# Correction to: Redefinition of the concept of fuzzy set based on vague partition from the perspective of axiomatization

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## 1 Correction to: Soft Comput https://doi.org/10.1007/s00500-017-2855-5

I recently found that several errors occur in the statement of Definition 5.2 in Section 5 in the paper “Redefinition of the concept of fuzzy set based on vague partition from the perspective of axiomatization”.

As it had been pointed out at the end of Section 4 of this paper that “the set of vague attribute values is defined as a free algebra on the elementary set of vague attribute values”, and fuzzy sets are mathematical formulation for vague attribute values, hence, the set of fuzzy sets in *U* can be seen as freely generated by a vague partition of *U*.

Based on this consideration, Definition 5.2 of this paper can be corrected as follows:

### Definition 5.2

*U*. The set \({\mathscr {F}}({\widetilde{U}})\) of fuzzy sets in

*U*with respect to \({\widetilde{U}}\) consists of the following elements:

- (1)
if there exists \(i \in {\overline{n}}\) such that \(\mu _{A}(x) = \mu _{A_{i}}(x)\) for all \(x \in U\), then \(A = \{(x, \mu _{A}(x)) \mid x \in U\} \in {\mathscr {F}}({\widetilde{U}})\);

- (2)
if \(\mu _{A}(x) = {\overline{\mu }}(x) = 1\) for all \(x \in U\), then \(A = \{(x, \mu _{A}(x)) \mid x \in U\} \in {\mathscr {F}}({\widetilde{U}})\);

- (3)
if \(\mu _{A}(x) = \underline{\mu }(x) = 0\) for all \(x \in U\), then \(A = \{(x, \mu _{A}(x)) \mid x \in U\} \in {\mathscr {F}}({\widetilde{U}})\);

- (4)
if \(A = \{(x, \mu _{A}(x)) \mid x \in U\} \in {\mathscr {F}}({\widetilde{U}})\) and \(r \in {\mathbb {Q}}^{+}\), then \(A^{r} = \{(x, (\mu _{A}(x))^{r}) \mid x \in U\} \in {\mathscr {F}}({\widetilde{U}})\);

- (5)
if \(A = \{(x, \mu _{A}(x)) \mid x \in U\} \in {\mathscr {F}}({\widetilde{U}})\), and

*N*is a strong negation on [0, 1], then \(A^{N} = \{(x, (\mu _{A}(x))^{N}) \mid x \in U\} \in {\mathscr {F}}({\widetilde{U}})\); - (6)
if \(A = \{(x, \mu _{A}(x)) \mid x \in U\}, B = \{(x, \mu _{B}(x)) \mid x \in U\} \in {\mathscr {F}}({\widetilde{U}})\), and \(\otimes \) is a triangular norm, then \(A \cap _{\otimes } B = \{(x, \mu _{A}(x) \otimes \mu _{B}(x)) \mid x \in U\} \in {\mathscr {F}}({\widetilde{U}})\);

- (7)
if \(A = \{(x, \mu _{A}(x)) \mid x \in U\}, B = \{(x, \mu _{B}(x)) \mid x \in U\} \in {\mathscr {F}}({\widetilde{U}})\), and \(\oplus \) is a triangular conorm, then \(A \cup _{\oplus } B = \{(x, \mu _{A}(x) \oplus \mu _{B}(x)) \mid x \in U\} \in {\mathscr {F}}({\widetilde{U}})\);

- (8)
\({\mathscr {F}}({\widetilde{U}})\) not include other elements.

In fact, \({\mathscr {F}}({\widetilde{U}})\) can be considered as a function space based on \({\widetilde{U}}\).

We apologize to the readers for any inconvenience these errors might have caused.