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

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.