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Practical additions to the axioms: c3. c is a continuous function c4. c is involutive

Complement. Usually,. and. Axioms (skeleton):. (boundary conditions). c1. (monotonicity). c2. A family of functions C satisfy c1,c2. Practical additions to the axioms: c3. c is a continuous function c4. c is involutive. 1. .9. .8. .7. .6. .5. .4. .3. .2. .1. 0.

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Practical additions to the axioms: c3. c is a continuous function c4. c is involutive

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  1. Complement Usually, and • Axioms (skeleton): (boundary conditions) c1. (monotonicity) c2. • A family of functions C satisfy c1,c2 • Practical additions to the axioms:c3. c is a continuous functionc4. c is involutive

  2. 1 .9 .8 .7 .6 .5 .4 .3 .2 .1 0 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 • Some examples: 1) satisfies c1,c2 t=threshold 2) satisfies c1,c2,c3 1 0 1

  3. More examples: 3) (M. Sugeno) 4) (R. Yager) 1 1 .9 .9 .8 .8 Classical complement .7 .7 .6 .6 .5 .5 .4 .4 .3 .3 .2 .2 .1 .1 0 0 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1

  4. c(a) Not increasing! e=c(e) Increasing! e a • Equilibrium of complement: for e for classical fuzzy complement • Every fuzzy complement hasat most one equilibrium.Illustration: has an equilibrium then • If • If satisfies c3 (is continuous) then (e is equilibrium, )

  5. Equilibrium of Sugeno-complement “ Yager “: 1 0 -1 0 1 2 3 4 5 6 w

  6. Dual point: If given a complement and any membership degree then the membership for which is ‘s dual point (with respect to ) • If satisfies c3 then • If (e is equilibrium) then • For • Example: Yager-c Involutive!

  7. Intersection (t-norm) • Axiomatic skeleton: t1. (boundary conditions) t2. (commutativity) t3. (monotonicity) t4. (associativity) • Some usual restrictions (practical motivation) t5. is a continuous function (idempotence) (subidempotence) t6a. t6b. t7.

  8. Union (t-conorm, s-norm) • Axiomatic skeleton: s1. (boundary conditions) s2. (commutativity) s3. (monotonicity) s4. (associativity) • Some usual restrictions: s5. is a continuous function s6a. s6b. s7. (idempotence) (superidempotence)

  9. Some examples: • Intersection: 1. 2. 3. 4. • Union: 1. 2. 3. 4.

  10. Some classes of fuzzy set unions and intersections Include algebraic norms: and

  11. Aggregation operations Very general axiomatic requirements: h1. (boundary conditions) h2. for arbitrary and (monotonicity) • Practical additions h3. h is a continuous function h4. h is symmetric for all the arguments : arbitrary permutation

  12. Union & intersection can be extended to n-ary operations because of associativity: For given ? Unions Intersections ? is the area of averaging operations • Generalized means: satisfies h1-h4

  13. Generalized means: • If h4 is not necessary (diff. importance of arguments) so that

  14. Various classes of aggregation operations Dombi Dombi Schweizer/Sklar Schweizer/Sklar Yager Yager Generalized means Intersection operations Averaging operations Union operations

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