We study some basic morphological operators acting on the lattice of all subgraphs of an arbitrary (unweighted) graph G. To this end, we consider two dual adjunctions between the edge set and the vertex set of G. This allows us (i) to recover the classical notion of a dilation/erosion of a subset of the vertices of G and (ii) to extend it to subgraphs of G. Afterward, we propose several new ope...

The starting point of the present study is the interpretation of intuitionistic linear logic in Petri nets proposed by U. Engberg and G. Winskel. We show that several categories of order algebras provide equivalent interpretations of this logic, and identify the category of the so called strongly coherent quantales arising in these interpretations. The equivalence of the interpretations is inti...

Proof theory for a logic with categorical semantics can be developed by the following methodology: define a sound and complete display calculus for an extension of the logic with additional adjunctions; translate this calculus to a shallow inference nested sequent calculus; translate this calculus to a deep inference nested sequent calculus; then prove this final calculus is sound with respect ...

Probabilistic Boolean Networks (PBNs) comprise a graphical model based on uncertain rule-based dependencies between nodes and have been proposed as a model for genetic regulatory networks. As with any algebraic structure, the characterization of important mappings between PBNs is critical for both theory and application. This paper treats the construction of mappings to alter PBN structure whil...

We study syntax-free models for name-passing processes. For interleaving semantics, we identify the indexing structure required of an early labelled transition system to support the usual π-calculus operations, defining Indexed Labelled Transition Systems. For noninterleaving causal semantics we define Indexed Labelled Asynchronous Transition Systems, smoothly generalizing both our interleaving...

We give a series of glueing constructions for categorical models of fragments of linear logic. Specifically, we consider the glueing of (i) symmetric monoidal closed categories (models of Multiplicative Intuitionistic Linear Logic), (ii) symmetric monoidal adjunctions (for interpreting the modality !) and (iii) -autonomous categories (models of Multiplicative Linear Logic); the glueing construc...

In this paper we use concepts from the lattice-based theory of morphological operators and fuzzy sets to develop generalized lattice image operators that are nonlinear convolutions that can be expressed as supremum (resp. infimum) of fuzzy intersection (resp. union) norms. Our emphasis and differences with many previous works is the construction of pairs of fuzzy dilation (sup of fuzzy intersec...

We introduce a syntax for quotient types in a predicate logic over a simply type theory. To illustrate its usefulness we construct in purely type theoretic terms (a) the free abelian group on a commutative monoid H, as quotient of H H; a special instance is the construction of Z from N; (b) the quotient poset of a preorder, (c) the abelian quotient of an arbitrary group, and (d) tensor products...

1 Preliminaries on monoids Definition 1.1. An Abelian monoid is a set with a binary, associative, and commutative operation which has a neutral element. It will often be called just a monoid in this manuscript because we will not deal with non-commutative monoids. A monoid M is called • finitely generated if there is a finite set of generators, or equivalently if there is a surjection of monoid...

We investigate the phenomenon that every monad is a linear state monad. We do this by studying a fully-complete state-passing translation from an impure call-by-value language to a new linear type theory: the enriched call-by-value calculus. The results are not specific to store, but can be applied to any computational effect expressible using algebraic operations, even to effects that are not ...