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Lattices with interior and closure operators and abstract approximation spaces. (English) Zbl 1248.06005
Peters, James F. (ed.) et al., Transactions on Rough Sets X. Berlin: Springer (ISBN 978-3-642-03280-6/pbk). Lecture Notes in Computer Science 5656. Journal Subline, 67-116 (2009).
Summary: The non-equational notion of abstract approximation space for roughness theory is introduced, and its relationship with the equational definition of lattice with Tarski interior and closure operations is studied. Their categorical isomorphism is proved, and the role of the Tarski interior and closure with an algebraic semantic of an S4-like model of modal logic is widely investigated.
A hierarchy of three particular models of this approach to roughness based on a concrete universe is described, listed from the stronger model to the weaker one: (1) the partition spaces, (2) the topological spaces by open basis, and (3) the covering spaces.
For the entire collection see [Zbl 1169.68304].

MSC:
06B75 Generalizations of lattices
03B45 Modal logic (including the logic of norms)
03G25 Other algebras related to logic
68T37 Reasoning under uncertainty in the context of artificial intelligence
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