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...

real numbers form the unique (up to an isomorphism) Dedekind-complete ordered field. Other common definitions of real numbers include equivalence classes...

is a property held by some algebraic structures, such as ordered or normed groups, and fields. The property, as typically construed, states that given...

numbers. Every ordered field contains an ordered subfield that is isomorphic to the rational numbers. Any Dedekind-complete ordered field is isomorphic...

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...

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...

mathematics, a non-Archimedean ordered field is an ordered field that does not satisfy the Archimedean property. Such fields will contain infinitesimal and...

an ordered exponential field is an ordered field together with a function which generalises the idea of exponential functions on the ordered field of...

mathematics Nested set collection Order polytope Ordered field – Algebraic object with an ordered structure Ordered group – Group with a compatible partial orderPages...

involved. More generally, this applies for an ordered field. For more information, see § Ordered fields. The property for the additive inverse states...

{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...

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...

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...

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...

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...

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...

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...

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...

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...

states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding. A finitary application...

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...

chosen a different free ultrafilter V, the quotient field A/U would be isomorphic as an ordered field to A/V. This question turns out to be equivalent to...

include both hyperreal cardinal and ordinal numbers, which is the largest ordered field. Vladimir Arnold wrote in 1990: Nowadays, when teaching analysis, it...

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...

mathematics, the field T L E {\displaystyle \mathbb {T} ^{LE}} of logarithmic-exponential transseries is a non-Archimedean ordered differential field which extends...

mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose...

In mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the...

theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least...

trace in order to be gauge-invariant. In quantum field theory it is useful to take the time-ordered product of operators. This operation is denoted by...

dependent variable falling into a higher category. Ordered logistic regressions have been used in multiple fields, such as transportation, marketing or disaster...