Using concept lattice theory to obtain the set of solutions of multi-adjoint relation equations

Díaz-Moreno, Juan Carlos; Medina, Jesús

doi:10.1016/j.ins.2014.01.006

Cited by 30 publications

(16 citation statements)

References 20 publications

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“…FREs have also been related to other mathematical frameworks, such as fuzzy formal concept analysis. ()…”

Section: Introductionmentioning

confidence: 99%

“…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%

“…FREs have also been related to other mathematical frameworks, such as fuzzy formal concept analysis. 17,18 A new type of (system of) fuzzy relation equation arises when equations contain unknown variables together with their logical negations simultaneously. These new kind of equations are called bipolar fuzzy relation equations.…”

Section: Introductionmentioning

confidence: 99%

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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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“…FREs have also been related to other mathematical frameworks, such as fuzzy formal concept analysis. ()…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Cornejo

Lobo

Medina

2019

Math Methods in App Sciences

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“…Multi‐adjoint relation equations (MARE) were introduced as a generalization of FRE to consider the flexibility provided by the multi‐adjoint philosophy, generalizing the recent framework, and important properties have been studied …”

Section: Introductionmentioning

confidence: 99%

Multi-adjoint Relation Equations: A Decision Support System for Fuzzy Logic

Cornejo

Díaz-Moreno

Medina

2017

Int. J. Intell. Syst.

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Fuzzy relation equations (FRE) are an important decision support system (DSS), for example, in fuzzy logic. FRE have recently been extended to a more general framework, called multiadjoint relation equations (MARE). This paper shows MARE as a fundamental DSS in multi‐adjoint logic programming. For that purpose, multi‐adjoint logic programs will be interpreted as a MARE, and the solvability of them will be given in terms of concept lattice theory. Furthermore, two approximations (optimistic and pessimistic approximations) of unsolvable equations will be obtained from a multiadjoint object‐oriented concept lattice. Finally, a real‐life example will be studied.

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“…De Baets [4] determined all minimal solutions and Wang [24] gave a formula of the number of minimal solutions under the condition that all elements of its righthand side have irredundant finite join-irreducible decompositions. And many other works about solving fuzzy relation equations were published (see [1,5,8,10,12,13,14,15,16,17,19,20,22,23]). Compared with the linear algebraic systems, the main difference with fuzzy relation equations is the operations and the fundamental domain.…”

Section: Introductionmentioning

confidence: 99%