[permalink] [id link]
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
.

** Antichain principle:
Every partially
ordered set has a maximal antichain
.

*
Every cofinal subset of a partially
ordered set must contain all maximal elements of that set
.
Every non-empty totally
ordered set is directed
.
Every subfield of an
ordered field is also an
ordered field in
the inherited order
.
Every ordered field contains an
ordered subfield that is isomorphic to
the rational
numbers.
Every ordered field is a formally real
field.
Every subfield of an
ordered field is also an
ordered field ( inheriting
the induced ordering ).
Every ordered field is a formally real
field, i
. e., 0 cannot
be written as a sum of nonzero squares
.

*
Every totally
ordered set with
the order topology is Tychonoff
.

*
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 totally
ordered set is a distributive lattice with max as join and min as meet
.
Every relatively atomic partially
ordered set with a least element is atomic
.
Every ordered basis lives in one equivalence class or another
.
Every time she entered, song typical of
the Brazilian northeast would play, stopping only when
ordered by Cirene herself
.
Every partially
ordered set is cofinal in itself
.
Every and field

**
Every field has an algebraic closure
.

**
Every field extension has a transcendence basis
.
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 field is, of course, one-dimensional over its center
.
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 local
field is isomorphic ( as a topological
field ) to one of
the following:
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
.

** Well-ordering theorem:
Every set
can be well-ordered
.
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 regular language is context-free because it
can be described by a context-free grammar
.
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 real number has a ( possibly infinite ) decimal representation ; i
. e., it
can be written as
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 positive integer n > 1
can be represented in exactly one way as a product of prime powers:
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 morpheme
can be classified as either free or bound
.
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 document window is an object with which
the user
can work
.
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 >.
0.343 seconds.