Get 20M+ Full-Text Papers For Less Than $1.50/day. Subscribe now for You or Your Team.

Learn More →

m-Polar Fuzzy Sets: An Extension of Bipolar Fuzzy Sets

m-Polar Fuzzy Sets: An Extension of Bipolar Fuzzy Sets Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 416530, 8 pages http://dx.doi.org/10.1155/2014/416530 Research Article 𝑚 -Polar Fuzzy Sets: An Extension of Bipolar Fuzzy Sets 1,2 1 1 3 Juanjuan Chen, Shenggang Li, Shengquan Ma, and Xueping Wang College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China School of Sciences, Xi’an University of Technology, Xi’an 710056, China College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China Correspondence should be addressed to Shenggang Li; shengganglinew@126.com Received 11 April 2014; Accepted 23 May 2014; Published 12 June 2014 Academic Editor: Jianming Zhan Copyright © 2014 Juanjuan Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Recently, bipolar fuzzy sets have been studied and applied a bit enthusiastically and a bit increasingly. In this paper we prove that bipolar fuzzy sets and [0, 1] -sets (which have been deeply studied) are actually cryptomorphic mathematical notions. Since researches or modelings on real world problems oen ft involve multi-agent, multi-attribute, multi-object, multi-index, multi-polar information, uncertainty, or/and limit process, we put forward (or highlight) the notion of 𝑚 -polar fuzzy set (actually, [0, 1] -set which can be seen as a generalization of bipolar fuzzy set, where 𝑚 is an arbitrary ordinal number) and illustrate how many concepts have been defined based on bipolar fuzzy sets and many results which are related to these concepts can be generalized to the case of 𝑚 -polar fuzzy sets.Wealsogiveexamplestoshowhow to apply 𝑚 -polar fuzzy sets in real world problems. 1. Introduction and Preliminaries any mappings. eTh set of all bipolar fuzzy sets on 𝑋 is denoted by BF(𝑋). Set theory and logic systems are strongly coupled in the Bipolar fuzzy sets are an extension of fuzzy sets whose development of modern logic. Classical logic corresponds to membership degree range is [−1, 1]. In a bipolar fuzzy set, the the crisp set theory, and fuzzy logic is associated with fuzzy membership degree 0 of an element means that the element set theory which was proposed by Zadeh in his pioneer work is irrelevant to the corresponding property, the member- [1]. ship degree (0, 1]of an element indicates that the element Den fi ition 1. An 𝐿 -subset (or an 𝐿 -set) on the set 𝑋 is a somewhat satisfies the property, and the membership degree synonym of a mapping 𝐴:𝑋 → 𝐿 ,where 𝐿 is a lattice (cf. [−1, 0)of an element indicates that the element somewhat [2]). When 𝐿=[0,1] (the ordinary closed unit interval with satisfies the implicit counter-property. eTh idea which lies the ordinary order relation), an 𝐿 -set on 𝑋 will be called a behind such description is connected with the existence of fuzzy set on 𝑋 (cf. [1]). “bipolar information” (e.g., positive information and nega- tive information) about the given set. Positive information eTh theory of fuzzy sets has become a vigorous area of represents what is granted to be possible, while negative research in different disciplines including medical and life information represents what is considered to be impossible. sciences, management sciences, social sciences, engineering, Actually, a wide variety of human decision making is based statistics, graph theory, artificial intelligence, pattern recog- on double-sided or bipolar judgmental thinking on a positive nition, robotics, computer networks, decision making, and side and a negative side. For instance, cooperation and automata theory. competition, friendship and hostility, common interests and An extension of fuzzy set, called bipolar fuzzy set, was conflict of interests, eeff ct and side eeff ct, likelihood and given by Zhang [3]in1994. unlikelihood, feedforward and feedback, and so forth are Den fi ition 2 (see Zhang [3]). A bipolar fuzzy set is a pair oen ft the two sides in decision and coordination. In the + − + − traditional Chinese medicine (TCM for short), “yin” and (𝜇 ,𝜇 ),where 𝜇 : 𝑋 → [0, 1] and 𝜇 :𝑋 → [−1,0] are 2 The Scientific World Journal “yang” are the two sides. Yin is the feminine or negative side yy of a system and yang is the masculine or positive side of a v u system. eTh coexistence, equilibrium, and harmony of the two sides are considered a key for the mental and physical u v health of a person as well as for the stability and prosperity of a social system. u Th s bipolar fuzzy sets indeed have potential xx impacts on many elds, fi including artificial intelligence, Figure 1 computer science, information science, cognitive science, decision science, management science, economics, neural science, quantum computing, medical science, and social yy science (cf. [4–45]). In recent years bipolar fuzzy sets seem to have been studied and applied a bit enthusiastically and a bit increasingly (cf. [4–45]). This is the chief motivation for us to introduce and study 𝑚 -polar fuzzy sets. u v The first object of this note is to answer the following xx question on bipolar fuzzy sets. Figure 2 Question 1. Is bipolar fuzzy set a very intuitive 𝐿 -set? The answer to Question 1 is positive. We will prove in this note that there is a natural one-to-one correspondence generalized to the case of 𝑚 -polar fuzzy sets (see Remarks 7 between BF(𝑋) and 2(𝑋) (for the set of all [0, 1] -sets on and 8 for details). 𝑋 ,see Theorem 5 ) which preserves all involved properties. Apart from the backgrounds (e.g., “multipolar informa- This makes the notion of bipolar fuzzy set more intuitive. tion”) of 𝑚 -polar fuzzy sets, the following question on further Since properties of 𝐿 -sets have already been studied very applications (particularly, further applications in real world deeply and exhaustively, this one-to-one correspondence problems) of 𝑚 -polar fuzzy sets should also be considered. may be beneficial for both researchers interested in above- mentioned papers and related elds fi (because they can use Question 3. How to nd fi further possible applications of 𝑚 - these properties directly and even cooperate with theoretical polar fuzzy sets in real world problems? fuzzy mathematicians for a possible higher-level research) Question 3 can be answered as in the case of bipolar fuzzy and theoretical fuzzy mathematicians as well (because coop- sets since researches or modelings on real world problems eration with applied fuzzy mathematicians and practitioners oeft n involve multiagent, multiattribute, multiobject, multi- probably makes their research more useful). index, multipolar information, uncertainty, or/and limits We notice that “multipolar information” (not just bipolar process. We will give examples to demonstrate it (see Exam- information which corresponds to two-valued logic) exists ples 9–14). becausedatafor arealworld problemare sometimesfrom 𝑛 agents (𝑛≥2 ). For example, the exact degree of Remark 3. In this note [0, 1] (𝑚 -power of [0, 1])iscon- telecommunication safety of mankind is a point in [0, 1] (𝑛≈ sidered a poset with the point-wise order ≤,where 𝑚 is an 7×10 ) because different person has been monitored different arbitrary ordinal number (we make an appointment that𝑚= times. There are many other examples: truth degrees of a logic {𝑛 | 𝑛 < }𝑚 when 𝑚>0 ), ≤ (which is actually very intuitive formula which are based on 𝑛 logic implication operators as illustrated below) is defined by 𝑥≤𝑦⇔𝑝 (𝑥) ≤ 𝑝 (𝑦)for 𝑖 𝑖 𝑚 𝑚 (𝑛≥2 ), similarity degrees of two logic formulas which are each 𝑖∈𝑚 (𝑥, 𝑦 ∈ [0, 1] ), and 𝑝 : [0, 1] → [0, 1] is the based on 𝑛 logic implication operators (𝑛≥2 ), ordering 𝑖 th projection mapping (𝑖∈𝑚 ). results of a magazine, ordering results of a university, and inclusion degrees (resp., accuracy measures, rough measures, (1) When𝑚=2 , [0, 1] is the ordinary closed unit square approximation qualities, fuzziness measures, and decision in Euclidean plane 𝑅 . eTh righter (resp., the upper) a performance evaluations) of a rough set. u Th s our second pointinthissquareis, thelargeritis. Let 𝑥 = ⟨0, 0⟩ = object of this note is to answer the following question on 0 (the smallest element of [0, 1] ), 𝑢 = ⟨0.25, 0.75⟩ , extensions of bipolar fuzzy sets. V = ⟨0.75, 0.25⟩,and𝑦=⟨1,1⟩ (the largest element of 2 2 [0, 1] ). Then 𝑥≤𝑧≤𝑦 for all 𝑧∈[0,1] (especially, Question 2. How to generalize bipolar fuzzy sets to multipo- 𝑥≤𝑢≤𝑦 and 𝑥≤ V ≤𝑦 hold). Notice that 𝑢≰ lar fuzzy sets and how to generalize results on bipolar fuzzy sets to the case of multipolar fuzzy sets? V ≰𝑢 because both 𝑝 (𝑢) = 0.25 ≤ 0.75 = 𝑝 (V)and 0 0 𝑝 (𝑢) = 0.75 ≥ 0.25 = 𝑝 (V)hold. eTh order relation The idea to answer Question 2 is from the