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NAFIPS 2005 - 2005 Annual Meeting of the North American Fuzzy Information Processing Society A type-2 fuzzy residuated algebra L. Di Lascio Dipartimento di Matematica e Informatica University ofSalerno 84040, via ponte don Melillo, Fisciano, Italy [email protected] Abstract - In this paper we introduce in a commutative monoid on type-2 fuzzy sets an adjoint pair, so we obtain an original fuzzy residuated algebra. I. INTRODUCTION The concept of type-2 fuzzy set was introduced by Zadeh [18] as an extension of the concept of type-I fuzzy set. Type-2 fuzzy sets have membership degrees that are themselves fuzzy sets. So a type-I fuzzy set is a special case of a type-2 fuzzy set. Their properties has been studied in [11, 12, 13, 14, 15, 16]. The activity research on type-2 fuzzy sets is nowadays flourishing. The book by Mendel [14] is the first and more exhaustive work about this field. In this paper we introduce a residuate commutative partially ordered monoid whose support set is the set of type-2 fuzzy sets having totally ordered triangular numbers on [0, 1] as grades of membership. This monoid, endowed with suitable functions useful for data modelling, allows to handle a large variety of applications [2-10]. The paper is organized as follows. In section 2 we recall basic mathematical concepts that we will use later in the paper. In section 3 we introduce the monoid and its algebraic properties. In section 4 the order relation is discussed and in the next section the residuation operation is introduced. II. PRELIMINARIES A commutative partially ordered monoid is a structure A= (A, *, e, <) such that (A, *, e) is a commutative monoid, where the element e is the unit, < is a partial ordering on A and for all a,b, c, d E A, ifa<b and c.dthen a*c <b*d. The structure AR = (A, *, -X, e, <) [1] is said to be a residuated commutative partially ordered monoid (rc-pomonoid, for short) if (A, *, e, .) is a commutative partially ordered monoid and for all a, b E A, c*a.b iff c.a->b. The binary operation -* on A is called residuum and the couple (*, -*) is also called adjoint pair. The residuum -> is antitone in the left argument, monotone in the right element and for any a, b E A results e - a = a. Let A be a non empty classical set. A fuzzy set s on A is a function s: A --> [0, 1]. If a E A then s(a) is said the membership degree of a to A. A triangular fuzzy number x=[a, b, c] on [0, 1] is a fuzzy set whose membership function is a triangle whose vertexes are the points (a, 0), (b, 1) and (c, 0). In the sequel we will use the following extended operations on the class of the [0,1]- A. Gisolfi Dipartimento di Matematica e Informatica University ofSalerno 84040, via ponte don Melillo, Fisciano, Italy gisolfi(unisa.it triangular fuzzy numbers: i) (x*[a,b,c]=[ax*a, a*b, ax*c] (product of a real number); ii) [a, b, c] + [d, e, f] = [a+d, b+e, c+f] (sum); iii) [a, b, c]ED[d, e, fl = [(a+d)/2, (b+e)/2, (c+f)/2] (arithmetical mean). A type-2 fuzzy set S2 on A is a function S2: A --> [0, 1][° 1]. III. THE COMMUTATIVE MONOID Suppose we have the following objects: i) U: a finite universe of the discourse of cardinality p; ii) Tr = {[0, 0, 0], [1, 1, 1]} u {[a, b, c]: {a, b, c} c [0, 1]}: a set of totally ordered triangular fuizzy numbers. We write [a, b, c] < [d, e, f] iff a<d, b<e, c<f. It is worth noting that the crisp numbers: [0, O, 0] and [1, 1, 1] belong to Tr; iii) F2 = {a: Yi m.. , wit np xi/ui}: class of the type-2 fuzzy sets U --> Tr, where xi E Tr, xi < xi+,, and {um, uml,., ul} belongs to the class of crisp partitions P(U) on U. In the sequel we call crisp parts the elements ui and fuzzy parts the elements xi; iv) S(U)= {[0= [0, O, 0]/U, (1, 0, 1)], [1=[l,l,l]/U, (0, 1, 1)]} u {[a, t]: a E F2, and t=(k, s, am, am-l, ..., a,) is a suitable tuple of positive integers, that satisfies the following constraints: j) if k = 1 the a1=1 for any i:1...m; jj) if k>1 the tuple (am, amnl, ..., a,) is symmetric with respect to the central values } jjj) s = 0 for 0, instead s=1 for any A . 0 and 1 in S(U). Moreover (k, s, am, .... a,) = (1, s, 1, 1, ..., 1) iff the related type-2 fuzzy set is not the product of other sets through the following operation. Given A=[Yi r p..1 xi/ui, (kA, SA, an, an-1, ..., a,)] and B=[i: m:p...I yi/Vi, (kB, SB, bm, bm-1 .., b1)] E S(U), the binary operation O on S(U)xS(U) is defined as follows [6, 7]: A 0 B = [Xi:n+m1... I Zi/Wi, (kA+kB, 1, Cn+m-I, ..., Cl)] where W = U (Uh n Vk) h=1... i k=i... 1 h.9 n,k S m fuzz parts (k = + SASB ) a h b k (k Axh + k B Yk) (kA + kB)Ci h=l1...i k=i ...1 h.5n, k .m ci = a hb k h =1I... i k=i...i h . n,k. m crisp parts It worth noting that A O 0=0 and A O1 = A. 0-7803-91 87-X/05/$20.00 ©2005 IEEE. 525

