Generalized Atanassov’s Intuitionistic Fuzzy Index. Construction Method Edurne Barrenechea Humberto Bustince Miguel Pagola Javier Fernandez Josean Sanz Dpt. Autom´ atica y Computaci ´ on, Universidad P´ ublica de Navarra Campus de Arrosad´ ıa, s/n 31006 Pamplona (Spain) Email: {edurne.barrenechea,bustince,miguel.pagola,fcojavier.fernandez,joseantonio.sanz}@unavarra.es Abstract— In this work we introduce the concept of Generalized Atanassov’s Intuitionistic Fuzzy Index. We characterize it in terms of fuzzy implication operators and we propose a construction method with automorphisms. Finally, we study some special properties of the generalized Atanassov’s intuitionistic fuzzy index. Keywords— Atanassov’s Intuitionistic Fuzzy Index, Fuzzy Impli- cation Operator, Automorphisms. 1 Introduction In 1983 Atanassov [1] introduced Atanassov’s intuitionistic fuzzy sets (A − IFSs) in such a way that each element of the set has two values assigned, the membership degree and the non-membership degree. In this direction, Atanassov de- fined these sets also indicating that the index of intuitionism of each element obtained by subtracting the sum of member- ship and non-membership from one (a subtraction that should be positive and less than or equal to one), it is a measure- ment of the effect of working with Atanassov’s intuitionistic fuzzy sets. We consider this intuitionistic index (or condition of intuitionism) a very important characteristic of A − IFSs since from it we can obtain very valuable information of each element and taking on advantage of this potentiality in differ- ent applications. For example, in image processing the task of divide into disjoint parts a digital image is denoted as seg- mentation. The most commonly used strategy for segmenting images is global thresholding that refers to the process of di- viding the pixels in an image on the basis of their intensity levels of gray. The experts have uncertainty when assigning the pixels either to the background or to the object trough the choice of the membership functions. Moreover, this choice has proven to be of uttermost importance regarding the algo- rithms performance. In order to overcome this problem, we consider using the Atanassov’s intuitionistic fuzzy index val- ues for representing the uncertainty of the expert in determin- ing that the pixel belongs to the background or that it belongs to the object. From this point of view, we can consider the expert provides the degree of membership of an element to an A − IFS and also the degree of intuitionism the expert has in given this membership degree (see [9]). This fact has led us to present the new concept of Generalized Atanassov’s In- tuitionistic Fuzzy Index that generalize the expression given by Atanassov. We also provide a characterization method by means of fuzzy implication operators. Moreover, we study a construction method using automorphisms that allow us to present simple expressions of said index. Pankowska and Wygralak ([16, 17, 21]) proposed another generalization of the intuitionistic index based on strong nega- tions and triangular norms. This approach is used to construct flexible algorithms of group decision making which involve relative scalar cardinalities defined by means of generalized sigma counts of fuzzy sets. 2 Preliminary definitions Let U be an ordinary finite non-empty set. An Atanassov’s intuitionistic fuzzy set (A − IFS) [1] in U is an expression A given by A = {(u, μ A (u),ν A (u))|u ∈ U } (1) where μ A : U −→ [0, 1] ν A : U −→ [0, 1] satisfy the condition 0 ≤ μ A (u)+ ν A (u) ≤ 1 for all u in U. The numbers μ A (u) and ν A (u) denote respectively the de- gree of membership and the degree of non-membership of the element u in set A. We will also use the notation A(u)= (μ A (u),ν A (u)). We will represent as A − IFSs(U ) the set of all the Atanassov’s intuitionistic fuzzy sets in U . Atanassov defined the Atanassov’s intuitionistic fuzzy index of the element u in A ∈ A − IFSs(U ) as: Π A (u)=1 − μ A (u) − ν A (u). (2) We know fuzzy sets are represented exclusively by the membership function degree, that is, A = {(u, μ A (u))|u ∈ U }. (3) Hereinafter, fuzzy sets have associated a non-membership de- gree given by one minus the membership degree: A = {(u, μ A (u),ν A (u))|u ∈ U } = {(u, μ A (u), 1 − μ A (u))|u ∈ U }. (4) Since μ A (u)+ ν A (u)= μ A (u)+1 − μ A (u)=1, in this sense fuzzy sets are considered as a particular case of Atanassov’s intuitionistic fuzzy sets. We will represent as FSs(U ) the set of all the fuzzy sets in U . We will call automorphism of the unit interval every func- tion ϕ : [0, 1] → [0, 1] that is continuous and strictly increas- ing such that ϕ(0) = 0 and ϕ(1) = 1. A function n : [0, 1] → [0, 1] such that n(0) = 1 and n(1) = 0 is called a strong negation whenever it is strictly de- creasing, continuous and involutive. Trillas ([19, 20]) proved that n : [0, 1] → [0, 1] is a strong negation if and only if there exists an automorphism ϕ of the unit interval such that ISBN: 978-989-95079-6-8 IFSA-EUSFLAT 2009 478