Results 11  20
of
85
SupLattice 2Forms and Quantales
 J. Algebra
, 2002
"... A 2form between two suplattices L and R is de ned to be a suplattice bimorphism LR ! 2. Such 2forms are equivalent to Galois connections, and we study them and their relation to quantales, involutive quantales and quantale modules. As examples we describe applications to C*algebras. ..."
Abstract

Cited by 9 (6 self)
 Add to MetaCart
A 2form between two suplattices L and R is de ned to be a suplattice bimorphism LR ! 2. Such 2forms are equivalent to Galois connections, and we study them and their relation to quantales, involutive quantales and quantale modules. As examples we describe applications to C*algebras.
A Topos for Algebraic Quantum Theory
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2009
"... The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*algebra of observables A induces a topos T (A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*algebra A. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum �(A) in T (A), which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topostheoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on �, and selfadjoint elements of A define continuous functions (more precisely, locale maps) from � to Scott’s interval domain. Noting that open subsets of �(A) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos T (A). These results were inspired by the topostheoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.
A Noncommutative Theory of Penrose Tilings
 International Journal of Theoretical Physics 44: 655689, 2005. [3] A. Palmigiano and
"... Considering quantales as generalised noncommutative spaces, we address as an example a quantale Pen based on the Penrose tilings of the plane. We study in general the representations of involutive quantales on those of binary relations, and show that in the case of Pen the algebraically irreducible ..."
Abstract

Cited by 8 (6 self)
 Add to MetaCart
Considering quantales as generalised noncommutative spaces, we address as an example a quantale Pen based on the Penrose tilings of the plane. We study in general the representations of involutive quantales on those of binary relations, and show that in the case of Pen the algebraically irreducible representations provide a complete classification of the set of Penrose tilings from which its representation as a quotient of Cantor space is recovered.
Classifying Toposes for First Order Theories
 Annals of Pure and Applied Logic
, 1997
"... By a classifying topos for a firstorder theory T, we mean a topos E such that, for any topos F , models of T in F correspond exactly to open geometric morphisms F ! E . We show that not every (infinitary) firstorder theory has a classifying topos in this sense, but we characterize those which ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
By a classifying topos for a firstorder theory T, we mean a topos E such that, for any topos F , models of T in F correspond exactly to open geometric morphisms F ! E . We show that not every (infinitary) firstorder theory has a classifying topos in this sense, but we characterize those which do by an appropriate `smallness condition', and we show that every Grothendieck topos arises as the classifying topos of such a theory. We also show that every firstorder theory has a conservative extension to one which possesses a classifying topos, and we obtain a Heytingvalued completeness theorem for infinitary firstorder logic.
Constructive complete distributivity IV
 Appl. Cat. Struct
, 1994
"... A complete lattice L is constructively completely distributive, (CCD), when the sup arrow from downclosed subobjects of L to L has a left adjoint. The Karoubian envelope of the bicategory of relations is biequivalent to the bicategory of (CCD) lattices and suppreserving arrows. There is a restrict ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
A complete lattice L is constructively completely distributive, (CCD), when the sup arrow from downclosed subobjects of L to L has a left adjoint. The Karoubian envelope of the bicategory of relations is biequivalent to the bicategory of (CCD) lattices and suppreserving arrows. There is a restriction to order ideals and "totally algebraic" lattices. Both biequivalences have left exact versions. As applications we characterize projective sup lattices and recover a known characterization of projective frames. Also, the known characterization of nuclear sup lattices in set as completely distributive lattices is extended to yet another characterization of (CCD) lattices in a topos. Research partially supported by grants from NSERC Canada. Diagrams typeset using Michael Barr's diagram package. AMS Subject Classification Primary: 06D10 Secondary 18B35, 03G10. Keywords: completely distributive, adjunction, projective, nuclear Introduction Idempotents do not split in the category of rel...
Tropological systems are points of quantales
 J. Pure Appl. Algebra
, 2002
"... We address two areas in which quantales have been used. One is of a topological nature, whereby quantales or involutive quantales are seen as generalized noncommutative spaces, and its main purpose so far has been to investigate the spectrum of noncommutative C*algebras. The other sees quantales as ..."
Abstract

Cited by 7 (6 self)
 Add to MetaCart
We address two areas in which quantales have been used. One is of a topological nature, whereby quantales or involutive quantales are seen as generalized noncommutative spaces, and its main purpose so far has been to investigate the spectrum of noncommutative C*algebras. The other sees quantales as algebras of abstract experiments on physical or computational systems, and has been applied to the study of the semantics of concurrent systems. We investigate connections between the two areas, in particular showing that concurrent systems, in the form of either settheoretic or localic tropological systems, can be identified with points of quantales by means of a suitable adjunction, which indeed holds for a much larger class of socalled “tropological models”. We show that in the case of tropological models in factor quantales, which still generalize tropological systems, the identification of models and (generalized) points preserves all the information needed for describing the observable behaviour of systems. We also define a notion of morphism of models that generalizes previous definitions of morphism of systems, and show that morphisms, too, can be defined in terms of either side of the adjunction, in fact giving us isomorphisms of categories. The relation between completeness notions for tropological systems and spatiality for quantales is also addressed, and a preliminary partial preservation result is obtained.
Tensor products of idempotent semimodules. An algebraic approach
 Mathematical Notes
, 1999
"... Abstract. We study idempotent analogs of topological tensor products in the sense of A. Grothendieck. The basic concepts and results are simulated on the algebraic level. This is one of a series of papers on idempotent functional analysis. Key words: idempotent functional analysis, idempotent semiri ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
Abstract. We study idempotent analogs of topological tensor products in the sense of A. Grothendieck. The basic concepts and results are simulated on the algebraic level. This is one of a series of papers on idempotent functional analysis. Key words: idempotent functional analysis, idempotent semiring, idempotent semimodule, tensor product, polylinear mapping, nuclear operator. Dedicated to S.G. Krein on the occasion of his 80th birthday
Forbidden Forests in Priestley Spaces
, 2001
"... We present a rst order formula characterizing the distributive lattices L whose Priestley spaces P(L) contain no copy of a forest T . For Heyting algebras L, prohibiting a poset T in P(L) is characterized by equations i T is a tree. We also give a condition characterizing the distributive lattic ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
We present a rst order formula characterizing the distributive lattices L whose Priestley spaces P(L) contain no copy of a forest T . For Heyting algebras L, prohibiting a poset T in P(L) is characterized by equations i T is a tree. We also give a condition characterizing the distributive lattices whose Priestley spaces contain no copy of a forest with a single additional point at the bottom.
From intuitionistic to pointfree topology: on the foundation of homotopy theory
, 2005
"... Brouwer’s pioneering results in topology, e.g. invariance of dimension, were developed within a classical framework of mathematics. Some years later he explained ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
Brouwer’s pioneering results in topology, e.g. invariance of dimension, were developed within a classical framework of mathematics. Some years later he explained