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Nuclear and Trace Ideals in Tensored *Categories
, 1998
"... We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed ..."
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Cited by 28 (10 self)
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We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored categories, all morphisms are nuclear, and in the tensored category of Hilbert spaces, the nuclear morphisms are the HilbertSchmidt maps. We also introduce two new examples of tensored categories, in which integration plays the role of composition. In the first, mor...
A NEW LATTICE CONSTRUCTION: THE BOX PRODUCT
, 2005
"... In a recent paper, the authors have proved that for lattices A and B with zero, the isomorphism Conc(A ⊗ B) ∼ = Conc A ⊗ Conc B, holds, provided that the tensor product satisfies a very natural condition (of being capped) implying that A ⊗ B is a lattice. In general, A ⊗ B is not a lattice; for i ..."
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Cited by 5 (1 self)
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In a recent paper, the authors have proved that for lattices A and B with zero, the isomorphism Conc(A ⊗ B) ∼ = Conc A ⊗ Conc B, holds, provided that the tensor product satisfies a very natural condition (of being capped) implying that A ⊗ B is a lattice. In general, A ⊗ B is not a lattice; for instance, we proved that M3 ⊗ F(3) is not a lattice. In this paper, we introduce a new lattice construction, the box product for arbitrary lattices. The tensor product construction for complete lattices introduced by G. N. Raney in 1960 and by R. Wille in 1985 and the tensor product construction of A. Fraser in 1978 for semilattices bear some formal resemblance to the new construction. For lattices A and B, while their tensor product A ⊗ B (as semilattices) is not always a lattice, the box product, A □B, is always a lattice. Furthermore, the box product and some of its ideals behave like an improved tensor product. For example, if A and B are lattices with unit, then the isomorphism Conc(A □B) ∼ = Conc A ⊗ Conc B holds. There are analogous results for lattices A and B with zero and for a bounded lattice A and an arbitrary lattice B. A joinsemilattice S with zero is called {0}representable, if there exists a lattice L with zero such that Conc L ∼ = S. The above isomorphism results yield the following consequence: The tensor product of two {0}representable semilattices is {0}representable.
A Duality Theory for Quantitative Semantics
 Computer Science Logic. 11th International Workshop, volume 1414 of Lecture Notes in Computer Science
, 1998
"... . A continuous predicate on a domain, or more generally a topological space, can be concretely described as an open or closed set, or less obviously, as the set of all predicates consistent with it. Generalizing this scenario to quantitative predicates, we obtain under certain wellunderstood hypothe ..."
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Cited by 4 (3 self)
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. A continuous predicate on a domain, or more generally a topological space, can be concretely described as an open or closed set, or less obviously, as the set of all predicates consistent with it. Generalizing this scenario to quantitative predicates, we obtain under certain wellunderstood hypotheses an isomorphism between continuous functions on points and supremum preserving functions on open sets, both with values in a fixed lattice. The functions on open sets provide a topological foundation for possibility theories in Artificial Intelligence, revealing formal analogies of quantitative predicates with continuous valuations. Three applications of this duality demonstrate its usefulness: we prove a universal property for the space of quantitative predicates, we characterize its infirreducible elements, and we show that bicontinuous lattices and Scottcontinuous maps form a cartesian closed category. 1 Introduction It is wellknown that a predicate p on a set X, i.e. a function p ...
Bicontinuous Function Spaces
, 1999
"... Given a sober space (X; O(X)) and a complete lattice L in its Scotttopology, we study the function space [X ! L] of all continuous maps f : X ! L, ordered pointwise. We show that this partial order is a bicontinuous lattice (i.e. the lattice and its order dual are continuous) if and only if L is bi ..."
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Cited by 2 (1 self)
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Given a sober space (X; O(X)) and a complete lattice L in its Scotttopology, we study the function space [X ! L] of all continuous maps f : X ! L, ordered pointwise. We show that this partial order is a bicontinuous lattice (i.e. the lattice and its order dual are continuous) if and only if L is bicontinuous, X is a continuous domain and O(X) is its Scotttopology. This extends known results on the continuity of the space [X ! L]. The techniques are novel in the theory of continuous lattices in that they employ a representation of the dual of [X ! L] as the lattice of maps preserving all suprema of type ¯: O(X) ! L op , where L op is the order dual of L. We specialize these results down to two classes of bicontinuous lattices: linear FSlattices and completely distributive lattices. 1 Introduction For a complete lattice L, a subset U is Scott open if it is an upper set that is inaccessible by directed suprema: if D ` L is directed and W D 2 U , then D " U 6= ;: This defines a...