Soft Computing

, Volume 22, Issue 6, pp 2079–2079

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

Correction

## 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. (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. (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. (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. (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. (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. (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. (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. (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.