Lattice properties of discrete fuzzy numbers under ... - EUSFLAT

IFSA-EUSFLAT 2009

Lattice properties of discrete fuzzy numbers under extended min and max Jaume Casasnovas

Juan Vicente Riera

Dept. Mathematics and Computer Science ,University of the Balearic Islands Palma de Mallorca, Spain Email: {jaume.casasnovas,jvicente.riera}@uib.es

Abstract— This paper proposes to study the lattice properties of two closed binary operations in the set of discrete fuzzy numbers. Using these operations to represent the meet and the join, we prove that the set of discrete fuzzy numbers whose support is a set of consecutive natural numbers is a distributive lattice. Finally, we demonstrate that the subsets of discrete fuzzy numbers, which have the same support, are distributive lattices too. Keywords— Fuzzy numbers, discrete fuzzy numbers, distributive lattice.

(A1 ,M AXw ,M INw ) and (FA ,M INw ,M AXw ) are distributive lattices, where FA is the subset of discrete fuzzy numbers whose support is the support of A, with A ∈ DF N .

2

Preliminaries

In this section, we recall some definitions and the main results about discrete fuzzy numbers which will be used later.

Definition 2.1 [13] A fuzzy subset u of R with membership mapping u : R → [0, 1] is called fuzzy number if its support 1 Introduction is an interval [a, b] and there exist real numbers s, t with a ≤ It is possible to approach the theory of fuzzy numbers in dif- s ≤ t ≤ b and such that: ferent directions: theoretical[8, 9, 10, 13], geometric [1, 2], 1. u(x)=1 with s ≤ x ≤ t applications in engineering [13], social science [12], lattice theory[13, 20], etc.. Voxman [18] introduced the concept 2. u(x) ≤ u(y) with a ≤ x ≤ y ≤ s of a discrete fuzzy number as a fuzzy subset of real num3. u(x) ≥ u(y) with t ≤ x ≤ y ≤ b bers with discrete support and analogous properties to a fuzzy number (convexity, normality). Also, like fuzzy numbers, it 4. u(x) is upper semi-continuous. is possible to consider discrete fuzzy numbers from different points of view: theoretical [16, 18], applications in engineer- We will denote the set of fuzzy numbers by F N . ing [11, 19], social sciences [17], etc. It is well known that, the arithmetic and lattice operations Definition 2.2 [18] A fuzzy subset u of R with membership such as maximum and minimum on fuzzy numbers can be ap- mapping u : R → [0, 1] is called discrete fuzzy number if proached either by the direct use of the membership function its support is finite, i.e., there are x1 , ..., xn ∈ R with x1 < (by the Zadeh’s extension principle) or by the equivalent use x2 < ... < xn such that supp(u) = {x1 , ..., xn }, and there are natural numbers s, t with 1 ≤ s ≤ t ≤ n such that: of the r-cuts representation, for instance, [13, 14, 20]. Nevertheless, in the discrete case using the same methods, 1. u(xi )=1 for any natural number and i with s ≤ i ≤ t this process can yield a fuzzy subset that does not satisfy the (core) conditions to be a discrete fuzzy number [3, 19]. In previous works [3, 4], we have presented a technique that allows us to 2. u(xi ) ≤ u(xj ) for each natural number i, j with 1 ≤ i ≤ obtain a closed addition on the set of discrete fuzzy numbers, j≤s DF N , and moreover, we focus on the addition of discrete 3. u(xi ) ≥ u(xj ) for each natural number i, j with t ≤ i ≤ fuzzy numbers whose support is an arithmetic sequence and j≤n even a subset of consecutive natural numbers. This type of numbers arise mainly when a fuzzy cardinality of a fuzzy set From now on, we will denote the set of discrete fuzzy num[5, 6] or a fuzzy multiset [15] is considered. bers by DF N and DF N (N) will stand for the set of discrete In [7] we define two closed binary operations in the set fuzzy numbers whose support is a subset of the set of Natural of discrete fuzzy numbers to obtain the maximum and the Numbers. Finally, a discrete fuzzy number will be denoted by minimum of discrete fuzzy numbers. We prove as well that dfn. in the set A1 , of discrete fuzzy numbers whose support is a In general, the operations on fuzzy numbers f, g can be apset of consecutive natural numbers, these operations coincide proached either by the direct use of their membership funcwith the function maximum and minimum obtained using the tion, µ (x), µ (x), as fuzzy subsets of R and the Zadeh’s exf g Zadeh’s extension principle. tension principle: The aim of this paper is to continue studying the properties of these operations and if it is possible to obtain a structure of lattice using them. We will see that, in general, O(f, g)(z) = sup{µf (x) ∧ µg (y)|O(x, y) = z} these binary operations only fulfill the associative, commutaor by the equivalent use of the α-cuts representation[13]: tive and idempotent laws. We show as well that the triplets ISBN: 978-989-95079-6-8

