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Topic: Stone duality

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	Stone duality - Wikipedia, the free encyclopedia
		Stone-type dualities also provide the foundation for pointless topology and are exploited in theoretical computer science for the study of formal semantics.
		Probably the most general duality which is classically referred to as "Stone duality" is the duality between the category Sob of sober spaces with continuous functions and the category SFrm of spatial frames with appropriate frame homomorphisms.
		Stone's representation for distributive lattices can be extended via an equivalence of coherent spaces and Priestley spaces (ordered topological spaces, that are compact and totally order-disconnected).
	en.wikipedia.org /wiki/Stone_duality (2055 words)

	Black Stone of Mecca - Crystalinks
		The Black Stone is in fact the cornerstone of the Ka'ba and is there as an emblem of the progeny of Abraham which was rejected by the Israelites and became the corner stone of the Kingdom of God.
		Touching or kissing the stone has a profound impact on the faithful as it is suppose to count in their favor on judgment day.
		The stone associated with Cybele's worship was, originally, probably at Pessinus but perhaps at Pergamum or on Mount Ida. What is certain is that in 204 BCE it was taken to Rome, where Cybele became 'Mother' to the Romans.
	www.crystalinks.com /blackstone.html (1452 words)

	Stone's representation theorem for Boolean algebras - Wikipedia, the free encyclopedia
		In mathematics, Stone's representation theorem for Boolean algebras, named in honor of Marshall H. Stone, is the duality between the category of Boolean algebras and the category of Stone spaces, i.e., totally disconnected compact Hausdorff topological spaces (also known as Boolean spaces).
		Stone's duality generalises to infinite sets of propositions the use of truth tables to characterise elements of finite Boolean algebras.
		In detail, the Stone space of a Boolean algebra A is the set of all 2-valued homomorphisms on A, with the topology of pointwise convergence of nets of such homomorphisms.
	en.wikipedia.org /wiki/Stone's_representation_theorem_for_Boolean_algebras (534 words)

	Abstract Stone Duality (Site not responding. Last check: 2007-11-02)
		Abstract Stone Duality (ASD) is a type theory in which the topology on a space is an exponential with a lambda-calculus, not an infinitary lattice.
		Stone also told us always to look for the topology on a mathematical construction, even one whose data come from discrete algebra [23].
		By Stone duality, the opposite of the category C of ``spaces'' is to be a category of ``algebras'', but defined by a monad over C rather than over sets.
	www.cs.man.ac.uk /~pt/ASD/manifesto.html (2776 words)

	Stone Duality - The Duality of Syntax and Semantics (Site not responding. Last check: 2007-11-02)
		Stone Duality - The Duality of Syntax and Semantics
		Topological interpretation of the representation theorems (Stone duality).
		The main idea is that modal logics are described by L-algebras where L is a functor on a category of algebras and Kripke models are described by T-coalgebras where T is a functor on the dual category of topological spaces.
	www.cs.nott.ac.uk /~txa/mgs/STD.html (127 words)

	Abstract Stone Duality (Site not responding. Last check: 2007-11-02)
		Abstract Stone Duality (ASD) is a revolutionary direct axiomatisation of general topology that is inherently computable.
		It illuminates the duality between open and closed concepts that is obscured by asymmetry in classical and intuitionistic topology.
		Dual to compactness is overtness, which replaces various metrical and enumerative conditions.
	www.cs.man.ac.uk /~pt/ASD (579 words)

	Fine Art photography and commentery on "Flesh and Stone" - Northstar Gallery
		All around Medusa's cavern were stone figures of men and animals which had risked a glimpse her and had been petrified with the sight.
		The upward-pointing stones acted as an intermediary between heaven and earth and the emblem of a supernatural presence that controlled and fertilized the surrounding land.
		The role of the stone statue as sacred Deity transitioned to becoming the image of the King who increasingly took on the identity of the Deity in the final stage of the Khmer Civilization.
	northstargallery.com /stone/commentarynsg.htm (3246 words)

	Subspaces in abstract Stone duality (Site not responding. Last check: 2007-11-02)
		By abstract Stone duality we mean that the topology or contravariant powerset functor, seen as a self-adjoint exponential $\Sigma^{(-)}$ on some category, is monadic.
		Paré showed that any elementary topos has this duality, and we prove it intuitionistically for the category of locally compact locales.
		Keywords: axiom of comprehension; subtype; typed lambda calculus; Stone duality; subspace topology; locally compact spaces; nucleus of a locale; injective object; monadic adjunction; Beck’s theorem.
	www.tac.mta.ca /tac/volumes/10/13/10-13abs.html (248 words)

