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Algebraic Methods for Rough Approximation Spaces by Lattice Interior–Closure Operations

  • Gianpiero Cattaneo
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

This chapter deals with the abstract approach to rough sets theory through the equational notion of closure operator in the context of lattice theory, with the associated notion of internal operator as not-closure-not. The involved lattice structures are not necessarily distributive to allow the development of rough theories in the so-called logical-algebraic context of Quantum Mechanics, based on non-distributive lattices of orthomodular type. The chapter is organized into four parts.

In Part I the more general lattice notion of closure operator is introduced, and the induced notion of interior operator is discussed. It is shown that this lattice approach of the inner-closure pairs is categorically equivalent to the non-equational abstract notion of approximation space based on the lower-upper approximation of each lattice element.

Part II deals with three variations of closure operators called respectively, from the more general to the stronger one, as Tarski, Kuratowski and Halmos closures. A characterization of this last is given in terms of Brouwer Zadeh lattice structure.

Part III provides an interpretation of the pairs of internal-closure operators in terms of pairs of necessity-possibility operators in the context of suitable modal logics. A Kripke-like semantic of such kinds of logics is also provided based on a set of possible worlds. The usual approach to the concrete theory of rough sets through Pawlak information systems is investigated in this context.

Part IV treats these internal-closure operators in context of Łukasiewicz algebraic structures, stronger than the Halmos ones. It is shown that the usual fuzzy sets theory is a model of such structures. Finally, the correlation between Łukasiewicz algebraic structures and Nelson algebras is discussed.

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Authors and Affiliations

  • Gianpiero Cattaneo
    • 1
  1. 1.Dipartimento di Informatica, Sistemistica e ComunicazioneUniversità di Milano–BicoccaMilanoItaly

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