The approach to process semantics using quantales and modules is topologized by considering tropological systems whose sets of states are replaced by locales and which satisfy a suitable stability axiom. A corresponding notion of localic suplattice (algebra for the lower powerlocale monad) is described, and it is shown that there are contravariant functors from sup-lattices to localic sup-latic...

We investigate those lax extensions of a Set-monad T = (T,m, e) to the category V-Rel of sets and V-valued relations for a quantale V = (V,⊗, k) that are fully determined by maps ξ : TV → V . We pay special attention to those maps ξ that make V a T-algebra and, in fact, (V,⊗, k) a monoid in the category SetT with its cartesian structure. Any such map ξ forms the main ingredient to Hofmann’s not...

We develop a general framework for various lattice-valued, probabilistic and approach uniform convergence spaces. To this end, we use the concept of $s$-stratified $LM$-filter, where $L$ and $M$ are suitable frames. A stratified $LMN$-uniform convergence tower is then a family of structures indexed by a quantale $N$. For different choices of $L,M$ and $N$ we obtain the lattice-valued, probabili...

the paper sets forth in detail categorically-algebraic or catalg foundations for the operations of taking the image and preimage of (fuzzy) sets called forward and backward powerset operators. motivated by an open question of s. e. rodabaugh, we construct a monad on the category of sets, the algebras of which generate the fixed-basis forward powerset operator of l. a. zadeh. on the next step, w...

Let $L$ be an integral and commutative quantale. In this paper, by fuzzifying the notion of generalized neighborhood systems, the notion of $L$-fuzzy generalized neighborhoodsystem is introduced and then a pair of lower and upperapproximation operators based on it are defined and discussed. It is proved that these approximation operators include generalized neighborhood system...

Linear logic is a logic of actions which seems well suited to various computer science applications. From its intrinsic ability to reeect computational resources, it is possible to reene diierent programming paradigms like formulae-as-types (proofs-as-programs) or formulae-as-states (proofs-as-computations) with a ner control on resource management. In the latter case, the correspondence betwee...

The original definition of a topological space given by Hausdorff used neighborhood systems. Lattice-valued maps appear in this context when you identify a topology with a monoid in the Kleisli category of the filter monad on SET. H?hle’s notion of a lattice-valued topology [2] uses the same idea and it’s inspired in the classical lattice-valued topologies. Ltopological spaces are motivated by ...

Effect systems are lightweight extensions to type that can verify a wide range of important properties with modest developer burden. But our general understanding effect is limited primarily where the order effects irrelevant. Understanding such in terms semilattice grounds essential issues and provides guidance when designing new systems. By contrast, sequential systems—where important—lack an...

The reticulation of an algebra was first defined for commutative rings by Simmons [19] and it was extended by Belluce to non-commutative rings [3]. As for the algebras of fuzzy logics, Belluce also constructed the reticulation of an MV-algebra [2], G. Georgescu defined the reticulation of a quantale [8] and L. Leuştean made this construction for BL-algebras [13, 14]. In each of the papers cited...

A notion of reticulation which provides topological properties on algebras has introduced on commutative rings in 1980 by Simmons in [5]. For a given commutative ring A, a pair (L, λ) of a bounded distributive lattice and a mapping λ : A → L satisfying some conditions is called a reticulation on A, and the map λ gives a homeomorphism between the topological space Spec(A) consisting of prime fil...