647

IFSA-EUSFLAT 2009

O(f, g)α = O(f α , g α ) = {O(x, y)|x ∈ f α , y ∈ g α }

3. For any r1 , r2 ∈ [0, 1] with 0 ≤ r1 ≤ r2 ≤ 1, if x ∈ Ar1 − Ar2 we have x < y for all y ∈ Ar2 , or x>y for all y ∈ Ar 2 

and O(f, g)(z) = sup{α ∈ [0, 1]|z ∈ O(f, g)α } Nevertheless, in the discrete case, this process can yield a fuzzy subset that does not satisfy the conditions to be a discrete fuzzy number [3, 19]. For example, let u = {0.3/1, 1/3, 0.5/7} and v = {0.4/2, 1/5, 1/6, 0.8/9} be two discrete fuzzy numbers. If we use the Zadeh’s extension principle to obtain their addition, it results the fuzzy subset S = {0.3/3, 0.4/5, 0.3/6, 0.3/7, 1/8, 1/9, 0.3/10, 0.8/12, 0.5/13, 0.5/16} which doesn’t fulfill the conditions to be a discrete fuzzy number, because the third property of the definition 2.2 fails. In a previous work [3, 4] we have presented an approach to a closed extended addition (⊕) of discrete fuzzy numbers after associating suitable non-discrete fuzzy numbers, which can be used like a carrier to obtain the desired addition. In a recent paper [4] we proved that a suitable carrier can be a discrete fuzzy number whose support is an arithmetic sequence and even a subset of consecutive natural numbers. Thus, we obtained the following results:

4. For any r0 ∈ [0, 1], there exists a real number r0 with   0 < r0 < r0 such that Ar0 = Ar0 ( i.e. Ar = Ar0 , for
any r ∈ [r0 , r0 ]) then there exists a unique u ∈ DF N such that ur = Ar for any r ∈ [0, 1].

3

Maximum and Minimum of discrete fuzzy numbers

It’s well known, for example [13], that the set of fuzzy numbers is a distributive lattice using the following operations M IN (u, v)(z) =

sup z=min(x,y)

M AX(u, v)(z) =

min(u(x), v(y)), ∀z ∈ R (1)

sup z=max(x,y)

min(u(x), v(y)), ∀z ∈ R (2)

for each couple u, v ∈ F N . If we use the same operations to obtain a similar structure in the set of discrete fuzzy numbers we see that this result is not possible. For instance, if we consider the discrete fuzzy numbers u = {0.3/1, 0.4/3, 1/4} and v = {0.5/2, 1/5, 1/6} and we use the previous definiProposition 2.3 [4] tion to calculate the M IN (u, v), we obtain the fuzzy subLet Ar be the set {f ∈ DFN (N), such that supp(f) set M = {0.3/1, 0.5/2, 0.4/3, 1/4} which doesn’t satisfy the is the set of terms of an arithmetic sequence with r as conditions of the definition 2.2. In [7], the authors study this common difference}. If f, g ∈ Ar . The following facts: drawback and we propose a new method to calculate them. Using this method, we will see later that, it is possible to pro1. f ⊕ g ∈ DF N (N) vide different subsets of the set DF N with a structure of dis2. f ⊕ g ∈ Ar tributive lattice.

