## A non-commutative and non-idempotent theory of quantale sets

### Authors:

- Ulrich Höhle,
- Tomasz Kubiak

### Abstract

In fuzzy set theory non-idempotency arises when the conjunction is interpreted by arbitrary t-norms. There are many instances in mathematics where set theory ought to be non-commutative and/or non-idempotent. The purpose of this paper is to combine both ideas and to present a theory of non-commutative and non-idempotent quantale sets (among other things, standard concepts like fuzzy preorders and fuzzy equivalence relations will be exhibited as special cases). More specifically, the category Q-Set is investigated where Q is an arbitrary involutive quantale. Objects - quantale sets - are pairs consisting of a set and a Q-valued equality with a suitable symmetry axiom. Three important properties of Q-Set are shown: it is complete, cocomplete and has the (epi, extremal mono)-factorization property. Its subcategory s-Q-Set of separated quantale sets is reflective in Q-Set and shares the same fundamental properties with Q-Set; in particular s-Q-Set is also a complete and cocomplete (epi, extremal mono)-category. The objects of the category Q-Set are interesting categories in their own right. Two categorical frameworks for objects of Q-Set are exhibited. First, it is shown that Q-valued equalities arise from Q-valued preorders (with self-adjoint extents) by symmetrization which leaves Q-valued equalities invariant. Here, sets with quantale preorders are shown to be B-categories with base B being a specific quantaloid. The second approach is based on involutive quantaloids - a combination of two well known things: quantaloids and ordered categories with involution. In this context quantale sets are precisely symmetric B-categories w.r.t. appropriately chosen quantaloid B with involution. Further, the Cauchy completion preserves the symmetry axiom for a large class of involutive quantales which include quantic frames - our non-commutative generalization of frames - and all left continuous t-norms. There exist at least two monads on Q-Set, the singleton monad and quasi-singleton monad, which are of special interest for fuzzy set theory: the Kleisli category associated with the singleton monad is the non-commutative and non-idempotent analogue of Higgs' topos, while the Eilenberg-Moore category of the quasi-singleton monad permits the internalization of Łukasiewicz' negation as truth arrow. Finally, an application of quantale sets to C ^{*}-algebras is given and the change of base is treated. © 2010 Elsevier B.V.

- Record ID
- UAMececa54a2a2148f580804812f72bcd1f
- Author
- Journal series
- Fuzzy Sets and Systems, ISSN 0165-0114
- Issue year
- 2011
- Vol
- 166
- Pages
- 1-43
- ASJC Classification
- ;
- DOI
- DOI:10.1016/j.fss.2010.12.001 opening in a new tab
- Language
- en English
- Score (nominal)
- 40
- Score source
- journalList
- Publication indicators
- = 41.000; = 37.000; : 2011 = 1.952; : 2011 = 1.759 (2) - 2011=1.988 (5)

- Uniform Resource Identifier
- https://researchportal.amu.edu.pl/info/article/UAMececa54a2a2148f580804812f72bcd1f/

* presented citation count is obtained through Internet information analysis and it is close to the number calculated by the Publish or Perishopening in a new tab system.

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