[IEEE NAFIPS 2005 - 2005 Annual Meeting of the North American Fuzzy Information Processing Society - Detroit, MI, USA (26-28 June 2005)] NAFIPS 2005 - 2005 Annual Meeting of the North

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Page 1: [IEEE NAFIPS 2005 - 2005 Annual Meeting of the North American Fuzzy Information Processing Society - Detroit, MI, USA (26-28 June 2005)] NAFIPS 2005 - 2005 Annual Meeting of the North

NAFIPS 2005 - 2005 Annual Meeting of the North American Fuzzy Information Processing Society

A type-2 fuzzy residuated algebra

L. Di LascioDipartimento di Matematica e Informatica

University ofSalerno84040, via ponte don Melillo, Fisciano, Italy

[email protected]

Abstract - In this paper we introduce in a commutativemonoid on type-2 fuzzy sets an adjoint pair, so we obtain anoriginal fuzzy residuated algebra.

I. INTRODUCTION

The concept of type-2 fuzzy set was introduced by Zadeh[18] as an extension of the concept of type-I fuzzy set. Type-2fuzzy sets have membership degrees that are themselves fuzzysets. So a type-I fuzzy set is a special case of a type-2 fuzzyset. Their properties has been studied in [11, 12, 13, 14, 15,16]. The activity research on type-2 fuzzy sets is nowadaysflourishing. The book by Mendel [14] is the first and moreexhaustive work about this field.

In this paper we introduce a residuate commutativepartially ordered monoid whose support set is the set of type-2fuzzy sets having totally ordered triangular numbers on [0, 1]as grades of membership. This monoid, endowed with suitablefunctions useful for data modelling, allows to handle a largevariety of applications [2-10].

The paper is organized as follows. In section 2 we recallbasic mathematical concepts that we will use later in the paper.In section 3 we introduce the monoid and its algebraicproperties. In section 4 the order relation is discussed and inthe next section the residuation operation is introduced.

II. PRELIMINARIES

A commutative partially ordered monoid is a structureA= (A, *, e, <) such that (A, *, e) is a commutative monoid,where the element e is the unit, < is a partial ordering on Aand for all a,b, c, d E A, ifa<b and c.dthen a*c <b*d. Thestructure AR = (A, *, -X, e, <) [1] is said to be a residuatedcommutative partially ordered monoid (rc-pomonoid, forshort) if (A, *, e, .) is a commutative partially ordered monoidand for all a, b E A, c*a.b iff c.a->b. The binary operation -*on A is called residuum and the couple (*, -*) is also calledadjoint pair. The residuum -> is antitone in the left argument,monotone in the right element and for any a, b E A results e- a = a.

