# 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

- Ganter, B.; Wille, R. Formal Concept Analysis: Mathematical Foundations; Springer: Berlin, Germany, 1999. [Google Scholar]
- Poelmans, J.; Ignatov, D.I.; Kuznetsov, S.O.; Dedene, G. Formal concept analysis in knowledge processing: A survey on applications. Expert Syst. Appl.
**2013**, 40, 6538–6560. [Google Scholar] [CrossRef] - Aswani Kumar, C.; Srinivas, S. Mining associations in health care data using formal concept analysis and singular value decomposition. J. Biol. Syst.
**2010**, 18, 787–807. [Google Scholar] [CrossRef] - Formica, A. Similarity reasoning in formal concept analysis: From one- to many-valued contexts. Knowl. Inf. Syst.
**2018**, 60, 715–739. [Google Scholar] [CrossRef] - Xu, W.; Li, W. Granular computing approach to two-way learning based on formal concept analysis in fuzzy datasets. IEEE Trans. Cybern.
**2016**, 46, 366–379. [Google Scholar] [CrossRef] - Tsang, E.C.C.; Fan, B.J.; Chen, D.G.; Xu, W.H.; Li, W.T. Multi-level cognitive concept learning method oriented to data sets with fuzziness: A perspective from features. Soft Comput.
**2020**, 24, 3753–3770. [Google Scholar] [CrossRef] - Fujita, H.; Gaeta, A.; Loia, V.; Orciuoli, F. Resilience analysis of critical infrastructures: A cognitive approach based on granular computing. IEEE Trans. Cybern.
**2019**, 49, 1835–1848. [Google Scholar] [CrossRef] [PubMed] - Düntsch, I.; Gediga, G. Modal-style operators in qualitative data analysis. In Proceedings of the 2002 IEEE International Conference on Data Mining (ICDM 2002), Maebashi, Japan, 9–12 December 2002; pp. 155–162. [Google Scholar]
- Yao, Y.Y. Concept lattices in rough set theory. In Proceedings of the 2004 IEEE Annual Meeting of the Fuzzy Information (NAFIPS 2004), Banff, AB, Canada, 27–30 June 2004; pp. 796–801. [Google Scholar]
- Burusco, A.; Fuentes-González, R. The study of the L-fuzzy concept lattice. Mathw. Soft Comput.
**1994**, 1, 209–218. [Google Scholar] - Belohlavek, R. Concept lattices and order in fuzzy logic. Knowl.-Based Syst.
**2004**, 128, 277–298. [Google Scholar] [CrossRef] - Qi, J.; Wei, L.; Yao, Y. Three-way formal concept analysis. In Proceedings of the 9th International Conference on Rough Sets and Knowledge Technology (RSKT 2014), Shanghai, China, 24–26 October 2014; pp. 732–741. [Google Scholar]
- Qi, J.; Qian, T.; Wei, L. The connections between three-way and classical concept lattices. Knowl.-Based Syst.
**2016**, 91, 143–151. [Google Scholar] [CrossRef] - Qian, T.; Wei, L.; Qi, J. A theoretical study on the object (property) oriented concept lattices based on three-way decisions. Soft Comput.
**2019**, 23, 9477–9489. [Google Scholar] [CrossRef] - Hu, Q.; Qin, K.Y.; Yang, L. A constructing approach to multi-granularity object-induced three-way concept lattices. Int. J. Approx. Reason.
**2022**, 150, 229–241. [Google Scholar] [CrossRef] - Wang, Z.; Qi, J.J.; Shi, C.J.; Ren, R.S.; Wei, L. Multiview granular data analytics based on three-way concept analysis. Appl. Intell. 2022. [CrossRef]
- Singh, P.K. Three-way fuzzy concept lattice representation using neutrosophic set. Int. J. Mach. Learn. Cybern.
**2017**, 8, 69–79. [Google Scholar] [CrossRef] [Green Version] - Yuan, K.; Xu, W.; Li, W.; Ding, W. An incremental learning mechanism for object classification based on progressive fuzzy three-way concept. Inf. Sci.
**2022**, 584, 127–147. [Google Scholar] [CrossRef] - Chen, J.; Mi, J.; Xie, B.; Lin, Y. A fast attribute reduction method for large formal decision contexts. Int. J. Approx. Reason.
**2019**, 106, 1–17. [Google Scholar] [CrossRef] - Qi, J.J. Attribute reduction in formal contexts based on a new discernibility matrix. J. Appl. Math. Comput.
**2009**, 30, 305–314. [Google Scholar] [CrossRef] - Zhang, W.X.; Wei, L.; Qi, J.J. Attribute reduction theory and approach to concept lattice. Sci. China. Ser. F Inf. Sci.
