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 a...

In this communication we generalize some recent results of Rump to categories enriched in a commutative quantale V. Using these results, show that every quantale-enriched multicategory admits an injective hull. Finally, expose connection between the Isbell adjunction and construction hulls for topological spaces made by Banaschewski 1973.

In this brief note we would like to give the construction of a free commutative unital associative Nijenhuis algebra on a commutative unital associative algebra based on an augmented modified quasi-shuffle product. —————————————

In this paper we prove that any quantale Q is (isomorphic to) a quantale of suitable relations on Q. As a consequence two isomorphism theorems are also shown with suitable sets of functions of Q into Q. These theorems are the mathematical background one needs in order to give natural and complete semantics for (non-commutative) Linear Logic using relations. 1991 Mathematics Subject Classificati...

We introduce the concept of a Girard couple, which consists of two (not necessarily unital) quantales linked by a strong form of duality. The two basic examples of Girard couples arise in the study of endomorphism quantales and of the spectra of operator algebras. We construct, for an arbitrary sup-lattice S, a Girard quantale whose right-sided part is isomorphic to S.

In this paper, the category of L-FINT is topological category over Set. We investigate the functorial relations between L-FTOP and LFINT where L is a strictly two-sided, commutative quantale lattice. Mathematics Subject Classification: 03E72, 54A40, 54B10

In this paper, we study the relations (L, )-fuzzy topologies and (L, )-filters on strictly two-sided, commutative quantale lattices (L, ) and (L, ∗). Mathematics Subject Classification: 03E72, 54A40, 54B10

In this paper, we study (L, )-fuzzy topologies induced by operations and (L, )-filters on strictly two-sided, commutative quantale lattices (L, ) and (L, ∗). Mathematics Subject Classification: 03E72, 54A40, 54B10

It is shown that every two-variable adjunction in categories enriched a commutative quantale serves as base for constructing Isbell adjunctions between functor categories, and Kan are precisely constructed from suitable associated adjunctions. Representation theorems established fixed points of these