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Every ordered field can be embedded into the surreal numbers.

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## Some Related Sentences

Every and ordered

** Zorn's lemma:

__Every__non-empty partially__ordered__set in which every chain ( i**.**e**.**totally__ordered__subset ) has an upper bound contains at least one maximal element**.**
The restricted principle "

__Every__partially__ordered__set has a maximal totally__ordered__subset " is also equivalent to AC over ZF**.**__Every__

__ordered__

**field**is a formally real

**field**, i

**.**e., 0 cannot

**be**written as a sum of nonzero squares

**.**

*

__Every__non-empty set of left ideals of R, partially__ordered__by inclusion, has a maximal element with respect to set inclusion**.**
*

__Every__totally__ordered__set that is a bounded lattice is also a Heyting algebra, where is equal to when, and 1 otherwise**.**__Every__time she entered, song typical of

**the**Brazilian northeast would play, stopping only when

__ordered__by Cirene herself

**.**

Every and field

__Every__

__field__has an algebraic extension which is algebraically closed ( called its algebraic closure ), but proving this in general requires some form of

**the**axiom of choice

**.**

__Every__time an MTA receives an email message, it adds a < tt > Received </ tt > trace header

__field__to

**the**top of

**the**header of

**the**message, thereby building a sequential record of MTAs handling

**the**message

**.**

__Every__instruction consists of a 9-bit opcode, a 4-bit register code, and a 23-bit effective address

__field__, which consists in turn of a 1-bit indirect bit, a 4-bit register code, and an 18-bit offset

**.**

__Every__

__field__theory of particle physics is based on certain symmetries of nature whose existence is deduced from observations

**.**

__Every__planetary body ( including

**the**Earth ) is surrounded by its own gravitational

__field__, which exerts an attractive force on all objects

**.**

__Every__prime ideal P in a Boolean ring R is maximal:

**the**quotient ring R / P is an integral domain and also a Boolean ring, so it is isomorphic to

**the**

__field__F < sub > 2 </ sub >, which shows

**the**maximality of P

**.**Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings

**.**

__Every__year,

**the**British Political Studies Association awards

**the**Walter Bagehot Prize for

**the**best dissertation in

**the**

__field__of government and public administration

**.**

__Every__polynomial in

**can**

**be**factorized

**into**polynomials that are irreducible over F

**.**This factorization is unique up to permutation of

**the**factors and

**the**multiplication of

**the**factors by nonzero constants from F ( because

**the**ring of polynomials over a

__field__is a unique factorization domain whose units are

**the**nonzero constant polynomials ).

__Every__quadratic form q in n variables over a

__field__of characteristic not equal to 2 is equivalent to a diagonal form

__Every__kind of material has unique magnetic properties, even those that we do not think of as being “ magnetic .” Different materials below

**the**ground

**can**cause local disturbances in

**the**Earth ’ s magnetic

__field__that are detectable with sensitive magnetometers

**.**

Every and can

__Every__such subset has a smallest element, so to specify our choice function we

__can__simply say that it maps each set to

**the**least element of that set

**.**

__Every__information exchange between living organisms — i

**.**e

**.**transmission of signals that involve a living sender and receiver

__can__

**be**considered a form of communication ; and even primitive creatures such as corals are competent to communicate

**.**

__Every__context-sensitive grammar which does not generate

**the**empty string

__can__

**be**transformed

**into**an equivalent one in Kuroda normal form

**.**

__Every__grammar in Chomsky normal form is context-free, and conversely, every context-free grammar

__can__

**be**transformed

**into**an equivalent one which is in Chomsky normal form

**.**

__Every__module over a division ring has a basis ; linear maps between finite-dimensional modules over a division ring

__can__

**be**described by matrices, and

**the**Gaussian elimination algorithm remains applicable

**.**

__Every__entire function

__can__

**be**represented as a power series that converges uniformly on compact sets

**.**

Group actions / representations:

__Every__group G__can__**be**considered as a category with a single object whose morphisms are**the**elements of G**.**A functor from G to Set is then nothing but a group action of G on a particular set, i**.**e**.**a G-set**.**__Every__sequence

__can__, thus,

**be**read in three reading frames, each of which will produce a different amino acid sequence ( in

**the**given example, Gly-Lys-Pro, Gly-Asn, or Glu-Thr, respectively ).

__Every__hyperbola is congruent to

**the**origin-centered East-West opening hyperbola sharing its same eccentricity ε ( its shape, or degree of " spread "), and is also congruent to

**the**origin-centered North-South opening hyperbola with identical eccentricity ε — that is, it

__can__

**be**rotated so that it opens in

**the**desired direction and

__can__

**be**translated ( rigidly moved in

**the**plane ) so that it is centered at

**the**origin

**.**

__Every__holomorphic function

__can__

**be**separated

**into**its real and imaginary parts, and each of these is a solution of Laplace's equation on R < sup > 2 </ sup >.

__Every__species

__can__

**be**given a unique ( and, one hopes, stable ) name, as compared with common names that are often neither unique nor consistent from place to place and language to language

**.**

__Every__vector v in determines a linear map from R to taking 1 to v, which

__can__

**be**thought of as a Lie algebra homomorphism

**.**

__Every__use of modus tollens

__can__

**be**converted to a use of modus ponens and one use of transposition to

**the**premise which is a material implication

**.**

__Every__adult, healthy, sane Muslim who has

**the**financial and physical capacity to travel to Mecca and

__can__make arrangements for

**the**care of his / her dependants during

**the**trip, must perform

**the**Hajj once in a lifetime

**.**

*

__Every__finite topological space gives rise to a preorder on its points, in which x ≤ y if and only if x belongs to every neighborhood of y, and every finite preorder__can__**be**formed as**the**specialization preorder of a topological space in this way**.**
*

__Every__preorder__can__**be**given a topology,**the**Alexandrov topology ; and indeed, every preorder on a set is in one-to-one correspondence with an Alexandrov topology on that set**.**__Every__binary relation R on a set S

__can__

**be**extended to a preorder on S by taking

**the**transitive closure and reflexive closure, R < sup >+=</ sup >.

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