**2005**, 48, 713–726. [Google Scholar] [CrossRef] - Ren, R.; Ling, W. The attribute reductions of three-way concept lattices. Knowl.-Based Syst.
**2016**, 99, 92–102. [Google Scholar] [CrossRef] - Li, T.J.; Wu, W.Z. Attribute reduction in formal contexts: A covering rough set approach. Fundam. Informaticae
**2011**, 111, 15–32. [Google Scholar] [CrossRef] - Niu, J.; Chen, D.; Li, J.; Wang, H. A dynamic rule-based classification model via granular computing q. Inf. Sci.
**2022**, 584, 325–341. [Google Scholar] [CrossRef] - Hu, Q.; Qin, K.Y.; Yang, H.; Xue, B.B. A novel approach to attribute reduction and rule acquisition of formal decision context. Appl. Intell. 2022. [CrossRef]
- Li, J.; Mei, C.; Lv, Y. Knowledge reduction in decision formal contexts. Knowl.-Based Syst.
**2011**, 24, 709–715. [Google Scholar] [CrossRef] - Li, J.H.; Mei, C.L.; Lv, Y.J. Incomplete decision contexts: Approximate concept construction, rule acquisition and knowledge reduction. Int. J. Approx. Reason.
**2013**, 54, 149–165. [Google Scholar] [CrossRef] - Chen, J.; Mi, J.; Lin, Y. A graph approach for fuzzy-rough feature selection. Fuzzy Sets Syst.
**2020**, 391, 96–116. [Google Scholar] [CrossRef] - Chen, J.K.; Mi, J.S.; Xie, B.; Lin, Y.J. Attribute reduction in formal decision contexts and its application to finite topological spaces. Int. J. Mach. Learn. Cybern.
**2021**, 12, 1–14. [Google Scholar] [CrossRef] - Qin, K.Y.; Li, B.; Pei, Z. Attribute reduction and rule acquisition of formal decision context based on object (property) oriented concept lattices. Int. J. Mach. Learn. Cybern.
**2019**, 10, 2837–2850. [Google Scholar] [CrossRef] - Tang, Y.; Pan, Z.; Pedrycz, W.; Ren, F.; Song, X. Viewpoint-based kernel fuzzy clustering with weight information granules. IEEE Trans. Emerg. Top. Comput. Intell. 2022. [CrossRef]
- Tang, Y.; Zhang, L.; Bao, G.; Ren, F.J.; Pedrycz, W. Symmetric implicational algorithm derived from intuitionistic fuzzy entropy. Iran. J. Fuzzy Syst.
**2022**, 19, 27–44. [Google Scholar] - Yang, J.Q.; Chen, C.H.; Li, J.Y.; Liu, D.; Li, T.; Zhan, Z.H. Compressed-encoding particle swarm optimization with fuzzy learning for large-scale feature selection. Symmetry
**2022**, 14, 1142. [Google Scholar] [CrossRef] - Ali, J.; Bashir, Z.; Tabasam, R. Weighted interval-valued dual-hesitant fuzzy sets and its application in teaching quality assessment. Soft Comput.
**2021**, 25, 3503–3530. [Google Scholar] [CrossRef] - Ali, J.; Garg, H. On spherical fuzzy distance measure and TAOV method for decision-making problems with incomplete weight information. Eng. Appl. Artif. Intell.
**2023**, 119, 105726. [Google Scholar] [CrossRef] - Zhai, Y.; Li, D. Knowledge structure preserving fuzzy attribute reduction in fuzzy formal context. Int. J. Approx. Reason.
**2019**, 115, 209–220. [Google Scholar] [CrossRef] - Singh, P.K.; Cherukuri, A.K.; Li, J. Concepts reduction in formal concept analysis with fuzzy setting using Shannon entropy. Int. J. Mach. Learn. Cybern.
**2017**, 8, 179–189. [Google Scholar] [CrossRef] - Lin, Y.; Li, J.; Liao, S.; Zhang, J.; Liu, J. Reduction of fuzzy-crisp concept lattice based on order-class matrix. J. Intell. Fuzzy Syst.
**2020**, 39, 8001–8013. [Google Scholar] [CrossRef] - Zhang, C.; Li, J.; Lin, Y. Matrix-based reduction approach for one-sided fuzzy three-way concept lattices. J. Intell. Fuzzy Syst.
**2021**, 40, 11393–11410. [Google Scholar] [CrossRef] - Xu, Z.S. An approach based on the uncertain LOWG and induced uncertain LOWG operators to group decision making with uncertain multiplicative linguistic preference relations. Decis. Support Syst.
