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Topic: Heyting algebra

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  Heyting algebra   (Site not responding. Last check: 2007-11-07)
Heyting algebras model intuitionistic logic, in which the law of excluded middle does not in general hold.
Formally, a Heyting algebra is a bounded lattice L such that for all a and b in L there is a greatest element x of L such that a ∧ x ≤ b.
Heyting algebras are always distributive; this is sometimes stated as an axiom, but in fact it follows from the existence of relative pseudo-complements.
www.theezine.net /h/heyting-algebra.html   (178 words)

 Heyting algebra - Encyclopedia Glossary Meaning Explanation Heyting algebra   (Site not responding. Last check: 2007-11-07)
Heyting algebras arise as models of intuitionistic logic, a logic in which the law of excluded middle does not in general hold.
A Heyting algebra H is a bounded lattice such that for all a and b in H there is a greatest element x of H such that
Arend Heyting (1898-1980) was himself interested in clarifying the foundational status of intuitionistic logic, in introducing this type of structures.
www.encyclopedia-glossary.com /en/Heyting-algebra.html   (791 words)

 Complete Heyting algebra   (Site not responding. Last check: 2007-11-07)
In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra which is complete as a lattice.
Locales and frames are especially important in the mathematical field of pointless topology, which indeed might roughly be described as the study of locales.
Finally, locales usually arise in the context of Stone duality and from the viewpoint of pointless topology it is desirable to obtain a category that covariantly corresponds to the category of topological spaces and continuous mappings.
www.worldhistory.com /wiki/C/Complete-Heyting-algebra.htm   (768 words)

 Constructive Mathematics
Heyting, working in intuitionistic algebra, concentrated on issues raised by considering algebraic structures over the real numbers, and so developed a handmaiden of analysis rather than a theory of discrete algebraic structures.
Paradoxically, it is in algebra where we are most likely to meet up with wildly nonconstructive arguments such as those that establish the existence of maximal ideals, and the existence of more than two automorphisms of the field of complex numbers.
It is important to realize that constructive algebra is algebra; in fact it is a generalization of classical algebra in that we do not assume the law of excluded middle, just as group theory is a generalization of abelian group theory in that the commutative law is not assumed.
www.math.fau.edu /Richman/HTML/CONSTRUC.HTM   (1293 words)

 Articles - Intuitionistic logic   (Site not responding. Last check: 2007-11-07)
The meet and join operations in the Boolean algebra are identified with the ∧ and ∨ logical connectives, so that the value of a formula of the form A ∧ B is the meet of the value of A and the value of B in the Boolean algebra.
A corresponding theorem is true for intuitionistic logic, but instead of assigning each formula a value from a Boolean algebra, one uses values from a Heyting algebra, of which Boolean algebras are a special case.
In this algebra, The ∧ and ∨ operations correspond to set intersection and union, and the value assigned to a formula A→B is the same as the value assigned to the formula ¬(A ∧ ¬B).
www.motionize.com /articles/Intuitionistic_logic   (1244 words)

 Lindenbaum algebras and linear logic
One can also say that a Heyting algebra is a bicartesian closed category such that for any two objects a and b, Hom(a,b) U Hom(b,a) has at most one element.
We get an algebraic structure (should we call it the "linear Lindenbaum algebra", Yuk, no!) on the set of equivalence classes under the equivalence A \equiv B iff - A -o B and - B -o A. In fact, this thing is simultaneously a lattice with an involution, and a commutative monoid.
By the way, the interpretation of the additives in the "linear Lindenbaum algebra" as lattice operations seem to suggest that Girard's naming of the constants is skewed.
www.seas.upenn.edu /~sweirich/types/archive/1992/msg00003.html   (854 words)