Let A be a non empty classical set. Afuzzy set s on A is afunction s: A --> [0, 1]. If a E A then s(a) is said themembership degree of a to A.

A triangular fuzzy number x=[a, b, c] on [0, 1] is a fuzzyset whose membership function is a triangle whose vertexesare the points (a, 0), (b, 1) and (c, 0). In the sequel we will usethe following extended operations on the class of the [0,1]-

A. GisolfiDipartimento di Matematica e Informatica

University ofSalerno84040, via ponte don Melillo, Fisciano, Italy

gisolfi(unisa.it

triangular fuzzy numbers: i) (x*[a,b,c]=[ax*a, a*b, ax*c](product ofa real number); ii) [a, b, c] + [d, e, f] = [a+d, b+e,c+f] (sum); iii) [a, b, c]ED[d, e, fl = [(a+d)/2, (b+e)/2, (c+f)/2](arithmetical mean).

A type-2fuzzy set S2 on A is a function S2: A --> [0, 1][° 1].

III. THE COMMUTATIVE MONOID

Suppose we have the following objects: i) U: a finiteuniverse of the discourse of cardinality p; ii) Tr = {[0, 0, 0],[1, 1, 1]} u {[a, b, c]: {a, b, c} c [0, 1]}: a set of totallyordered triangular fuizzy numbers. We write [a, b, c] < [d, e, f]iff a<d, b<e, c<f. It is worth noting that the crisp numbers: [0,O, 0] and [1, 1, 1] belong to Tr; iii) F2 = {a: Yi m.. , wit npxi/ui}: class of the type-2 fuzzy sets U --> Tr, where xi E Tr,xi < xi+,, and {um, uml,., ul} belongs to the class of crisppartitions P(U) on U. In the sequel we call crisp parts theelements uiand fuzzy parts the elements xi; iv) S(U)= {[0= [0,O, 0]/U, (1, 0, 1)], [1=[l,l,l]/U, (0, 1, 1)]} u {[a, t]: a E F2,and t=(k, s, am, am-l, ..., a,) is a suitable tuple of positiveintegers, that satisfies the following constraints: j) if k = 1 thea1=1 for any i:1...m; jj) if k>1 the tuple (am, amnl, ..., a,) issymmetric with respect to the central values } jjj) s = 0 for 0,instead s=1 for any A . 0 and 1 in S(U). Moreover (k, s, am, ....a,) = (1, s, 1, 1, ..., 1) iff the related type-2 fuzzy set is not theproduct of other sets through the following operation.Given A=[Yi r p..1 xi/ui, (kA, SA, an, an-1, ..., a,)] and B=[i: m:p...Iyi/Vi, (kB, SB, bm, bm-1 .., b1)] E S(U), the binary operation O onS(U)xS(U) is defined as follows [6, 7]:

A 0 B = [Xi:n+m1... I Zi/Wi, (kA+kB, 1, Cn+m-I, ..., Cl)]

whereW = U (Uh n Vk)

h=1... ik=i... 1h.9 n,k S m

fuzz parts

(k= +SASB ) a h b k (k Axh + k BYk)(kA + kB)Ci h=l1...ik=i ...1h.5n, k.m

ci = a hb kh =1I... ik=i...ih . n,k. m

crisp parts

It worth noting that A O0=0 and A O1 = A.

0-7803-91 87-X/05/$20.00 ©2005 IEEE. 525

Page 2: [IEEE NAFIPS 2005 - 2005 Annual Meeting of the North American Fuzzy Information Processing Society - Detroit, MI, USA (26-28 June 2005)] NAFIPS 2005 - 2005 Annual Meeting of the North

The indices ah e bk represent the number of sets that havegenerated the ith class of A and B, respectively. The indiceskA e kB represent, in turn, the number of sets that havegenerated the classes of A and B, respectively. Moreover, wenote that the operation for zi represents essentially a meanamong the type 2 fuzzy sets, where each fuzzy set takes aweight in some way related to the changes induced by thecomposition. We emphasize this fact saying that these indicesinclude the computational history of our type 2 fuzzy sets. Theoperation C is well defined: i) (wn+m-1, wn+m 2 m.., wI) E P(U);ii) the tuple (cm,,I. ...... cl) is symmetric with respect to thecentral values and strictly increasing until these values; iii)AOBe S(U); iii) the elements zi are triangular fuzzy numberson [O,1].

