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Page "Ordered field" ¶ 2
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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 is a formally real field.
Every subfield of an ordered field is also an ordered field ( inheriting the induced ordering ).
Every ordered field can be embedded into the surreal numbers.
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 contains
** Every unital ring other than the trivial ring contains a maximal ideal.
: Every non-empty set A contains an element B which is disjoint from A.
Every computer contains an internal clock that regulates the rate at which instructions are executed and synchronizes all the various computer components.
Cantor points out that his constructions prove more — namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers.
Every organic compound contains at least one atom of carbon.
The revised edition of Robyn was released in the UK in April 2007, and contains two new tracks —" With Every Heartbeat " ( a collaboration with Kleerup ) and " Cobrastyle " ( a cover of a 2006 single by Swedish rockers Teddybears )— alongside slightly altered versions of two of the original songs.
# Every simple path from a given node to any of its descendant leaves contains the same number of black nodes.
Every altar in every Orthodox church contains relics, usually of martyrs.
Every synset contains a group of synonymous words or collocations ( a collocation is a sequence of words that go together to form a specific meaning, such as " car pool "); different senses of a word are in different synsets.
Every PostScript printer contains a RIP in its firmware.
* Every connected graph G admits a spanning tree, which is a tree that contains every vertex of G and whose edges are edges of G.
Every one has higher glucose levels for one or two hours after eating food which contains some types of carbohydrates.
Every bag of blood ( 450 – 500 ml ) contains 200 – 250 milligrams of iron.
Every temple of The Church of Jesus Christ of Latter-day Saints ( LDS Church ) contains a baptismal font on twelve oxen that is modeled after the molten sea.
Every issue of GAMES Magazine contains a large crossword with a double clue list, under the title The World's Most Ornery Crossword ; both lists are straight and arrive at the same solution, but one list is significantly more challenging than the other.
Every gap sequence that contains 1 yields a correct sort ; however, the properties of thus obtained versions of Shellsort may be very different.
Every prostaglandin contains 20 carbon atoms, including a 5-carbon ring.
Every device contains at least one swap chain.
Every living organism contains DNA, RNA, and proteins.
Every game in the Mario Party series contains 50 to 80 mini-games of a few different types.
The offset part of the logical address contains an offset inside the segment, i. e. the physical address can be calculated as ( if the address line A20 is enabled ), respectively ( segment_part × 16 + offset ) mod 2 < sup > 20 </ sup > ( if A20 is off ) Every segment has a size of 2 < sup > 16 </ sup > bytes.
Every chapter contains a comprehensive compilation all of the published examples of the reaction organized in tables according to the structure of the starting material.
Every Riemann surface is a two-dimensional real analytic manifold ( i. e., a surface ), but it contains more structure ( specifically a complex structure ) which is needed for the unambiguous definition of holomorphic functions.

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