Deschrijver, Glad

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Ghent University Academic Bibliography

Additive generators in interval-valued and intuitionistic fuzzyset theoryGlad DeschrijverFuzziness and Uncertainty Modeling Research UnitDepartment of Mathematics and Computer Science, Ghent UniversityKrijgslaan 281 (S9), B–9000 Gent, BelgiumE-mail: Glad.Deschrijver@UGent.beHomepage: http://www.fuzzy.UGent.beAbstractIntuitionistic fuzzy sets in the sense of Atanassov and interval-valued fuzzy sets canbe seen as L-fuzzy sets w.r.t. a special lattice LI . Deschrijver [2] introduced additiveand multiplicative generators on LI based on a special kind of addition introduced in[3]. Actually, many other additions can be introduced. In this paper we investigateadditive generators on LI as far as possible independently of the addition. For somespecial additions we investigate which t-norms can be generated by continuous additivegenerators which are a natural extension of an additive generator on the unit interval.Keywords: intuitionistic fuzzy set, interval-valued fuzzy set, additive generator, additionon LI , representable1 IntroductionTriangular norms on ([0, 1],≤) were introduced in [18] and play an important role in fuzzyset theory (see e.g. [12, 14] for more details). Generators are very useful in the constructionof t-norms: any generator on ([0, 1],≤) can be used to generate a t-norm. Generators playalso an important role in the representation of continuous Archimedean t-norms on ([0, 1],≤).Moreover, some properties of t-norms which have a generator can be related to properties oftheir generator. See e.g. [9, 13, 14, 15, 16] for more information about generators on the unitinterval.Interval-valued fuzzy set theory [11, 17] is an extension of fuzzy theory in which to eachelement of the universe a closed subinterval of the unit interval is assigned which approximatesthe unknown membership degree. Another extension of fuzzy set theory is intuitionistic fuzzyset theory introduced by Atanassov [1]. In [6] it is shown that intuitionistic fuzzy set theoryis equivalent to interval-valued fuzzy set theory and that both are equivalent to L-fuzzy settheory in the sense of Goguen [10] w.r.t. a special lattice LI . In [2] additive and multiplicativegenerators on LI are investigated based on a special kind of addition introduced in [3]. In [8]another addition was defined. In fact, many more additions can be introduced. Therefore, inthis paper we will investigate additive generators on LI as far as possible independently ofthe addition. For some special additions we will investigate which t-norms can be generated1by continuous additive generators which are a natural extension of an additive generator onthe unit interval.2 The lattice LIDefinition 2.1 We define LI = (LI ,≤LI ), whereLI = {[x1, x2] | (x1, x2) ∈ [0, 1]2 and x1 ≤ x2},[x1, x2] ≤LI [y1, y2] ⇐⇒ (x1 ≤ y1 and x2 ≤ y2), for all [x1, x2], [y1, y2] in LI .Similarly as Lemma 2.1 in [6] it can be shown that LI is a complete lattice.Definition 2.2 [11, 17] An interval-valued fuzzy set on U is a mapping A : U → LI .Definition 2.3 [1] An intuitionistic fuzzy set on U is a setA = {(u, µA(u), νA(u)) | u ∈ U},where µA(u) ∈ [0, 1] denotes the membership degree and νA(u) ∈ [0, 1] the non-membershipdegree of u in A and where for all u ∈ U , µA(u) + νA(u) ≤ 1.An intuitionistic fuzzy set A on U can be represented by the LI -fuzzy set A given byA : U → LI :u 7→ [µA(u), 1− νA(u)],In Figure 1 the set LI is shown. Note that to each element x = [x1, x2] of LI correspondsa point (x1, x2) ∈ R2.