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Topic: Galois connection

Related Topics

Closure operator

Injective function

Galois group

Surjection

Galois theory

Supremum

Infimum

Greatest element

Algebraic geometry

Evariste Galois

Regular polygon

Constructible polygon

Symmetry group

Bourg la Reine

Cauchy

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	Science Fair Projects - Galois connection
		Galois connections generalize the correspondence between subgroups and subfields investigated in Galois theory.
		Galois connections also provide an interesting class of mappings between posets which can be used to obtain categories of posets.
		Galois connections may be used to describe many forms of abstraction in the theory of abstract interpretation of programming languages.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Galois_connection (2348 words)

	Evariste Galois
		A tragic hero to mathematicians, Galois brief, chaotic, and dire life on the fringes of the mathematical and political establishments lends easily to the great myth of Galois as the genius oppressed by the hostile, ignorant power structure of his day.
		However, Galois was arrested again on Bastille Day 1831 for possessing a weapon and wearing the uniform of the Guard, which had been ordered disbanded by a fearful Louis-Philippe.
		Using this theory, Galois gave an elegant proof of the mathematical theorem that states that the general quintic polynomial is not solvable, a problem of much interest in his day.
	www.iscid.org /encyclopedia/Evariste_Galois (1902 words)

	Galois theory Summary
		Galois theory stemmed from an undertaking by Galois for a deeper understanding of the essential conditions an equation must satisfy in order for it to be solvable by radicals.
		Galois theory depends on the concept of a group since it is the mathematical interpretation of group theory.
		Further abstraction of Galois theory is achieved by the theory of Galois connections.
	www.bookrags.com /Galois_theory (2469 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		Connections including Galois connections are introduced and basic properties are given.
		Connections are "systems" consisting of two partially ordered sets and two order-preserving maps between them.
		We "extend" this Galois connection to a Galois correspondence and then modify the Galois correspondence to form a correspondence between generalized complete partial orders and generalized Scott topologies, an interesting result for programming language semantics.
	www.mth.uct.ac.za /colloquium/2004/col06-04.html (145 words)

	Our Name (Site not responding. Last check: 2007-10-20)
		Among his many contributions, Galois founded abstract algebra and group theory, which are fundamental to computer science, physics, coding theory and cryptography.
		Galois' contributions are even more remarkable in light of the fact that many were captured as hastily scribbled notes on the eve of his untimely death in a duel.
		Today, a "Galois connection" is a way of solving challenging mathematical problems by translating them into different mathematical domains, creating a new solution paradigm by making the original problem amenable to a number of mathematical techniques.
	www.galois.com /ourname.php (159 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		The general concept of a Galois connnection is one of the most important mathematical concepts and is used in almost all branches of Mathematics.
		The leading idea on which Galois connections are based is to switch from one context A to another context B which is more known and better understood.
		The most famous and perhaps oldest example is the Galois connection between certain extensions of a ground field and subgroups of the group of all relative automorphisms of this field.
	users.math.uni-potsdam.de /~denecke/galoisconf1.htm (216 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		Galois extensions of commutative rings by profinite families of groups.
		Galois theory and a general notion of central extension (with G. Kelly).
		Galois theory and a new homotopy double groupoid of a map of spaces (with R. Brown).
	www.rmi.acnet.ge /atestacia/staff/janel-list.htm (567 words)

	Earliest Known Uses of Some of the Words of Mathematics (G)
		The attribution 'Galois Connection' is simply because classical Galois Theory, as developed by Artin in the 1930's, establishes a correspondence between subfields of an algebraic number field and subgroups of the group of automorphisms of that field which is a dual lattice isomorphism between the lattice of normal subfields and the lattice of normal subgroups.
		The Galois Connection is then an order reversing correspondence between the posets which is a lattice dual isomorphism between the posets of 'closed' elements.
		The modern definition of a group is somewhat different from that of Galois, for whom the term denoted a subgroup of the group of permutations of the roots of a given polynomial.
	members.aol.com /jeff570/g.html (6517 words)

