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Topic: Dedekind completion

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Dedekind cut

Supremum

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	Dedekind cut - Wikipedia, the free encyclopedia
		For example it is shown that the typical Dedekind cut in the real numbers is either a pair with A the interval (−∞, a), in which case B must be [ a, +∞); or a pair with A the interval (−∞, a ], in which case B must be (a, +∞).
		Regard one Dedekind cut { A, B } as less than another Dedekind cut { C, D } if A is a proper subset of C, or, equivalently D is a proper subset of B.
		The Dedekind cut is named after Richard Dedekind, who invented this construction in order to represent the real numbers as Dedekind cuts of the rational numbers.
	en.wikipedia.org /wiki/Dedekind_cut (511 words)

	Articles - Complete lattice (Site not responding. Last check: 2007-10-22)
		Complete lattices must not be confused with complete partial orders (cpos), which constitute a strictly more general class of partially ordered sets.
		Using the standard definition from universal algebra, a free complete lattice over a generating set S is a complete lattice L together with a function i:Sâ†’L, such that any function f from S to the underlying set of some complete lattice M can be factored uniquely through a morphism fÂ° from L to M.
		When this completion is applied to a poset that already is a complete lattice, then the result is a complete lattice of sets which is isomorphic to the original one.
	www.izeez.com /articles/Complete_lattice (2059 words)

	PlanetMath: real number (Site not responding. Last check: 2007-10-22)
		More technically, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section).
		This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way.
		This sense of completeness is most closely related to the construction of the reals from surreal numbers, since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield.
	planetmath.org /encyclopedia/RealNumber.html (861 words)

	Real number - Wikipedia, the free encyclopedia
		Since it can be proved that any uniformly complete Archimedean field must also be Dedekind complete (and vice versa, of course), this justifies using "the" in the phrase "the complete Archimedean field".
		This sense of completeness is most closely related to the construction of the reals from Cauchy sequences (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals) and forms the uniform completion of it in a standard way.
		But the original use of the phrase "complete Archimedean field" was by David Hilbert, who meant still something else by it.
	en.wikipedia.org /wiki/Real_number (1943 words)

	Station Information - Dedekind cut
		A Dedekind cut in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that whenever a is in A and x ≤ a, then x is in A as well), B is closed upwards.
		If the linearly ordered set S does not enjoy the least-upper-bound property, then the set of Dedekind cuts is strictly bigger than S.
		A construction very similar to Dedekind cuts is used for the construction of surreal numbers.
	www.stationinformation.com /encyclopedia/d/de/dedekind_cut.html (299 words)

	Encyclopedia: Dedekind cut (Site not responding. Last check: 2007-10-22)
		Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 â€“ February 12, 1916) was a German mathematician and Ernst Eduard Kummers closest follower in arithmetic.
		In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound.
		A typical Dedekind cut of the rational numbers is given by Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 â€“ February 12, 1916) was a German mathematician and Ernst Eduard Kummers closest follower in arithmetic.
	www.nationmaster.com /encyclopedia/Dedekind-cut (1079 words)

	Dedekind cut - TheBestLinks.com - Dedekind completion, Cauchy sequence, Interval, Mathematics, ... (Site not responding. Last check: 2007-10-22)
		Dedekind cut - TheBestLinks.com - Dedekind completion, Cauchy sequence, Interval, Mathematics,...
		Dedekind completion, Dedekind cut, Cauchy sequence, Interval, Mathematics...
		For example it is shown that the typical Dedekind cut in the real numbers is either a pair with A the interval (-∞,a), in which case B must be [a,+∞); or a pair with A the interval (-∞,a], in which case B must be (a,+∞).
	www.thebestlinks.com /Dedekind_completion.html (535 words)

	Real number - FreeEncyclopedia (Site not responding. Last check: 2007-10-22)
		The order is Dedekind complete[?], i.e., every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R.
		Construction by Dedekind cuts -- A Dedekind cut in an ordered field is a partition of it, (A, B), such that A is nonempty and closed downwards, B is nonempty and closed upwards, and A has no maximum.
		Occasionally, formal elements +∞ and -∞ are added to the reals to form the extended real number line, a compact space which is not a field anymore but retains many of the properties of the real numbers.
	openproxy.ath.cx /re/Real_number.html (2236 words)

	[No title]
		complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section).
		completeness (topology); the description in the section Completeness above is a special case.
		matrices) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvalues are real and they form a real associative algebra.
	en-cyclopedia.com /wiki/Real_number (1750 words)

	Dedekind cut - Encyclopedia Glossary Meaning Explanation Dedekind cut (Site not responding. Last check: 2007-10-22)
		Dedekind cut - Encyclopedia Glossary Meaning Explanation Dedekind cut.
		Here you will find more informations about Dedekind cut.
		For example it is shown that the typical Dedekind cut in the real numbers is either a pair with A the interval (−∞, a), in which case B must be
	www.encyclopedia-glossary.com /en/Dedekind-cut.html (529 words)

