Information Quanta and Approximation Operators: Once More Around the Track

The present paper investigates topological variations of information structures (together with the associated operators) from the theory of Information Quanta, namely Information Quantum Relational Systems (IQRSs) and Property Systems, with special emphasis put on approximation spaces from Rough Set Theory (RST) and contexts from Formal Concept Analysis (FCA), respectively. The main novelty of this study comes from the influence of the theory of Heyting-Brouwer Algebras (HBAs) – some of that influence is reflected in the choice of Alexandroff topological spaces as the main source of both IQRSs and Property Systems. It allows us to establish some new results concerning RST and FCA, and to extend RST by a new type of rough sets, i.e. Heyting-Brouwer rough sets, defined with respect to bitopological spaces. Furthermore, we provide two interpretations of operators from RST and FCA, the first one in terms of HBAs and the second one (by the extended Gödel translation from intuitionistic logic to S4) in terms of temporal logic S4.t.