[0, 0][1, 1][0, 1]x1x2x = [x1, x2]x1x2Figure 1: The grey area is LI .In the sequel, if x ∈ LI , then we denote its bounds by x1 and x2, i.e. x = [x1, x2]. Thesmallest and the largest element of LI are given by 0LI = [0, 0] and 1LI = [1, 1]. We define2the relation LI by x LI y ⇐⇒ (x1 < y1 and x2 < y2), for x, y in LI . We define forfurther usage the setsD = {[x1, x1] | x1 ∈ [0, 1]};L¯I = {[x1, x2] | (x1, x2) ∈ R2 and x1 ≤ x2};D¯ = {[x1, x1] | x1 ∈ R};L¯I+ = {[x1, x2] | (x1, x2) ∈ [0, +∞[2 and x1 ≤ x2};D¯+ = {[x1, x1] | x1 ∈ [0, +∞[};L¯I∞,+ = {[x1, x2] | (x1, x2) ∈ [0, +∞]2 and x1 ≤ x2};D¯∞,+ = {[x1, x1] | x1 ∈ [0, +∞]}.Note that for any non-empty subset A of LI it holds thatsup A = [sup{x1 | x1 ∈ [0, 1] and (∃x2 ∈ [x1, 1])([x1, x2] ∈ A)},sup{x2 | x2 ∈ [0, 1] and (∃x1 ∈ [0, x2])([x1, x2] ∈ A)}].Definition 2.4 [4] A t-norm on LI is a commutative, associative, increasing mapping T :(LI)2 → LI which satisfies T (1LI , x) = x, for all x ∈ LI . A t-conorm on LI is a commutative,associative, increasing mapping S : (LI)2 → LI which satisfies S(0LI , x) = x, for all x ∈ LI .In [4, 5, 7] the following classes of t-norms on LI are introduced: let T and T ′ be t-normson ([0, 1],≤), then the mappings TT,T ′ , TT , TT,t and T′T given by, for all x, y in LI ,TT,T ′(x, y) = [T (x1, y1), T′(x2, y2)], (t-representable t-norms)TT (x, y) = [T (x1, y1), max(T (x1, y2), T (x2, y1))], (pseudo-t-representable t-norms)TT,t(x, y) = [T (x1, y1), max(T (t, T (x2, y2)), T (x1, y2), T (x2, y1))],T ′T (x, y) = [min(T (x1, y2), T (x2, y1)), T (x2, y2)],are t-norms on LI . The corresponding classes of t-conorms are given by, for all x, y in LI ,SS,S′(x, y) = [S(x1, y1), S′(x2, y2)], (t-representable t-conorms)SS(x, y) = [min(S(x1, y2), S(x2, y1)), S(x2, y2)], (pseudo-t-representable t-conorms)SS,t(x, y) = [min(S(1− t, S(x1, y1)), S(x1, y2), S(x2, y1)), S(x2, y2)],S ′S(x, y) = [S(x1, y1), max(S(x1, y2), S(x2, y1))].If for a mapping f on [0, 1] and a mapping F on LI it holds that F (D) ⊆ D¯, andF ([a, a]) = [f(a), f(a)], for all a ∈ LI , then we say that F is a natural extension of f to LI .E.g. TT,T , TT , TT,t and T′T are all natural extensions of T to LI .3 Additive generators on LIIn order to investigate additive generators on LI , a suitable addition on L¯I is needed. Weassume from now on that ⊕ : (L¯I ∪ L¯I∞,+)2 → L¯I satisfies the following natural properties,for all a, b in L¯I ∪ L¯I∞,+,(i) ⊕ is commutative,3(ii) ⊕ is associative,(iii) ⊕ is increasing,(iv) 0LI ⊕ a = a,Note that from (iii) and (iv) it follows that a ⊕ b ≥LI a, if b ≥LI 0LI , for all a, b inL¯I ∪ L¯I∞,+.Definition 3.1 Let f : LI → L¯I∞,+ be a strictly decreasing function. The pseudo-inversef(−1) : L¯I∞,+ → LI of f is defined by, for all y ∈ L¯I∞,+,f(−1)(y) =sup{x | x ∈ LI and f(x) LI y}, if y LI f(0LI );sup({0LI} ∪ {x | x ∈ LI and (f(x))1 > y1and (f(x))2 ≥ (f(0LI ))2}), if y2 ≥ (f(0LI ))2;sup({0LI} ∪ {x | x ∈ LI and (f(x))2 > y2and (f(x))1 ≥ (f(0LI ))1}), if y1 ≥ (f(0LI ))1.Definition 3.2 A mapping f : LI → L¯I∞,+ satisfying the following conditions:(AG.1) f is strictly decreasing;(AG.2) f(1LI ) = 0LI ;(AG.3) f is right-continuous in 0LI ;(AG.4) f(x)⊕ f(y) ∈ R(f), for all x, y in LI , whereR(f) = rng(f) ∪ {x | x ∈ L¯I∞,+ and [x1, (f(0LI ))2] ∈ rng(f) and x2 ≥ (f(0LI ))2}∪ {x | x ∈ L¯I∞,+ and [(f(0LI ))1, x2] ∈ rng(f) and x1 ≥ (f(0LI ))1}∪ {x | x ∈ L¯I∞,+ and x ≥LI f(0LI )};(AG.5) f(−1)(f(x)) = x, for all x ∈ LI ;is called an additive generator on LI .Theorem 3.1 Let f be an additive generator on ([0, 1],≤) and f : LI → L¯I∞,+ the mappingdefined by, for all x ∈ LI ,f(x) = [f(x2), f(x1)].Then, for all y ∈ L¯I∞,+,f(−1)(y) = [f (−1)(y2), f(−1)(y1)]. (1)Lemma 3.2 Let f : LI → L¯I∞,+ be a mapping satisfying (AG.1), (AG.2), (AG.3) and (AG.5).Then, for all x ∈ LI such that x1 > 0, it holds that (f(x))2 < (f(0LI ))2 and (f(x))1 < (f(0LI ))1.Lemma 3.3 Let f : LI → L¯I∞,+ be a mapping satisfying (AG.1), (AG.2), (AG.3) and (AG.5).Then (f([0, 1]))1 = (f(0LI ))1 or (f([0, 1]))2 = (f(0LI ))2.Corollary 3.4 Let f : LI → L¯I∞,+ be a mapping satisfying (AG.1), (AG.2), (AG.3), (AG.5)and f(D) ⊆ D¯∞,+. Then (f([0, 1]))2 = (f(0LI ))2.4Theorem 3.5 Let f : LI → L¯I∞,+ be a continuous mapping satisfying (AG.1), (AG.2),(AG.3), (AG.5) and f(D) ⊆ D¯∞,+. Then there exists an additive generator f on ([0, 1],≤)such that, for all a ∈ L¯I∞,+,f(−1)(a) = [f (−1)(a2), f(−1)(a1)].The following theorem can be shown independently of the addition ⊕ used in (AG.4).Theorem 3.6 A mapping f : LI → L¯I∞,+ is a continuous additive generator on LI such thatf(D) ⊆ D¯∞,+ if and only if there exists a continuous additive generator f on ([0, 1],≤) suchthat, for all a ∈ LI ,f(a) = [f(a2), f(a1)]. (2)4 Additive generators and t-norms on LIWe will give a sufficient condition for ⊕ under which an additive generator associated to ⊕generates a t-norm. First we give a lemma.Lemma 4.1 Let f be an additive generator on LI associated to ⊕. If, for all x, y, a in L¯I+such that x ≤LI a and y ≤LI a⊕ a,y2 ≥ a2 =⇒((x⊕ y)1 = (x⊕ [y1, a2])1 or min((x⊕ y)1, (x⊕ [y1, a2])1) ≥ a1)(3)andy1 ≥ a1 =⇒((x⊕ y)2 = (x⊕ [a1, y2])2 or min((x⊕ y)2, (x⊕ [a1, y2])2) ≥ a2)(4)then, for all x ∈ LI and y ∈ R(f), we have that f(x)⊕ f(f(−1)(y)) ∈ R(f) andf(−1)(f(x)⊕ f(f(−1)(y))) = f(−1)(f(x)⊕ y).Using Lemma 4.1, the following theorem can be shown.Theorem 4.2 Let f be an additive generator on LI associated to ⊕. If (3) and (4) hold, forall x, y, a in LI such that x ≤LI a and y ≤LI a⊕ a, then the mapping T : (LI)2 → LI definedby, for all x, y in LI ,T (x, y) = f(−1)(f(x)⊕ f(y)),is a t-norm on LI .Theorem 4.3 Let f be an additive generator on ([0, 1],≤). Then the mapping f : LI → L¯I∞,+defined by, for all x ∈ LI ,f(x) = [f(x2), f(x1)],is an additive generator on LI associated to ⊕ if and only if, for all x, y in LI ,f(x)⊕ f(y) ∈ (rng(f) ∪ [f(0), +∞])2.Theorem 3.6 and Theorem 3.1 show that no matter which operation ⊕ is used in (AG.4),a continuous additive generator f on LI satisfying f(D) ⊆ D¯∞,+ satisfies (2) and its pseudo-inverse satisfies (1). Therefore it depends on the operation ⊕ which classes of t-norms on LIcan have continuous additive generators that are a natural extension of additive generatorson ([0, 1],≤).54.1 Additive generators based on ⊕LIStarting from the observation that the  Lukasiewicz t-conorm SW is given by SW (x, y) =min(1, x + y), for all x, y in [0, 1], and that the pseudo-t-representable t-conorm SSW is givenby SSW (x, y) = [min(1, x1 + y2, x2 + y1), min(1, x2 + y2)], for all x, y in LI , the followingdefinition of addition on L¯I is introduced in such a way that SSW (x, y) = inf(1LI , x ⊕LI y),for all x, y in LI .Definition 4.1 [3] We define the addition on L¯I ∪ L¯I∞,+ by, for all x, y in L¯I ∪ L¯I∞,+,x⊕LI y = [min(x1 + y2, x2 + y1), x2 + y2],where, for all x ∈ R, x +∞ = +∞ and +∞+∞ = +∞.The other arithmetic operations introduced in [3] allow to write also some other importantoperations on LI , such as the  Lukasiewicz t-norm, the product t-norm on LI and their residualimplications, using a similar algebraic formula as their counterparts on ([0, 1],≤).Theorem 4.4 [2] Let f be any generator on LI associated to ⊕LI . Then the mapping T :(LI)2 → LI defined by, for all x, y in LI ,T (x, y) = f(−1)(f(x)⊕LI f(y)),is a t-norm on LI .Theorem 4.5 [2] Let f be a continuous additive generator on LI associated to ⊕LI for whichf(D) ⊆ D¯∞,+. Then there exists a t-norm T on ([0, 1],≤) such that, for all x, y in LI ,f(−1)(f(x)⊕LI f(y)) = TT (x, y).Thus, using ⊕LI , only pseudo-t-representable t-norms on LI can have continuous additivegenerators f for which f(D) ⊆ D¯∞,+. Other natural extensions of t-norms on ([0, 1],≤) whichhave a continuous generator f cannot have a continuous additive generator on LI that is anatural extension of