 CONK! Encyclopedia: Arend_Heyting   (Site not responding. Last check: 2007-11-07)
Arend Heyting (May 9, 1898 – July 9, 1980) was a Dutch mathematician and logician.
Brouwer, and did much to put intuitionistic logic on a footing where it could become part of mathematical logic (which in a definite sense ran counter to some of the initial intentions of its founder).
This article about a mathematician is a stub.
www.conk.com /search/encyclopedia.cgi?q=Arend_Heyting   (69 words)

 A Logical System for Multicriteria Decision Analysis
Boolean algebra [15], Heyting algebra and Brouwer algebra [1,2,7,13,17,18], Heyting-Brouwer (semi-Boolean) algebra [19], D-algebra [20]).
The class of D-algebras [20] is a minimal extension of the union between the class of Heyting algebras and the class of Brouwer algebras.
A Heyting lattice is a relatively pseudo-complemented lattice (A, Ù
www.ici.ro /ici/revista/sic1998_3/art07.html   (4621 words)

 Glossary of order theory - Wikpedia   (Site not responding. Last check: 2007-11-07)
A Boolean algebra is a distributive lattice with least element 0 and greatest element 1, in which every element x has a complement ¬x, such that x ^ ¬x = 0 and x v ¬x = 1.
A Heyting algebra that is a complete lattice is called a complete Heyting algebra.
The incidence algebra of a poset is the associative algebra of all scalar-valued functions on intervals, with addition and scalar multiplication defined pointwise, and multiplication defined as a certain convolution; see incidence algebra for the details.
www.bostoncoop.net /~tpryor/wiki/index.php?title=Glossary_of_order_theory   (2673 words)

 [No title]   (Site not responding. Last check: 2007-11-07)
Namely, to each finitely generated projective Heyting algebra there corresponds a projective formula; to non-isomorphic finitely generated projective algebras there correspond nonequivalent projective formulas, but there can be non-equvalent projective formulas which correspond to isomorphic projective algebras.
To a fixed n-generated projective Heyting algebra H there correspond as many projective formulas as there are di®erent retractions between H and \Phi_n (\Phi_n being the free n-generated Heyting algebra and H its retract).
We have a one-to-one correspondence between projective formulas and (H; ir) couples, where H is a projective algebra, and i; r the retractions.
www.illc.uva.nl /staging/Publications/ResearchReports/MoL-2001-09.abstract.txt   (624 words)

 Re: Brouwerian Algebra
Graham Solomon replied: > A 'Brouwerian algebra' is a name sometimes used for the dual to a > Heyting algebra.
In a Heyting algebra the "implication" is a relative pseudo- complement and that seems to be fairly standard terminology.
On the possibility of having two negations, it should be said that, at least in part in response to cricism from Diederik Batens, Priest has suggested having a second negation that behaves intuitionistically at least to the extent that ex falso quodlibet holds.
philo.at /phlo/199910/msg00023.html   (298 words)

 :: more focus on the constructive discussion ::
This is not meant to affirm a logical linguistic programme, only to show that there are Heyting algebras associated to the fields around such a programme from several different directions, and the theory of computation in particular has strong foundations in structures that are associated with these algebras.
There are many interesting relationships between Heyting algebras and other logics known, such as "Combining possibilities and negations" by Greg Restall "A semantical study of orthologics" by Miyazaki Yutaka "Algebras and frames for modal logics" And so on...
Their are many beautiful models with more exotic structures, but Heyting algebras seem to have this special universality principle that somehow seems connected to the Church-Turing thesis.
www.talkabouteducation.com /group/sci.edu/messages/26039.html   (1379 words)

 [No title]   (Site not responding. Last check: 2007-11-07)
Using algebraic techniques (as opposed to techniques from proof-theory) we show that every such Heyting algebra is in fact co-Heyting, improving on a result of Ghilardi who showed that Heyting algebras free on a finite set of generators are co-Heyting.
Along the way we give a new and simple proof of the finite model property.\bibpar Our main technical tool is a representation of finitely presented Heyting algebras in terms of a colimit of finite distributive lattices.
As applications we construct explicitly the minimal join-irreducible elements (the atoms) and the maximal join-irreducible elements of a finitely presented Heyting algebras in terms of a given presentation.
www.brics.dk /upd/BRICS/RS/98/30/BRICS-RS-98-30.bib   (178 words)