	Amazon.com: Lectures on Boolean Algebras: Books: Paul R. Halmos (Site not responding. Last check: 2007-11-02)
		A Boolean algebra's Stone space is the space of all of its 2-valued homomorphisms with the topology of pointwise convergence of nets of such homomorphisms.
		That every Boolean space is the Stone space of some Boolean algebra (namely, the Boolean algebra of all of its clopen subsets) is one of the important facts of "Stone's duality".
		Halmos never mentions the phrase "Stone space", but he proves the basic facts about "Stone's duality": that the category of Boolean algebras and Boolean homomorphisms is the opposite of the category of Boolean spaces and continuous functions.
	www.amazon.com /Lectures-Boolean-Algebras-Paul-Halmos/dp/0387900942 (695 words)

	AMCA: Duality as a Unifying Framework by Ingrid Rewitzky, Hilary Priestley, Mai Gehrke, Achim Jung, Marcello Bonsangue
		Duality theory as unifying framework for modal logics Marcello Bonsangue (LIACS, Leiden University) The aim of this talk is to describe a framework for the use of dualities for a coalgebraic semantics of modal logic.
		A hierarchy of Stone dualities Achim Jung (Department of Computer Science, University of Birmingham) Stone's classical representation theorem (1936/37) associates with every Boolean algebra a compact totally disconnected space (now commonly referred to as a "Stone space").
		A duality for binary multirelations Ingrid Rewitzky (Department of Mathematics and Applied Mathematics, UCT) Specifications involving angelic and demonic nondeterminism may be thought of as a contract between two agents, both of which are free to make various choices.
	at.yorku.ca /c/a/o/p/07.htm (954 words)

	\bf The Duality Between Aglebraic Posets and Bialgebraic Frames: A Lattice Theoretic Perspective
		Bialgebraic (that is, algebraic and dually algebraic), distributive lattices should be familiar enough to algebraists, but this may not be the case with domains.
		Coherent lattices form an important class of mathematical objects, due to their role in Stone-type dualities and to the fact that they possess logical presentations involving only finite conjunctions and disjunctions (see Johnstone [13]).
		Theorem 5 The category PCoh is dually equivalent to the category PCohFrm, and the category Coh is dually equivalent to the category CohFrm.
	www.mtsu.edu /~jhart/ALGFRM.html (9751 words)

	The Philosophers' Stone and its SECRET explained by CLAUS FURSTNER . The Philosophers' Stone Art Studio.
		The Philosophers' Stone is a visual symbol combining the four principle Concepts of ancient philosophy.
		The Philosophers' Stone was first used around 3000 BC in the ancient philosophy of Hermetic Alchemy, where it symbolised the transformation from base metal (the 'worldly') to gold ('spiritual fulfilment').
		The concept of duality is shown in the inner circle.
	members.ozemail.com.au /~clauspat/stone.htm (364 words)

	Geometric and Higher Order Logic in terms of Abstract Stone Duality (Site not responding. Last check: 2007-11-02)
		In topology, the lattice duals of these equations also hold, and are related to the Phoa principle in synthetic domain theory.
		Our treatment of overt discrete spaces and open maps is precisely dual to that of compact Hausdorff spaces and proper maps.
		The category of overt discrete spaces forms a pretopos and the paper concludes with a converse of Paré's theorem (that the contravariant powerset functor is monadic) that characterises elementary toposes by means of the monadic and Euclidean properties together with all quantifiers, making no reference to subsets.
	www.tac.mta.ca /tac/volumes/7/n15/7-15abs.html (243 words)

	STONE PONY. We offers stone pony and ear piercing for boys
		Your stone pony favorite Officially licensed and imported from Italy.
		The Marybeth what stone pony does he meant to the perineum is official team colors.
		This stone pony info 16g nipple jewelry pairs beautifully with silver bracelet and creative gem stone stone pony platinum antique diamond rings is wonderful custom pink base.
	goldjewelry.1colony.com /stone-pony.html (372 words)

	CiteULike: Tag duality (Site not responding. Last check: 2007-11-02)
		Duality and Self-Duality (Energy Reflection Symmetry) of Quasi-Exactly Solvable Periodic Potentials
		A duality theory for some non-convex functions of matrices
		Duality, achievable rates, and sum-rate capacity of Gaussian MIMO broadcast channels
	www.citeulike.org /tag/duality (333 words)

	Category Theory (Stanford Encyclopedia of Philosophy)
		Dualities play an important role in mathematics and they can be described with the help of equivalences between categories.
		Then, the Pontryagin duality theorem amounts to the claim that the category C is equivalent to the category C°, that is, to the opposite category.
		Categorical study of duality theorems is still a very active and significant field, and is largely inspired by Stone's result.
	plato.stanford.edu /entries/category-theory (11769 words)

	Domains and Lambda Calculi (book announcement)
		The chapter is also a case study of Stone duality: the D-infinity models can also be constructed out of certain theories of `types', or functional characters.
		We interpret CCS using a domain equation which involves the convex powerdomain and relate the denotational semantics to the operational one.
		Chapter 10 presents Stone duality (originally the correspondence between Boolean algebras and certain topological spaces), applied to domains.
	www.cis.upenn.edu /~bcpierce/types/archives/1997-98/msg00317.html (1798 words)