answer to 1 1 𝑚 2 Question 1, intuitiveness of the point-wise order on [0, 1] ≤ on [0, 1] can be illustrated in at least two ways (see (see Remark 3), and the proven corresponding results on Figure 1). bipolar fuzzy sets. We put forward the notion of 𝑚 -polar (2) When 𝑚>2 , the order relation ≤ on [0, 1] can be fuzzy set (an extension of bipolar fuzzy set) and point out illustrated in at least one way (see Figure 2 for the case that many concepts which have been defined based on bipolar 𝑚=4 ,where 𝑥≤𝑢≤𝑦 , 𝑥≤ V ≤𝑦 ). fuzzy sets and many results related to these concepts can be The Scientific World Journal 3 + − 2. Main Results Example 6. Let (𝜇 ,𝜇 )be a bipolar fuzzy set, where 𝑋= {𝑢, V,𝑤,𝑥,𝑦,𝑧} is a six-element set and 𝜇 : 𝑋 → [0, 1] and In this section we will prove that a bipolar fuzzy set is just 𝜇 :𝑋 → [−1,0] are defined by a very specific 𝐿 -set, that is, [0, 1] -set. We also put forward (or highlight) the notion of 𝑚 -polar fuzzy set (which is still a 0.4 0.5 0.3 1 1 0.6 𝜇 ={ , , , , , }, special 𝐿 -set, i.e., [0, 1] -set, although it is a generalization of 𝑢 V 𝑤 𝑥 𝑦 𝑧 bipolar fuzzy set) and point out that many concepts which (6) −0.3 −0.6 −1 −0.2 −1 −0.5 have been defined based on bipolar fuzzy sets and results − 𝜇 ={ , , , , , }. related to these concepts can be generalized to the case of 𝑚 - 𝑢 V 𝑤 𝑥 𝑦 𝑧 polar fuzzy sets. Then the corresponding 2-polar fuzzy set on 𝑋 is Den fi ition 4. An 𝑚 -polar fuzzy set (or a [0, 1] -set) on 𝑋 is 𝑚 0.4, 0.3 0.5, 0.6 0.3, 1 1, 0.2 ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ exactly a mapping 𝐴:𝑋 → [0,1] .Theset of all 𝑚 -polar 𝐴 ={ , , , , 𝑢 V 𝑤 𝑥 fuzzy sets on 𝑋 is denoted by 𝑚(𝑋) . (7) ⟨1, 1⟩ ⟨0.6, 0.5⟩ , }. eTh following theorem shows that bipolar fuzzy sets and 𝑦 𝑧 2-polar fuzzy sets are cryptomorphic mathematical notions and that we can obtain concisely one from the corresponding In therestofthisnote, we investigatethe possible one. applications of 𝑚 -polar fuzzy sets. First we consider the theoretic applications of 𝑚 -polar fuzzy sets. More precisely, + − Theorem 5. Let 𝑋 be a set. For each bipolar fuzzy set (𝜇 ,𝜇 ) we will give some remarks to illustrate how many concepts on 𝑋 ,denfi e a 2-polar fuzzy set which have been den fi ed based on bipolar fuzzy sets and resultsrelated to theseconceptscan be generalizedtothe case + − 2 of 𝑚 -polar fuzzy sets (see the following Remarks 7 and 8). 𝜑 (𝜇 ,𝜇 )=𝐴 :𝑋󳨀→ [0, 1] (1) Remark 7. The notions of bipolar fuzzy graph (see [ 4, 45]) on 𝑋 by putting and fuzzy graph (see [46, 47]) canbegeneralized to the convenient (because it allows a computing in computers) and + − 𝐴 (𝑥 )=⟨𝜇 (𝑥 ),−𝜇 (𝑥 )⟩ (∀𝑥 ∈ 𝑋 ). (2) intuitive notion of 𝑚 -polar fuzzy graph. An 𝑚 -polar fuzzy graph with an underlying pair (𝑉, 𝐸) (where 𝐸⊆𝑉 × 𝑉 is symmetric; i.e., it satisfies ⟨𝑥, 𝑦⟩ ∈ 𝐸 ⇔ ⟨𝑦, 𝑥⟩ ∈ 𝐸 ) Then we obtain a one-to-one correspondence is denfi edtobeapair 𝐺 = (𝐴, ) 𝐵 ,where 𝐴:𝑉 → [0, 1] (i.e., an 𝑚 -polar fuzzy set on 𝑉 )and 𝐵:𝐸 → 𝜑:𝐵𝐹 (𝑋 )󳨀→ 2 (𝑋 ); (3) [0, 1] (i.e., an 𝑚 -polar fuzzy set on 𝐸 )satisfy 𝐵(⟨𝑥, 𝑦⟩) ≤ inf{𝐴(𝑥), 𝐴(𝑦)} (∀⟨𝑥, 𝑦⟩ ∈ 𝐸) ; 𝐴 is called the 𝑚 -polar fuzzy vertex set of 𝑉 and 𝐵 is called the 𝑚 -polar fuzzy edge set of its inverse mapping 𝜓 : 2(𝑋) → 𝐹(𝑋) 𝐵 is given by 𝜓(𝐴) = + − + 𝐸 .An 𝑚 -polar fuzzy graph 𝐺 = (𝐴, ) 𝐵 with an underlying (𝜇 ,𝜇 ) (∀𝐴 ∈ 2(𝑋)), 𝜇 (𝑥) = 𝑝 ∘ (𝑥) 𝐴 (∀𝑥 ∈ 𝑋) ,and 𝐴 𝐴 𝐴 pair (𝑉, 𝐸) and satisfying 𝐵(⟨𝑥, 𝑦⟩) = (⟨𝑦, 𝐵 𝑥⟩) (∀⟨𝑥, 𝑦⟩ ∈ 𝐸) 𝜇 (𝑥) = −𝑝 ∘ (𝑥) 𝐴 (∀𝑥 ∈ 𝑋) . 𝐴 1 and 𝐵(⟨𝑥, 𝑥⟩) = 0 (∀𝑥 ∈ 𝑉) is called a simple 𝑚 -polar fuzzy graph, where 0 is the smallest element of [0, 1] .An 𝑚 -polar Proof. Obviously, both 𝜑 and 𝜓 are mappings. For each + − fuzzy graph 𝐺 = (𝐴, ) 𝐵 with an underlying pair (𝑉, 𝐸) and (𝜇 ,𝜇 )∈BF(𝑋), satisfying 𝐵(⟨𝑥, 𝑦⟩) = inf{𝐴(𝑥), 𝐴(𝑦)} (∀⟨𝑥, 𝑦⟩ ∈ 𝐸) is called astrong 𝑚 -polar fuzzy graph. The complement of a strong 𝑚 - + − [𝜓 ∘ 𝜑 (𝜇 ,𝜇 )](𝑥 ) polar fuzzy graph 𝐺 = (𝐴, ) 𝐵 (which has an underlying pair + − + − (𝑉, 𝐸) )isastrong 𝑚 -polar fuzzy graph 𝐺 = (𝐴, 𝐵) with an =⟨𝑝 ∘𝜑(𝜇 ,𝜇 )(𝑥 ),−𝑝 ∘𝜑(𝜇 ,𝜇 )(𝑥 )⟩ 0 1 (4) underlying pair (𝑉, 𝐸) ,where 𝐵:𝐸 → [0,1] is defined by + − + − =⟨𝑝 (⟨𝜇 (𝑥 ),𝜇 (𝑥 )⟩) , −𝑝 (⟨𝜇 (𝑥 ),𝜇 (𝑥 )⟩)⟩ (⟨𝑥, 𝑦⟩ ∈ 𝐸 , 𝑖∈𝑚 ) 0 1 + − + − =⟨𝜇 (𝑥 ),−𝜇 (𝑥 )⟩=(𝜇 ,𝜇 )(𝑥 )( ∀𝑥 ∈ 𝑋 ), 𝑝 ∘ 𝐵(⟨𝑥,𝑦⟩) (8) 0, 𝑝 ∘𝐵(⟨𝑥,𝑦⟩) > 0, + − + − which means [𝜓 ∘ (𝜇 𝜑 ,𝜇 )] = (𝜇 ,𝜇 ).Again,for each 𝐴∈ ={ inf {𝑝 ∘𝐴 (𝑥 ),𝑝 ∘ 𝐴 (𝑦)} , 𝑝 ∘𝐵(⟨𝑥,𝑦⟩) = 0. 𝑖 𝑖 𝑖 2(𝑋) and each 𝑥∈𝑋 , Give two 𝑚 -polar fuzzy graphs (withunderlyingpairs + − [𝜑 ∘ 𝜓 (𝐴 )] (𝑥 )=𝜑(𝜇 ,𝜇 )(𝑥 ) 𝐴 𝐴 (𝑉 ,𝐸 )and (𝑉 ,𝐸 ),resp.) 𝐺 =(𝐴 ,𝐵 )and 𝐺 =(𝐴 ,𝐵 ). 1 1 2 2 1 1 1 2 2 2 (5) A homomorphism from 𝐺 to 𝐺 is a mapping 𝑓: 𝑉 →𝑉 + − 1 2 1 2 =⟨𝜇 (𝑥 ),−𝜇 (𝑥 )⟩=𝐴 (𝑥 ), 𝐴 𝐴 which satisfies 𝐴 (𝑥) ≤ 𝐴 (𝑓(𝑥)) (∀𝑥 ∈ 𝑉 )and 𝐵 (⟨𝑥, 𝑦⟩) ≤ 1 2 1 1 𝐵 (⟨𝑓(𝑥), 𝑓(𝑦)⟩) (∀⟨𝑥, 𝑦⟩ ∈ 𝐸 ). An isomorphism from 2 1 which means 𝜑∘𝜓()𝐴 = 𝐴 . 𝐺 to 𝐺 is a bijective mapping 𝑓: 𝑉 →𝑉 which 1 2 1 2 4 The Scientific World Journal satisfies 𝐴 (𝑥) = 𝐴 (𝑓(𝑥)) (∀𝑥 ∈ 𝑉 )and 𝐵 (⟨𝑥, 𝑦⟩) = set on 𝑄×𝑋×𝑄 .Moreover, if𝐵:𝑄 → [0,1] is an 𝑚 -polar 1 2 1 1 𝐵 (⟨𝑓(𝑥), 𝑓(𝑦)⟩) (∀⟨𝑥, 𝑦⟩ ∈ 𝐸 ). A weak isomorphism from fuzzy set on 𝑄 satisfying 2 1 𝐺 to 𝐺 is a bijective mapping 𝑓: 𝑉 →𝑉 which is a 1 2 1 2 homomorphism and satisfies 𝐴 (𝑥) = 𝐴 (𝑓(𝑥)) (∀𝑥 ∈ 𝑉 ). 1 2 1 𝐵(𝑞) ≥ inf {𝐵(),𝐴( 𝑝 ⟨,𝑥𝑝 ,𝑞⟩)} Astrong 𝑚 -polar fuzzy graph 𝐺 is called self-complementary (11) if 𝐺≃ 𝐺 (i.e., there exists an isomorphism between 𝐺 and its (∀ ⟨𝑝, 𝑥, 𝑞⟩ ∈ 𝑄 × 𝑋 × 𝑄) , complement 𝐺 ). It is not difficult to verify the following conclusions (some then 𝑀 = (,𝑄 ,𝑋 𝐴, 𝐵) is called an 𝑚 -polar subsystem of 𝑀 . of which generalize the corresponding results in [1, 45]). ∗ Furthermore, let 𝑋 be the set of all words of elements of 𝑋 of finite length and 𝜆 be the empty word in 𝑋 (cf. [28]). en Th (1) In a self-complementary strong 𝑚 -polar fuzzy graph ∗ ∗ 𝑚 one can define a 𝑚 -polar fuzzy set 𝐴 :𝑄×𝑋 ×𝑄 → [0, 1] 𝐺 = (𝐴, ) 𝐵 (with an underlying pair (𝑉, 𝐸) ), we have on 𝑄×𝑋 ×𝑄 by putting 𝑝 ∘ 𝐵 (⟨𝑥, 𝑦⟩) 1, if𝑞=𝑝, = inf {𝑝 ∘𝐴 (𝑥 ),𝑝 ∘ 𝐴 (𝑦)} − 𝑝 ∘𝐵(⟨𝑥,𝑦⟩) 𝐴 (⟨𝑞, 𝜆, 𝑝⟩) = { 𝑖 𝑖 𝑖 0, if 𝑞 =𝑝̸, (𝑖 ∈ ,𝑚 ⟨𝑥, 𝑦⟩ ∈ 𝐸) , ∗ ∗ ∗ 𝐴 (⟨𝑞, x,𝑥,𝑝⟩) = sup {𝐴 (⟨𝑞, 𝑥, 𝑟⟩) , 𝐴 (⟨𝑟, 𝑥, 𝑝⟩)} (9) 𝑟∈𝑄 ∑ 𝑝 ∘𝐵(⟨𝑥,𝑦⟩) 𝑥 =𝑦̸ (∀ ⟨𝑞, x,𝑥,𝑝⟩ ∈ 𝑄 × 𝑋 ×𝑋×𝑄), (12) = ∑ inf {𝑝 ∘𝐴 (𝑥 ),𝑝 ∘ 𝐴 (𝑦)} (𝑖∈𝑚 ). 𝑖 𝑖 𝑥 =𝑦̸ where 1 is the biggest element of [0, 1] . The following conclusions hold. (2) A strong 𝑚 -polar fuzzy graph 𝐺 = (𝐴, ) 𝐵 (with an underlying pair (𝑉, 𝐸) ) is self-complementary if and (1) An 𝑚 -polar fuzzy set 𝐴:𝐺 → [0,1] is an 𝑚 -polar only if it satisfies fuzzy subgroup of a group (𝐺, ∘) if and only if 𝐴 = [𝑎] {𝑥 ∈ 𝐺 | (𝑥) 𝐴 ≥ 𝑎} is 0 or 𝐴 is a subgroup of [𝑎] 𝑝 ∘𝐵(⟨𝑥,𝑦⟩) = inf {𝑝 ∘𝐴 (𝑥 ),𝑝 ∘𝐴(𝑦)} 𝑖 𝑖 𝑖 (𝐺, ∘) (∀𝑎 ∈ [0, 1] ). (10) (∀𝑖 ∈ 𝑚, ∀ ⟨𝑥, 𝑦⟩ ∈ 𝐸) . (2) An 𝑚 -polar fuzzy set 𝐴:𝐺 → [0,1] is an 𝑚 -polar fuzzy subalgebra of a 𝐾 -algebra (,∘𝐺 ,𝑒,⊙) if and only if𝐴 ={𝑥∈𝐺|𝐴(𝑥) ≥𝑎} is0 or𝐴 is a subalgebra (3) If 𝐺 and 𝐺 are strong 𝑚 -polar fuzzy graphs, then [𝑎] [𝑎] 1 2 of (𝐺, ∘, ,𝑒 ⊙) (∀𝑎 ∈ [0, 1] ). 𝐺 ≃𝐺 if and only if 𝐺 ≃ 𝐺 . 1 2 1 2 (4) Let 𝐺 and 𝐺 be strong 𝑚 -polar fuzzy graphs. If there (3) An 𝑚 -polar fuzzy set 𝐴:𝑋 → [0,1] is an 𝑚 - 1 2 is a weak isomorphism from 𝐺 to 𝐺 , then there is a polar fuzzy subincline (resp., an 𝑚 -polar fuzzy ideal, 1 2 weak isomorphism from 𝐺 to 𝐺 . an 𝑚 -polar fuzzy filter) of an incline (𝑋, +, ∗) if and 2 1 only if 𝐴 is a subincline (resp., ideal, filter) of [𝑎] Remark 8. The fuzzifications or bipolar fuzzifications of (𝑋, +, ∗) (∀𝑎 ∈ [0, 1] ). some algebraic concepts (such as group, 𝐾 -algebra, incline algebra (cf. [48]), ideal, filter, and finite state machine) can (4) Let 𝑀 = (,𝑋𝑄 ,𝐴) be an 𝑚 -polar fuzzy finite be generalized to the case of 𝑚 -polar fuzzy sets. An 𝑚 - state machine and 𝐵:𝑄 → [0,1] be an polar fuzzy set 𝐴:𝐺 → [0,1] is called an 𝑚 -polar 𝑚 -polar fuzzy set on 𝑄 .Then (,𝑋𝑄 ,𝐴,𝐵) is an −1 𝑚 -polar subsystem of 𝑀 if and only if 𝐵(𝑞) ≥ fuzzy subgroup of a group (𝐺, ∘) if it satisfies 𝐴(𝑥 ∘ 𝑦 )≥ ∗ ∗ inf{𝐵(𝑝), 𝐴 (⟨𝑝, x,𝑞⟩)} (∀⟨,𝑝 x,𝑞⟩ ∈ 𝑄 × 𝑋 ×𝑄) . inf{𝐴(𝑥), 𝐴(𝑦)} (∀𝑥, 𝑦 ∈ ) 𝐺 .An 𝑚 -polar fuzzy set 𝐴:𝐺 → Please see [49, 50] for more results. [0, 1] is called an 𝑚 -polar fuzzy subalgebra of a 𝐾 -algebra (,∘𝐺 ,𝑒,⊙) if it satisfies 𝐴(𝑥 ∘ 𝑦) ≥ inf{𝐴(𝑥), 𝐴(𝑦)} (∀𝑥, 𝑦 ∈ 𝐺) .An 𝑚 -polar fuzzy set 𝐴:𝑋 → [0,1] is called an Next we consider the applications of 𝑚 -polar fuzzy sets in 𝑚 -polar fuzzy subincline of an incline (𝑋, +, ∗) if it satisfies real world problems. (𝑥∗𝑦) ≥ inf{𝐴(𝑥), 𝐴(𝑦)} (∀𝑥, 𝑦 ∈ ) 𝐺 ;itiscalledan 𝑚 - polar fuzzy ideal (resp., an 𝑚 -polar fuzzy filter) of (𝑋, +, ∗) Example 9. Let 𝑋 be a set consisting of five patients 𝑥 , if it is an 𝑚 -polar fuzzy subincline of (𝑋, +, ∗) and satisfies 𝑦 , 𝑧 , 𝑢 ,and V (thus 𝑋 = {𝑥,𝑦,𝑧,𝑢, V}). They have diag- 𝐴(𝑥) ≥ (𝑦) 𝐴 whenever 𝑥≤𝑦 (resp., satisfies 𝐴(𝑥) ≤ (𝑦) 𝐴 nosis data consisting of three aspects, diagnosis datum of whenever 𝑥≤𝑦 ). An 𝑚 -polar fuzzy nfi itestate machineisa 𝑥 is (𝑥) = ⟨0.49, 0.46, 0.51⟩,where datum 0.5 represents triple 𝑀 = (,𝑋𝑄 ,𝐴) ,where 𝑄 and 𝑋 are ni fi te nonempty “normal” or “OK.” Suppose 𝐴(𝑦) = ⟨0.45, 0.42, 0.59⟩ , sets (called the set of states and the set of input symbols, 𝐴(𝑧) = ⟨0.50, 0.40, 0.54⟩ , 𝐴(𝑢) = ⟨0.40, 0.49, 0.60⟩ ,and resp.) and 𝐴:𝑄 × 𝑋 × 𝑄 → [0,1] is any 𝑚 -polar fuzzy 𝐴( V) = ⟨0.51, 0.52, 0.50⟩.Then we obtain a 3-polar fuzzy The Scientific World Journal 5 3 𝑛 set 𝐴:𝑋 → [0,1] which can describe the situation; this 𝐴:𝑋={𝑥 ,𝑥 ,...,𝑥 }→[0,1] be a mapping satisfying 1 2 𝑛 3-polar fuzzy set can also be written as follows: 𝑝 ∘𝐴(𝑥 )=𝐵(𝑥 )(if 𝑗=𝑖 )or 0 (otherwise) (𝑖 = 1,2,...,𝑛). 