 Boolean algebra - Metaweb   (Site not responding. Last check: 2007-11-07)
This is or is not a placeholder for Boolean algebra
In mathematics and computer science, Boolean algebras, or Boolean lattices, are algebraic structures which "capture the essence" of the logical operations AND, OR and NOT as well as the set theoretic operations union, intersection and complement.
Specifically, Boolean algebra was an attempt to use algebraic techniques to deal with expressions in the propositional calculus.
www.metaweb.com /wiki/wiki.phtml?title=Boolean_algebra&printable=yes   (243 words)

 Heyting algebra   (Site not responding. Last check: 2007-11-07)
The Lindenbaum algebra of propositional intuitionistic logic is a Heyting algebra.
A Heyting algebra, from the logical standpoint, is essentially a generalization of the usual system of truth values.
The usual two-valued logic system is the simplest example of a Heyting algebra, one in which the elements of the algebra are
www.worldhistory.com /wiki/H/Heyting-algebra.htm   (816 words)

 [No title]   (Site not responding. Last check: 2007-11-07)
ABSTRACT: We show that every finitely presented Heyting algebra is a bi-Heyting algebra, i.e., there is an operation `suplement' defined dual to the notion of implication.
Our main technique is algebraic: We use a filtration of a given (f.p.) Heyting algebras by finite distributive lattices.
Although not part of this talk it is worthwhile to note that the class of Heyting algebras that are bi-Heyting is much much larger: It includes all (!) free Heyting algebras.
www.math.mcgill.ca /rags/seminar/Butz1.txt   (129 words)

 Read about Intuitionistic logic at WorldVillage Encyclopedia. Research Intuitionistic logic and learn about ...   (Site not responding. Last check: 2007-11-07)
The valuation can then be extended to formulæ by matching the propositional connectives with their corresponding operations in the algebra.
A valid sentence is then one which has valuation 1 in any valuation on any Heyting algebra.
It can be shown that we need in fact consider only the Heyting algebra given by the open sets of the real plane with its usual topology—intuitionistic validities correspond precisely to Heyting formulae which evaluate to the entire plane for any assignment of open subsets to the variables.
encyclopedia.worldvillage.com /s/b/Intuitionistic_logic   (1008 words)

 :: towards a constructive education :: (news server friendly)
But they are all Heyting, and this substructure common to all semantic theory allows for a rich functor structure of equivalences and dualities amongst these logics.
Applications of Heyting semantics have been detailed for antigen-antibody interactions or the deviation of a cancer process, but one of the most fascinating applications for me has always been the analysis of quantum propositions and the evolution of quantum systems.
And when I say Heyting here, prior, and to come, I also intend co-Heyting, for obviously there are semantics mentioned who have more reasonable interpretations in the dual algebras, but that is of course a contravariant functor away.
www.talkabouteducation.com /group/sci.edu/messages/26012.html   (2359 words)

 A Logical System for Multicriteria Decision Analysis
A many-valued space over a biresiduated algebra is a set equipped with an equivalence function and a distance function such that these functions are complementary.
A general description of the connection between some basic algebraic structures from the category of biresiduated algebras, is given.
The structures of Boolean algebra, Heyting algebra and Brouwer algebra are related to the structure of D-algebra as follows:
www.ici.ro /ici/revista/sic1999_4/art10.html   (2617 words)

