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

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