	A PARADIGM FOR PROGRAM SEMANTICS: POWER STRUCTURES AND DUALITY (Site not responding. Last check: 2007-11-02)
		At the heart of their method is the notion of a power construction, and an invocation of duality theory.
		Specifically, the duality theory at work in this book is "Priestley duality", which identifies certain topological spaces ("Priestley spaces") as the duals of distributive lattices.
		The importance of the book lies in its demonstration that, although there are many variations of each of the four versions of program semantics, in principle they may be thought of as intertranslatable.
	math.sun.ac.za /~rewitzky/projects/duality/psps.html (251 words)

	Concurrency Abstracts (via CobWeb/3.1 planetlab-3.cs.princeton.edu) (Site not responding. Last check: 2007-11-02)
		The dual of a true concurrency schedule appears to be a false concurrency automaton, a paradox we resolved in a previous paper by extending the latter to higher dimensions.
		This perspective dualizes the alternative view of Chu spaces as generalized topological spaces, and has the advantage of substituting the intuitions of formal language theory for those of topology.
		The symmetry is with the dual notion of state space, a poset with all nonempty meets representing choice and a bottom representing the start state.
	boole.stanford.edu.cob-web.org:8888 /abstracts.html (9716 words)

	Modal Logic, Stone Duality and Coalgebras
		These spaces can be traced back to the work of M. Stone (1937) on the representation of distributive lattices by spectral spaces, of L. Nachbin (1950) on compact ordered topological spaces, and of H. Priestley (1972) on representation of distributive lattices by ordered Stone spaces.
		An immediate corollary is that there is a strongly complete logic for T-coalgebras for any functor T on Stone spaces that is determined by its action on finite Stone spaces.
		Abstract: A theory T is said to be a conservative extension of a theory T′ if any consequence of T, which only uses symbols from T′, is a consequence of T′ as well.
	www.cs.le.ac.uk /events/ml06/index.html (1697 words)

	Atlas: Stone duality for Dedekind sigma-complete lattice-ordered groups with order-unit by Daniele Mundici (Site not responding. Last check: 2007-11-02)
		Atlas: Stone duality for Dedekind sigma-complete lattice-ordered groups with order-unit by Daniele Mundici
		Building on the Goodearl-Handelman-Lawrence functional representation theorem, we provide a purely topological representation (specifically, a categorical duality) for a large class of Dedekind sigma-complete lattice-ordered groups G with order-unit u, including all G where u has a finite index of nilpotence.
		Our duality is a far-reaching generalization of the well known Stone duality between sigma-complete Boolean algebras and basically disconnected compact Hausdorff spaces.
	atlas-conferences.com /cgi-bin/abstract/capw-82 (129 words)

	[No title]
		Another category with this property is the category of locally compact locales, where Sigma is the Sierpinski space and "powerset" means "lattice of open sets", equipped with the Scott topology.
		This is a way of re-axiomatising general topology without the arbitrary unions in the traditional definition and in locale theory: the category of "frames" is to be both dual to the category of "spaces" and algebraic over it.
		Subject: categories: re: An Abstract Stone Duality This is a reaction to Paul Taylor's advertisement for his paper on Abstract Stone Duality.
	www.mta.ca /~cat-dist/catlist/1999/abs-stone (1464 words)

	@CAT 2002-2003
		Abstract: The quadrality square combines (involutive) duality in linear structures such as the categories of abelian groups and sup-lattices with Girard's linear decomposition of cartesian closed categories and Stone duality for those categories.
		in which the Stone duality at the bottom is Pare's theorem.
		One is naturally led to examine the extension's dual to see if it is geometric.
	www.mscs.dal.ca /~pare/Sem02-03.html (1493 words)

	Atlas: $MV$-Algebras: Convergence and Duality by Roman Fric (Site not responding. Last check: 2007-11-02)
		For semisimple MV-algebras (also called archimedean) we define an initial sequential convergence and prove a Stone-type duality in which sequentially continuous MV-algebra homomorphiams are dual to generalized measurable maps.
		The duality can be viewed as a fuzzyfication of the boolean case (cf.
		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 # caeq-10.
	atlas-conferences.com /c/a/e/q/10.htm (187 words)

	Stone's representation theorem for Boolean al... - Wikipedia, the free encyclopedia
		Please search for Stone's representation theorem for Boolean al...
		Start the Stone's representation theorem for Boolean al...
		Look for "Stone's representation theorem for Boolean al...
	en.wikipedia.org /wiki/Stone's_representation_theorem_for_Boolean_al... (180 words)

	Marta Bunge - Relative stone duality
		Closely related to it is a relative Stone Duality.
		We prove that there is a duality between these categories and that it restricts to an equivalence between suitably defined categories
		is Sets, this reduces to the usual Stone Duality.
	www.cms.math.ca /Events/winter99/abstracts/node39.html (111 words)

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