Definition 3.1 [7] Let u, v be two dfn. For each α ∈ [0, 1], let’s consider the α-cut sets: uα = Remark 2.4 [4] Note that the set A1 is the set of discrete {xα , ..., xα },v α = {y α , ..., y α } for u and v respectively and p 1 1 fuzzy numbers whose support is a set of consecutive natural the set supp(u) ! supp(v) k= {x ∨ y|x ∈ supp(u), y ∈ numbers. supp(v)}. Let’s define the set: hold.

Finally, we will use a kind of representation in the study of discrete fuzzy numbers:

Aα = {z ∈ supp(u)

Theorem 2.5 [19] Let u be a dfn and let ur be the r-cut ={x ∈ R|u(x) ≥ r} for any r ∈ (0, 1]. Let’s u0 denote the support of u. Then the following statements (1)-(4) hold:

min(uα ∨ v α ) ≤ z ≤ max(uα ∨ v α )} =

1. ur is a nonempty finite subset of R, for any r ∈ [0, 1] 2. ur2 ⊂ ur1 for any r1 , r2 ∈ [0, 1] with 0 ≤ r1 ≤ r2 ≤ 1

= {z ∈ supp(u)

supp(v) such that

supp(v) such that

(min uα ∨ min v α ) ≤ z ≤ (max uα ∨ max v α )} i.e.:

3. For any r1 , r2 ∈ [0, 1] with 0 ≤ r1 ≤ r2 ≤ 1, if x ∈ α α α Aα = {z ∈ supp(u) supp(v)|(xα 1 ∨y1 ) ≤ z ≤ (xp ∨yk )} ur1 − ur2 we have x < y for all y ∈ ur2 , or x>y for all y ∈ u r2 Proposition 3.2 [7] The finite set Aα , as defined above,  4. For any r0 ∈ [0, 1], there exist some real numbers r0 with satisfies the properties 1,2,3 and 4 of theorem 2.5 and  0 < r0
< r0 such that ur0 = ur0 ( i.e. ur = ur0 for any a discrete fuzzy number, M AXw (u, v), whose α-cuts are the finite set Aα exists and M AXw (u, v)(z) = r ∈ [r0
, r0 ]). sup {α ∈ [0, 1] such that z ∈ Aα }. r And conversely, if for any r ∈ [0, 1], there exist A ⊂ R satisfying the following conditions (1)-(4): Proposition 3.3 [7] If u, v ∈ A1 , then M AX(u, v), defined through the extension principle, coincides with M AXw (u, v). 1. Ar is a nonempty finite for any r ∈ [0, 1] So, if u, v ∈ A1 , M AX(u, v) is a discrete fuzzy number and M AX(u, v) ∈ A1 . 2. Ar2 ⊂ Ar1 , for any r ∈ [0, 1] with 0 ≤ r1 ≤ r2 ≤ 1 ISBN: 978-989-95079-6-8

648

IFSA-EUSFLAT 2009 Definition 3.4 [7] Let u, v be two dfn. For each α ∈ [0, 1], let’s consider the α-cut sets: uα = α α α α {xα 1 , ..., xp },v = # {y1 , ..., yk } for u and v respectively and the set supp(u) supp(v) = {x ∧ y|x ∈ supp(u), y ∈ supp(v)}. Let’s define the set: B α = {z ∈ supp(u)

:

supp(A)

α

α

(supp(A)

α α α supp(v)|(xα 1 ∧y1 ) ≤ z ≤ (xp ∧yk )}

do not hold. For example, if we consider the sets u = {−10, −9, −8, −7, −6, −1}, v α = {0, 1, 2, 3, 4} and the usual product of real numbers as binary operation then min(uα · v α ) = −40 and min(uα ) · min(v α ) = −10 · 0 = 0, and max(uα ·v α ) = 0 and max(uα )·max(v α ) = −1·4 = −4 Proposition 3.6 [7] The finite set B α , as defined above, satisfies the properties 1,2,3 and 4 of Proposition 2.5 and a discrete fuzzy number, M INw (u, v), whose αcuts are the finite set B α exists and M INw (u, v)(z) = sup {α ∈ [0, 1] such that z ∈ B α }. Proposition 3.7 [7] If u, v ∈ A1 , then M IN (u, v), defined through the extension principle, coincides with M INw (u, v). So, if u, v ∈ A1 , M IN (u, v) is a discrete fuzzy number and M IN (u, v) ∈ A1 .