	Issa: Integrated System for Structural Analysis (Site not responding. Last check: 2007-10-20)
		After characterizing the Galois lattice, it is possible to calculate either partial orders, or frequent sequential patterns, or also a new notion of association rules with order by just traversing its nodes.
		The standard Galois connection for a binary relation maps each family of objects to the set of all items that are present in all of them, and each set of items to the set of objects where they are present.
		Formally, the closed sets of objects and the closed sets of sequences of the same node are linked by the Galois connection, and because of this, the system receives the name of Galois lattice.
	www.lsi.upc.edu /~abifet/ISSA/ISSA2.html (2200 words)

	PlanetMath:
		Galois criterion for solvability of a polynomial by radicals owned by djao
		Galois group of a cover (in deck transformation) owned by mathcam
		Galois group of the compositum of two Galois extensions owned by alozano
	planetmath.org /encyclopedia/G (1568 words)

	PRG Technical Report TR-6-00
		A Galois connection is a `natural' way of relating two partially-ordered spaces (for example semantic models of different expressive power).
		The most common form of Galois connection, which reflects one model being finer than the other, is that of a Galois embedding.
		This letter shows a published claim, `two partially-ordered spaces related by a Galois embedding in each direction are isomorphic', to be false in general but true in the finite case.
	web.comlab.ox.ac.uk /oucl/publications/tr/tr-7-00.html (95 words)

	Algebraic K-theory, groups and categories
		Categorical Galois theory (called CGT below for short) was developed by G. Janelidze in 1984-90, and one of the major objectives of this project was to investigate its various connections with higher-dimensional homotopical algebra developed by R. Brown, T. Porter and other members of the Bangor team.
		In connection with this a categorical reformulation of commutator theory with many new results was given in [34].
		The connection between abstract Galois theories of Ligon and of Chase and Sweedler was investigated.
	www.bangor.ac.uk /~mas010/intasrep.html (7592 words)

	Common Notions: Towards a Lattice-Theoretic Turn in Social Epistemology
		Our goal here is to bring a formal frame for studying both networks empirically as well as to point out stylized facts that would explain their reciprocal influence and the emergence of clusters of agents, which may also be regarded as "cultural cliques".
		We show how to apply the Galois lattice theory to the modeling of the coevolution of social and conceptual networks, and the characterization of cultural communities.
		While I don't want to suggest for a moment that the stuff about Galois lattices is window-dressing, the intuitive idea behind what Roth and Bourgine are doing is simple and compelling, and I think can be accurately presented without an excursion through higher mathematics.
	www.cscs.umich.edu /~crshalizi/weblog/368.html (1617 words)

	News \| TimesDaily.com \| TimesDaily \| Florence, AL
		Any Galois connection gives rise to closure operators and to inverse order-preserving bijections between the corresponding closed elements.
		As is the case for Galois groups, the real interest lies often in refining a correspondence to a duality (i.e.
		A treatment of Galois theory along these lines by Kaplansky was influential in the recognition of the general structure here.
	www.timesdaily.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=adjoint_functors (3484 words)

	resume 2001.25 (Site not responding. Last check: 2007-10-20)
		On the roles of Galois connections in classification
		Galois connections (or residuated mappings) are of growing interest in various domains related with or relevant from classification.
		We partially revisit them in a common frame provided by a recent study about Galois connections between closure spaces.
	mse.univ-paris1.fr /Cahiers2001/2001025B.htm (165 words)

	Atlas: A Galois connection between sets of relations and sets of surjective functions by Ferdinand Borner (Site not responding. Last check: 2007-10-20)
		We investigate a Galois Connection Inv-sPol between sets of relations and sets of surjective functions on a finite basic set A. This connection is obtained from the Galois connection Inv-Pol by restricting the set of all functions on A to the set of all surjective functions.
		The Galois closed sets of functions can be represented by surjectively generated clones, and the Galois closed sets of relations are relational clones, closed under the additional operation of universal quantification.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cake-47.
	atlas-conferences.com /cgi-bin/abstract/cake-47 (249 words)

	Atlas: Sequential Convergence via Galois Connections by Gonçalo Gutierres (Site not responding. Last check: 2007-10-20)
		Topological sequential spaces are the fixed points of a Galois connection between collections of open sets and sequential convergence structures.
		The fixed points of these other Galois connections are not topological spaces in general, but they can be embedded into larger topological categories, such as pretopological, pseudotopological or convergence spaces.
		In this talk, we try to characterize the sequential convergences which are fixed points of these connections as well as their restrictions to topological spaces.
	atlas-conferences.com /cgi-bin/abstract/cate-09 (140 words)

