Status, Abstract

Rudolf Berghammer

Institut für Informatik
Christian-Albrechts-Universität zu Kiel
Olshausenstr. 40, 24098 Kiel, Germany

Contact, closure, topology, and the linking of row and column types of relations

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Journal of Logic and Algebraic Programming 80 (6) 2011, 339-361

Extended version

Gunther Schmidt

Fakultät für Informatik
Universität der Bundeswehr München
85577 Neubiberg, Germany

Forming closures of subsets of a set X is a technique that plays an important role in many scientific disciplines and there are many cryptomorphic mathematical structures that describe closures and their construction. One of them was introduced by Aumann in the year 1970 under the name contact relation. Using relation algebra, we generalize Aumann's notion of a contact relation between X and its powerset P(X) and that of a closure operation on P(X) from powersets to general membership relations and their induced partial orders. We also investigate the relationship between contacts and closures in this general setting and present some applications. In particular, we investigate the connections between contacts, closures and topologies and use contacts to establish a one-to-one correspondence between the column intersections space and the row intersections space of arbitrary relations.