supp(A)

:

supp(A) = supp(A)

Example 3.8 If we use the method, as explained above, to calculate the minimum of u = {0.3/1, 0.4/3, 1/4} and v = {0.5/2, 1/5, 1/6}, we obtain the following discrete fuzzy number M INw (u, v) = {0.3/1, 0.5/2, 0.5/3, 1/4} where B 0.3 = {z ∈ {1, 2, 3, 4} such that 1 ≤ z ≤ 4} = {1, 2, 3, 4} = {z ∈ {1, 2, 3, 4} such that 2 ≤ z ≤ 4} = {2, 3, 4} = {z ∈ {1, 2, 3, 4} such that 2 ≤ z ≤ 4} = {2, 3, 4}

B 1 = {z ∈ {1, 2, 3, 4} such that 4 ≤ z ≤ 4} = {4} Proposition 3.9 Let A, B and C be three dfn such that their supports are supp(A),supp(B) and supp(C) respectively. The following properties hold:

supp(A) = supp(A)

4. If the supports of A and B are the same or supp(B) ⊆ supp(A) or A, B ∈ A1 then the next absorption laws : supp(A) (supp(A) supp(B)) = supp(A) supp(A)

(supp(A)

:

supp(B)) = supp(A)

hold. 5. If the supports of A, B and C are the same or A, B, C ∈ A1 then the following distributive properties : supp(A) (supp(B) supp(C)) = = (supp(A)

:

and

supp(B))

(supp(A)

:

supp(C))

: (supp(B) supp(C)) = : supp(B)) (supp(A) supp(C))

supp(A) = (supp(A)

We have seen, in the previous propositions, that the operations M AXw (u, v) and M INw (u, v) are discrete fuzzy numbers.

ISBN: 978-989-95079-6-8

supp(C) =

3. Idempotence

supp(A)

α

B

:

and

• max(uα ∗ v α ) = max uα ∗ max v α

0.5

supp(B))

(supp(A) supp(B)) supp(C) = ! ! = supp(A) (supp(B) supp(C))

• min(uα ∗ v α ) = min uα ∗ min v α

B

:

and

Remark 3.5 In general, if ”∗” is a binary operation the equalities

0.4

supp(A)

# # = supp(A) (supp(B) supp(C))

α

i.e.: :

supp(B) = supp(B)

2. Associativity

(min u ∧ min v ) ≤ z ≤ (max u ∧ max v )}

B α = {z ∈ supp(u)

and

supp(v) such that

min(uα ∧ v α ) ≤ z ≤ max(uα ∧ v α )} = : = {z ∈ supp(u) supp(v) such that α

1. Commutativity : : supp(A) supp(B) = supp(B) supp(A)

hold. 6. For any A, B ∈ DF N then : supp(A) (supp(B) ⊆ (supp(A)

:

supp(B))

(supp(A)

and supp(A) ⊆ (supp(A)

(supp(B) supp(B))

supp(C))) ⊆

:

:

:

supp(C))) ⊆

(supp(A)

7. For any A, B ∈ DF N then : supp(A) ⊆ supp(A) (supp(A) supp(A) ⊆ supp(A)

supp(C))

(supp(A)

supp(C))

supp(B)) :

supp(B))

649

IFSA-EUSFLAT 2009 Proof From the definition 2.2, the support of a discrete fuzzy number is a finite linearly ordered subset of real numbers. Moreover, it is well known that the set of real numbers is a distributive lattice with the usual operations maximum(max) and minimum(min). Let X, Y and Z denote the supports of A, B, C ∈ DF N respectively. 1. Using the commutative property of the real functions maximum and minimum the proof is trivial. 2. Associativity # # If z ∈ (X Y ) Z then z = min(x, c) where x = min(a, b), a ∈ X, b ∈ Y and c ∈ Z. So z = min(min(a, b), c). Using the associativity of the function minimum, z # = min(min(a, b), c) = min(a, min(b, c)). # Then z ∈ X (Y Z). Therefore : : : : (X Y ) Z ⊆ X (Y Z)

