Nuclearity in the category of complete semilattices

Higgs, Denis; Rowe, Katelyn

doi:10.1016/0022-4049(89)90028-5

Cited by 24 publications

(25 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Overall, this set of results yields an important clarification of the algebra used in our previous works [ 6 , 21 , 22 ] and, more importantly, new characterizations of completely distributive lattices adding up to the existing ones, see e.g. [ 7 , 11 , 15 , 16 , 23 ]. These characterizations strongly rely on the algebra of quantales and residuated lattices thus on relation algebra, in a wider sense.…”

Section: Introductionsupporting

confidence: 60%

The Involutive Quantaloid of Completely Distributive Lattices

Santocanale¹

2020

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Let L be a complete lattice and let be the unital quantale of join-continuous endo-functions of L . We prove that has at most two cyclic elements, and that if it has a non-trivial cyclic element, then L is completely distributive and is involutive (that is, non-commutative cyclic -autonomous). If this is the case, then the dual tensor operation corresponds, via Raney’s transforms, to composition in the (dual) quantale of meet-continuous endo-functions of L . Let be the category of sup-lattices and join-continuous functions and let be the full subcategory of whose objects are the completely distributive lattices. We argue that is itself an involutive quantaloid, thus it is the largest full-subcategory of with this property. Since is closed under the monoidal operations of , we also argue that if is involutive, then is completely distributive as well; consequently, any lattice embedding into an involutive quantale of the form has, as its domain, a distributive lattice.

show abstract

Section: Introductionsupporting

confidence: 60%

The Involutive Quantaloid of Completely Distributive Lattices

Santocanale¹

2020

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…The key idea, expressed in our abstract definition of nuclear ideals, is that certain morphisms can be thought of as behaving like "matrices". Our work naturally follows on from the development of Higgs and Rowe [37], the fundamental difference being that we have no closed structure. Crudely speaking, Higgs and Rowe generalize Banach space theory while we generalize Hilbert space theory.…”

Section: Discussionmentioning

confidence: 99%

Nuclear and trace ideals in tensored ∗-categories

Abramsky

Blute

Panangaden

1999

Journal of Pure and Applied Algebra

112

View full text Add to dashboard Cite

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 Hilbert-Schmidt maps.We also introduce two new examples of tensored * -categories, in which integration plays the role of composition. In the first, morphisms are a special class of distributions, which we call tame distributions. We also introduce a category of probabilistic relations.Finally, we extend the recent work of Joyal, Street and Verity on traced monoidal categories to this setting by introducing the notion of a trace ideal. We establish a close correspondence between nuclear ideals and trace ideals in a tensored * -category, suggested by the correspondence between Hilbert-Schmidt operators and trace operators on a Hilbert space.

show abstract

“…Nuclearity was first introduced by Grothendieck in Grothendieck ( 1955 ). Later it was taken up by Rowe and Higgs in multiple papers (Rowe 1988 ; Higgs and Rowe 1989 ). More recently Abramsky et al.…”

Section: Tracing On Finite Objectsmentioning

confidence: 99%