A non-commutative and non-idempotent theory of quantale sets
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 frames to build non-commutative topologies upon. A short appendix provides the necessary results on Cauchy completion of quantaloid-enriched categories, which is used in the paper as a tool for both building new fuzzy preorders from old ones and constructing the above-mentioned monads on the category \(Q\)-\textbf{Set}.
scientific article; zbMATH DE number 5882955
scientific article