𝑗 𝑖 𝑖 Then a cooperative game model V ∘𝐴 : 𝑋 → 𝑅 is established, ⟨0.49, 0.46, 0.51⟩ ⟨0.45, 0.42, 0.59⟩ ⟨0.50, 0.40, 0.54⟩ where V : [0, 1] →𝑅 is defined by 𝐴={ , , , 𝑥 𝑦 𝑧 𝑛 𝑛 V (⟨𝑠 ,𝑠 ,...,𝑠 ⟩) =∑𝑔 (𝑠 )− 𝑘( ∑𝐵(𝑥 )) 1 2 𝑛 𝑖 𝑖 𝑖 ⟨0.40, 0.49, 0.60⟩ ⟨0.51, 0.52, 0.50⟩ , }. 𝑖=1 𝑖=1 (16) 𝑢 V (∀ ⟨𝑠 ,𝑠 ,...,𝑠 ⟩∈ [0, 1] ), 1 2 𝑛 (13) and the function 𝑔 : [0, 1] → 𝑅 is continuously monotonic Example 10. 𝑚 -polar fuzzy sets can be used in decision increasing with 𝑔 (0) = (0𝑖 = 1,2,...,𝑛) .Obviously,the gain making. In many decision making situations, it is necessary of agent 𝑥 (with participation level 𝐵(𝑥 ))is 𝑖 𝑖 to gather the group consensus. This happens when a group of friends decides which movie to watch, when a company V ∘𝐴(𝑥 )=𝑔 ∘𝐵(𝑥 )− 𝑘( ∑𝐵(𝑥 )) , (17) 𝑖 𝑖 𝑖 𝑖 decides which product design to manufacture, and when a 𝑛 𝑖=1 democratic country elects its leaders. For instance, we con- and the total gain is sider here only the case of election. Let 𝑋 = {𝑥,𝑦,𝑧,...,𝑢, V} be the set of voters and 𝐶={𝑐 ,𝑐 ,𝑐 ,𝑐 } be the set of all 1 2 3 4 𝑛 𝑛 𝑛 the four candidates. Suppose the voting is weighted. For each ∑V ∘𝐴(𝑥 )=∑𝑔 ∘𝐵(𝑥 )−𝑘( ∑𝐵(𝑥 )) . (18) 𝑖 𝑖 𝑖 𝑖 candidate𝑐∈𝐶 ,avoterin {𝑥, 𝑦, }𝑧 cansendavaluein [0, 1] to 𝑖=1 𝑖=1 𝑖=1 𝑐 ,but avoter in𝑋−{𝑥,𝑦,𝑧} can only send a value in [0.1, 0.8] (2) er Th e are two goods, denoted 𝑔 and 𝑔 ,and three 1 2 to 𝑐 .Suppose 𝐴(𝑥) = ⟨0.9, 0.4, 0.01, 0.1⟩ (which means the agents 𝑎 , 𝑏 ,and 𝑐 with endowments (𝜀, 𝜀), (1 − 𝜀, 0),and preference degrees of 𝑥 corresponding to 𝑐 ,𝑐 ,𝑐 ,and 𝑐 1 2 3 4 2 (0, 1 − 𝜀) (0 < 𝜀 ≤ 1).Let V : [0, 1] →𝑅 be any mapping are 0.9, 0.4, 0.01,and 0.1,resp.), 𝐴(𝑦) = ⟨0.2, 0.3, 0.8, 0.1⟩ , satisfying V(⟨0, 0⟩) =. 0en Th the corresponding cooperative 𝐴(𝑧) = ⟨0.8, 0.9, 0.8, 0.2⟩, . . . , (𝑢) 𝐴 = ⟨0.6, 0.8, 0.8, 0.1⟩ ,and game model is V ∘ 𝐴 : 𝑋 = {𝑎, 𝑏, 𝑐} → 𝑅 ,where 𝐴( V) = ⟨0.7, 0.8, 0.4, 0.2⟩.Then we obtain a 4-polar fuzzy set 𝐴:𝑋 → [0,1] which can describe the situation; this 4- ⟨𝜀, 𝜀 ⟩ ⟨1−𝜀,0 ⟩ ⟨0, 1 − 𝜀 ⟩ 𝐴={ , , }, polar fuzzy set can also be written as follows: 𝑎 𝑏 𝑐 (19) V 𝜀, 𝜀 V 1−𝜀,0 V 0, 1 − 𝜀 (⟨ ⟩) (⟨ ⟩) (⟨ ⟩) ⟨0.9, 0.4, 0.01, 0.1⟩ ⟨0.2, 0.3, 0.8, 0.1⟩ V ∘𝐴 = { , , }. 𝐴={ , , 𝑎 𝑏 𝑐 𝑥 𝑦 Example 12. 𝑚 -polar fuzzy sets can be used to den fi e ⟨0.8, 0.9, 0.8, 0.2⟩ ⟨0.6, 0.8, 0.8, 0.1⟩ ,..., , (14) weighted games. A weighted game is a 4-tuple (𝑋, P,𝑊,Δ) , 𝑧 𝑢 where 𝑋={𝑥 ,𝑥 ,...,𝑥 } is the set of 𝑛 players or voters 1 2 𝑛 (𝑛≥2 ), P is a collection of fuzzy sets on 𝑋 (called coalitions) ⟨0.7, 0.8, 0.4, 0.2⟩ }. such that (P,≤)is upper set (i.e., a fuzzy set 𝑄 on 𝑋 belongs to P if𝑄≥𝑃 for some𝑃∈ P), 𝑊: 𝑋 → [0,1] is an 𝑚 -polar fuzzy set on 𝑋 (called voting weights), andΔ⊆[0,+∞) −{0} Example 11. 𝑚 -polar fuzzy sets canbeusedincooperative games (cf. [51]). Let𝑋={𝑥 ,𝑥 ,...,𝑥 } be the set of 𝑛 agents (called quotas). Imagine a situation: three people, 𝑥 , 𝑦 ,and 1 2 𝑛 or players (𝑛≥1 ), 𝑚 = {0,1,...,𝑚 − 1} be the set of the 𝑧 ,votefor aproposalonreleasing of astudent. Supposethat grand coalitions, and 𝐴:𝑋 → [0,1] be an 𝑚 -polar fuzzy 𝑥 casts 200 US Dollars and lose 80 hairs on her head votes set, where 𝑝 ∘ (𝑥) 𝐴 is the degree of player 𝑥 participating in each, 𝑦 casts 60000 US Dollars and 100 grams Cordyceps coalition 𝑖 (𝑥∈𝑋,𝑖∈𝑚 ). Again let V : [0, 1] →𝑅 (the set of sinensis votes each, 𝑧 casts 100000 US Dollars and 100 grams allrealnumbers)beamappingsatisfying V(0)=0.Then the gold votes each. Then an associated weighted game model is mapping V ∘𝐴 : 𝑋 → 𝑅 is called a cooperative game, where (𝑋, P,𝑊,Δ) ,where 𝑋={𝑥,𝑦,}𝑧 and P is a collection of fuzzy sets on 𝑋 with (P,≤)an upper set, 𝑚=4 , V ∘ (𝑥) 𝐴 represents the amount of money obtained by player 𝑥 under the coalition participating ability 𝐴(𝑥) (𝑥∈𝑋 ). ⟨200/160200, 80/80, 0, 0⟩ (1) (a public good game; compare with [51,Example 6.5]) 𝑊={ , Suppose 𝑛 agents 𝑥 ,𝑥 ,...,𝑥 want to create a facility for 1 2 𝑛 joint use. eTh cost of the facility depends on the sum of ⟨60000/160200, 0, 100/100, 0⟩ the participation levels (or degrees) of the agents and it is described by (20) ⟨100000/160200, 0, 0, 100/100⟩ }, 𝑘( ∑𝐵(𝑥 )) , (15) 𝑖=1 Δ={⟨200000, 0, 0, 0⟩, ⟨100000, 300, 0, 0⟩, where 𝑘:[0,𝑛] → 𝑅 is a continuous monotonic increasing function with 𝑘(0) = 0and 𝐵:𝑋 → [0,1] is a mapping. Let ⟨0, 0, 500, 0⟩, ⟨0, 0, 0, 65000⟩}. 6 The Scientific World Journal (1) If the situation is a little simple, 𝑥 casts [100, 300] US 3. Conclusion Dollars (i.e., the cast is between 100 US Dollars and In this note, we show that the enthusiastically studied notion 300 US Dollars, where [100, 300] is an interval num- of bipolarfuzzy setisactuallyasynonymofa [0, 1] -set (we ber which can be looked as a point [0, +∞))votes call it 2-polar fuzzy set), and thus we highlight the notion of each, 𝑦 casts [50000, 70000] US Dollars votes each, 𝑧 𝑚 -polar fuzzy set (actually a [0, 1] -set,𝑚≥2 ). The 𝑚 -polar casts [90000, 110000] US Dollars votes each, and quota is fuzzy sets not only have real backgrounds (e.g., “multipolar [100000, 120000]. eTh n the corresponding weighted game information” exists) but also have applications in both theory model is (𝑋, P,,𝑊 [100000, 120000]) ,where P is a collection and real world problems (which have been illustrated by of fuzzy sets on 𝑋 with (P,≤)an upper set, 𝑚=1 ,and examples). 400/320400 120000/320400 200000/320400 𝑊={ , , }. 𝑥 𝑦 𝑧 Conflict of Interests (21) eTh authors declare that there is no conflict of interests (2) If the situation is more simple, 𝑥 casts 200 US Dollars regarding the publication of this paper. votes each, 𝑦 casts 60000 US Dollars votes each, 𝑧 casts 100000 US Dollars votes each, and quota is 110000.Then the corresponding weighted game model is (𝑋, P,,𝑊 110000) , Acknowledgments where P = {{𝑥, },𝑧 {𝑦, },𝑧 {𝑥, 𝑦, }} 𝑧 , 𝑚=1 ,and This work was supported by the International Science and 200/160200 60000/160200 100000/160200 Technology Cooperation Foundation of China (Grant no. 𝑊={ , , }. 𝑥 𝑦 𝑧 2012DFA11270) and the National Natural Science Foundation (22) of China (Grant no. 11071151). Notice that the subset {𝑥, }𝑧 ⊆ 𝑋 is exactly a fuzzy set 𝐴: References 𝑋→[0,1] on 𝑋 defined by 𝐴(𝑥) = (𝑧) 𝐴 = 1 and 𝐴(𝑦) = 0 . [1] L. A. Zadeh, “Fuzzy sets,” Information and Control,vol.8,no. 3, Example 13. 𝑚 -polar fuzzy sets can be used as a model for pp. 338–353, 1965. clustering or classification. Consider a set 𝑋 consisting of 𝑛 [2] U. Hoh ¨ le and S. E. Rodabaugh, Eds., Mathematics of Fuzzy Sets: students 𝑥 ,𝑥 ,...,𝑥 (𝑛 ≥ 2)in Chinese middle school. 1 2 𝑛 Logic, Topology, and Measure eTh ory , eTh Handbooks of Fuzzy For a student 𝑥∈𝑋 , we use integers 𝑥 (resp., 𝑥 ,...,𝑥 ) 1 2 6 Sets Series, Kluwer Academic, Dordrecht, eTh Netherlands, in [0, 100] to denote the average score of Mathematics (resp., Physics, Chemistry, Biology, Chinese, and English), and [3] W.R.Zhang,“Bipolarfuzzy sets andrelations:acomputational framework for cognitive modeling and multiagent decision 𝐴 (𝑥 )=⟨𝑥 × 0.01, 𝑥 × 0.01, 𝑥 × 0.01, 1 2 3 analysis,” in Proceedings of the Industrial Fuzzy Control and (23) Intelligent Systems Conference, and the NASA Joint Technology 𝑥 × 0.01, 𝑥 × 0.01, 𝑥 × 0.01⟩ . 4 5 6 Workshop on Neural Networks and Fuzzy Logic and Fuzzy Information Processing Society Biannual Conference,pp. 305– eTh nweobtaina 6-polar fuzzy set model 𝐴:𝑋 → [0,1] , 309, SanAntonio,Tex,USA,December1994. which can be used for clustering or classification of these [4] M.Akram,“Bipolarfuzzygraphs,” Information Sciences,vol.181, students. no. 24, pp. 5548–5564, 2011. Example 14. 𝑚 -polar fuzzy sets can be used to den fi e multi- [5] M. Akram, “Bipolar fuzzy graphs with applications,” Knowledge- valued relations. Based Systems,vol.39, pp.1–8,2013. [6] M. Akram, W. Chen, and Y. Yin, “Bipolar fuzzy Lie superalge- (1) Consider a set 𝑋 consisting of 𝑛 net users (resp., bras,” Quasigroups and Related Systems,vol.20, no.2,pp. 139– patients) 𝑥 ,𝑥 ,...,𝑥 (𝑛 ≥ 2). For net users (resp., 1 2 𝑛 156, 2012. patients) 𝑥, 𝑦 ∈ 𝑋 ,weuse (𝑥,𝑦,𝑗) to denote the [7] M.Akram,S.G.Li, andK.P.Shum, “Antipodal bipolarfuzzy similarity between 𝑥 and 𝑦 in 𝑗 th aspect (1≤ graphs,” Italian Journal of Pure and Applied Mathematics,vol. 