, i.e., it’s the set of natural numbers z such that z ∈ {min(x1 , max(y1 , z1 )), · · · , min(xn , max(ym , zk ))} = (using the distributive property of natural numbers with the usual order) = {z ∈ {max(min(x1 , y1 ), min(x1 , z1 )), · · · · · · , max(min(xn , ym ), min(xn , zk ))}} Where the previous set is an interval of consecutive natural numbers. On the other hand, : : (supp(A) supp(B)) (supp(A) supp(C)) = = {z ∈ {{min(x1 , y1 ), · · · , min(xn , ym )} {min(x1 , z1 ), · · · , min(xn , zk )}} = = {z ∈ {max(min(x1 , y1 ), min(x1 , z1 )), · · · · · · , max(min(xn , ym ), min(xn , zk ))}.

# # If z ∈ X (Y Z) then z = min(a, x) where x = min(b, c), a ∈ X, b ∈ Y and c ∈ Z. So z = min(a, min(b, c)). Using the associativity of the function minimum, z = # min(a, # min(b, c)) = min(min(a, b), c). Then z ∈ (X Y ) Z. Therefore : : : : X (Y Z) ⊆ (X Y) Z

The proof of the second distributive law is analogous. # ! 6. If z ∈ X (Y Z)) then z = min(a, max(b, c)) where a ∈ X, b ∈ Y and c ∈ Z. Using the distributive property of the minimum and the maximum of real numbers, z = max(min(a, # ! b), # min(a, c)). Then # we ! obtain that#z ∈ ! (X # Y ) (X Z) and so X (Y Z)) ⊆ (X Y ) (X Z).

The proof of the other associative law is analogous.

7. If z ∈ X then z = max(z, min(z, b)) for all b ∈ Y . Because if min(z, b) = b then max(z, b) = z. And, if ! min(z, #b) = z then max(z, z) = z. So, X ⊆ X (X Y )

3. Idempotence # If z ∈ X X then z = min(a, a
) with a, a
∈ X. Therefore z = a ∈#X or z = a
∈ X. This means that z ∈ X and so X # X ⊆ X. On the other hand, it is evident that X ⊆ X X because the function minimum is idempotent and then for each z ∈ X, z = min(z, z). The proof of the other idempotence law is similar. 4. Absorption # ! If z ∈ X (X Y ) then z = min(a, max(a
, b)) where a, a
∈ X and b ∈ Y . Then if z = a or z = a
obviously z ∈ X. But if z = b we have that a
≤ b ≤ a. Using the hypothesis #of the ! proposition we obtain that b ∈ X. Therefore X (X Y ) ⊆ X.

Analogously for the other inclusion.

Remark 3.10 In general, the absorption and the distributive laws of the supports do not hold. For example, if X = {4, 7, 9},Y = {4, 6, 7} and Z = {8, 9, 10} represent the supports of the discrete fuzzy numbers A, B and C respectively, then # ! a) X (X Y ) = {4, 6, 7, 9} but X = {4, 7, 9}. # ! # ! # b) X (Y Z) = {4, 7, 8, 9}, but (X Y ) (X Z) = {4, 6, 7, 8, 9}. The following theorem establishes some properties of the binary operations M AXw and M INw using the same conditions and results that the previous proposition 3.9:

If z ∈ X then z = min(z, max(z, b)) for all b ∈ Y . Because if max(z, b) = b then min(z, b) = z. And, Theorem 3.11 Let M AXw and M INw be the binary operations on DF N defined by the propositions 3.2 and 3.6, respecif # max(z, ! b) = z then min(z, z) = z. So, X ⊆ tively. Then, for any, A, B, C ∈ DF N the following properX (X Y ) ties hold: 5. Distributivity M INw (A, B) = M INw (B, A) 1. Commutativity: M AXw (A, B) = M AXw (B, A) Let’s A, B, C ∈ A1 where supp(A) = {x1 , · · · , xn },supp(B) = {y1 , · · · , ym } 2. Associativity: and supp(C) = {z1 , · · · , zk } respectively. Then : M INw (M INw (A, B), C) = supp(A) (supp(B) supp(C)) = M INw (A, M INw (B, C)) is the set of natural numbers z such that they belong to and the set M AXw (M AXw (A, B), C) = : = M AXw (A, M AXw (B, C)) {x1 , · · · , xn } {max(y1 , z1 ), · · · , max(ym , zk )} ISBN: 978-989-95079-6-8