Proposition 1: The structure S1(U) = (S(U), C, 1) is acommutative monoid.Proof. [6]

IV. THE COMMUTATIVE PARTIALLY ORDERED MONOID

Now we introduce on S1(U) an order relation. We say thatA .B ifand only ifexists C e S1(U) such that A = B O C. Wedenote the element C by B a A. In particular we let 0 b 0 = 1.If neither A< B neither B.A, we write A B.Algorithms for calculating C such that A = B C C is given in[6, 7], where we prove:

Proposition 2: If exists BI A and BO (BI A) = A then thesolution of the equation A=BOX is X=B1 A.

We prove the properties of the binary relation: i) Reflexivity:A<A since A=AO1; ii) Antisymmetricity if A<B and B<Athen A= B C C and B =AO C' for some C and C', then A=A CC C C'. It follows C C C' = 1, since A k A = 1, for any A ES,(U). Since X E S1(U), such that A C X = 1, doesn't exist,from C C C' = 1 follows C = C' = 1. Finally A = B C 1 = B; iii)Transitivity: if A<B and B<C then A= B C D and B = C C Efor some D and E. Then A= COEOD, so A<C.It is immediate to verify that 1 is the top element of S1(U) and0 is the bottom. From the definition we deduce the followingresults: A b A = L, i A = A, A I 0 = 0. Moreover 0 4 Aneither Ab 1 is not defined, ifA.1 and A.O, respectively.The basic properties of the order relation and the operation tare as follows: PO: A O B < A; PI: If AOC = BOC and C .0,thenA=B;P2:i)IfA<BthenAOC<BOC; ii)IfA YC<BOCandC0, thenA<B;P3:A<BiffA<B ItA;P4:A<Biff(B t A) IA= B; P5: IfA < B< C0 then CJ A < CIB;P6:IfC<A<BthenB dIC<AJIC.

We obtain:Proposition 3: S2(U) = (S(U), <, C, 1)) is a partially orderedcommutative monoid.Proof. If A<B and C<D, then exist H, K E S,(U) such thatA=BOH and C=DDOK. We obtain: AOC= BODOHOK < BOD, byPO .

V. THE RESIDUATED COMMUTATIVE PARTIALLY ORDEREDMONOID

The operation AIbB is not defined whenever the equationAOX=B doesn't have solution. Now we extend this operationso that it is the residuum of the operation C on S2(U).We define

A I BA -+ B = 1

B

if B < Aif A < B

if A I |BProposition 5: A -+ B = sup c E sZu) {C: AOC < B}.Proof: By hypothesis, for any A, B, C, such H E S2(U) thatAOC=BOH exists. Let us consider three cases. Case 1): A<B,therefore sup C = 1. Case 2): B < A. In this case B = AOK, forsome K E S2(U) and A -4 B = A2LB. We obtain K=A4IB,therefore C = HO(AIbB) < ALB, by PO. Case 3): Al lB. IfAOC=BOH=AOB, we deduce C=B. If were AOB<AOC (andAOB<BOH), we could deduce sup C =1, but this implies A<B.If it is AOC<AOB and BOH<AOB, with C H, we deduce supC = B. The other cases (C<H and H<C) are in contradictionwith the premise A B. .

Corollary 6: A O C < B if and only ifC < A - B.

Proposition 7: S3(U) = (S2(U), <, 0, -* 1) is a rc-pomonoid..VI. CONCLUSIONS

In this paper we have presented a new residuatecommutative partially ordered monoid defined on a specificclass of type-2 fuzzy sets. This monoid has been used inseveral application fields and allows to develop a novelapproach to modeling in soft computing. In [8] we deal withthe factors that affect the presence of a diabetical neuropathy;in [4, 10] an user model for adaptive hypermedial systems isillustrated. This user model is the principal module of ourFuzzy Adaptive E-Learning System [5]; in [3] we havepresented a model of formative evaluation that utilizes theconcept of matrix of educational goals according to theselected didactic taxonomy; in [7] we have analized theclassification of 38 youngsters (13, 14 and 15 years hold)according to their physical constitution. Our results areidentical to those obtained in [17]; in [9] we have anapplication concerning "reading the hands" in order to getpredictions about the health condition of an individual; in [2]the monoid, endowed with suitable linguistic labels, has givenresults similar to those obtained by means of well knownmodels for fuzzy decision making.

