# The Linguistic Concept’s Reduction Methods under Symmetric Linguistic-Evaluation Information

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## Abstract

**:**

## 1. Introduction

- The formal concepts in classical formal concept analysis only express whether objects share attributes or whether attributes are shared by objects. In real life, it is not enough to study only these two relations between objects and attributes. Likewise, formal contexts with linguistic information face the same problem. Thus, other relationships that may exist between objects and linguistic concepts need to be discussed as well.
- The construction of conceptual knowledge is inherently a complex challenge. There is easily a large amount of redundant information in the process of knowledge processing, resulting in a high amount of computational complexity. Therefore, there is an urgent need to propose a reduction method that can reduce the complexity of linguistic concept knowledge.
- Scholars have achieved many results in fuzzy formal concept analysis. A large amount of fuzzy information can exist in a linguistic environment, so studying the fuzzy linguistic concept formal context is necessary based on the challenges presented above.

- Based on three-way concept lattice and modal operators with possibilities and necessity, a fuzzy-object-induced three-way attribute-oriented linguistic (FOEAL) concept lattice is proposed to express more information in a fuzzy linguistic concept formal context.
- A novel linguistic-concept granular-reduction method based on the FOEAL lattice is designed to preserve granular concept information, which reduces the scale of conceptual knowledge in a linguistic environment.
- In order to highlight the importance of linguistic-concept information, an entropy-reduction method based on the FOEAL lattice is also presented. We further verified that the set of this entropy reduction is consistent with those of the granular reduction.
- The examples of the student-debate competition can confirm the rationality, and the comparative analysis provides strong evidence for the effectiveness of the proposed method.

## 2. Preliminaries

#### 2.1. Basic Notions on Concept Lattice

**Definition 1.**

**Definition 2.**

**Definition 3.**

**Definition 4.**

**Definition 5.**

**Definition 6.**

**Definition 7.**

#### 2.2. Linguistic Term Set

- (1)
- order relation: ${s}_{\alpha}\ge {s}_{\beta}$, if $\alpha \ge \beta $,
- (2)
- negation operator: $Neg({s}_{\alpha})={s}_{\beta}$, where $\beta =g-\alpha $,
- (3)
- maximization operator: max$\{{s}_{\alpha},{s}_{\beta}\}={s}_{\alpha}$, if $\alpha \ge \beta $,
- (4)
- minimization operator: min$\{{s}_{\alpha},{s}_{\beta}\}={s}_{\beta}$, if $\alpha \ge \beta $.

#### 2.3. Linguistic Concept Lattice

**Definition 8.**

#### 2.4. Linguistic Concept Reduction

## 3. Fuzzy-Object-Induced Three-Way Attribute-Oriented Linguistic Concept Lattice

#### 3.1. The Construction of a Fuzzy-Object-Induced Three-Way Attribute-Oriented Linguistic Concept Lattice

**Definition 9.**

**Definition 10.**

**Definition 11.**

**Definition 12.**

**Definition 13.**

**Example 1.**

#### 3.2. The Granular Reduction of Fuzzy Linguistic Concept Formal Context

**Definition 14.**

**Definition 15.**

**Theorem 1.**

**Proof.**

- (1)
- core linguistic concept set ${C}_{r}$: ${C}_{r}=\cap Red({\mathcal{K}}_{AG})$,
- (2)
- relatively necessary linguistic concept set ${K}_{r}$: ${K}_{r}=\cup Red({\mathcal{K}}_{AG})-\cap Red({\mathcal{K}}_{AG})$,
- (3)
- unnecessary linguistic concept set ${I}_{r}$: ${I}_{r}=G-\cup Red({\mathcal{K}}_{AG})$.

**Definition 16.**

**Definition 17.**

**Theorem 2.**

**Proof.**

**Example 2.**

## 4. The Relation between Granular Reduction and Entropy Reduction in the Fuzzy Linguistic Concept Formal Context

**Definition 18.**

**Theorem 3.**

- 1.
- $H({C}_{{s}_{\alpha}})\le H({B}_{{s}_{\alpha}})$ and ${x}^{\u2aaf{C}_{{s}_{\alpha}}\u2ab0{C}_{{s}_{\alpha}}}\supseteq {x}^{\u2aaf{B}_{{s}_{\alpha}}\u2ab0{B}_{{s}_{\alpha}}}$;
- 2.
- if $H({C}_{{s}_{\alpha}})=H({B}_{{s}_{\alpha}})$, then ${x}^{\u2aaf{C}_{{s}_{\alpha}}\u2ab0{C}_{{s}_{\alpha}}}={x}^{\u2aaf{B}_{{s}_{\alpha}}\u2ab0{B}_{{s}_{\alpha}}}$.