 Abstracts   (Site not responding. Last check: 2007-11-07)
We develop a self-dual algebra such that every point in the algebra is representable by some formula in the logic.
In particular, we show that this algebra is an instantiation of the Chu construction applied to a Heyting algebra, the second Dialectica construction applied to a Heyting algebra, and as an algebra for the study of recursion and corecursion.
In particular, we show that this algebra is an instantiation of the $\Chu$ construction applied to a pseudo-Boolean algebra, the second Dialectica construction applied to a pseudo-Boolean algebra, and as posetal case of a suggestion by Pitts for the study of recursion and corecursion.
www-formal.stanford.edu /annap/www/abstracts.html   (1829 words)

 Thematic Afternoon on Constructivism
heyting stipulated in he last will that the first Arend Heyting lecture should be devoted to workconnected with Heytings own work on intuitionism.
In "Untersuchungen über intuitionistische Algebra", Heyting reduces the question of whether a nonzero polynomial with coefficients in a field divides another polynomial, to whether a finite number of field elements are zero.
He has published extensively on algebra, and has been a leading researcher in the field of Constructive Mathematics in teh spirit of Errett Bishop, especially algebra.
staff.science.uva.nl /~anne/heyting.html   (685 words)

 M567: Boolean Algebra
of ideals of the Boolean algebra B is not a Boolean algebra.
Show that a subset of a Boolean algebra is an ideal of the Boolean algebra if and only if it is an ideal of the corresponding Boolean ring.
Show that a subset of a Boolean algebra is a prime ideal of the Boolean algebra if and only if it is a prime ideal of the corresponding Boolean ring.
orion.math.iastate.edu /jdhsmith/class/M567S05.htm   (880 words)

 Re: Brouwerian Algebra   (Site not responding. Last check: 2007-11-07)
Contrast that with heyting negation which is the largest element disjoint from x.
I do think bi-heyting algebras, with their two negations, have some nice features of some philosophical interest.
They might be useful for handling vagueness, and they may be just the right tool for the formal treatment of predicate negation and predicate term negation.
philo.at /phlo/199910/msg00024.html   (239 words)

Characterize the definable elements in the provability algebra of PA.
Formulate some general (algebraic) sufficient conditions for the well-foundedness of graded provability algebras.
Develop a definability theory for graded provability algebras that would allow the box operator to be applicable to (definably) infinite sets of elements.
www.phil.uu.nl /~lev/problems.html   (1032 words)

 Sheaves in Geometry and Logic : A First Introduction to Topos Theory (Universitext) Review and price   (Site not responding. Last check: 2007-11-07)
Replacing the propositional calculus with the (Heyting) intuitionistic propositional calculus results in a different representation by a Heyting algebra.
From the standpoint of ordinary topology, the Heyting algebra is significant in that the algebra of open sets is not Boolean, i.e.
A Grothendieck topology on a category is thus a function that assigns to each object in the category a collection of sieves on the object (this function must have certain properties which are discussed by the authors).
www.wi-fitechnology.com /Wi-Fi-Products-0387977104.html   (1323 words)

 CST LECTURES: Lecture   (Site not responding. Last check: 2007-11-07)
But when D is an infinite set then we cannot generally expect to be able to compute the truth values of the quantified sentences, even when we know the truth value of each P(a) for a:D. Moreover, in constructive mathematics the truth value of the quantified sentences is not even determined in general.
Classical propositional logic also can be given a semantics in any Boolean algebra, each formula A being assigned a value in the Boolean algebra so that certain conditions hold.
The open sets of a topological space on a set X of points form a complete Heyting algebra, when partially ordered by the subset relation.
www.cs.man.ac.uk /~petera/Padua_Lectures/lect6.html   (930 words)

 Cogprints - The Algebraic Structure of Sets of Regions   (Site not responding. Last check: 2007-11-07)
Heyting algebras, co-Heyting algebras, and bi-Heyting algebras are structures having considerable potential for the theoretical basis of these ontologies.
The main evidence is a proof that elements of certain Heyting algebras provide models of the Region-Connection Calculus developed by Cohn et al.
This theory uses modal operators which are related to the algebraic operations present in a bi-Heyting algebra.
cogprints.org /517   (206 words)

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