𝑗≤𝑚, 𝑚≥2 ), and let 𝐴(𝑥, 𝑦) = (𝑦, 𝐴 𝑥) = 31, pp. 425–438, 2013. ⟨(𝑥,𝑦,1),(𝑥,𝑦,2),...,(𝑥,𝑦,𝑚)⟩ .Then we obtain an [8] M.Akram,A.B.Saeid, K. P. Shum, andB.L.Meng, “Bipolar 𝑚 -polar fuzzy set 𝐴:𝑋 → [0,1] ,which is a fuzzy K-algebras,” International Journal of Fuzzy Systems,vol. multivalued similarity relation. 12, no. 3, pp. 252–259, 2010. (2) Consider a set 𝑋 consisting of 𝑛 people 𝑥 ,𝑥 , 1 2 [9] L. Amgoud, C. Cayrol, M. C. Lagasquie-Schiex, and P. Livet, ...,𝑥 (𝑛≥2) in a social network. For 𝑥, 𝑦 ∈ 𝑛 “On bipolarity in argumentation frameworks,” International 𝑋 ,weuse (𝑥,𝑦,𝑗) to denote the degree of connec- Journal of Intelligent Systems,vol.23,no.10,pp.1062–1093,2008. tion between 𝑥 and 𝑦 in 𝑗 th aspect (1≤𝑗≤ [10] H. Y. Ban, M. J. Kim, and Y. J. Park, “Bipolar fuzzy ideals with 𝑚, 𝑚 ≥ 2 ), and let 𝐴(𝑥, 𝑦) = (𝑦, 𝐴 𝑥) = ⟨(𝑥, 𝑦, 1) , operators in semigroups,” Annals of Fuzzy Mathematics and (𝑥,𝑦,2),...,(𝑥,𝑦,𝑚)⟩ .Then we obtain an 𝑚 -polar Informatics,vol.4,no. 2, pp.253–265,2012. fuzzy set 𝐴:𝑋 → [0,1] ,which is amultivalued [11] S. Benferhat, D. Dubois, S. Kaci, and H. Prade, “Bipolar possi- social graph (or multivalued social network) model. bility theory in preference modeling: representation, fusion and The Scientific World Journal 7 optimal solutions,” Information Fusion,vol.7,no. 1, pp.135–150, [29] Y. B. Jun, H. S. Kim, and K. J. Lee, “Bipolar fuzzy translation 2006. in BCK/BCI-algebra,” Journal of the Chungcheong Mathematical Society,vol.22, no.3,pp. 399–408, 2009. [12] S. Bhattacharya and S. Roy, “Study on bipolar fuzzy-rough control theory,” International Mathematical Forum,vol.7,no. [30] Y. B. Jun and C. H. Park, “Filters of BCH-algebras based on 41, pp. 2019–2025, 2012. bipolar-valued fuzzy sets,” International Mathematical Forum, vol. 4, no.13, pp.631–643,2009. [13] I. Bloch, “Dilation and erosion of spatial bipolar fuzzy sets,” in Applications of Fuzzy Sets eo Th ry ,F.Masulli, S. Mitra, andG. [31] S. Kaci, “Logical formalisms for representing bipolar prefer- Pasi, Eds., vol. 4578 of Lecture Notes in Computer Science,pp. ences,” International Journal of Intelligent Systems,vol.23, no. 385–393, Springer, Berlin, Germany, 2007. 9, pp. 985–997, 2008. [14] I. Bloch, “Bipolar fuzzy spatial information: geometry, mor- [32] K. J. Lee, “Bipolar fuzzy subalgebras and bipolar fuzzy ideals of BCK/BCI-algebras,” Bulletin of the Malaysian Mathematical phology, spatial reasoning,” in Methods for Handling Imperfect Spatial Information, R. Jeansoulin, O. Papini, H. Prade, and Sciences Society,vol.32, no.3,pp. 361–373, 2009. S. Schockaert, Eds., vol. 256 of Studies in Fuzziness and Soft [33] K. J. Lee and Y. B. Jun, “Bipolar fuzzy a-ideals of BCI-algebras,” Computing, pp. 75–102, Springer, Berlin, Germany, 2010. Communications of the Korean Mathematical Society,vol.26,no. [15] I. Bloch, “Lattices of fuzzy sets and bipolar fuzzy sets, and math- 4, pp. 531–542, 2011. ematical morphology,” Information Sciences,vol.181,no. 10,pp. [34] K. M. Lee, “Comparison of interval-valued fuzzy sets, intuition- 2002–2015, 2011. istic fuzzy sets, and bipolar-valued fuzzy sets,” Journal of Fuzzy Logic Intelligent Systems,vol.14, no.2,pp. 125–129, 2004. [16] I. Bloch, “Mathematical morphology on bipolar fuzzy sets: gen- eral algebraic framework,” International Journal of Approximate [35] R. Muthuraj and M. Sridharan, “Bipolar anti fuzzy HX group Reasoning,vol.53, no.7,pp. 1031–1060, 2012. and its lower level sub HX groups,” JournalofPhysicalSciences, vol. 16, pp. 157–169, 2012. [17] I. Bloch and J. Atif, “Distance to bipolar information from morphological dilation,” in Proceedings of the 8th Conference of [36] S. Narayanamoorthy and A. Tamilselvi, “Bipolar fuzzy line theEuropeanSociety forFuzzy Logicand Technology,pp. 266– graph of a bipolar fuzzy hypergraph,” Cybernetics and Informa- 273, 2013. tion Technologies,vol.13, no.1,pp. 13–17, 2013. [18] J. F. Bonnefon, “Two routes for bipolar information processing, [37] E. Raufaste and S. Vautier, “An evolutionist approach to and a blind spot in between,” International Journal of Intelligent information bipolarity: representations and aeff cts in human Systems,vol.23, no.9,pp. 923–929, 2008. cognition,” International Journal of Intelligent Systems,vol.23, no. 8, pp. 878–897, 2008. [19] P. Bosc and O. Pivert, “On a fuzzy bipolar relational algebra,” Information Sciences,vol.219,pp. 1–16,2013. [38] A. B. Saeid, “BM-algebras defined by bipolar-valued sets,” Indian Journal of Science and Technology,vol.5,no. 2, pp.2071– [20] D. Dubois, S. Kaci, and H. Prade, “Bipolarity in reasoning 2078, 2012. and decision, an introduction,” in Proceedings of the Interna- tional Conference on Information Processing and Management of [39] S. Samanta and M. Pal, “Irregular bipolar fuzzy graphs,” Inter- Uncertainty, pp. 959–966, 2004. national Journal of Applications of Fuzzy Sets,vol.2,no. 2, pp. 91–102, 2012. [21] D. Dubois and H. Prade, “An overview of the asymmetric bipolar representation of positive and negative information in [40] H. L. Yang,S.G.Li, Z. L. Guo, andC.H.Ma, “Transformationof possibility theory,” Fuzzy Sets and Systems,vol.160,no. 10,pp. bipolar fuzzy rough set models,” Knowledge-Based Systems,vol. 1355–1366, 2009. 27, pp. 60–68, 2012. [22] U. Dudziak and B. Pe¸kala, “Equivalent bipolar fuzzy relations,” [41] H. L. Yang,S.G.Li, S. Y. Wang,and J. Wang,“Bipolarfuzzy Fuzzy Sets and Systems,vol.161,no. 2, pp.234–253,2010. rough set model on two different universes and its application,” Knowledge-Based Systems,vol.35, pp.94–101, 2012. [23] H. Fargier and N. Wilson, “Algebraic structures for bipo- lar constraint-based reasoning,” in Symbolic and Quantitative [42] W. R. Zhang, “Equilibrium relations and bipolar fuzzy cluster- Approaches to Reasoning with Uncertainty,vol.4724of Lecture ing,” in Proceedings of the 18th International Conference of the Notes in Computer Science, pp. 623–634, Springer, Berlin, North American Fuzzy Information Processing Society (NAFIPS Germany, 2007. '99), pp. 361–365, June 1999. [24] M. Grabisch, S. Greco, and M. Pirlot, “Bipolar and bivari- [43] W. R. Zhang, Ed., YinYang Bipolar Relativity: A Unifying eTh ory ate models in multicriteria decision analysis: descriptive and of Nature, Agents and Causality with Applications in Quantum constructive approaches,” International Journal of Intelligent Computing, Cognitive Informatics and Life Sciences,IGI Global, Systems,vol.23, no.9,pp. 930–969, 2008. 2011. [25] M. M. Hasankhani and A. B. Saeid, “Hyper MV-algebras [44] W. R. Zhang, “Bipolar quantum logic gates and quantum defined by bipolar-valued fuzzy sets,” Annals of West University cellular combinatorics—a logical extension to quantum entan- of Timisoara-Mathematics,vol.50, no.1,pp. 39–50, 2012. glement,” Journal of Quantum Information Science,vol.3,no. 2, pp. 93–105, 2013. [26] C. Hudelot, J. Atif, and I. Bloch, “Integrating bipolar fuzzy mathematical morphology in description logics for spatial [45] H. L. Yang,S.G.Li, W. H. Yang,and Y. Lu,“Noteson‘bipolar reasoning,” Frontiers in Articia fi l Intelligence and Applications , fuzzy graphs’,” Information Sciences,vol.242,pp. 113–121,2013. vol. 215, pp. 497–502, 2010. [46] A. Rosenfeld, “Fuzzy graphs,” in Fuzzy Sets and eTh ir Applica- [27] Y. B. Jun, M. S. Kang, and H. S. Kim, “Bipolar fuzzy hyper BCK- tions to Cognitive and Decision Process,L.A.Zadeh,K.S.Fu, ideals in hyper BCK-algebras,” Iranian Journal of Fuzzy Systems, and M. Shimura, Eds., pp. 77–95, Academic Press, New York, vol. 8, no.2,pp. 105–120, 2011. NY, USA, 1975. [28] Y. B. Jun and J. Kavikumar, “Bipolar fuzzy finite state machines,” [47] R. T. Yeh and S. Y. Bang, “Fuzzy relations, fuzzy graphs and their Bulletin of the Malaysian Mathematical Sciences Society,vol.34, application to clustering analysis,” in Fuzzy Sets and eTh ir Appli- no. 1, pp. 181–188, 2011. cations to Cognitive and Decision Process,L.A.Zadeh,K.S.Fu, 8 The Scientific World Journal and M. Shimura, Eds., pp. 338–353, Academic Press, New York, NY, USA, 1975. [48] Z. Q. Cao, K. H. Kim, and F. W. Roush, Incline Algebra and Applications, Ellis Horwood Series in Mathematics and Its Applications, Halsted Press, Chichester, UK; John Wiley & Sons, New York, NY, USA, 1984. [49] Y. M. Li, “Finite automata theory with membership values in lattices,” Information Sciences,vol.181,no. 5, pp.1003–1017,2011. [50] J. H. Jin, Q. G. Li, and Y. M. Li, “Algebraic properties of L- fuzzy finite automata,” Information Sciences,vol.234,pp. 182– 202, 2013. [51] L. J. Xie and M. Grabisch, “The core of bicapacities and bipolar games,” Fuzzy Sets and Systems,vol.158,no. 9, pp.1000–1012, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Scientific World Journal Pubmed Central