**Proof.**

**Definition 19.**

- (1)
- core linguistic concept set ${C}_{t}$: ${C}_{t}=\cap Red({\mathcal{K}}_{AE})$,
- (2)
- relatively necessary linguistic concept set ${K}_{t}$: ${K}_{t}=\cup Red({\mathcal{K}}_{AE})-\cap Red({\mathcal{K}}_{AE})$,
- (3)
- unnecessary linguistic concept set ${I}_{t}$: ${I}_{t}=G-\cup Red({\mathcal{K}}_{AE})$.

**Theorem 4.**

**Proof.**

**Theorem 5.**

**Proof.**

**Remark 1.**

**Theorem 6.**

**Proof.**

**Theorem 7.**

**Proof.**

**Theorem 8.**

**Proof.**

**Example 3.**

## 5. Comparative Analysis

- In Ref. [30]: Utilizing the object-oriented concept lattice ${L}_{O}(G,M,I)$ and the attribute-oriented concept lattice ${L}_{A}(G,M,I)$, Qin et al. introduced a technique for attribute reduction that maintains decision rules.
- In Ref. [39]: Zhang et al. put forward a new fuzzy three-way concept lattice, denoted by $OFTL(G,M,\tilde{I})$ and $AFTL(G,M,\tilde{I})$, which takes into account the fuzziness of objects and attributes, respectively. Furthermore, they presented a granular matrix-based reduction method to handle fuzzy data in a fuzzy formal context.
- In Ref. [22]: Ren et al. developed four techniques for attribute reduction that preserve lattice structure, granular information and join (meet)-irreducible elements, utilizing three-way concept lattices $OEL(G,M,I)$ and $AEL(G,M,I)$.
- In Ref. [48]: Zou et al. presented a linguistic concept lattice $LL(G,{L}_{{s}_{\alpha}},I)$, and further studied a multi-granularity linguistic-concept reduction algorithm based on the similarity relations in an incomplete linguistic concept formal context, which can deal with different types of linguistic information.

**Remark 2.**

- (1)
- To accurately represent the uncertainty and complexity of real-world situations, we introduced a fuzzy linguistic concept formal context that establishes a fuzzy relation between objects and linguistic concepts. This approach generates a FOEAL lattice that aligns more closely with human cognition.
- (2)
- By combining the advantages of $OEL(G,M,I)$ and ${L}_{A}(G,M,I)$, we propose a FOEAL lattice in a fuzzy linguistic concept formal context, which can not only show the idea of three divisions in three-way decisions, but also express the complementary structure of a linguistic concept lattice compared with symmetric linguistic-evaluation information.
- (3)
- In view of the validity and simplicity of granular reduction in formal concept analysis, two linguistic-concept-reduction methods preserving granular information and information entropy, granular reduction and entropy reduction based on the FOEAL lattice, are given to reduce the scale of linguistic concepts.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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a | b | c | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{a}}_{{\mathit{s}}_{\mathbf{0}}}$ | ${\mathit{a}}_{{\mathit{s}}_{\mathbf{1}}}$ | ${\mathit{a}}_{{\mathit{s}}_{\mathbf{2}}}$ | ${\mathit{a}}_{{\mathit{s}}_{\mathbf{3}}}$ | ${\mathit{a}}_{{\mathit{s}}_{\mathbf{4}}}$ | ${\mathit{b}}_{{\mathit{s}}_{\mathbf{0}}}$ | ${\mathit{b}}_{{\mathit{s}}_{\mathbf{1}}}$ | ${\mathit{b}}_{{\mathit{s}}_{\mathbf{2}}}$ | ${\mathit{b}}_{{\mathit{s}}_{\mathbf{3}}}$ | ${\mathit{b}}_{{\mathit{s}}_{\mathbf{4}}}$ | ${\mathit{c}}_{{\mathit{s}}_{\mathbf{0}}}$ | ${\mathit{c}}_{{\mathit{s}}_{\mathbf{1}}}$ | ${\mathit{c}}_{{\mathit{s}}_{\mathbf{2}}}$ | ${\mathit{c}}_{{\mathit{s}}_{\mathbf{3}}}$ | ${\mathit{c}}_{{\mathit{s}}_{\mathbf{4}}}$ | |

