On relationship between modified sets, topological spaces and rough sets. (English) Zbl 0828.04002

Kortelainen, Jari

Fuzzy Sets Syst. 61, No. 1, 91-95 (1994).

The author starts from a binary relation \(R_{\mathcal H}\) on a set \(X\) and then defines a so-called modifier \({\mathcal H}\) by: \[ {\mathcal H} (A) = \biggl \{y \mid (\exists x \in A) \bigl( (x,y) \in R_{\mathcal H} \bigr) \biggr\}. \] When the relation \(R\) is reflexive, i.e. \(R\) is called an accessibility relation, then \({\mathcal H}\) is called a weakening modifier. To every weakening modifier \({\mathcal H}\) a so-called substantiating modifier \({\mathcal H}^*\) is defined by \({\mathcal H}^* (A) = \overline {{\mathcal H} (\overline A)}\). The main results concern the interaction between set-theoretic union, intersection, inclusion on the one hand and weakening and substantiating modifiers on the other hand. The proofs are very simple. Finally it is shown that certain modifiers are topological closure operators and that rough sets can be obtained as modifiers.
This paper can be described as old wine in new bottles. Indeed \({\mathcal H}(A)\) is nothing but the direct image of \(A\) under the binary relation \(R_{\mathcal H}\) and \({\mathcal H}^* (A)\) is easily identified as the superdirect image of \(A\) under a reflexive relation \(R_{\mathcal H}\). Hence all the main results were well known.

Reviewer: E.Kerre (Gent)

MSC:

03E72

Theory of fuzzy sets, etc.

Keywords:

accessibility relation; weakening modifier; substantiating modifier; union; intersection; inclusion; topological closure operators; rough sets; direct image; superdirect image

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