Fuzzy Adjunctions revisited
Abstract
En este trabajo se intenta obtener la noción de adjunción más débil entre estructuras difusas. Este trabajo continúa la línea de investigación en el estudio y construcción d adjunciones que han realizado los autores en contribuciones anteriores. Nos centraremos ahora en la noción de relación difusa que es en cierto sentido interpretable como una función difusa. Existen varios trabajos en la literatura relacionados con este tema. Entre todos ellos, trabajaremos con un enfoque próximo al de Ciric et al cuando definen las denominadas funciones parciales difusas. El nuevo concepto estudiado es el de relaciones difusas funcionales y la construcción de adjunciones entre ellas.
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Fuzzy adjunctions revisited I.P. Cabrera1, P. Cordero1, B. De Baets2, F. Garc´ ıa-Pardo1, M. Ojeda-Aciego1 1Universidad de M´ alaga, Depto. Matem´ atica Aplicada, Spain Blv. Louis Pasteur 35, 29071 M´ alaga, Spain {ipcabrera,cordero,fgarciap,aciego}@uma.es 2KERMIT. Department of Mathematical Modelling, Statistics and Bioinformatics. Ghent University, Coupure links 653, 9000 Gent, Belgium [email protected] Extended abstract In this work we are aiming at obtaining the weakest notion of adjunction between fuzzy structures. This topic continues our research line on the study and construction of adjunctions [2,5,6]. We will focus on the notion of fuzzy relation which, in some sense, can be interpreted as a fuzzy function. A number of papers dealing with this problem can be found in the literature [3,4,7] and among all these approaches, we will work with the notion of functional fuzzy relation, which is equivalent to what ´ Ciri´ c calls partial fuzzy function (but actually does not coincide with the original notion of partial fuzzy function given by Demirci). Definition 1 A fuzzy relation µ∈LA×Bis said to be functional if for each a1,a2∈A and b1,b2∈B the following inequality holds µ(a1,b1)⊗µ(a2,b1)⊗µ(a1,b2)≤µ(a2,b2)(1) A functional fuzzy relation µinduces two fuzzy equivalence relations, ≈µon Band ≈µ−1 on A, defined, respectively by (b1≈µb2) = (b1=b2)∨_ a∈A µ(a,b1)⊗µ(a,b2) (a1≈µ−1a2) = (a1=a2)∨_ b∈B µ(a1,b)⊗µ(a2,b) for all b1,b2∈Band a1,a2∈A. The two fuzzy equivalences above allow to consider Aand Bas fuzzy structures hA,≈Aiand hB,≈Biwhere the fuzzy equivalences ≈Aand ≈Bare the induced by µ. In order to consider an adequate notion of fuzzy adjunction between fuzzy structures, it is necessary to fix fuzzy preorders on both of them. A fuzzy relation ρA:A×A→Lis said to be a fuzzy preorder compatible with hA,≈Aiif and only if the two following conditions hold (a1≈Aa2)≤ρA(a1,a2)and ρA(a1,a2)⊗ρA(a2,a3)≤ρA(a1,a3)
In this abstract setting, it makes sense to study suitable notions of isotonicity for a functional fuzzy relation which generalize the well-known definitions of isotonicity for compatible crisp mappings and for perfect fuzzy functions. A possible definition could be µ(a1,b1)⊗µ(a2,b2)⊗ρA(a1,a2)≤ρB(b1,b2)(2) for all b1,b2∈Band a1,a2∈A. Similarly, another related notion which has to be extended is the inflationary property, whose definition for an internal functional fuzzy relation ϕ:A×A→Lin this setting could be ϕ(a1,a2)≤ρA(a1,a2). What would be a desirable notion of adjunction in this extended framework? An adjunction between fuzzy preorders hA,ρAiand hB,ρBican be defined as a pair (µ,ν)of functional fuzzy relations. In principle, we can find the three possibilities below: •µ(a1,b1)⊗ν(b2,a2)≤ρB(b1,b2)↔ρA(a1,a2). •ν(b2,a2)⊗ρA(a2,a2) = µ(a1,b1)⊗ρB(b1,b2). •µand νare isotone, ν◦µinflationary and µ◦νdeflationary. It has to be studied which of the definitions above behaves as expected in relation to closure operators and closure systems. References [1] R. Belohlavek. Fuzzy Relational Systems: Foundations and Principles. Kluwer Academic Publishers, Norwell, MA, USA. 2002. [2] I.P. Cabrera, P. Cordero, F. Garca, M. Ojeda-Aciego, B. De Baets. On the construction of adjunctions between a fuzzy preposet and an unstructured set. Submitted for publication, 2016. [3] M. ´ Ciri´ c, J. Ignjatovi´ c, S. Bogdanovi´ c: Uniform fuzzy relations and fuzzy functions. Fuzzy Sets and Systems 160(8):1054–1081, 2009. [4] M. Demirci. Fuzzy functions and their applications. J. Mathematical Analisis Applications, 252:495–517, 2000. [5] F. Garca-Pardo, I.P. Cabrera, P. Cordero, and M. Ojeda-Aciego. On the construction of fuzzy Galois connections. In XVII Spanish Conference on Fuzzy Logic and Technology, ESTYLF, pages 99-102, 2014. [6] F. Garca-Pardo, I.P. Cabrera, P. Cordero, M. Ojeda-Aciego, and F.J. Rodrguez. On the definition of suitable orderings to generate adjunctions over an unstructured codomain. Information Sciences, 286:173-187, 2014. [7] S. Gottwald. Fuzzy Sets and Fuzzy Logic. Foundations of Application—from a Mathematical Point of View. Vieweg, Braunschweig, Wiesbaden 1993.