${x}_{1}$ | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 |

${x}_{2}$ | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 |

${x}_{3}$ | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 |

${x}_{4}$ | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 |

${x}_{5}$ | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |

$\mathit{G}/\mathit{M}$ | a | b | c | ||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{a}}_{{\mathit{s}}_{\mathbf{0}}}$ | ${\mathit{a}}_{{\mathit{s}}_{\mathbf{1}}}$ | ${\mathit{a}}_{{\mathit{s}}_{\mathbf{2}}}$ | ${\mathit{b}}_{{\mathit{s}}_{\mathbf{0}}}$ | ${\mathit{b}}_{{\mathit{s}}_{\mathbf{1}}}$ | ${\mathit{b}}_{{\mathit{s}}_{\mathbf{2}}}$ | ${\mathit{c}}_{{\mathit{s}}_{\mathbf{0}}}$ | ${\mathit{c}}_{{\mathit{s}}_{\mathbf{1}}}$ | ${\mathit{c}}_{{\mathit{s}}_{\mathbf{2}}}$ | |

${x}_{1}$ | 0.1 | 0.2 | 0.7 | 0.2 | 0.2 | 0.6 | 0.7 | 0.2 | 0.1 |

${x}_{2}$ | 0.7 | 0.1 | 0.2 | 0.2 | 0.2 | 0.6 | 0.4 | 0 | 0.6 |

${x}_{3}$ | 0.3 | 0.2 | 0.5 | 0.4 | 0.5 | 0.1 | 0.1 | 0.6 | 0.3 |

${x}_{4}$ | 0.1 | 0.2 | 0.7 | 0.3 | 0.2 | 0.5 | 0.8 | 0 | 0.2 |

**Table 3.**The complementary fuzzy linguistic concept formal context $(G,{L}_{{s}_{\alpha}},{\tilde{I}}^{c})$.

$\mathit{G}/\mathit{M}$ | a | b | c | ||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{a}}_{{\mathit{s}}_{\mathbf{0}}}$ | ${\mathit{a}}_{{\mathit{s}}_{\mathbf{1}}}$ | ${\mathit{a}}_{{\mathit{s}}_{\mathbf{2}}}$ | ${\mathit{b}}_{{\mathit{s}}_{\mathbf{0}}}$ | ${\mathit{b}}_{{\mathit{s}}_{\mathbf{1}}}$ | ${\mathit{b}}_{{\mathit{s}}_{\mathbf{2}}}$ | ${\mathit{c}}_{{\mathit{s}}_{\mathbf{0}}}$ | ${\mathit{c}}_{{\mathit{s}}_{\mathbf{1}}}$ | ${\mathit{c}}_{{\mathit{s}}_{\mathbf{2}}}$ | |

${x}_{1}$ | 0.9 | 0.8 | 0.3 | 0.8 | 0.8 | 0.4 | 0.3 | 0.8 | 0.9 |

${x}_{2}$ | 0.3 | 0.9 | 0.8 | 0.8 | 0.8 | 0.4 | 0.6 | 1 | 0.4 |

${x}_{3}$ | 0.7 | 0.8 | 0.5 | 0.6 | 0.5 | 0.9 | 0.9 | 0.4 | 0.7 |

${x}_{4}$ | 0.9 | 0.8 | 0.3 | 0.7 | 0.8 | 0.5 | 0.2 | 1 | 0.8 |

**Table 4.**FOEAL concepts of Table 2.

Extent | Intent |
---|---|

$\{{x}_{1}\}$ | $(\{({a}_{{s}_{2}},0.7),({b}_{{s}_{2}},0.6),({c}_{{s}_{0}},0.7)\},\{({a}_{{s}_{0}},0.9),({a}_{{s}_{1}},0.8),({b}_{{s}_{0}},0.8),({b}_{{s}_{1}},0.8),({c}_{{s}_{1}},0.8),({c}_{{s}_{2}},0.9)\})$ |