	Information on Real number (Site not responding. Last check: 2007-10-22)
		The order is Dedekind completion, i.e., every Empty set subset S of R with an Upper bound in R has a Supremum (also called supremum) in R.
		More technically, the reals are Completeness (topology) (in the sense of Metric space or Uniform space, which is a different sense than the Dedekind completeness of the order in the previous section).
		Hermitian on a Hilbert space (for example, self-adjoint square complex Matrix (math)) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their Eigenvector are real and they form a real Associative algebra.
	www.information-resource.net /search/Real_number.html (1980 words)

	Real number (Site not responding. Last check: 2007-10-22)
		More technically, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completenessof the order in the previous section).
		This sense of completeness is mostclosely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (therationals) and then forms the Dedekind-completion of it in a standard way.
		This sense of completeness ismost closely related to the construction of the reals from surrealnumbers, since that construction starts with a proper class that contains every ordered field (the surreals) and then selectsfrom it the largest Archimedean subfield.
	www.therfcc.org /real-number-3907.html (1793 words)

	Algebra univers. (Site not responding. Last check: 2007-10-22)
		Monk [1970] extended the notion of the completion of a Boolean algebra to Boolean algebras with operators.
		Under the assumption that the operators of such an algebra $ \cal A $ are completely additive, he showed that the completion of $ \cal A $ always exists and is unique up to isomorphisms over $ \cal A $.
		Moreover, strictly positive equations are preserved under completions a strictly positive equation that holds in $ \cal A $ must hold in the completion of $ \cal A $.
	202.38.126.65 /mirror/mathlib.nankai.edu.cn/link/service/journals/00012/bibs/9041001/90410047.htm (208 words)

	Articles in Conference Proceedings
		We prove the following fact: the MacNeille-Dedekind completion of a free modal μ-algebra is a complete modal algebra, hence a modal μ-algebra (i.e.
		This is not the case for non distributive lattices, as the second author has shown that the alternation-depth hierarchy is infinite.
		We show on the other hand that the existence of a term which does not preserve the order is an essential condition for the least prefixed point to be definable by equations.
	www.lif.univ-mrs.fr /~lsantoca/respapers/proceedings-abstracts.html (852 words)

	ABSTRACTS DROSTE
		We construct a complete subfield F of P(R), isomorphic to P(R), of pairwise non-Borel-isomorphic rigid strong Blackwell subsets of R such that there are only 'very few' measurable functions between any two members of F.
		Using combinatorial methods, we prove that in each of these lattices the partially ordered subset of all those elements which are finitely generated as normal subgroups is a lattice in which infima and suprema of subsets of cardinality $\leq\aleph$, always exist; two infinite distributive identities are also shown to hold.
		We characterize the structure of the normal subgroup lattice of 2-transitive automorphism groups $A(\Omega)$ of infinite chains $(\Omega,\leq)$ by the structure of the Dedekind completion $(\bar\Omega,\leq)$ of the chain $(\Omega,\leq)$.
	www.math.tu-dresden.de /alg/droabal.html (3181 words)

	[No title]
		Also, by \cite{rob}, the complete theory (in for example $\L_{\div}$) of an algebraically closed non-trivially valued field $K$ is determined by the pair $(\char(K),\char(k))$.
		Part (i) is made explicit in \cite{mmd}, but follows quickly from the model completeness and existence of prime models in \cite{rob}.
		Suppose that $T$ is an arbitrary complete theory, $M$ is a model of $T$, and $p$ is a type over $M$ defined over $B \subset M$.
	www.amsta.leeds.ac.uk /Pure/staff/macpherson/hdm22.tex (8837 words)

	ORDINAL REAL NUMBERS 2
		It is surprising that in one of the consequences of the theory of ordinal real numbers, it is proved a far more advanced and complete result for the whole category of order types that has as corollary the previous important and elementary arithmetisation.
		But as an erroneous application of terms R is also the minimal Cauchy complete field of characteristic w and this also applies for the fields Rα in the sense that a completion of a linearly ordered field of characteristic α must be the field Rα.
		In other words the fields of ordinal real numbers are Archemidean complete fields (although they may be non-Archemidean).But this is a characteristic property of the fields of transfinite real numbersb of Glayzal.
	www.softlab.ntua.gr /~kyritsis/PapersInMaths/InfinityandStochastics/OR2.htm (2766 words)

	Atlas: Canonical completions and MacNeille completions by John Harding (Site not responding. Last check: 2007-10-22)
		Two common methods to complete a Boolean algebra are the MacNeille completion and the canonical completion.
		The MacNeille completion is the familiar generalization of Dedekind's completion by cuts.
		The canonical completion is obtained by embedding a Boolean algebra into the power set of its Stone space.
	atlas-conferences.com /cgi-bin/abstract/caoc-02 (217 words)

	anguelov (Site not responding. Last check: 2007-10-22)
		The order completion method for solving nonlinear partial differential equations is based on the order completion of spaces of usual piecewise smooth functions defined on a set
		The application of Hausdorff continuous interval-valued functions is motivated by their exceptional property of being a Dedekind order complete set.
		which is order complete and contains the sets of piecewise smooth functions used in the order completion method.
	academic.sun.ac.za /maths/SAMS02/Abstracts/anguelov (122 words)