m-Polar Fuzzy Sets: An Extension of Bipolar Fuzzy Sets

The Scientific World Journal , Volume 2014 – Jun 12, 2014

Loading next page...
 
/lp/pubmed-central/m-polar-fuzzy-sets-an-extension-of-bipolar-fuzzy-sets-AoyGrd5eKY

References (51)

Publisher
Pubmed Central
Copyright
Copyright © 2014 Juanjuan Chen et al.
ISSN
2356-6140
eISSN
1537-744X
DOI
10.1155/2014/416530
Publisher site
See Article on Publisher Site

Abstract

Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 416530, 8 pages http://dx.doi.org/10.1155/2014/416530 Research Article 𝑚 -Polar Fuzzy Sets: An Extension of Bipolar Fuzzy Sets 1,2 1 1 3 Juanjuan Chen, Shenggang Li, Shengquan Ma, and Xueping Wang College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China School of Sciences, Xi’an University of Technology, Xi’an 710056, China College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China Correspondence should be addressed to Shenggang Li; shengganglinew@126.com Received 11 April 2014; Accepted 23 May 2014; Published 12 June 2014 Academic Editor: Jianming Zhan Copyright © 2014 Juanjuan Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Recently, bipolar fuzzy sets have been studied and applied a bit enthusiastically and a bit increasingly. In this paper we prove that bipolar fuzzy sets and [0, 1] -sets (which have been deeply studied) are actually cryptomorphic mathematical notions. Since researches or modelings on real world problems oen ft involve multi-agent, multi-attribute, multi-object, multi-index, multi-polar information, uncertainty, or/and limit process, we put forward (or highlight) the notion of 𝑚 -polar fuzzy set (actually, [0, 1] -set which can be seen as a generalization of bipolar fuzzy set, where 𝑚 is an arbitrary ordinal number) and illustrate how many concepts have been defined based on bipolar fuzzy sets and many results which are related to these concepts can be generalized to the case of 𝑚 -polar fuzzy sets.Wealsogiveexamplestoshowhow to apply 𝑚 -polar fuzzy sets in real world problems. 1. Introduction and Preliminaries any mappings. eTh set of all bipolar fuzzy sets on 𝑋 is denoted by BF(𝑋). Set theory and logic systems are strongly coupled in the Bipolar fuzzy sets are an extension of fuzzy sets whose development of modern logic. Classical logic corresponds to membership degree range is [−1, 1]. In a bipolar fuzzy set, the the crisp set theory, and fuzzy logic is associated with fuzzy membership degree 0 of an element means that the element set theory which was proposed by Zadeh in his pioneer work is irrelevant to the corresponding property, the member- [1]. ship degree (0, 1]of an element indicates that the element Den fi ition 1. An 𝐿 -subset (or an 𝐿 -set) on the set 𝑋 is a somewhat satisfies the property, and the membership degree synonym of a mapping 𝐴:𝑋 → 𝐿 ,where 𝐿 is a lattice (cf. [−1, 0)of an element indicates that the element somewhat [2]). When 𝐿=[0,1] (the ordinary closed unit interval with satisfies the implicit counter-property. eTh idea which lies the ordinary order relation), an 𝐿 -set on 𝑋 will be called a behind such description is connected with the existence of fuzzy set on 𝑋 (cf. [1]). “bipolar information” (e.g., positive information and nega- tive information) about the given set. Positive information eTh theory of fuzzy sets has become a vigorous area of represents what is granted to be possible, while negative research in different disciplines including medical and life information represents what is considered to be impossible. sciences, management sciences, social sciences, engineering, Actually, a wide variety of human decision making is based statistics, graph theory, artificial intelligence, pattern recog- on double-sided or bipolar judgmental thinking on a positive nition, robotics, computer networks, decision making, and side and a negative side. For instance, cooperation and automata theory. competition, friendship and hostility, common interests and An extension of fuzzy set, called bipolar fuzzy set, was conflict of interests, eeff ct and side eeff ct, likelihood and given by Zhang [3]in1994. unlikelihood, feedforward and feedback, and so forth are Den fi ition 2 (see Zhang [3]). A bipolar fuzzy set is a pair oen ft the two sides in decision and coordination. In the + − + − traditional Chinese medicine (TCM for short), “yin” and (𝜇 ,𝜇 ),where 𝜇 : 𝑋 → [0, 1] and 𝜇 :𝑋 → [−1,0] are 2 The Scientific World Journal “yang” are the two sides. Yin is the feminine or negative side yy of a system and yang is the masculine or positive side of a v u system. eTh coexistence, equilibrium, and harmony of the two sides are considered a key for the mental and physical u v health of a person as well as for the stability and prosperity of a social system. u Th s bipolar fuzzy sets indeed have potential xx impacts on many elds, fi including artificial intelligence, Figure 1 computer science, information science, cognitive science, decision science, management science, economics, neural science, quantum computing, medical science, and social yy science (cf. [4–45]). In recent years bipolar fuzzy sets seem to have been studied and applied a bit enthusiastically and a bit increasingly (cf. [4–45]). This is the chief motivation for us to introduce and study 𝑚 -polar fuzzy sets. u v The first object of this note is to answer the following xx question on bipolar fuzzy sets. Figure 2 Question 1. Is bipolar fuzzy set a very intuitive 𝐿 -set? The answer to Question 1 is positive. We will prove in this note that there is a natural one-to-one correspondence generalized to the case of 𝑚 -polar fuzzy sets (see Remarks 7 between BF(𝑋) and 2(𝑋) (for the set of all [0, 1] -sets on and 8 for details). 𝑋 ,see Theorem 5 ) which preserves all involved properties. Apart from the backgrounds (e.g., “multipolar informa- This makes the notion of bipolar fuzzy set more intuitive. tion”) of 𝑚 -polar fuzzy sets, the following question on further Since properties of 𝐿 -sets have already been studied very applications (particularly, further applications in real world deeply and exhaustively, this one-to-one correspondence problems) of 𝑚 -polar fuzzy sets should also be considered. may be beneficial for both researchers interested in above- mentioned papers and related elds fi (because they can use Question 3. How to nd fi further possible applications of 𝑚 - these properties directly and even cooperate with theoretical polar fuzzy sets in real world problems? fuzzy mathematicians for a possible higher-level research) Question 3 can be answered as in the case of bipolar fuzzy and theoretical fuzzy mathematicians as well (because coop- sets since researches or modelings on real world problems eration with applied fuzzy mathematicians and practitioners oeft n involve multiagent, multiattribute, multiobject, multi- probably makes their research more useful). index, multipolar information, uncertainty, or/and limits We notice that “multipolar information” (not just bipolar process. We will give examples to demonstrate it (see Exam- information which corresponds to two-valued logic) exists ples 9–14). becausedatafor arealworld problemare sometimesfrom 𝑛 agents (𝑛≥2 ). For example, the exact degree of Remark 3. In this note [0, 1] (𝑚 -power of [0, 1])iscon- telecommunication safety of mankind is a point in [0, 1] (𝑛≈ sidered a poset with the point-wise order ≤,where 𝑚 is an 7×10 ) because different person has been monitored different arbitrary ordinal number (we make an appointment that𝑚= times. There are many other examples: truth degrees of a logic {𝑛 | 𝑛 < }𝑚 when 𝑚>0 ), ≤ (which is actually very intuitive formula which are based on 𝑛 logic implication operators as illustrated below) is defined by 𝑥≤𝑦⇔𝑝 (𝑥) ≤ 𝑝 (𝑦)for 𝑖 𝑖 𝑚 𝑚 (𝑛≥2 ), similarity degrees of two logic formulas which are each 𝑖∈𝑚 (𝑥, 𝑦 ∈ [0, 1] ), and 𝑝 : [0, 1] → [0, 1] is the based on 𝑛 logic implication operators (𝑛≥2 ), ordering 𝑖 th projection mapping (𝑖∈𝑚 ). results of a magazine, ordering results of a university, and inclusion degrees (resp., accuracy measures, rough measures, (1) When𝑚=2 , [0, 1] is the ordinary closed unit square approximation qualities, fuzziness measures, and decision in Euclidean plane 𝑅 . eTh righter (resp., the upper) a performance evaluations) of a rough set. u Th s our second pointinthissquareis, thelargeritis. Let 𝑥 = ⟨0, 0⟩ = object of this note is to answer the following question on 0 (the smallest element of [0, 1] ), 𝑢 = ⟨0.25, 0.75⟩ , extensions of bipolar fuzzy sets. V = ⟨0.75, 0.25⟩,and𝑦=⟨1,1⟩ (the largest element of 2 2 [0, 1] ). Then 𝑥≤𝑧≤𝑦 for all 𝑧∈[0,1] (especially, Question 2. How to generalize bipolar fuzzy sets to multipo- 𝑥≤𝑢≤𝑦 and 𝑥≤ V ≤𝑦 hold). Notice that 𝑢≰ lar fuzzy sets and how to generalize results on bipolar fuzzy sets to the case of multipolar fuzzy