Minimal solutions of general fuzzy relation equations on linear carriers. An algebraic characterization

Díaz-Moreno, Juan Carlos; Medina, Jesús; Turunen, Esko

doi:10.1016/j.fss.2016.02.004

Cited by 37 publications

(22 citation statements)

References 20 publications

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“…Since A i • R = B i is a fuzzy set on Y whose membership degrees are fully defined, we can prove that (11), (12) and (13). Property (11) provides the general relation of the inferred outputs A i • R and the consequents B i , that they can be incomparable only at the values y ∈ Y such that (A i • R )(y) = .…”

Section: Sup-t Systemmentioning

confidence: 90%

“…Sketch of the proof: Using (16)- (17) we can prove (22) in a similar way to the proof of (13). In particular, we first prove ↓ R ⊆R S .…”

Section: Sup-t Systemmentioning

confidence: 94%

“…Property (11) provides the general relation of the inferred outputs A i • R and the consequents B i , that they can be incomparable only at the values y ∈ Y such that (A i • R )(y) = . Property (13) shows that in case of solvability of system (5), the inferred outputs are greater than or equal to the consequents and moreover, modelR is always greater than or equal to ↓ R for any solution R of the system. This property may help in narrowing the set of solutions, serving to find an approximate solution of system (5).…”

Section: Sup-t Systemmentioning

confidence: 99%

“…Apart from the fuzzy control, the systems of fuzzy relational equations have been widely applied in many other areas covering the duality of optimization problems with generalized fuzzy relational equations [16], the linear optimization problems [15], or the posynomial geometric programming [25]. Up to the present time, the topic has been continuously updated both on theoretical and applied aspects, see [12,8,13,5] for the most relevant results.…”

Section: Introductionmentioning

confidence: 99%

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Solvability of Fuzzy Relational Equations Employing Undefined Values

Cao

2019

Proceedings of the 2019 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and

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The paper aims at studying the solvability of fuzzy relational equations where the input fuzzy sets may have undefined values. The appropriate operations dealing undefined values are taken from the algebras built in partial fuzzy set theory, in particular, the Bochvar operations, the Sobociński operations, the Kleene operations and the lower estimation operations are used for the investigation. We first elaborate new models using distinct operations handling undefined values and mirroring the shapes of the implicative model and Mamdani-Assilian model. Then the solvability criterions for such new systems of fuzzy relational equations are set up, and consequently, we show a way of checking whether the systems are solvable or not.

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Section: Sup-t Systemmentioning

confidence: 90%

“…Sketch of the proof: Using (16)- (17) we can prove (22) in a similar way to the proof of (13). In particular, we first prove ↓ R ⊆R S .…”

Section: Sup-t Systemmentioning

confidence: 94%

Section: Sup-t Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Solvability of Fuzzy Relational Equations Employing Undefined Values

Cao

2019

Proceedings of the 2019 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and

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“…1,2 In the literature, we can find many papers dealing with the resolution of (systems of) FREs defined with either the max-min composition, [3][4][5][6] the max-product composition, 7-10 the max-Archimedean t-norm composition, [11][12][13] or other different compositions. [14][15][16][17][18][19][20][21][22][23][24] In what regards to the applied perspective, it is important to emphasize that FREs have increased the range of applications of fuzzy sets theory. The compression and decompression of images and videos, 25 the modelling of fuzzy inference systems in fuzzy control, 26 and the representation of restrictions in optimization problems, [27][28][29][30] among others, are some of the most recent applications of FREs based on max-t-norm compositions.…”

Section: Introductionmentioning

confidence: 99%

Bipolar fuzzy relation equations systems based on the product t‐norm

Cornejo

Lobo

Medina

2019

Math Methods in App Sciences

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Bipolar fuzzy relation equations arise as a generalization of fuzzy relation equations considering unknown variables together with their logical connective negations. The occurrence of a variable and the occurrence of its negation simultaneously can give very useful information for certain frameworks where the human reasoning plays a key role. Hence, the resolution of bipolar fuzzy relation equations systems is a research topic of great interest. This paper focuses on the study of bipolar fuzzy relation equations systems based on the max‐product t‐norm composition. Specifically, the solvability and the algebraic structure of the set of solutions of these bipolar equations systems will be studied, including the case in which such systems are composed of equations whose independent term be equal to 0. As a consequence, this paper complements the contribution carried out by the authors on the solvability of bipolar max‐product fuzzy relation equations.

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