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

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

Let $U = [a, b] \subset {\mathbb {R}}$ and ${\widetilde{U}} = \{\mu _{A_{1}}(x), \ldots , \mu _{A_{n}}(x)\}, n \in {\mathbb {N}}^{+},$ a vague partition of 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.