L can be well-ordered)
{\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...
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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...
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and Boolean algebras. These lattice-like structures all admit order-theoretic as well as algebraic descriptions. The sub-field of abstract algebra that...
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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...
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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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The Order of Australia is an Australian honour that recognises Australian citizens and other persons for outstanding achievement and service. It was established...
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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...
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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...
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