sets? V ≰𝑢 because both 𝑝 (𝑢) = 0.25 ≤ 0.75 = 𝑝 (V)and 0 0 𝑝 (𝑢) = 0.75 ≥ 0.25 = 𝑝 (V)hold. eTh order relation The idea to answer Question 2 is from the answer to 1 1 𝑚 2 Question 1, intuitiveness of the point-wise order on [0, 1] ≤ on [0, 1] can be illustrated in at least two ways (see (see Remark 3), and the proven corresponding results on Figure 1). bipolar fuzzy sets. We put forward the notion of 𝑚 -polar (2) When 𝑚>2 , the order relation ≤ on [0, 1] can be fuzzy set (an extension of bipolar fuzzy set) and point out illustrated in at least one way (see Figure 2 for the case that many concepts which have been defined based on bipolar 𝑚=4 ,where 𝑥≤𝑢≤𝑦 , 𝑥≤ V ≤𝑦 ). fuzzy sets and many results related to these concepts can be The Scientific World Journal 3 + − 2. Main Results Example 6. Let (𝜇 ,𝜇 )be a bipolar fuzzy set, where 𝑋= {𝑢, V,𝑤,𝑥,𝑦,𝑧} is a six-element set and 𝜇 : 𝑋 → [0, 1] and In this section we will prove that a bipolar fuzzy set is just 𝜇 :𝑋 → [−1,0] are defined by a very specific 𝐿 -set, that is, [0, 1] -set. We also put forward (or highlight) the notion of 𝑚 -polar fuzzy set (which is still a 0.4 0.5 0.3 1 1 0.6 𝜇 ={ , , , , , }, special 𝐿 -set, i.e., [0, 1] -set, although it is a generalization of 𝑢 V 𝑤 𝑥 𝑦 𝑧 bipolar fuzzy set) and point out that many concepts which (6) −0.3 −0.6 −1 −0.2 −1 −0.5 have been defined based on bipolar fuzzy sets and results − 𝜇 ={ , , , , , }. related to these concepts can be generalized to the case of 𝑚 - 𝑢 V 𝑤 𝑥 𝑦 𝑧 polar fuzzy sets. Then the corresponding 2-polar fuzzy set on 𝑋 is Den fi ition 4. An 𝑚 -polar fuzzy set (or a [0, 1] -set) on 𝑋 is 𝑚 0.4, 0.3 0.5, 0.6 0.3, 1 1, 0.2 ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ exactly a mapping 𝐴:𝑋 → [0,1] .Theset of all 𝑚 -polar 𝐴 ={ , , , , 𝑢 V 𝑤 𝑥 fuzzy sets on 𝑋 is denoted by 𝑚(𝑋) . (7) ⟨1, 1⟩ ⟨0.6, 0.5⟩ , }. eTh following theorem shows that bipolar fuzzy sets and 𝑦 𝑧 2-polar fuzzy sets are cryptomorphic mathematical notions and that we can obtain concisely one from the corresponding In therestofthisnote, we investigatethe possible one. applications of 𝑚 -polar fuzzy sets. First we consider the theoretic applications of 𝑚 -polar fuzzy sets. More precisely, + − Theorem 5. Let 𝑋 be a set. For each bipolar fuzzy set (𝜇 ,𝜇 ) we will give some remarks to illustrate how many concepts on 𝑋 ,denfi e a 2-polar fuzzy set which have been den fi ed based on bipolar fuzzy sets and resultsrelated to theseconceptscan be generalizedtothe case + − 2 of 𝑚 -polar fuzzy sets (see the following Remarks 7 and 8). 𝜑 (𝜇 ,𝜇 )=𝐴 :𝑋󳨀→ [0, 1] (1) Remark 7. The notions of bipolar fuzzy graph (see [ 4, 45]) on 𝑋 by putting and fuzzy graph (see [46, 47]) canbegeneralized to the convenient (because it allows a computing in computers) and + − 𝐴 (𝑥 )=⟨𝜇 (𝑥 ),−𝜇 (𝑥 )⟩ (∀𝑥 ∈ 𝑋 ). (2) intuitive notion of 𝑚 -polar fuzzy graph. An 𝑚 -polar fuzzy graph with an underlying pair (𝑉, 𝐸) (where 𝐸⊆𝑉 × 𝑉 is symmetric; i.e., it satisfies ⟨𝑥, 𝑦⟩ ∈ 𝐸 ⇔ ⟨𝑦, 𝑥⟩ ∈ 𝐸 ) Then we obtain a one-to-one correspondence is denfi edtobeapair 𝐺 = (𝐴, ) 𝐵 ,where 𝐴:𝑉 → [0, 1] (i.e., an 𝑚 -polar fuzzy set on 𝑉 )and 𝐵:𝐸 → 𝜑:𝐵𝐹 (𝑋 )󳨀→ 2 (𝑋 ); (3) [0, 1] (i.e., an 𝑚 -polar fuzzy set on 𝐸 )satisfy 𝐵(⟨𝑥, 𝑦⟩) ≤ inf{𝐴(𝑥), 𝐴(𝑦)} (∀⟨𝑥, 𝑦⟩ ∈ 𝐸) ; 𝐴 is called the 𝑚 -polar fuzzy vertex set of 𝑉 and 𝐵 is called the 𝑚 -polar fuzzy edge set of its inverse mapping 𝜓 : 2(𝑋) → 𝐹(𝑋) 𝐵 is given by 𝜓(𝐴) = + − + 𝐸 .An 𝑚 -polar fuzzy graph 𝐺 = (𝐴, ) 𝐵 with an underlying (𝜇 ,𝜇 ) (∀𝐴 ∈ 2(𝑋)), 𝜇 (𝑥) = 𝑝 ∘ (𝑥) 𝐴 (∀𝑥 ∈ 𝑋) ,and 𝐴 𝐴 𝐴 pair (𝑉, 𝐸) and satisfying 𝐵(⟨𝑥, 𝑦⟩) = (⟨𝑦, 𝐵 𝑥⟩) (∀⟨𝑥, 𝑦⟩ ∈ 𝐸) 𝜇 (𝑥) = −𝑝 ∘ (𝑥) 𝐴 (∀𝑥 ∈ 𝑋) . 𝐴 1 and 𝐵(⟨𝑥, 𝑥⟩) = 0 (∀𝑥 ∈ 𝑉) is called a simple 𝑚 -polar fuzzy graph, where 0 is the smallest element of [0, 1] .An 𝑚 -polar Proof. Obviously, both 𝜑 and 𝜓 are mappings. For each + − fuzzy graph 𝐺 = (𝐴, ) 𝐵 with an underlying pair (𝑉, 𝐸) and (𝜇 ,𝜇 )∈BF(𝑋), satisfying 𝐵(⟨𝑥, 𝑦⟩) = inf{𝐴(𝑥), 𝐴(𝑦)} (∀⟨𝑥, 𝑦⟩ ∈ 𝐸) is called astrong 𝑚 -polar fuzzy graph. The complement of a strong 𝑚 - + − [𝜓 ∘ 𝜑 (𝜇 ,𝜇 )](𝑥 ) polar fuzzy graph 𝐺 = (𝐴, ) 𝐵 (which has an underlying pair + − + − (𝑉, 𝐸) )isastrong 𝑚 -polar fuzzy graph 𝐺 = (𝐴, 𝐵) with an =⟨𝑝 ∘𝜑(𝜇 ,𝜇 )(𝑥 ),−𝑝 ∘𝜑(𝜇 ,𝜇 )(𝑥 )⟩ 0 1 (4) underlying pair (𝑉, 𝐸) ,where 𝐵:𝐸 → [0,1] is defined by + − + − =⟨𝑝 (⟨𝜇 (𝑥 ),𝜇 (𝑥 )⟩) , −𝑝 (⟨𝜇 (𝑥 ),𝜇 (𝑥 )⟩)⟩ (⟨𝑥, 𝑦⟩ ∈ 𝐸 , 𝑖∈𝑚 ) 0 1 + − + − =⟨𝜇 (𝑥 ),−𝜇 (𝑥 )⟩=(𝜇 ,𝜇 )(𝑥 )( ∀𝑥 ∈ 𝑋 ), 𝑝 ∘ 𝐵(⟨𝑥,𝑦⟩) (8) 0, 𝑝 ∘𝐵(⟨𝑥,𝑦⟩) > 0, + − + − which means [𝜓 ∘ (𝜇 𝜑 ,𝜇 )] = (𝜇 ,𝜇 ).Again,for each 𝐴∈ ={ inf {𝑝 ∘𝐴 (𝑥 ),𝑝 ∘ 𝐴 (𝑦)} , 𝑝 ∘𝐵(⟨𝑥,𝑦⟩) = 0. 𝑖 𝑖 𝑖 2(𝑋) and each 𝑥∈𝑋 , Give two 𝑚 -polar fuzzy graphs (withunderlyingpairs + − [𝜑 ∘ 𝜓 (𝐴 )] (𝑥 )=𝜑(𝜇 ,𝜇 )(𝑥 ) 𝐴 𝐴 (𝑉 ,𝐸 )and (𝑉 ,𝐸 ),resp.) 𝐺 =(𝐴 ,𝐵 )and 𝐺 =(𝐴 ,𝐵 ). 1 1 2 2 1 1 1 2 2 2 (5) A homomorphism from 𝐺 to 𝐺 is a mapping 𝑓: 𝑉 →𝑉 + − 1 2 1 2 =⟨𝜇 (𝑥 ),−𝜇 (𝑥 )⟩=𝐴 (𝑥 ), 𝐴 𝐴 which satisfies 𝐴 (𝑥) ≤ 𝐴 (𝑓(𝑥)) (∀𝑥 ∈ 𝑉 )and 𝐵 (⟨𝑥, 𝑦⟩) ≤ 1 2 1 1 𝐵 (⟨𝑓(𝑥), 𝑓(𝑦)⟩) (∀⟨𝑥, 𝑦⟩ ∈ 𝐸 ). An isomorphism from 2 1 which means 𝜑∘𝜓()𝐴 = 𝐴 . 𝐺 to 𝐺 is a bijective mapping 𝑓: 𝑉 →𝑉 which 1 2 1 2 4 The Scientific World Journal satisfies 𝐴 (𝑥) = 𝐴 (𝑓(𝑥)) (∀𝑥 ∈ 𝑉 )and 𝐵 (⟨𝑥, 𝑦⟩) = set on 𝑄×𝑋×𝑄 .Moreover, if𝐵:𝑄 → [0,1] is an 𝑚 -polar 1 2 1 1 𝐵 (⟨𝑓(𝑥), 𝑓(𝑦)⟩) (∀⟨𝑥, 𝑦⟩ ∈ 𝐸 ). A weak isomorphism from fuzzy set on 𝑄 satisfying 2 1 𝐺 to 𝐺 is a bijective mapping 𝑓: 𝑉 →𝑉 which is a 1 2 1 2 homomorphism and satisfies 𝐴 (𝑥) = 𝐴 (𝑓(𝑥)) (∀𝑥 ∈ 𝑉 ). 1 2 1 𝐵(𝑞) ≥ inf {𝐵(),𝐴( 𝑝 ⟨,𝑥𝑝 ,𝑞⟩)} Astrong 𝑚 -polar fuzzy graph 𝐺 is called self-complementary (11) if 𝐺≃ 𝐺 (i.e., there exists an isomorphism between 𝐺 and its (∀ ⟨𝑝, 𝑥, 𝑞⟩ ∈ 𝑄 × 𝑋 × 𝑄) , complement 𝐺 ). It is not difficult to verify the following conclusions (some then 𝑀 = (,𝑄 ,𝑋 𝐴, 𝐵) is called an 𝑚 -polar subsystem of 𝑀 . of which generalize the corresponding results in [1, 45]). ∗ Furthermore, let 𝑋 be the set of all words of elements of 𝑋 of finite length and 𝜆 be the empty word in 𝑋 (cf. [28]). en Th (1) In a self-complementary strong 𝑚 -polar fuzzy graph ∗ ∗ 𝑚 one can define a 𝑚 -polar fuzzy set 𝐴 :𝑄×𝑋 ×𝑄 → [0, 1] 𝐺 = (𝐴, ) 𝐵 (with an underlying pair (𝑉, 𝐸) ), we have on 𝑄×𝑋 ×𝑄 by putting 𝑝 ∘ 𝐵 (⟨𝑥, 𝑦⟩) 1, if𝑞=𝑝, = inf {𝑝 ∘𝐴 (𝑥 ),𝑝 ∘ 𝐴 (𝑦)} − 𝑝 ∘𝐵(⟨𝑥,𝑦⟩) 𝐴 (⟨𝑞, 𝜆, 𝑝⟩) = { 𝑖 𝑖 𝑖 0, if 𝑞 =𝑝̸, (𝑖 ∈ ,𝑚 ⟨𝑥, 𝑦⟩ ∈ 𝐸) , ∗ ∗ ∗ 𝐴 (⟨𝑞, x,𝑥,𝑝⟩) = sup {𝐴 (⟨𝑞, 𝑥, 𝑟⟩) , 𝐴 (⟨𝑟, 𝑥, 𝑝⟩)} (9) 𝑟∈𝑄 ∑ 𝑝 ∘𝐵(⟨𝑥,𝑦⟩) 𝑥 =𝑦̸ (∀ ⟨𝑞, x,𝑥,𝑝⟩ ∈ 𝑄 × 𝑋 ×𝑋×𝑄), (12) = ∑ inf {𝑝 ∘𝐴 (𝑥 ),𝑝 ∘ 𝐴 (𝑦)} (𝑖∈𝑚 ). 𝑖 𝑖 𝑥 =𝑦̸ where 1 is the biggest element of [0, 1] . The following conclusions hold. (2) A strong 𝑚 -polar fuzzy graph 𝐺 = (𝐴, ) 𝐵 (with an underlying pair (𝑉, 𝐸) ) is self-complementary if and (1) An 𝑚 -polar fuzzy set 𝐴:𝐺 → [0,1] is an 𝑚 -polar only if it satisfies fuzzy subgroup of a group (𝐺, ∘) if and only if 𝐴 = [𝑎] {𝑥 ∈ 𝐺 | (𝑥) 𝐴 ≥ 𝑎} is 0 or 𝐴 is a subgroup of [𝑎] 𝑝 ∘𝐵(⟨𝑥,𝑦⟩) = inf {𝑝 ∘𝐴 (𝑥 ),𝑝 ∘𝐴(𝑦)} 𝑖 𝑖 𝑖 (𝐺, ∘) (∀𝑎 ∈ [0, 1] ). (10) (∀𝑖 ∈ 𝑚, ∀ ⟨𝑥, 𝑦⟩ ∈ 𝐸) . (2) An 𝑚 -polar fuzzy set 𝐴:𝐺 → [0,1] is an 𝑚 -polar fuzzy subalgebra of a 𝐾 -algebra (,∘𝐺 ,𝑒,⊙) if and only if𝐴 ={𝑥∈𝐺|𝐴(𝑥) ≥𝑎} is0 or𝐴 is a subalgebra (3) If 𝐺 and 𝐺 are strong 𝑚 -polar fuzzy graphs, then [𝑎] [𝑎] 1 2 of (𝐺, ∘, ,𝑒 ⊙) (∀𝑎 ∈ [0, 1] ). 𝐺 ≃𝐺 if and only if 𝐺 ≃ 𝐺 . 1 2 1 2 (4) Let 𝐺 and 𝐺 be strong 𝑚 -polar fuzzy graphs. If there (3) An 𝑚 -polar fuzzy set 𝐴:𝑋 → [0,1] is an 𝑚 - 1 2 is a weak isomorphism from 𝐺 to 𝐺 , then there is a polar fuzzy subincline (resp., an 𝑚 -polar fuzzy ideal, 1 2 weak isomorphism from 𝐺 to 𝐺 . an 𝑚 -polar fuzzy filter) of an incline (𝑋, +, ∗) if and 2 1 only if 𝐴 is a subincline (resp., ideal, filter) of [𝑎] Remark 8. The fuzzifications or bipolar fuzzifications of (𝑋, +, ∗) (∀𝑎 ∈ [0, 1] ). some algebraic concepts (such as group, 𝐾 -algebra, incline algebra (cf. [48]), ideal, filter, and finite state machine) can (4) Let 𝑀 = (,𝑋𝑄 ,𝐴) be an 𝑚 -polar fuzzy finite be generalized to the case of 𝑚 -polar fuzzy sets. An 𝑚 - state machine and 𝐵:𝑄 → [0,1] be an polar fuzzy set 𝐴:𝐺 → [0,1] is called an 𝑚 -polar 𝑚 -polar fuzzy set on 𝑄 .Then (,𝑋𝑄 ,𝐴,𝐵) is an −1 𝑚 -polar subsystem of 𝑀 if and only if 𝐵(𝑞) ≥ fuzzy subgroup of a group (𝐺, ∘) if it satisfies 𝐴(𝑥 ∘ 𝑦 )≥ ∗ ∗ inf{𝐵(𝑝), 𝐴 (⟨𝑝, x,𝑞⟩)} (∀⟨,𝑝 x,𝑞⟩ ∈ 𝑄 × 𝑋 ×𝑄) . inf{𝐴(𝑥), 𝐴(𝑦)} (∀𝑥, 𝑦 ∈ ) 𝐺 .An 𝑚 -polar fuzzy set 𝐴:𝐺 → Please see [49, 50] for more results. [0, 1] is called an 𝑚 -polar fuzzy subalgebra of a 𝐾 -algebra (,∘𝐺 ,𝑒,⊙) if it satisfies 𝐴(𝑥 ∘ 𝑦) ≥ inf{𝐴(𝑥), 𝐴(𝑦)} (∀𝑥, 𝑦 ∈ 𝐺) .An 𝑚 -polar fuzzy set 𝐴:𝑋 → [0,1] is called an Next we consider the applications of 𝑚 -polar fuzzy sets in 𝑚 -polar fuzzy subincline of an incline (𝑋, +, ∗) if it satisfies real world problems. (𝑥∗𝑦) ≥ inf{𝐴(𝑥), 𝐴(𝑦)} (∀𝑥, 𝑦 ∈ ) 𝐺 ;itiscalledan 𝑚 - polar fuzzy ideal (resp., an 𝑚 -polar fuzzy filter) of (𝑋, +, ∗) Example 9. Let 𝑋 be a set consisting of five patients 𝑥 , if it is an 𝑚 -polar fuzzy subincline of (𝑋, +, ∗) and satisfies 𝑦 , 𝑧 , 𝑢 ,and V (thus 𝑋 = {𝑥,𝑦,𝑧,𝑢, V}). They have diag- 𝐴(𝑥) ≥ (𝑦) 𝐴 whenever 𝑥≤𝑦 (resp., satisfies 𝐴(𝑥) ≤ (𝑦) 𝐴 nosis data consisting of three aspects, diagnosis datum of whenever 𝑥≤𝑦 ). An 𝑚 -polar fuzzy nfi itestate machineisa 𝑥 is (𝑥) = ⟨0.49, 0.46, 0.51⟩,where datum 0.5 represents triple 𝑀 = (,𝑋𝑄 ,𝐴) ,where 𝑄 and 𝑋 are ni fi te nonempty “normal” or “OK.” Suppose 𝐴(𝑦) = ⟨0.45, 0.42, 0.59⟩ , sets (called the set of states and the set of input symbols, 𝐴(𝑧) = ⟨0.50, 0.40, 0.54⟩ , 𝐴(𝑢) = ⟨0.40, 0.49, 0.60⟩ ,and resp.) and 𝐴:𝑄 × 𝑋 × 𝑄 → [0,1] is any 𝑚 -polar fuzzy 𝐴( V) = ⟨0.51, 0.52, 0.50⟩.Then we obtain a 3-polar fuzzy The Scientific World Journal 5 3 𝑛 set 𝐴:𝑋 → [0,1] which can describe the situation; this 𝐴:𝑋={𝑥 ,𝑥 ,...,𝑥 }→[0,1] be a mapping satisfying 1 2 𝑛 3-polar fuzzy set can also be written as follows: 𝑝 ∘𝐴(𝑥 )=𝐵(𝑥 )(if 𝑗=𝑖 )or 0 (otherwise) (𝑖 = 1,2,...,𝑛). 