	[No title]
		An embedding $\varphi:(C,\leq)\longrightarrow (S,\leq)$ is said to be {\em complete} if it preserves all suprema and infima of subsets of $C$ which happen to exist in $(C,\leq)$.
		The sets $S_1$ and $S_2$ lie in the Dedekind completion $\overline{S}_0$ of $S_0$, and we have $z \in \At(\overline{S}_0)$ and $z^3 \in \At(S_0)$.
		By Theorem~\ref{main2}(d), the Dedekind completion of the latter contains elements of character $(\aleph_0,\aleph_1)$ but not elements of character $(\aleph_1,\aleph_0)$, so is not isomorphic to its reverse ordering.
	www.maths.leeds.ac.uk /Pure/preprints/hdm/hdm3 (2523 words)

	Algebra univers. (Site not responding. Last check: 2007-10-22)
		We show that the permutations of $ \bar{\mit\Omega} $ which commute with the tyings are exactly those in the closure of G in the full automorphism group $ \bar{\mit\Omega} $ with respect to the coarse stabilizer topology.
		We show that each o-primitive component of $ \G^: $ consists of those permutations of the closure of the corresponding G component which respect the orbits of the points which are "tied" to nonsingleton o-blocks.
		Finally, we show that any two representations of the same lattice-ordered group which are complete and without dead segments give rise to the same $ \G^: $, and that in this case $ \G^: $ is the $\alpha$ -completion of G.
	202.38.126.65 /mirror/mathlib.nankai.edu.cn/link/service/journals/00012/bibs/7037001/70370024.htm (203 words)

	When the Maximum Ring of Quotients of C(X) is Uniformly Complete (ResearchIndex)
		Abstract: A Tychono space X such that the maximum ring of quotients of C(X) is uniformly complete is called a uniform quotients space.
		It is shown that this condition is equivalent to the Dedekind-MacNeille completion of C(X) being a ring of quotients of C(X), in the sense of Utumi.
		1 On Banach and Dedekind completeness of spaces of continuous..
	citeseer.ist.psu.edu /491745.html (406 words)

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	explanation-guide.info /meaning/Dedekind-cut.html (55 words)

	SDRL: Systems Design Research Lab
		The aim of this talk is to show how techniques from the formal logic world can sometimes be applied directly to specific mathematical problems studied completely independently in the world of combinatorics.
		This Goedel's program was accomplished in a recently found Logic of Proofs (LP) which enjoys a natural provability semantics, is complete, and able to realize the whole of S4.
		There are a number of completeness theorems, the most interesting of which is a consequence of the remarkable fact that every non-trivial algebra of this finitely presented equational theory has a simple quotient and that any simple algebra appears uniquely as a subalgebra of the intended model.
	www.cis.upenn.edu /sdrl/seminars.php3 (2937 words)

	RealNumber (Site not responding. Last check: 2007-10-22)
		uniform spaces, which is a different sense than the Dedekind completeness of the
		topology); the description in the Completeness section above is a special case.
		This sense of completeness is most closely related to the construction of the reals from
	202.41.85.103 /manuals/planetmath/entries/12/RealNumber/RealNumber.html (721 words)

	Representations of lattices (Site not responding. Last check: 2007-10-22)
		The poset induced by irreducible elements (Dedekind-MacNeille Completion); Irreducible Bipartite poset (Maximal antichains lattice) and its complementary (Galois or Concepts lattice).
		A different representation due to Duquenne 1991, is the core of a lattice, which use join or meet-irreducible elements and some extra elements.
		Indeed, all the following problems are equivalents: 2-dimension of a poset P, 2-dimension the Dedekind-MacNeille completion of P, Edge-covering of a bipartite graph by complete bipartite graphs, Coloration of a graph, Ambiguous rank of a matrix....
	www.isima.fr /~nourine/Science/Recherches/representation.htm (327 words)

	Publications of G.Ya.Lozanovsky
		[7] (with B.Z. Vulikh) Metric completeness of normed and countably normed lattices (Russian).
		[14] (with V.A. Solov'ev) The monotone extension of a Banach norm from a vector lattice to its Dedekind completion (Russian).
		USSR Sbornik 13 (1971), 323-343, as The representation of completely linear and of regular functionals
	www.mathsoc.spb.ru /pers/lozanovs/bib.html (784 words)

	TOPPS Bibliography
		Dedekind completion as a method for constructing new Scott domains.
		The language is Turing complete and the theory has power like set theory.
		For size-relation graphs that can loop (i.e., they give rise to an infinite multipath), and that satisfy SCT, it is usually easy to find a "progress point"- some graph whose infinite occurrence in a multipath causes infinite descent.
	www.diku.dk /topps/bibliography/2002.html (4805 words)

	Real number
		It is easy to see that every convergent sequence is a Cauchy sequence.
		Now the important fact about the real numbers is that the converse is true: :'''Every Cauchy sequence of real numbers is convergent.''' That is, the reals are complete.
		Occasionally, formal elements +∞ and -∞ are added to the reals to form the extended real number line, a compact space which is not a field but retains many of the properties of the real numbers.
	www.datamass.net /re/real-number.html (2063 words)

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