REFERENCES[1] Birkhoff, G. Lattice Theory, 1st ed. Amer. Math. Soc. Coll. Pub. AMS,

Providence, RL 1940 (3rd edition 1967)

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[2] Di Lascio L., Gisolfi A., U. Cortes, An algebraic approach to FuzzyDecision Making and Fuzzy Screening, (submitted to FuzzyOptimization and Decision Making, in revision), 2004

[3] Di Lascio L., Gisolfi A., Nappi A., Fuzzy Formative Evaluation for E-learning Applications, submitted to Computers & Education (inrevision), 2004

[4] Di Lascio L., Fischetti E., Gisolfi A., Loia V., Nappi A., Linguisticresources and fuzzy algebraic computing in adaptive hypermediasystems, in E. Damiani, L. Jain (Eds.), Soft Computing And SoftwareEngineering, Springer Verlag, 2004

[5] Di Lascio L., Fischetti E., Gisolfi A. Nappi A., A fuizzy approach to thedevelopment of e-learning systems, The Proceedings of MDMPEC03,Salerno, Italy, 2003

[6] Di Lascio L., Gisolfi A., Ciamillo P., An algebraic approach to SoftComputing (sub. to Soft Computing), 2003

[7] Di Lascio L., Gisolfi A., Rosa G., A commutative l-monoid forclassifications with fuzzy attributes, Int. J. Of Approximate Reasoning,pp 26, 1-46, 2002

[8] Di Lascio L., Gisolfi A., Albunia A., Galardi G., Meschi F., A fuzzy-based methodology for the analysis of diabetic nuropathy, Fuzzy Setsand Systems, 129, 203 -228, 2002

[9] Di Lascio L. Fischetti E. Gisolfi A., An algebraic tool for classificationin fuzzy environments, in G. Gerla, A. Di Nola (Eds.), Fuzzy Logic andSoft Computing, Springer-Verlag, Berlin, 2001

[11] Dubois D., Prade H., Fuzzy Sets and Systems: Theory and Applications.New York: Academic, 1980.

[12] Karnik N.N., Mendel J. M, Operations on type-2 fuzzy sets, Int. J. FuzzySets Syst., vol. 122, pp. 327-348, 2001.

[13] Karnik, N. N., Mendel J. M. and Q. Liang , Type-2 Fuzzy LogicSystems, IEEE Trans. on Fuzzy Systems, vol. 7, pp. 643-658, Dec.1999.

[14] Mendel J. M., Uncertain Rule-Based Fuzzy Logic Systems: Introductionand New Directions, Prentice Hall, Upper Saddle River, NJ, 2001.

[15] Mizumoto M. and Tanaka K, Fuzzy sets of type 2 under algebraicproduct and algebraic sum, Fuzzy Sets Syst., vol. 5, pp. 277-290, 1981.

[16] Mizumoto M., Tanaka K, Some properties of fuzzy sets of type 2,Inform. Contr., vol. 31, pp. 312-340, 1976.

[17] Sato M, Sato Y., Jain L. C., Fuzzy Clustering Models and Applications,Studies in Fuzzyness and Soft Computing, vol. 9, Physica-Verlag, NY,1997

[18] Zadeh L.A., The Concept of a Linguistic Variable and its Application toApproximate Reasoning-I, IL Ill. Information Sciences I 8 - II 8 - m 9.pp 199-249 pp 301-357; pp 43-80, 1975

[10] Di Lascio L., Fischetti E., Gisolfi A., A fuzzy-based approach tostereotype selection in hypermedia, User Modelling and User-AdaptedInteraction, 9: 285-320, 1999

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