650

IFSA-EUSFLAT 2009 M INw (A, A) = A M AXw (A, A) = A

3. Idempotence:

3. We want to demonstrate that M INw (A, A) = A

If the supports of A and B are the same or A, B, C ∈ A1 then By definition 3.4

M INw (A, M AXw (A, B)) = A M AXw (A, M INw (A, B)) = A

4. Absorption:

M INw (A, A)α = : = {z ∈ supp(A) supp(A) such that

5. Distributivity: M INw (A, M AXw (B, C)) =

( min Aα ∧ min Aα ) ≤ z ≤ ( max Aα ∧ max Aα )} =

= M AXw (M INw (A, B), M INw (A, C))

α α α = {z ∈ supp(A)|(xα 1 ∧ x1 ) ≤ z ≤ (xp ∧ xp )} = α = {z ∈ supp(A)|xα 1 ≤ z ≤ xp } =

And M AXw (A, M INw (B, C)) =

= Aα

= M INw (M AXw (A, B), M AXw (A, C))

It is the same for the other idempotence law.

Proof Let A, B and C be three dfn. Let’s consider the αα α = {y1α , ..., ykα }, C α = cut sets: Aα = {xα 1 , ..., xp }, B {w1α , ..., wlα } for A, B and C respectively. 1. We want to show that M INw (A, B) = M INw (B, A) It is enough to prove that the discrete fuzzy numbers M INw (A, B) and M INw (B, A) are the same α-cut sets for each α ∈ [0, 1]. By definition 3.4 α

M INw (A, B) = : = {z ∈ supp(A) supp(B) such that

4. We want to demonstrate that M INw (A, M AXw (A, B)) = A By definition 3.4 M INw (A, M AXw (A, B))α = : = {z ∈ supp(A) supp(M AXw (A, B)) such that α

min A ∧ min M AXw (A, B)

= {z ∈ supp(A)

α

:

α

α

≤ z ≤ max A ∧ max M AXw (A, B) } =

supp(M AXw (A, B)) such that

α α α α α xα 1 ∧ (x1 ∨ y1 ) ≤ z ≤ xp ∧ (xp ∨ yk }) = ( min Aα ∧ min B α ) ≤ z ≤ ( max Aα ∧ max B α )} = : ( using the absorption law of real numbers) α α α = {z ∈ supp(A) supp(B)|(xα 1 ∧y1 ) ≤ z ≤ (xp ∧yk )} = : : = {z ∈ supp(A) supp(M AXw (A, B)) such that α α = {z ∈ supp(A) supp(B)|(y1α ∧xα 1 ) ≤ z ≤ (yp ∧xk )} = α α α xα 1 ≤ z ≤ xp } = {z ∈ supp(A) such that x1 ≤ z ≤ xp } = = M IN (B, A)α w

= Aα . (The previous equality is only possible if we use the same hypotheses and results of the proposition 3.9)

Analogously for the other commutative law. 2. We want to see that

5. We want to show the first distributive law, ie,

M INw (M INw (A, B), C) = M INw (A, M INw (B, C)) By definition 3.4

By definition 3.4

M INw (M INw (A, B), C)α = : = {z ∈ supp(M INw (A, B)) supp(C) such that α

min M INw (A, B) ∧ min C

α

α

∧

y1α )

∧

w1α

= {z ∈ (supp(A) (xα 1

∧

w1α

α

≤z≤

:

(xα p

supp(B))

supp(C) such that

∧ :

ykα )

∧

wlα }

=

supp(C) such that

∧ ykα ) ∧ wlα } = : : = {z ∈ (supp(A) supp(B)) supp(C) such that ∧

y1α )

:

M INw (A, M AXw (B, C))α = : = {z ∈ supp(A) supp(M AXw (B, C)) such that

≤ z ≤ max M INw (A, B) ∧ max C } =

= {z ∈ supp(M INw (A, B)) (xα 1

≤z≤

(xα p

α α α α α xα 1 ∧ (y1 ∧ w1 ) ≤ z ≤ xp ∧ (yk ∧ wl }) = : : = {z ∈ supp(A) (supp(B) supp(C)) such that

xα 1

∧

(y1α

M INw (A, M AXw (B, C)) = M AXw (M INw (A, B), M INw (A, C))

