Distributive contact join-semilattices

Contact algebra is one of the main tools in region-based theory of space. In \cite{dmvw1, dmvw2,iv,i1} it is generalized by dropping the operation Boolean complement. Furthermore we can generalize contact algebra by dropping also the operation meet. We call the obtained structure a distributive contact join-semilattice (DCJS). We obtain representation theorems for DCJS and the universal theory of DCJS which is decidable

Contact semilattices

We devise exact conditions under which a join semilattice with a weak contact relation can be semilattice embedded into a Boolean algebra with an overlap contact relation, equivalently, into a distributive lattice with additive contact relation. A similar characterization is proved with respect to Boolean algebras and distributive lattices with weak contact, not necessarily additive, nor overlap.Comment: v3: noticed that former Condition (D2-) is pleonastic; added two new equivalent conditions in Theorem 3.2. We realized all this after the paper has been published: variations with respect to the published version are printed in a blue character. v2: solved a problem left open in v1; added a counterexample; a few fixe

Residuation algebras with functional duals

We employ the theory of canonical extensions to study residuation algebras whose associated relational structures are functional, i.e., for which the ternary relations associated to the expanded operations admit an interpretation as (possibly partial) functions. Providing a partial answer to a question of Gehrke, we demonstrate that functionality is not definable in the language of residuation algebras (or even residuated lattices), in the sense that no equational or quasi-equational condition in the language of residuation algebras is equivalent to the functionality of the associated relational structures. Finally, we show that the class of Boolean residuation algebras such that the atom structures of their canonical extensions are functional generates the variety of Boolean residuation algebras

Pi-Calculus in Logical Form

Abramsky’s logical formulation of domain theory is extended to encompass the domain theoretic model for picalculus processes of Stark and of Fiore, Moggi and Sangiorgi. This is done by defining a logical counterpart of categorical constructions including dynamic name allocation and name exponentiation, and showing that they are dual to standard constructs in functor categories. We show that initial algebras of functors defined in terms of these constructs give rise to a logic that is sound, complete, and characterises bisimilarity. The approach is modular, and we apply it to derive a logical formulation of pi-calculus. The resulting logic is a modal calculus with primitives for input, free output and bound output

Coalgebras and Their Logics

Transition systems pervade much of computer science. This article outlines the beginnings of a general theory of specification languages for transition systems. More specifically, transition systems are generalised to coalgebras. Specification languages together with their proof systems, in the following called (logical or modal) calculi, are presented by the associated classes of algebras (e.g., classical propositional logic by Boolean algebras). Stone duality will be used to relate the logics and their coalgebraic semantics