Abstract
[4] placed an approximation space (U,≡ ) in a type-lowering retraction with its power set 2U such that the ≡ -exact subsets of U comprise the kernel of the retraction, where ≡ is the equivalence relation of set-theoretic indiscernibility within the resulting universe of exact sets. Since a concept thus forms a set just in case it is ≡ -exact, set-theoretic comprehension in (U,≡ ) is governed by the method of upper and lower approximations of Rough Set Theory. Some central features of this universe were informally axiomatized in [3] in terms of the notion of a Proximal Frege Structure and its associated modal Boolean algebra of exact sets. The present essay generalizes the axiomatic notion of a PFS to tolerance (reflexive, symmetric) relations, where the universe of exact sets forms a modal ortho-lattice. An example of this general notion is provided by the tolerance relation of “matching” over U.
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Apostoli, P.J., Kanda, A. (2005). Proximity Spaces of Exact Sets. In: Ślęzak, D., Wang, G., Szczuka, M., Düntsch, I., Yao, Y. (eds) Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing. RSFDGrC 2005. Lecture Notes in Computer Science(), vol 3641. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11548669_10
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