We revisit sheaves on locales by placing them in the context of the theory of Hilbert quantale modules. The local homeomorphisms p : X → B are identified with the locales X that are Hilbert B-modules equipped with a natural notion of basis. These modules form a full subcategory B-HMB of the category of Hilbert B-modules where all the homomorphisms are adjointable: for each homomorphism h there ...

In this work, we describe an adjunction between the comma category of Set-based monads under the V -powerset monad and the category of associative lax extensions of Set-based monads to the category of V -relations. In the process, we give a general construction of the Kleisli extension of a monad to the category of V-relations.

Following the point-set lattice-theoretic (poslat) approach to topology of S. E. Rodabaugh [11], we introduced a more general way to do categorical (fuzzy) topology under the name of categorically-algebraic (catalg) topology [12], which unified many lattice-valued topological settings, and provided the means of intercommunication between different such frameworks. Moreover, motivated by the res...

The Morita equivalence of m-regular involutive quantales in the context of the theory of Hilbert A-modules is presented. The corresponding fundamental representation theorems are shown. We also prove that two commutative m-regular involutive quantales are Morita equivalent if and only if they are isomorphic. In the paper [5] F. Borceux and E.M. Vitale made a first step in extending the theory o...

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

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

A quantum causal topology is presented. This is modeled after a non-commutative scheme type of theory for the curved finitary spacetime sheaves of the non-abelian incidence Rota algebras that represent ‘gravitational quantum causal sets’. The finitary spacetime primitive algebra scheme structures for quantum causal sets proposed here are interpreted as the kinematics of a curved and reticular l...

We define the Lukasiewicz Transform as a residuated map and a homomorphism between semimodules over the semiring reducts of an MV-algebra. Then we describe the “ Lukasiewicz Transform Based” ( LTB) algorithm for image processing. We also propose a strong extension of such concepts and results in the framework of quantales and quantale modules. 1 The Lukasiewicz transform and its applications Th...

1.1 Relationship between Petri net and linear logic Petri nets were first introduced by Petri in his seminal Ph.D. thesis, and both the theory and the applications of his model have flourished in concurrency theory (Reisig & Rozenberg, 1998a; Reisig & Rozenberg, 1998b). The relationships between Petri nets and linear logics have been studied by many researchers (Engberg & Winskel, 1997; Farwer,...

Addressing the problem in the title, we diagnose the essential diiculty as the implicit assumption of two-valued logic in the event structure view of event occurrence as a true-or-false proposition. Taking the 3-valued logic of before-during-after as the common underlying logic of both event structures and higher dimensional automata (HDA's) uniies them into a single model having the same proce...