an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields...
14 KB (2,208 words) - 10:00, 10 June 2024
Real number (redirect from Complete ordered field)
real numbers form the unique (up to an isomorphism) Dedekind-complete ordered field. Other common definitions of real numbers include equivalence classes...
61 KB (8,184 words) - 14:03, 14 June 2024
Total order (redirect from TotalOrderedSet)
numbers. Every ordered field contains an ordered subfield that is isomorphic to the rational numbers. Any Dedekind-complete ordered field is isomorphic...
21 KB (3,094 words) - 17:19, 9 April 2024
mathematics, a non-Archimedean ordered field is an ordered field that does not satisfy the Archimedean property. Such fields will contain infinitesimal and...
4 KB (474 words) - 05:05, 2 March 2024
Archimedean property (redirect from Nonarchimedean ordered field)
is a property held by some algebraic structures, such as ordered or normed groups, and fields. The property, typically construed, states that given two...
16 KB (2,386 words) - 11:48, 4 May 2024
an ordered exponential field is an ordered field together with a function which generalises the idea of exponential functions on the ordered field of...
6 KB (799 words) - 11:55, 12 February 2022
mathematics Nested set collection Order polytope Ordered field – Algebraic object with an ordered structure Ordered group – Group with a compatible partial orderPages...
40 KB (5,395 words) - 04:34, 3 June 2024
Linearly ordered group – Group with translationally invariant total order; i.e. if a ≤ b, then ca ≤ cb Ordered field – Algebraic object with an ordered structure...
8 KB (1,120 words) - 11:18, 1 April 2024
an ordered field, with the usual ordering ≥. The Artin–Schreier theorem states that a field can be ordered if and only if it is a formally real field, which...
86 KB (10,288 words) - 20:18, 2 May 2024
rationals and reals in fact form ordered fields.) The complex numbers, in contrast, do not form an ordered ring or field, because there is no inherent order...
4 KB (557 words) - 16:10, 27 August 2023
Rational number (redirect from Rational field)
{Q} } is an ordered field that has no subfield other than itself, and is the smallest ordered field, in the sense that every ordered field contains a unique...
24 KB (3,494 words) - 09:10, 16 June 2024
Inequality (mathematics) (section Ordered fields)
involved. More generally, this applies for an ordered field. For more information, see § Ordered fields. The property for the additive inverse states...
28 KB (3,582 words) - 05:15, 28 May 2024
Well-order (redirect from Well-ordered set)
well order, well ordered, and well ordering. Every non-empty well-ordered set has a least element. Every element s of a well-ordered set, except a possible...
12 KB (1,882 words) - 01:42, 10 June 2024
complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ordered field exists...
30 KB (4,086 words) - 05:04, 2 June 2024
first-order language of fields is true in F if and only if it is true in the reals. There is a total order on F making it an ordered field such that, in this...
21 KB (2,974 words) - 05:47, 20 April 2024
Surreal number (category Real closed field)
they form an ordered field. If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that...
75 KB (11,406 words) - 06:19, 20 May 2024
Weak ordering (redirect from Ordered partition of a set)
generalization of totally ordered sets (rankings without ties) and are in turn generalized by (strictly) partially ordered sets and preorders. There are...
30 KB (4,360 words) - 19:22, 2 February 2024
In statistics, the ordered logit model (also ordered logistic regression or proportional odds model) is an ordinal regression model—that is, a regression...
8 KB (1,122 words) - 06:52, 20 May 2024
equipped with a (not necessarily unique) ordering that makes it an ordered field. The definition given above is not a first-order definition, as it requires...
3 KB (449 words) - 14:52, 3 January 2023
nonstandard reals (usually denoted as *R), denote an ordered field that is a proper extension of the ordered field of real numbers R and satisfies the transfer...
62 KB (7,755 words) - 07:51, 25 May 2024
Infinitesimal (section The Levi-Civita field)
include both hyperreal cardinal and ordinal numbers, which is the largest ordered field. Vladimir Arnold wrote in 1990: Nowadays, when teaching analysis, it...
37 KB (5,086 words) - 17:42, 25 May 2024
In mathematics, the Levi-Civita field, named after Tullio Levi-Civita, is a non-Archimedean ordered field; i.e., a system of numbers containing infinite...
8 KB (1,213 words) - 19:31, 11 June 2024
mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose...
19 KB (2,467 words) - 21:05, 28 May 2024
Characteristic (algebra) (redirect from Characteristic of a field)
fields that are widely used in number theory. They have absolute values which are very different from those of complex numbers. For any ordered field...
10 KB (1,266 words) - 15:15, 15 April 2024
states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding. The theorem was conjectured...
14 KB (1,739 words) - 21:56, 26 April 2024
In mathematics, a Euclidean field is an ordered field K for which every non-negative element is a square: that is, x ≥ 0 in K implies that x = y2 for...
4 KB (435 words) - 16:59, 18 July 2021
article, only the case of scalars in an ordered field is considered. A subset C of a vector space V over an ordered field F is a cone (or sometimes called a...
24 KB (3,204 words) - 06:45, 23 May 2024
Non-Archimedean geometry (category Fields of geometry)
ordered field, or a subset thereof. The aforementioned Dehn plane takes the self-product of the finite portion of a certain non-Archimedean ordered field...
2 KB (321 words) - 15:21, 4 February 2024
Order topology (redirect from Linearly ordered topological space)
totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If X is a totally ordered set,...
15 KB (2,107 words) - 17:53, 7 June 2024
Lattice (order) (section As partially ordered set)
sub-field of abstract algebra that studies lattices is called lattice theory. A lattice can be defined either order-theoretically as a partially ordered set...
40 KB (5,663 words) - 17:53, 6 June 2024