$\{{x}_{2}\}$ | $(\{({a}_{{s}_{0}},0.7),({b}_{{s}_{2}},0.6),({c}_{{s}_{0}},0.6)\},\{({a}_{{s}_{1}},0.9),({a}_{{s}_{2}},0.8),({b}_{{s}_{0}},0.8),({b}_{{s}_{1}},0.8),({c}_{{s}_{0}},0.6),({c}_{{s}_{1}},1)\})$ |

$\{{x}_{3}\}$ | $(\left\{({c}_{{s}_{1}},0.6)\right\},\{({a}_{{s}_{0}},0.7),({a}_{{s}_{1}},0.8),({b}_{{s}_{0}},0.6),({b}_{{s}_{2}},0.9),({c}_{{s}_{0}},0.9),({c}_{{s}_{2}},0.7)\})$ |

$\{{x}_{4}\}$ | $(\{({a}_{{s}_{2}},0.7),({c}_{{s}_{0}},0.8)\},\{({a}_{{s}_{0}},0.9),({a}_{{s}_{1}},0.8),({b}_{{s}_{0}},0.7),({b}_{{s}_{1}},0.8),({c}_{{s}_{1}},1),({c}_{{s}_{2}},0.8)\})$ |

$\{{x}_{1},{x}_{2}\}$ | $(\{({a}_{{s}_{0}},0.7),({a}_{{s}_{2}},0.7),({b}_{{s}_{2}},0.6),({c}_{{s}_{0}},0.7)\},\{({a}_{{s}_{0}},0.9),({a}_{{s}_{1}},0.9),({a}_{{s}_{2}},0.8),({b}_{{s}_{0}},0.8),({b}_{{s}_{1}},0.8)$, $({c}_{{s}_{0}},0.6),({c}_{{s}_{1}},1),({c}_{{s}_{2}},0.9)\})$ |

$\{{x}_{1},{x}_{3}\}$ | $(\{({a}_{{s}_{2}},0.7),({b}_{{s}_{2}},0.6),({c}_{{s}_{0}},0.7),({c}_{{s}_{1}},0.6)\},\{({a}_{{s}_{0}},0.9),({a}_{{s}_{1}},0.8),({b}_{{s}_{0}},0.8),({b}_{{s}_{1}},0.8),({b}_{{s}_{2}},0.9)$, $({c}_{{s}_{0}},0.9),({c}_{{s}_{1}},0.8),({c}_{{s}_{2}},0.9)\})$ |

$\{{x}_{1},{x}_{4}\}$ | $(\{({a}_{{s}_{2}},0.7),({b}_{{s}_{2}},0.6),({c}_{{s}_{0}},0.8)\},\{({a}_{{s}_{0}},0.9),({a}_{{s}_{1}},0.8),({b}_{{s}_{0}},0.8),({b}_{{s}_{1}},0.8),({c}_{{s}_{1}},1),({c}_{{s}_{2}},0.9)\})$ |

$\{{x}_{2},{x}_{3}\}$ | $(\{({a}_{{s}_{0}},0.7),({b}_{{s}_{2}},0.6),({c}_{{s}_{0}},0.6),({c}_{{s}_{1}},0.6)\},\{({a}_{{s}_{0}},0.7),({a}_{{s}_{1}},0.9),({a}_{{s}_{2}},0.8),({b}_{{s}_{0}},0.8),({b}_{{s}_{1}},0.8)$, $({b}_{{s}_{2}},0.9),({c}_{{s}_{0}},0.9),({c}_{{s}_{1}},1),({c}_{{s}_{2}},0.7)\})$ |

$\{{x}_{2},{x}_{4}\}$ | $(\{({a}_{{s}_{0}},0.7),({a}_{{s}_{2}},0.7),({b}_{{s}_{2}},0.6),({c}_{{s}_{0}},0.8)\},\{({a}_{{s}_{0}},0.9),({a}_{{s}_{1}},0.9),({a}_{{s}_{2}},0.8),({b}_{{s}_{0}},0.8),({b}_{{s}_{1}},0.8)$, $({c}_{{s}_{0}},0.6),({c}_{{s}_{1}},1),({c}_{{s}_{2}},0.8)\})$ |