𝑗 𝑖 𝑖 Then a cooperative game model V ∘𝐴 : 𝑋 → 𝑅 is established, ⟨0.49, 0.46, 0.51⟩ ⟨0.45, 0.42, 0.59⟩ ⟨0.50, 0.40, 0.54⟩ where V : [0, 1] →𝑅 is defined by 𝐴={ , , , 𝑥 𝑦 𝑧 𝑛 𝑛 V (⟨𝑠 ,𝑠 ,...,𝑠 ⟩) =∑𝑔 (𝑠 )− 𝑘( ∑𝐵(𝑥 )) 1 2 𝑛 𝑖 𝑖 𝑖 ⟨0.40, 0.49, 0.60⟩ ⟨0.51, 0.52, 0.50⟩ , }. 𝑖=1 𝑖=1 (16) 𝑢 V (∀ ⟨𝑠 ,𝑠 ,...,𝑠 ⟩∈ [0, 1] ), 1 2 𝑛 (13) and the function 𝑔 : [0, 1] → 𝑅 is continuously monotonic Example 10. 𝑚 -polar fuzzy sets can be used in decision increasing with 𝑔 (0) = (0𝑖 = 1,2,...,𝑛) .Obviously,the gain making. In many decision making situations, it is necessary of agent 𝑥 (with participation level 𝐵(𝑥 ))is 𝑖 𝑖 to gather the group consensus. This happens when a group of friends decides which movie to watch, when a company V ∘𝐴(𝑥 )=𝑔 ∘𝐵(𝑥 )− 𝑘( ∑𝐵(𝑥 )) , (17) 𝑖 𝑖 𝑖 𝑖 decides which product design to manufacture, and when a 𝑛 𝑖=1 democratic country elects its leaders. For instance, we con- and the total gain is sider here only the case of election. Let 𝑋 = {𝑥,𝑦,𝑧,...,𝑢, V} be the set of voters and 𝐶={𝑐 ,𝑐 ,𝑐 ,𝑐 } be the set of all 1 2 3 4 𝑛 𝑛 𝑛 the four candidates. Suppose the voting is weighted. For each ∑V ∘𝐴(𝑥 )=∑𝑔 ∘𝐵(𝑥 )−𝑘( ∑𝐵(𝑥 )) . (18) 𝑖 𝑖 𝑖 𝑖 candidate𝑐∈𝐶 ,avoterin {𝑥, 𝑦, }𝑧 cansendavaluein [0, 1] to 𝑖=1 𝑖=1 𝑖=1 𝑐 ,but avoter in𝑋−{𝑥,𝑦,𝑧} can only send a value in [0.1, 0.8] (2) er Th e are two goods, denoted 𝑔 and 𝑔 ,and three 1 2 to 𝑐 .Suppose 𝐴(𝑥) = ⟨0.9, 0.4, 0.01, 0.1⟩ (which means the agents 𝑎 , 𝑏 ,and 𝑐 with endowments (𝜀, 𝜀), (1 − 𝜀, 0),and preference degrees of 𝑥 corresponding to 𝑐 ,𝑐 ,𝑐 ,and 𝑐 1 2 3 4 2 (0, 1 − 𝜀) (0 < 𝜀 ≤ 1).Let V : [0, 1] →𝑅 be any mapping are 0.9, 0.4, 0.01,and 0.1,resp.), 𝐴(𝑦) = ⟨0.2, 0.3, 0.8, 0.1⟩ , satisfying V(⟨0, 0⟩) =. 0en Th the corresponding cooperative 𝐴(𝑧) = ⟨0.8, 0.9, 0.8, 0.2⟩, . . . , (𝑢) 𝐴 = ⟨0.6, 0.8, 0.8, 0.1⟩ ,and game model is V ∘ 𝐴 : 𝑋 = {𝑎, 𝑏, 𝑐} → 𝑅 ,where 𝐴( V) = ⟨0.7, 0.8, 0.4, 0.2⟩.Then we obtain a 4-polar fuzzy set 𝐴:𝑋 → [0,1] which can describe the situation; this 4- ⟨𝜀, 𝜀 ⟩ ⟨1−𝜀,0 ⟩ ⟨0, 1 − 𝜀 ⟩ 𝐴={ , , }, polar fuzzy set can also be written as follows: 𝑎 𝑏 𝑐 (19) V 𝜀, 𝜀 V 1−𝜀,0 V 0, 1 − 𝜀 (⟨ ⟩) (⟨ ⟩) (⟨ ⟩) ⟨0.9, 0.4, 0.01, 0.1⟩ ⟨0.2, 0.3, 0.8, 0.1⟩ V ∘𝐴 = { , , }. 𝐴={ , , 𝑎 𝑏 𝑐 𝑥 𝑦 Example 12. 𝑚 -polar fuzzy sets can be used to den fi e ⟨0.8, 0.9, 0.8, 0.2⟩ ⟨0.6, 0.8, 0.8, 0.1⟩ ,..., , (14) weighted games. A weighted game is a 4-tuple (𝑋, P,𝑊,Δ) , 𝑧 𝑢 where 𝑋={𝑥 ,𝑥 ,...,𝑥 } is the set of 𝑛 players or voters 1 2 𝑛 (𝑛≥2 ), P is a collection of fuzzy sets on 𝑋 (called coalitions) ⟨0.7, 0.8, 0.4, 0.2⟩ }. such that (P,≤)is upper set (i.e., a fuzzy set 𝑄 on 𝑋 belongs to P if𝑄≥𝑃 for some𝑃∈ P), 𝑊: 𝑋 → [0,1] is an 𝑚 -polar fuzzy set on 𝑋 (called voting weights), andΔ⊆[0,+∞) −{0} Example 11. 𝑚 -polar fuzzy sets canbeusedincooperative games (cf. [51]). Let𝑋={𝑥 ,𝑥 ,...,𝑥 } be the set of 𝑛 agents (called quotas). Imagine a situation: three people, 𝑥 , 𝑦 ,and 1 2 𝑛 or players (𝑛≥1 ), 𝑚 = {0,1,...,𝑚 − 1} be the set of the 𝑧 ,votefor aproposalonreleasing of astudent. Supposethat grand coalitions, and 𝐴:𝑋 → [0,1] be an 𝑚 -polar fuzzy 𝑥 casts 200 US Dollars and lose 80 hairs on her head votes set, where 𝑝 ∘ (𝑥) 𝐴 is the degree of player 𝑥 participating in each, 𝑦 casts 60000 US Dollars and 100 grams Cordyceps coalition 𝑖 (𝑥∈𝑋,𝑖∈𝑚 ). Again let V : [0, 1] →𝑅 (the set of sinensis votes each, 𝑧 casts 100000 US Dollars and 100 grams allrealnumbers)beamappingsatisfying V(0)=0.Then the gold votes each. Then an associated weighted game model is mapping V ∘𝐴 : 𝑋 → 𝑅 is called a cooperative game, where (𝑋, P,𝑊,Δ) ,where 𝑋={𝑥,𝑦,}𝑧 and P is a collection of fuzzy sets on 𝑋 with (P,≤)an upper set, 𝑚=4 , V ∘ (𝑥) 𝐴 represents the amount of money obtained by player 𝑥 under the coalition participating ability 𝐴(𝑥) (𝑥∈𝑋 ). ⟨200/160200, 80/80, 0, 0⟩ (1) (a public good game; compare with [51,Example 6.5]) 𝑊={ , Suppose 𝑛 agents 𝑥 ,𝑥 ,...,𝑥 want to create a facility for 1 2 𝑛 joint use. eTh cost of the facility depends on the sum of ⟨60000/160200, 0, 100/100, 0⟩ the participation levels (or degrees) of the agents and it is described by (20) ⟨100000/160200, 0, 0, 100/100⟩ }, 𝑘( ∑𝐵(𝑥 )) , (15) 𝑖=1 Δ={⟨200000, 0, 0, 0⟩, ⟨100000, 300, 0, 0⟩, where 𝑘:[0,𝑛] → 𝑅 is a continuous monotonic increasing function with 𝑘(0) = 0and 𝐵:𝑋 → [0,1] is a mapping. Let ⟨0, 0, 500, 0⟩, ⟨0, 0, 0, 65000⟩}. 6 The Scientific World Journal (1) If the situation is a little simple, 𝑥 casts [100, 300] US 3. Conclusion Dollars (i.e., the cast is between 100 US Dollars and In this note, we show that the enthusiastically studied notion 300 US Dollars, where [100, 300] is an interval num- of bipolarfuzzy setisactuallyasynonymofa [0, 1] -set (we ber which can be looked as a point [0, +∞))votes call it 2-polar fuzzy set), and thus we highlight the notion of each, 𝑦 casts [50000, 70000] US Dollars votes each, 𝑧 𝑚 -polar fuzzy set (actually a [0, 1] -set,𝑚≥2 ). The 𝑚 -polar casts [90000, 110000] US Dollars votes each, and quota is fuzzy sets not only have real backgrounds (e.g., “multipolar [100000, 120000]. eTh n the corresponding weighted game information” exists) but also have applications in both theory model is (𝑋, P,,𝑊 [100000, 120000]) ,where P is a collection and real world problems (which have been illustrated by of fuzzy sets on 𝑋 with (P,≤)an upper set, 𝑚=1 ,and examples). 400/320400 120000/320400 200000/320400 𝑊={ , , }. 𝑥 𝑦 𝑧 Conflict of Interests (21) eTh authors declare that there is no conflict of interests (2) If the situation is more simple, 𝑥 casts 200 US Dollars regarding the publication of this paper. votes each, 𝑦 casts 60000 US Dollars votes each, 𝑧 casts 100000 US Dollars votes each, and quota is 110000.Then the corresponding weighted game model is (𝑋, P,,𝑊 110000) , Acknowledgments where P = {{𝑥, },𝑧 {𝑦, },𝑧 {𝑥, 𝑦, }} 𝑧 , 𝑚=1 ,and This work was supported by the International Science and 200/160200 60000/160200 100000/160200 Technology Cooperation Foundation of China (Grant no. 𝑊={ , , }. 𝑥 𝑦 𝑧 2012DFA11270) and the National Natural Science Foundation (22) of China (Grant no. 11071151). Notice that the subset {𝑥, }𝑧 ⊆ 𝑋 is exactly a fuzzy set 𝐴: References 𝑋→[0,1] on 𝑋 defined by 𝐴(𝑥) = (𝑧) 𝐴 = 1 and 𝐴(𝑦) = 0 . [1] L. A. Zadeh, “Fuzzy sets,” Information and Control,vol.8,no. 3, Example 13. 𝑚 -polar fuzzy sets can be used as a model for pp. 338–353, 1965. clustering or classification. Consider a set 𝑋 consisting of 𝑛 [2] U. Hoh ¨ le and S. E. Rodabaugh, Eds., Mathematics of Fuzzy Sets: students 𝑥 ,𝑥 ,...,𝑥 (𝑛 ≥ 2)in Chinese middle school. 1 2 𝑛 Logic, Topology, and Measure eTh ory , eTh Handbooks of Fuzzy For a student 𝑥∈𝑋 , we use integers 𝑥 (resp., 𝑥 ,...,𝑥 ) 1 2 6 Sets Series, Kluwer Academic, Dordrecht, eTh Netherlands, in [0, 100] to denote the average score of Mathematics (resp., Physics, Chemistry, Biology, Chinese, and English), and [3] W.R.Zhang,“Bipolarfuzzy sets andrelations:acomputational framework for cognitive modeling and multiagent decision 𝐴 (𝑥 )=⟨𝑥 × 0.01, 𝑥 × 0.01, 𝑥 × 0.01, 1 2 3 analysis,” in Proceedings of the Industrial Fuzzy Control and (23) Intelligent Systems Conference, and the NASA Joint Technology 𝑥 × 0.01, 𝑥 × 0.01, 𝑥 × 0.01⟩ . 4 5 6 Workshop on Neural Networks and Fuzzy Logic and Fuzzy Information Processing Society Biannual Conference,pp. 305– eTh nweobtaina 6-polar fuzzy set model 𝐴:𝑋 → [0,1] , 309, SanAntonio,Tex,USA,December1994. which can be used for clustering or classification of these [4] M.Akram,“Bipolarfuzzygraphs,” Information Sciences,vol.181, students. no. 24, pp. 5548–5564, 2011. Example 14. 𝑚 -polar fuzzy sets can be used to den fi e multi- [5] M. Akram, “Bipolar fuzzy graphs with applications,” Knowledge- valued relations. Based Systems,vol.39, pp.1–8,2013. [6] M. Akram, W. Chen, and Y. Yin, “Bipolar fuzzy Lie superalge- (1) Consider a set 𝑋 consisting of 𝑛 net users (resp., bras,” Quasigroups and Related Systems,vol.20, no.2,pp. 139– patients) 𝑥 ,𝑥 ,...,𝑥 (𝑛 ≥ 2). For net users (resp., 1 2 𝑛 156, 2012. patients) 𝑥, 𝑦 ∈ 𝑋 ,weuse (𝑥,𝑦,𝑗) to denote the [7] M.Akram,S.G.Li, andK.P.Shum, “Antipodal bipolarfuzzy similarity between 𝑥 and 𝑦 in 𝑗 th aspect (1≤ graphs,” Italian Journal of Pure and Applied Mathematics,vol. 