**2006**, 2, 488–499. [Google Scholar] [CrossRef] - Seiti, H.; Hafezalkotob, A.; Herrera-Viedma, E. A novel linguistic approach for multi-granular information fusion and decision-making using risk-based linguistic D numbers. Inf. Sci.
**2020**, 530, 43–65. [Google Scholar] [CrossRef] - Rodríguez, R.M.; Labella, L.; Sesma-Sara, M.; Bustince, H.; Martínez, L. A cohesion-driven consensus reaching process for large scale group decision making under a hesitant fuzzy linguistic term sets environment. Comput. Ind. Eng.
**2021**, 155, 107–158. [Google Scholar] [CrossRef] - Song, Y.M.; Hu, J. Vector similarity measures of hesitant fuzzy linguistic term sets and their applications. PLoS ONE
**2017**, 12, e0189579. [Google Scholar] [CrossRef] [Green Version] - Ali, J.; Naeem, M.; Mahmood, W. Generalized q-rung picture linguistic aggregation operators and their application in decision making. J. Intell. Fuzzy Syst.
**2023**, 44, 4419–4443. [Google Scholar] [CrossRef] - Ali, J.; Bashir, Z.; Rashid, T. WASPAS-based decision making methodology with unknown weight information under uncertain evaluations. Expert Syst. Appl.
**2021**, 168, 114–143. [Google Scholar] [CrossRef] - Yang, L.; Wang, Y.; Li, H. Research on the disease intelligent diagnosis model based on linguistic truth-valued concept lattice. Complexity
**2021**, 2021, 1–11. [Google Scholar] [CrossRef] - Cui, H.; Yue, G.L.; Zou, L.; Liu, X.; Deng, A.S. Multiple multidimensional linguistic reasoning algorithm based on property-oriented linguistic concept lattice. Int. J. Approx. Reason.
**2020**, 131, 80–92. [Google Scholar] [CrossRef] - Zou, L.; Pang, K.; Song, X.Y.; Kang, N.; Liu, X. A knowledge reduction approach for linguistic concept formal context. Inf. Sci.
**2020**, 524, 165–183. [Google Scholar] [CrossRef] - Leung, Y.; Wu, W.Z.; Zhang, W.X. Knowledge acquisition in incomplete information systems: A rough set approach. Eur. J. Oper. Res.
**2006**, 168, 164–180. [Google Scholar] [CrossRef] - Zadeh, L.A. The Concept of a linguistic variable and its application to approximate reasoning-I. Inf. Sci.
**1975**, 8, 199–249. [Google Scholar] [CrossRef] - Herrera, F.; Herrera, V.E.; Verdegay, J.L. A model of monsensus in group decision making under linguistic assessments. Fuzzy Sets Syst.
**1996**, 78, 73–87. [Google Scholar] [CrossRef] - Shao, M.W.; Wu, W.Z.; Wang, X.Z.; Wang, C.Z. Knowledge reduction methods of covering approximate spaces based on concept lattice. Knowl.-Based Syst.
**2020**, 191, 1–9. [Google Scholar] [CrossRef] - Li, L.; Zhang, D. 0-1 linear integer programming method for granule knowledge reduction and attribute reduction in concept lattices. Soft Comput.
**2019**, 23, 383–391. [Google Scholar] [CrossRef] - Bartl, E.; Konecny, J. L-Concept lattices with positive and negative attributes: Modeling uncertainty and reduction of size. Inf. Sci.
**2019**, 472, 163–179. [Google Scholar] [CrossRef] - Wang, P.; Wu, W.; Zhong, H.M. Information flow-based second-order cone programming model for big data using rough concept lattice. Neural Comput. Appl.
**2022**, 35, 2257–2266. [Google Scholar] [CrossRef] - Hao, F.; Gao, J.; Bisogni, C.; Loia, V.; Pei, Z.; Nasridinov, A. Exploring invariance of concept stability for attribute reduction in three-way concept lattice. Soft Comput.
**2023**, 27, 723–735. [Google Scholar] [CrossRef] - Liu, G.L.; Xie, Y.H.; Gao, X.W. Three-way reduction for formal decision contexts. Inf. Sci.
**2022**, 615, 39–57. [Google Scholar] [CrossRef] - Benitez-Caballero, M.J.; Medina, J.; Ramirez-Poussa, E. Characterizing one-sided formal concept analysis by multi-adjoint concept lattices. Mathematics
**2022**, 10, 1020. [Google Scholar] [CrossRef] - Antoni, L.; Cornejo, M.E.; Medina, J.; Ramírez-Poussa, E. Attribute classification and reduct computation in multi-adjoint concept lattices. IEEE Trans. Fuzzy Syst.
**2021**, 29, 1121–1132. [Google Scholar] [CrossRef]

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