$\{{x}_{3},{x}_{4}\}$ | $(\{({a}_{{s}_{2}},0.7),({c}_{{s}_{0}},0.8),({c}_{{s}_{1}},0.6)\},\{({a}_{{s}_{0}},0.9),({a}_{{s}_{1}},0.8),({b}_{{s}_{0}},0.7),({b}_{{s}_{1}},0.8),({b}_{{s}_{2}},0.9),({c}_{{s}_{0}},0.9)$, $({c}_{{s}_{1}},1),({c}_{{s}_{2}},0.8)\})$ |

$\{{x}_{1},{x}_{2},{x}_{3}\}$ | $(\{({a}_{{s}_{0}},0.7),({a}_{{s}_{2}},0.7),({b}_{{s}_{2}},0.6),({c}_{{s}_{0}},0.7),({c}_{{s}_{1}},0.6)\},\{({a}_{{s}_{0}},0.9),({a}_{{s}_{1}},0.9),({a}_{{s}_{2}},0.8),({b}_{{s}_{0}},0.8)$, $({b}_{{s}_{1}},0.8),({b}_{{s}_{2}},0.9),({c}_{{s}_{0}},0.9),({c}_{{s}_{1}},1),({c}_{{s}_{2}},0.9)\})$ |

$\{{x}_{1},{x}_{2},{x}_{4}\}$ | $(\{({a}_{{s}_{0}},0.7),({a}_{{s}_{2}},0.7),({b}_{{s}_{2}},0.6),({c}_{{s}_{0}},0.8)\},\{({a}_{{s}_{0}},0.9),({a}_{{s}_{1}},0.9),({a}_{{s}_{2}},0.8),({b}_{{s}_{0}},0.8),({b}_{{s}_{1}},0.8)$, $({c}_{{s}_{0}},0.6),({c}_{{s}_{1}},1),({c}_{{s}_{2}},0.9)\})$ |

$\{{x}_{1},{x}_{3},{x}_{4}\}$ | $(\{({a}_{{s}_{2}},0.7),({b}_{{s}_{2}},0.7),({c}_{{s}_{0}},0.8),({c}_{{s}_{1}},0.6)\},\{({a}_{{s}_{0}},0.9),({a}_{{s}_{1}},0.8),({b}_{{s}_{0}},0.8),({b}_{{s}_{1}},0.8),({b}_{{s}_{2}},0.9)$, $({c}_{{s}_{0}},0.9),({c}_{{s}_{1}},1),({c}_{{s}_{2}},0.9)\})$ |

$\{{x}_{2},{x}_{3},{x}_{4}\}$ | $(\{({a}_{{s}_{0}},0.7),({a}_{{s}_{2}},0.7),({b}_{{s}_{2}},0.6),({c}_{{s}_{0}},0.8),({c}_{{s}_{1}},0.6)\},\{({a}_{{s}_{0}},0.9),({a}_{{s}_{1}},0.9),({a}_{{s}_{2}},0.8),({b}_{{s}_{0}},0.8)$, $({b}_{{s}_{1}},0.8),({b}_{{s}_{2}},0.9),({c}_{{s}_{0}},0.9),({c}_{{s}_{1}},1),({c}_{{s}_{2}},0.8)\})$ |

$\{{x}_{1},{x}_{2},{x}_{3},{x}_{4}\}$ | $(\{({a}_{{s}_{0}},0.7),({a}_{{s}_{2}},0.7),({b}_{{s}_{2}},0.6),({c}_{{s}_{0}},0.8),({c}_{{s}_{1}},0.6)\},\{({a}_{{s}_{0}},0.9),({a}_{{s}_{1}},0.9),({a}_{{s}_{2}},0.8),({b}_{{s}_{0}},0.8)$, $({b}_{{s}_{1}},0.8),({b}_{{s}_{2}},0.9),({c}_{{s}_{0}},0.9),({c}_{{s}_{1}},1),({c}_{{s}_{2}},0.9)\})$ |

∅ | $(\varnothing ,\varnothing )$ |

${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | ${\mathit{x}}_{4}$ | |
---|---|---|---|---|