	Domain Theory in Abstract Interpretation
		The lack of a Galois connection in relating abstract and concrete domains is typically due to the possible lack of best approximations for concrete domain objects, e.g.
		Although this is true, in particular in practical applications, we believe that Galois connections still represent the essence of any approximation process.
		It is known that any Galois connection may be lifted to a Galois insertion by identifying in an equivalence class those values of the abstract domain with the same concrete meaning.
	www.cs.ucy.ac.cy /compulog/newpage114.htm (4650 words)

	Practical Foundations of Mathematics
		Galois connections Evariste Galois's name was given to Definition 3.6.1(b), by Ø ystein Ore, not because he spent his short life (1811-32) considering such definitional minutiae, but because the correspondence between intermediate fields and subgroups of the Galois group of a field extension (Example 3.8.15(j)) was the first such situation known.
		A Galois connection is often presented as the lower set
		For us, the most important example of a Galois connection will be that defining the factorisation of functions into epis and monos in Section 5.7, which we shall use to study the existential quantifier in Sections 5.8 and 9.3.
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/s38.html (1882 words)

	EMail Msg <9311092334.AA02203@rodin.wustl.edu>
		Speaking slightly imprecisely for brevity: in all systems I've seen, the more some individuals number, the fewer the predicates (descriptions or containing sets) they share in common; dually, the more the predicates (or containing sets), the fewer the individuals which they all apply to or contain.
		In pure First-Order systems based on finite sets, the Galois connection between the powerset of monadic predicates and the powerset of individuals is a dual isomorphism (of Boolean lattices).
		In higher-order, approximative and intensional systems, there is still a Galois connection, but it is no longer a dual isomorphism.
	www-ksl.stanford.edu /email-archives/interlingua.messages/419.html (1273 words)

	Tutorial on Logical Relation and Relational Parametricity - Seminars/Workshops - ROPAS
		Abstract: In the mid 80's, one of the main issues in the theory of abstract interpretation was how to handle higher-order functions properly.
		While Cousot's original approach to design abstract domains with Galois connection is conceptually appealing, it has difficulties to handle higher-order functions.
		To see the difficulties, suppose that C and A be concrete and abstract interpretations of type t such that P(C) and A are related by Galois connection.
	ropas.snu.ac.kr /seminar/20020711.html (362 words)

	Search Result
		Roland Carl Backhouse: Pair algebras and Galois connections.
		Sergei O. Kuznetsov: Galois Connections in Data Analysis: Contributions from the Soviet Era and Modern Russian Research.
		Ronald Brown, George Janelidze: Galois Theory and a New Homotopy Double Groupoid of a Map of Spaces.
	www.informatik.uni-trier.de /ley/dbbin/dblpquery.cgi?query=Search&return=100&title=Galois (1486 words)

	Mathematical Biosciences Institute
		Next, it is shown that (i) swarm cognition mechanism parameters have been tuned by natural selection to provide a balance between speed and accuracy of choice, and (ii) the key component of swarm cognition, accurate group memory, is a result of this same balance.
		In a Galois connection between two sets, the increase in size of one set corresponds to the decrease in size of the other set and vice versa.
		Interaction strengths along specific connections were sensitive to local geographic conditions and parameterized against reported data on the time and spatial location of detected rabid animals.
	www.mbi.osu.edu /sciprograms/seminars2005.html (8687 words)

	ICML98 - Submission #103 (Site not responding. Last check: 2007-10-20)
		Structural Machine Learning with Galois Lattice and Graphs Michel Liquiere LIRMM, 161 Rue Ada, 34392 Montpellier Cedex 5 France Jean Sallantin LIRMM, 161 Rue Ada, 34392 Montpellier Cedex 5 France Abstract The main objective of this paper is to define a formal approach to learning from examples, when examples are described by labelled graphs.
		For this purpose, this paper use a formal model based on the use of lattice theory and, more precisely, in the use of Galois connection.
		The advantage of this formalization is that we can now use Galois lattice model with structural description of the examples and concepts in order to enlarge the domain of formal conceptual analysis.
	www.cs.wisc.edu /icml98/papers/paper103.html (165 words)

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