∧

w1α )

≤z≤

xα p

∧

(ykα

∧

wlα }) α

= M INw (A, M INw (B, C))

The proof of the other associative law is similar. ISBN: 978-989-95079-6-8

=

α

min A ∧ min M AXw (B, C)

= {z ∈ supp(A) x1 ∧

α (y1

α

:

∨

α

α

≤ z ≤ max A ∧ max M AXw (A, B) } =

supp(M AXw (B, C)) such that

α w1 )

α

α

α

≤ z ≤ xp ∧ (yk ∨ wl )} =

( using the distributive law of real numbers) : = {z ∈ supp(A) supp(M AXw (B, C)) such that α

α

α

α

α

α

α

α

(x1 ∧ y1 ) ∨ ((x1 ∧ w1 ) ≤ z ≤ (xp ∧ yk ) ∨ (xp ∧ wl )} =

= {z ∈ supp(A) α (x1

∧

α y1 )

∨

α ((x1

= {z ∈ (supp(A) such that

α (x1

∧

α y1 )

∨

:

∧

:

(supp(B)

α w1 )

≤z≤

α (xp

supp(B))

α ((x1

∧

α w1 )

≤z≤

supp(C)) such that α

α

α

∧ yk ) ∨ (xp ∧ wl )} =

(supp(A) α (xp

∧

α yk )

∨

:

supp(C))

α (xp

α

∧ wl )} =

= M AXw (M INw (A, B), M INw (A, C))α Analogously for the other distributive law.


651

IFSA-EUSFLAT 2009 Corollary 3.12 The triplet ( A1 ,M INw ,M AXw ) is a distributive lattice, in which M INw and M AXw represent the meet and join, respectively.

References [1] J.J. Buckley and Y. Qu. Solving systems of linear fuzzy equations. Fuzzy Sets and Systems, 43(2): 33–43, 1991.

Remark 3.13 The lattice ( A1 ,M INw ,M AXw ) can also be expressed as the pair ( A1 , ), where is a partial ordering defined as: A B if and only if M INw (A, B) = A or, alternatively, A B if and only if M AXw (A, B) = B for any A, B ∈ A1 .

[2] J.J. Buckley and E.Eslami. Fuzzy plane geometry I: Points and lines. Fuzzy Sets and Systems, 86(2): 179–188, 1997.

Remark 3.14 We can also define the partial ordering in terms of α-cuts: A B if and only if min(Aα , B α ) = Aα A B if and only if max(Aα , B α ) = B α for any A, B ∈ A1 and α ∈ (0, 1], where Aα and B α are α α α α finite sets ( say , Aα = {xα 1 , · · · , xp },B = {y1 , · · · , yk }). Then, : min(Aα , B α ) = {z ∈ supp(A) supp(B) such that

[5] J. Casasnovas and J.Torrens. An Axiomatic Approach to the fuzzy cardinality of finite fuzzy sets. Fuzzy Sets and Systems, 133: 193–209, 2003.

α α α α min(xα 1 , y1 ) ≤ z ≤ min(xp , yk )} = A and

max(Aα , B α ) = {z ∈ supp(A)

supp(B) such that

α α α α max(xα 1 , y1 ) ≤ z ≤ max(xp , yk )} = B

[3] J. Casasnovas and J. Vicente Riera. On the addition of discrete fuzzy numbers. Wseas Transactions on Mathematics 5(5): 549– 554, 2006. [4] J. Casasnovas and J. Vicente Riera. Discrete fuzzy numbers defined on a subset of natural numbers. Theoretical Advances and Applications of Fuzzy Logic and Soft Computing:Advances in Soft Computing , Springer, 42: 573–582, 2007.