${x}_{1}$ | ∅ | ${a}_{{s}_{0}}{a}_{{s}_{2}}{c}_{{s}_{0}}{c}_{{s}_{2}}$ | ${a}_{{s}_{0}}{a}_{{s}_{2}}{b}_{{s}_{0}}{b}_{{s}_{1}}{b}_{{s}_{2}}{c}_{{s}_{0}}{c}_{{s}_{1}}{c}_{{s}_{2}}$ | ${b}_{{s}_{0}}{b}_{{s}_{2}}{c}_{{s}_{2}}$ |

${x}_{2}$ | ${a}_{{s}_{0}}{a}_{{s}_{1}}{a}_{{s}_{2}}{c}_{{s}_{0}}{c}_{{s}_{1}}{c}_{{s}_{2}}$ | ∅ | ${a}_{{s}_{0}}{a}_{{s}_{1}}{a}_{{s}_{2}}{b}_{{s}_{0}}{b}_{{s}_{1}}{b}_{{s}_{2}}{c}_{{s}_{1}}{c}_{{s}_{2}}$ | ${a}_{{s}_{0}}{a}_{{s}_{1}}{a}_{{s}_{2}}{b}_{{s}_{0}}{b}_{{s}_{2}}{c}_{{s}_{0}}{c}_{{s}_{2}}$ |

${x}_{3}$ | ${b}_{{s}_{2}}{c}_{{s}_{0}}{c}_{{s}_{1}}$ | ${a}_{{s}_{0}}{b}_{{s}_{2}}{c}_{{s}_{0}}{c}_{{s}_{1}}{c}_{{s}_{2}}$ | ∅ | ${b}_{{s}_{2}}{c}_{{s}_{0}}{c}_{{s}_{1}}$ |

${x}_{4}$ | ${c}_{{s}_{0}}{c}_{{s}_{1}}$ | ${a}_{{s}_{0}}{a}_{{s}_{2}}{c}_{{s}_{0}}{c}_{{s}_{2}}$ | ${a}_{{s}_{0}}{a}_{{s}_{2}}{b}_{{s}_{0}}{b}_{{s}_{1}}{c}_{{s}_{0}}{c}_{{s}_{1}}{c}_{{s}_{2}}$ | ∅ |

Methods | The Type of Concept Lattice | Concept Extent | Concept Intent | Reduction Methods | Reduction Conditions for Preservation | Linguistic Information | Fuzzy Information |
---|---|---|---|---|---|---|---|

Ref. [30] | ${L}_{O}(G,M,I)$ | X | B | 1 | decision rules | × | × |

Ref. [39] | $OFTL(G,M,\tilde{I})$ | X | ($\tilde{B}$, $\tilde{C}$) | 1 | granular matrix | × | √ |

Ref. [22] | $OEL(G,M,I)$ | X | (B, C) | 4 | lattice structure/ granular information/ join (meet)-irreducible elements | × | × |

Ref. [48] | $LL(G,{L}_{{s}_{\alpha}},I)$ | X | ${B}_{{s}_{\alpha}}$ | 2 | multi-granularity similarity relations/ binary relation | √ | × |

Our methods | $FOEAL(G,{L}_{{s}_{\alpha}},\tilde{I})$ | X | (${\tilde{B}}_{{s}_{\alpha}}$, ${\tilde{C}}_{{s}_{\alpha}}$) | 2 | granular concept/ entropy information | √ | √ |

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## Share and Cite

**MDPI and ACS Style**

Cui, H.; Deng, A.; Yue, G.; Zou, L.; Martinez, L. The Linguistic Concept’s Reduction Methods under Symmetric Linguistic-Evaluation Information. *Symmetry* **2023**, *15*, 813.
https://doi.org/10.3390/sym15040813

**AMA Style**

Cui H, Deng A, Yue G, Zou L, Martinez L. The Linguistic Concept’s Reduction Methods under Symmetric Linguistic-Evaluation Information. *Symmetry*. 2023; 15(4):813.
https://doi.org/10.3390/sym15040813

**Chicago/Turabian Style**

Cui, Hui, Ansheng Deng, Guanli Yue, Li Zou, and Luis Martinez. 2023. "The Linguistic Concept’s Reduction Methods under Symmetric Linguistic-Evaluation Information" *Symmetry* 15, no. 4: 813.
https://doi.org/10.3390/sym15040813