A non-commutative and non-idempotent theory of quantale sets
type: Item
instance of: scholarly article
title: A non-commutative and non-idempotent theory of quantale sets
zbMATH Open document ID: 1226.06011
author: Ulrich Höhle
author: Tomasz Kubiak
published in: Fuzzy Sets and Systems
publication date: 2011-05-02T00:00:00Z
review text: Motivated by the theory of lattice-valued sets, the paper introduces the concept of quantale set, which is based in an arbitrary involutive quantale. The involution in question is used to generalize the symmetry axiom of lattice-valued equality relations. The main purpose of the manuscript is to address various issues related to non-idempotency and non-commutativity arising in the modern set theory. The authors study both the features of the new notion and its relationships to the similar concepts already existing in the literature through the properties of the category \(Q\)-\textbf{Set} of quantale sets based in a fixed involutive quantale \(Q\). In particular, they introduce an extension of lattice-valued preorder relations in the form of quantale preordered sets (omitting the symmetry axiom), whose category forms a supercategory of \(Q\)-\textbf{Set} with convenient properties. On the other hand, separated quantale sets provide an epireflective subcategory of \(Q\)-\textbf{Set}. These notions are also considered as quantaloid-enriched categories, where it is shown that quantale sets require involutive quantaloids, whereas quantale preordered ones do not. Two particular monads on \(Q\)-\textbf{Set} are constructed, i.e., singleton and quasi-singleton monads, and it is shown that the former is a submonad of the latter. The respective Kleisli category of the singleton monad is claimed to be a non-commutative and non-idempotent generalization of Higgs's topos, which is, however, not a topos. Motivated by the important issue of the change of base, at the end of the paper, the authors construct an adjunction between the categories of quantale sets based in two different quantales. For convenience of the reader, the paper starts with an extensive glimpse into the theory of quantales. Moreover, to address the issue of topological representation of non-commutative \(C^*\)-algebras, the authors introduce the concept of quantic frame as a non-commutative generalization of …
Mathematics Subject Classification ID: 06F07
Mathematics Subject Classification ID: 03E72
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 18C20
Mathematics Subject Classification ID: 46L85
zbMATH DE Number: 5882955
zbMATH Keywords: \(C^*\)-algebra
zbMATH Keywords: fuzzy subset
zbMATH Keywords: monad
zbMATH Keywords: quantale set
zbMATH Keywords: quantaloid
zbMATH Keywords: quantaloid-enriched category
zbMATH Keywords: quantic frame
zbMATH Keywords: topological category
zbMATH Keywords: topos
zbMATH Keywords: involutive quantale
reviewed by: Sergejs Solovjovs
MaRDI profile type: Publication
full work available at URL: j.fss.2010.12.001
OpenAlex ID: W2005254643
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A non-commutative and non-idempotent theory of quantale sets (DOI)
DOI: 10.1016/J.FSS.2010.12.001