𝑗≤𝑚, 𝑚≥2 ), and let 𝐴(𝑥, 𝑦) = (𝑦, 𝐴 𝑥) = 31, pp. 425–438, 2013. ⟨(𝑥,𝑦,1),(𝑥,𝑦,2),...,(𝑥,𝑦,𝑚)⟩ .Then we obtain an [8] M.Akram,A.B.Saeid, K. P. Shum, andB.L.Meng, “Bipolar 𝑚 -polar fuzzy set 𝐴:𝑋 → [0,1] ,which is a fuzzy K-algebras,” International Journal of Fuzzy Systems,vol. multivalued similarity relation. 12, no. 3, pp. 252–259, 2010. (2) Consider a set 𝑋 consisting of 𝑛 people 𝑥 ,𝑥 , 1 2 [9] L. Amgoud, C. Cayrol, M. C. Lagasquie-Schiex, and P. Livet, ...,𝑥 (𝑛≥2) in a social network. For 𝑥, 𝑦 ∈ 𝑛 “On bipolarity in argumentation frameworks,” International 𝑋 ,weuse (𝑥,𝑦,𝑗) to denote the degree of connec- Journal of Intelligent Systems,vol.23,no.10,pp.1062–1093,2008. tion between 𝑥 and 𝑦 in 𝑗 th aspect (1≤𝑗≤ [10] H. Y. Ban, M. J. Kim, and Y. J. Park, “Bipolar fuzzy ideals with 𝑚, 𝑚 ≥ 2 ), and let 𝐴(𝑥, 𝑦) = (𝑦, 𝐴 𝑥) = ⟨(𝑥, 𝑦, 1) , operators in semigroups,” Annals of Fuzzy Mathematics and (𝑥,𝑦,2),...,(𝑥,𝑦,𝑚)⟩ .Then we obtain an 𝑚 -polar Informatics,vol.4,no. 2, pp.253–265,2012. fuzzy set 𝐴:𝑋 → [0,1] ,which is amultivalued [11] S. Benferhat, D. Dubois, S. Kaci, and H. Prade, “Bipolar possi- social graph (or multivalued social network) model. bility theory in preference modeling: representation, fusion and The Scientific World Journal 7 optimal solutions,” Information Fusion,vol.7,no. 1, pp.135–150, [29] Y. B. Jun, H. S. Kim, and K. J. Lee, “Bipolar fuzzy translation 2006. in BCK/BCI-algebra,” Journal of the Chungcheong Mathematical Society,vol.22, no.3,pp. 399–408, 2009. [12] S. Bhattacharya and S. Roy, “Study on bipolar fuzzy-rough control theory,” International Mathematical Forum,vol.7,no. [30] Y. B. Jun and C. H. Park, “Filters of BCH-algebras based on 41, pp. 2019–2025, 2012. bipolar-valued fuzzy sets,” International Mathematical Forum, vol. 4, no.13, pp.631–643,2009. [13] I. Bloch, “Dilation and erosion of spatial bipolar fuzzy sets,” in Applications of Fuzzy Sets eo Th ry ,F.Masulli, S. Mitra, andG. [31] S. Kaci, “Logical formalisms for representing bipolar prefer- Pasi, Eds., vol. 4578 of Lecture Notes in Computer Science,pp. ences,” International Journal of Intelligent Systems,vol.23, no. 385–393, Springer, Berlin, Germany, 2007. 9, pp. 985–997, 2008. [14] I. Bloch, “Bipolar fuzzy spatial information: geometry, mor- [32] K. J. Lee, “Bipolar fuzzy subalgebras and bipolar fuzzy ideals of BCK/BCI-algebras,” Bulletin of the Malaysian Mathematical phology, spatial reasoning,” in Methods for Handling Imperfect Spatial Information, R. Jeansoulin, O. Papini, H. Prade, and Sciences Society,vol.32, no.3,pp. 361–373, 2009. S. Schockaert, Eds., vol. 256 of Studies in Fuzziness and Soft [33] K. J. Lee and Y. B. Jun, “Bipolar fuzzy a-ideals of BCI-algebras,” Computing, pp. 75–102, Springer, Berlin, Germany, 2010. Communications of the Korean Mathematical Society,vol.26,no. [15] I. Bloch, “Lattices of fuzzy sets and bipolar fuzzy sets, and math- 4, pp. 531–542, 2011. ematical morphology,” Information Sciences,vol.181,no. 10,pp. [34] K. M. Lee, “Comparison of interval-valued fuzzy sets, intuition- 2002–2015, 2011. istic fuzzy sets, and bipolar-valued fuzzy sets,” Journal of Fuzzy Logic Intelligent Systems,vol.14, no.2,pp. 125–129, 2004. [16] I. Bloch, “Mathematical morphology on bipolar fuzzy sets: gen- eral algebraic framework,” International Journal of Approximate [35] R. Muthuraj and M. Sridharan, “Bipolar anti fuzzy HX group Reasoning,vol.53, no.7,pp. 1031–1060, 2012. and its lower level sub HX groups,” JournalofPhysicalSciences, vol. 16, pp. 157–169, 2012. [17] I. Bloch and J. Atif, “Distance to bipolar information from morphological dilation,” in Proceedings of the 8th Conference of [36] S. Narayanamoorthy and A. Tamilselvi, “Bipolar fuzzy line theEuropeanSociety forFuzzy Logicand Technology,pp. 266– graph of a bipolar fuzzy hypergraph,” Cybernetics and Informa- 273, 2013. tion Technologies,vol.13, no.1,pp. 13–17, 2013. [18] J. F. Bonnefon, “Two routes for bipolar information processing, [37] E. Raufaste and S. Vautier, “An evolutionist approach to and a blind spot in between,” International Journal of Intelligent information bipolarity: representations and aeff cts in human Systems,vol.23, no.9,pp. 923–929, 2008. cognition,” International Journal of Intelligent Systems,vol.23, no. 8, pp. 878–897, 2008. [19] P. Bosc and O. Pivert, “On a fuzzy bipolar relational algebra,” Information Sciences,vol.219,pp. 1–16,2013. [38] A. B. Saeid, “BM-algebras defined by bipolar-valued sets,” Indian Journal of Science and Technology,vol.5,no. 2, pp.2071– [20] D. Dubois, S. Kaci, and H. Prade, “Bipolarity in reasoning 2078, 2012. and decision, an introduction,” in Proceedings of the Interna- tional Conference on Information Processing and Management of [39] S. Samanta and M. Pal, “Irregular bipolar fuzzy graphs,” Inter- Uncertainty, pp. 959–966, 2004. national Journal of Applications of Fuzzy Sets,vol.2,no. 2, pp. 91–102, 2012. [21] D. Dubois and H. Prade, “An overview of the asymmetric bipolar representation of positive and negative information in [40] H. L. Yang,S.G.Li, Z. L. Guo, andC.H.Ma, “Transformationof possibility theory,” Fuzzy Sets and Systems,vol.160,no. 10,pp. bipolar fuzzy rough set models,” Knowledge-Based Systems,vol. 1355–1366, 2009. 27, pp. 60–68, 2012. [22] U. Dudziak and B. Pe¸kala, “Equivalent bipolar fuzzy relations,” [41] H. L. Yang,S.G.Li, S. Y. Wang,and J. Wang,“Bipolarfuzzy Fuzzy Sets and Systems,vol.161,no. 2, pp.234–253,2010. rough set model on two different universes and its application,” Knowledge-Based Systems,vol.35, pp.94–101, 2012. [23] H. Fargier and N. Wilson, “Algebraic structures for bipo- lar constraint-based reasoning,” in Symbolic and Quantitative [42] W. R. Zhang, “Equilibrium relations and bipolar fuzzy cluster- Approaches to Reasoning with Uncertainty,vol.4724of Lecture ing,” in Proceedings of the 18th International Conference of the Notes in Computer Science, pp. 623–634, Springer, Berlin, North American Fuzzy Information Processing Society (NAFIPS Germany, 2007. '99), pp. 361–365, June 1999. [24] M. Grabisch, S. Greco, and M. Pirlot, “Bipolar and bivari- [43] W. R. Zhang, Ed., YinYang Bipolar Relativity: A Unifying eTh ory ate models in multicriteria decision analysis: descriptive and of Nature, Agents and Causality with Applications in Quantum constructive approaches,” International Journal of Intelligent Computing, Cognitive Informatics and Life Sciences,IGI Global, Systems,vol.23, no.9,pp. 930–969, 2008. 2011. [25] M. M. Hasankhani and A. B. Saeid, “Hyper MV-algebras [44] W. R. Zhang, “Bipolar quantum logic gates and quantum defined by bipolar-valued fuzzy sets,” Annals of West University cellular combinatorics—a logical extension to quantum entan- of Timisoara-Mathematics,vol.50, no.1,pp. 39–50, 2012. glement,” Journal of Quantum Information Science,vol.3,no. 2, pp. 93–105, 2013. [26] C. Hudelot, J. Atif, and I. Bloch, “Integrating bipolar fuzzy mathematical morphology in description logics for spatial [45] H. L. Yang,S.G.Li, W. H. Yang,and Y. Lu,“Noteson‘bipolar reasoning,” Frontiers in Articia fi l Intelligence and Applications , fuzzy graphs’,” Information Sciences,vol.242,pp. 113–121,2013. vol. 215, pp. 497–502, 2010. [46] A. Rosenfeld, “Fuzzy graphs,” in Fuzzy Sets and eTh ir Applica- [27] Y. B. Jun, M. S. Kang, and H. S. Kim, “Bipolar fuzzy hyper BCK- tions to Cognitive and Decision Process,L.A.Zadeh,K.S.Fu, ideals in hyper BCK-algebras,” Iranian Journal of Fuzzy Systems, and M. Shimura, Eds., pp. 77–95, Academic Press, New York, vol. 8, no.2,pp. 105–120, 2011. NY, USA, 1975. [28] Y. B. Jun and J. Kavikumar, “Bipolar fuzzy finite state machines,” [47] R. T. Yeh and S. Y. Bang, “Fuzzy relations, fuzzy graphs and their Bulletin of the Malaysian Mathematical Sciences Society,vol.34, application to clustering analysis,” in Fuzzy Sets and eTh ir Appli- no. 1, pp. 181–188, 2011. cations to Cognitive and Decision Process,L.A.Zadeh,K.S.Fu, 8 The Scientific World Journal and M. Shimura, Eds., pp. 338–353, Academic Press, New York, NY, USA, 1975. [48] Z. Q. Cao, K. H. Kim, and F. W. Roush, Incline Algebra and Applications, Ellis Horwood Series in Mathematics and Its Applications, Halsted Press, Chichester, UK; John Wiley & Sons, New York, NY, USA, 1984. [49] Y. M. Li, “Finite automata theory with membership values in lattices,” Information Sciences,vol.181,no. 5, pp.1003–1017,2011. [50] J. H. Jin, Q. G. Li, and Y. M. Li, “Algebraic properties of L- fuzzy finite automata,” Information Sciences,vol.234,pp. 182– 202, 2013. [51] L. J. Xie and M. Grabisch, “The core of bicapacities and bipolar games,” Fuzzy Sets and Systems,vol.158,no. 9, pp.1000–1012,

Journal

The Scientific World JournalPubmed Central

Published: Jun 12, 2014

There are no references for this article.