We define strict closure space algebras as Boolean algebras augmented with a unary operation Int that satisfies conditions analogous to those of the interior operation from strict closure spaces. We also consider a first order modal logic SCS in which the necessity operator satisfies analogous conditions. Sikorski proved that every interior algebra can be represented as an algebra of subsets of...

We deene the notion of weak relative pseudo-complement, and we show that, for an arbitrary lattice, the property of weak relative pseudo-complementation is strictly weaker than relative pseudo-complementation, but stronger than pseudo-complementation. Our main interest for this notion is in relation with the theory of closure operators. We prove that if a complete lattice L is completely inf-di...

We study relations between predicate transformers and mul-tifunctions in a topological setting based on closure operators. We give topological deenitions of safety and liveness predicates and using these predicates we deene predicate transformers. State transformers are mul-tifunctions with values in the collection of xed points of a closure operator. We derive several isomorphisms between pred...

We consider a specific class of combinatorial search or optimization problems where the search space gives rise to a closure operator and essentially the hulls are the only relevant subsets that must be checked in a brute force approach. We suggest that such a closure property can help to reduce time complexities. Moreover we propose two types of (structural) parameterization of instance classe...

After a brief review of hidden algebra, we give behavioral speciications for set theory and closure operators, and then use these to give a behavioral speciication of an abstract protocol to support distributed cooperative work structured by dependencies in such a way as to form what we call a weak closure operator. We give some correctness proofs for this protocol, and then describe a concrete...

We study a cooperation structure in which the feasible coalitions are those belonging to a partition closure system. First, we develop the relation between the dividends of Harsanyi in the restricted game and the worth function in the original game. Next, we obtain an method for computing the Shapley and Banzhaf values of the closure-restricted games using the dividends formula. Finally, we ana...

For simplicity, we adopt the following rules: i, x, I denote sets, A, M denote many sorted sets indexed by I, f denotes a function, and F denotes a many sorted function indexed by I. The scheme MSSUBSET deals with a set A , a non-empty many sorted set B indexed by A , a many sorted set C indexed by A , and a unary predicate P , and states that: If for every many sorted set X indexed by A holds ...

Relativizing an idea from multiplicity theory, we say that an element x of a von Neumann algebra M is n-divisible if W (x) ∩ M unitally contains a factor of type In. We decide the density of the n-divisible operators, for various n, M, and operator topologies. The most sensitive case is σ-strong density in II1 factors, which is closely related to the McDuff property. We make use of Voiculescu’s...