[6] J. Casasnovas and J.Torrens. Scalar cardinalities of finite fuzzy sets for t-norms and t-conorms. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems,11 : 599–615, 2003. [7] J. Casasnovas, J. Vicente Riera. ”Maximum and Mininum of Discrete Fuzzy numbers”. Frontiers in Artificial Intelligence and Applications: artificial intelligence research and development,IOS Press,163: 273–280, 2007. [8] D. Dubois. A new definition of the fuzzy cardinality of finite sets preserving the classical additivity property. Bull.Stud.Fuzziness Appl.(BUSEFAL), 5: 11–12, 1981. [9] D.Dubois and H.Prade. Fuzzy Sets and Systems:Theory and Applications. Academic Press, New York, 1980.

[10] D.Dubois and H.Prade (Eds.).Fundamentals of Fuzzy Sets. The Example 3.15 handbooks of Fuzzy Sets Series, Kluwer, Boston,2000. Let u = {0.4/1, 1/2, 0.8/3, 0.6/4, 0.5/5, 0.4/6, 0.3/7} and v = {0.3/4, 0.6/5, 0.7/6, 0.8/7, 1/8, 0.8/9} be two discrete [11] M.Hanns. On the implementation of fuzzy arithmetical operations for engineering problems.Fuzzy Information Processing fuzzy numbers. It is easy to prove that u v because Society(NAFIPS):462–466, 1999. M AXw (u, v) = v or equivalently M INw (u, v) = u. [12] Wen-Ling Hung and Miin-Shen Yang, Fuzzy clustering on LRtype fuzzy numbers with an application in Taiwanese tea evalCorollary 3.16 For each finite subset X of real numbers, let’s uation. Fuzzy Sets and Systems ,150: 561–577, 2005. consider the subset FX of DF N such that any element of FX has as support X. Then, the triplet (FX ,M INw ,M AXw ) is [13] George J. Klir and Yuan Bo. Fuzzy sets and fuzzy logic ( Theory and applications). Prentice Hall, 1995. a distributive lattice, in which M INw and M AXw represent the meet and join, respectively. [14] G. Mayor et al. Multi-dimensional Aggregation of Fuzzy Numbers Through the Extension Principle. Data Mining, Rought sets and Granular Computing, Eds Lin, Yao, Zadeh, PhysicaVerlag: 350-363, 2002.

Example 3.17 Let u = {0.3/3, 0.4/5, 1/7, 0.7/8} and v = {0.3/3, 1/5, 0.7/7, 0.6/8} be two discrete fuzzy numbers. It is easy to prove that v u because M AXw (u, v) = u or [15] S. Miyamoto. Fuzzy Multisets and Their Generalizations. Multisets Processing(LNCS), 2235 : 225–235, 2001. equivalently M INw (u, v) = v. [16] Daniel Rocacher, Patrick Bosc. The set of fuzzy rational num-

bers and flexible querying. Fuzzy Sets and Systems, 155(3): Remark 3.18 It is possible to find two discrete fuzzy num317-339, 2005. bers A, B ∈ A1 such that M AXw (A, B) = A and M AXw (A, B) = B or equivalenty M INw (A, B) = A and [17] D. Tadic. Fuzzy Multi-criteria approach to ordering policy M INw (A, B) = B. This means that the two discrete fuzzy ranking in a supply chain. Yugoslav Journal of Operations Research: 243–258, 2005. numbers, A and B, are not comparable. For example, let’s consider A = {0.3/4, 0.5/5, 0.8/6, 1/7, 0.9/8, 0.7/9}, B = [18] W. Voxman. Canonical representations of discrete fuzzy numThen, {0.5/6, 1/7, 0.9/8, 0.6/9, 0.5/10} ∈ A1 . bers. Fuzzy Sets and Systems, 54: 457–466, 2001. M INw (A, B) = {0.3/4, 0.5/5, 0.8/6, 1/7, 0.9/8, 0.6/9} = [19] Guixiang Wang,Cong Wu, Chunhui Zhao. Representation and A, B. Operations of discrete fuzzy numbers. Southeast Asian Bulletin of Mathematics 28: 1003–1010, 2005.

Acknowledgment We would like to express our thanks to anonymous reviewers who have contributed to improve this article. ISBN: 978-989-95079-6-8


≤). [20] Kun-lun Zhang, Kaoru Hirota. On fuzzy number lattice (R, Fuzzy Sets and Systems, 92: 113–122, 1997.

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