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Modal-Like Operators in Boolean Lattices, Galois Connections and Fixed Points

Järvinen J., Kondo M., Kortelainen J.

Fundamenta Informaticae

2007

Vol. 76, nr 1-2

129-145

In this work, four modal-like operators on Boolean lattices are introduced and their theory is presented from lattice-theoretical, topological and algebraic point of view. It is also shown how rough set approximation operators, modal operators in temporal logic, and linguistic modifiers determined by L-sets can be interpreted as modal-like operators.

On the evaluation of compatibility with gradual rules in information systems : a topological approach

Kortelainen J.

Control and Cybernetics

1999

Vol. 28, no 1

121-131

The paper presents an approach to detecting dependence of a decision-maker's actions (or decisions) on conditions with respect to a certain type of fuzzy rules, namely gradual rules. Gradual rules are of the form "the more c is C, the more d is D", where C and D are L-fuzzy sets on the domain of the attributes c and d with membership values possibly in different lattices, respectively. The decision-maker's actions and conditions for those are found in a data base-like system, called an information system. We use the information system to define relations corresponding to L-fuzzy sets, and we define weakening modifiers by these relations. These modifiers are closure operators in the corresponding topologies, and we can define fineness relation between L-fuzzy sets. This relation is useful when comparing different decision-maker's actions. We define strong degree and topological degree of depedence of the attributes (with respect to a gradual rule), and we study some properties of those. Finally, we present a small application.