f .4.2 Additive generators based on ⊕tLINow we discuss a second type of addition on L¯I which was introduced in [8]. Similarly as for⊕LI , we have that SSW ,t(x, y) = inf(1LI , x⊕tLIy), for all x, y in LI .Definition 4.2 [8] Let t ∈ [0, 1]. Then we define the t-addition on L¯I ∪ L¯I∞,+ by, for all x, yin L¯I ∪ L¯I∞,+,x⊕tLIy = [min(1− t + x1 + y1, x1 + y2, x2 + y1), x2 + y2].Theorem 4.6 Let t ∈ [0, 1] and f be any generator on LI associated to ⊕tLI. Then themapping T : (LI)2 → LI defined by, for all x, y in LI ,T (x, y) = f(−1)(f(x)⊕tLIf(y)),is a t-norm on LI .6Theorem 4.7 Let t ∈ [0, 1] and f be a continuous additive generator on LI associated to ⊕tLIfor which f(D) ⊆ D¯∞,+. Then there exists a t-norm T on ([0, 1],≤) such that, for all x, y inLI ,f(−1)(f(x)⊕tLIf(y)) = TT,f−1(1−t)(x, y).Similarly as for ⊕LI , from Theorem 4.7 it follows that a t-norm T on LI which is a naturalextension of a t-norm on ([0, 1],≤) generated by a continuous additive generator f can onlyhave a continuous additive generator associated to ⊕tLIwhich is a natural extension of f , ifT belongs to the class of t-norms TT,t.4.3 Additive generators based on ⊕′LIFinally, we introduce the following addition on L¯I .Definition 4.3 We define the addition on L¯I ∪ L¯I∞,+ by, for all x, y in L¯I ∪ L¯I∞,+,x⊕′LIy = [x1 + y1, max(x1 + y2, x2 + y1)].This addition is closely related to the t-conorm S ′SW : for all x, y in LI , S ′SW (x, y) =inf(1LI , x⊕′LIy).Theorem 4.8 Let f be any generator on LI associated to ⊕′LIsuch that f(0LI ) ∈ D¯∞,+. Thenthe mapping T : (LI)2 → LI defined by, for all x, y in LI ,T (x, y) = f(−1)(f(x)⊕′LIf(y)),is a t-norm on LI .Theorem 4.9 Let f be a continuous additive generator on LI associated to ⊕′LIfor whichf(D) ⊆ D¯∞,+. Then there exists a t-norm T on ([0, 1],≤) such that, for all x, y in LI ,f(−1)(f(x)⊕′LIf(y)) = T ′T (x, y).Similarly as for the two other additions, only t-norms on LI belonging to the class oft-norms T ′T can have continuous additive generators f associated to ⊕′LIwhich are a naturalextension of a continuous additive generator on ([0, 1],≤).5 ConclusionIn [3, 8] two kinds of arithmetic operations on LI are introduced. In [2] one of these kindsof operations is used to construct additive generators on LI . Since these are not the onlypossible ways to define addition, substraction, multiplication and division on LI , we developeda new theory of additive generators on LI as much as possible independently of the additionneeded. We found a sufficient condition for ⊕ such that additive generators associated to⊕ generate t-norms on LI . We showed that continuous additive generators on LI which area natural extension to LI of a generator on ([0, 1],≤) can be represented in a unique wayby the generator on ([0, 1],≤). As a consequence, the choice of the operation ⊕ determineswhich classes of t-norms on LI can have continuous additive generators which form a naturalextension of a generator on the unit interval.7References[1] K. T. Atanassov, Intuitionistic fuzzy sets, Physica-Verlag, Heidelberg/New York, 1999.[2] G. Deschrijver, Additive and multiplicative generators in interval-valued fuzzy set theory,submitted.[3] G. Deschrijver, Algebraic operators in interval-valued fuzzy set theory, submitted.[4] G. Deschrijver, C. Cornelis and E. E. Kerre, On the representation of intuitionistic fuzzyt-norms and t-conorms, IEEE Transactions on Fuzzy Systems, 12(1) (2004) 45–61.[5] G. Deschrijver and E. E. 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Additive generators in interval-valued fuzzy and intuitionistic fuzzy set theory

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Additive generators in interval-valued fuzzy and intuitionistic fuzzy set theory

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