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On triangular norm-based generalized cardinals and singular fuzzy sets
Fuzzy Sets and Systems 133 (2003) 211 – 226
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On triangular norm-based generalized cardinals and singular fuzzy sets Krzysztof Dyczkowski, Maciej Wygralak ∗ Adam Mickiewicz University, Faculty of Mathematics and Computer Science, Matejki 48=49, 60-769 Pozna&n, Poland Received 6 August 2001; received in revised form 6 June 2002; accepted 14 June 2002
Abstract There are three basic types of triangular norm-based generalized cardinals of fuzzy sets, namely generalized FGCounts, FLCounts and FECounts. All of them are convex fuzzy sets of usual cardinal numbers. Our attention will be focused on generalized FECounts. If nonstrict Archimedean triangular norms are involved, generalized FECounts of many fuzzy sets become the zero function. Those singular fuzzy sets, being totally dissimilar to any set of any cardinality, and their properties are the subject of this paper. A relationship between singularity and fuzziness measures suited to fuzzy sets with complements induced by a strict negation will also be discussed. c 2002 Elsevier Science B.V. All rights reserved. Keywords: Cardinalities of fuzzy sets; Triangular norms; Singular fuzzy sets; Fuzziness measures
1. Introduction Two dominating streams are present in contemporary cardinality theory of fuzzy sets. The :rst one comprises so-called scalar approaches in which the cardinality of a fuzzy set is a single ordinary cardinal number or a nonnegative real number (see e.g. [3,11,14] for details and further references). An axiomatic theory of scalar cardinalities of fuzzy sets with triangular norms is constructed in [17,18]. Our attention in this paper will be focused on the other stream which o?ers a “fuzzy” perception of the cardinality of a fuzzy set. It is then a convex fuzzy set of usual cardinal numbers, a weighted family of usual cardinals, called a generalized cardinal number. This perception is more advanced and complicated from both the conceptual and operational viewpoints, but at that price it gives us in comparison with any scalar approach a considerably more complete and adequate cardinal ∗
Corresponding author. Tel.: +48-61-866-6615; fax: +48-61-866-2992. E-mail address: [email protected] (M. Wygralak).
c 2002 Elsevier Science B.V. All rights reserved. 0165-0114/02/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 3 3 2 - 9
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information. Generalized cardinals will be denoted by lowercase letters ; ; ; : : : from the beginning of the Greek alphabet. If does express the cardinality of a fuzzy set A : M → [0; 1], we write |A| = and we say that the cardinality of A is equal to . Reviews of generalized cardinals of fuzzy sets with the standard operations ∧ := min and ∨ := max can be found in [3,14–16]; as usual, := stands for “equals by de:nition”. A theory of generalized cardinals for fuzzy sets with triangular norms is constructed and developed in [19–21]. It o?ers three basic classes of such cardinals, namely generalized (or triangular norm-based) FGCounts, FLCounts and FECounts. Their short presentation is placed in Section 2. Our attention in the next sections will be restricted to generalized FECounts. If one uses them and nonstrict Archimedean triangular norms are involved, many fuzzy sets appear to be singular: their generalized FECounts become the zero function. This singularity of a fuzzy set thus means its total dissimilarity to any set of any cardinality. The phenomenon of singularity is the main subject of this paper. It will be investigated in Sections 3 and 4. The :rst formulation of the problem and some initial results can be found in [4,19]. Our further discussion requires more notation and terminology. First of all, it will be restricted to :nite fuzzy sets with triangular norm-based operations. The reason in the context of generalized cardinals is the lack of fully satisfactory extensions of triangular norms di?ering from ∧ to in:nitely, including uncountably, many arguments (cf. [9]). This restriction to :nite fuzzy sets is easy to accept because just they play the central role in applications. So, from now on, the phrase “fuzzy set” will mean “:nite fuzzy set” if not emphasized otherwise. The universe M is however assumed to be of a quite arbitrary cardinality, except for the last part of Section 4 in which M is :nite. The family of all :nite crisp sets in M will be denoted by FCS. FFS symbolizes the family of all (:nite) fuzzy sets in M . Their generalized cardinals are just some convex fuzzy sets in the set N of all nonnegative integers. The following convenient vector notation will be used for those cardinals: = (a0 ; a1 ; : : : ; ak ; (a))
with k ∈ N;
which means that (i) = ai for i6k and (i) = a for i¿k. Moreover, (a0 ; a1 ; : : : ; ak ) := (a0 ; a1 ; : : : ; ak ; (0)): So, = (0; 0:3; 0:6; 0:4; 0:1) is the notation for such that (0) = 0; (1) = 0:3; (2) = 0:6; (3) = 0:4; (4) = 0:1 and (i) = 0 for i¿4. Throughout, let A ∈ FFS with n := |supp(A)| and m := |core(A)|. Generally, single capitals in italic will denote fuzzy sets in M , whereas single bold italicized capitals will be symbols of sets. The characteristic function of D ⊂ M is denoted by 1D . Let At := {x ∈ M : A(x)¿t} with t ∈ (0; 1] (t-cut t set of A), A := {x ∈ M : A(x)¿t} with t ∈ [0; 1) (sharp t-cut set of A), and [A]i := {t: |At |¿i} with i ∈ N. We see that [A]0 = 1 and [A]i = 0 for i¿n, whereas each [A]i with 0¡i6n is the ith element in the nonincreasingly ordered sequence of all positive membership values A(x), including their possible repetitions. The de:nition of [A]i remains reasonable if A is in:nite and=or i is a trans:nite cardinal. Let us recollect a few de:nitions and facts from the theory of triangular norms which will be useful in the next sections. An exhaustive presentation of that theory can be found in [6,9]. A binary operation t in the closed unit interval [0; 1] is called a triangular norm (t-norm, in short) if t is commutative, associative, nondecreasing in the :rst and, hence, in each argument, and has 1 as unit. A binary operation s in [0; 1] which ful:ls the :rst three properties and has 0 as unit is
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said to be a triangular conorm (t-conorm). A continuous t-norm t is called Archimedean if ata¡a for each a ∈ (0; 1). One says that an Archimedean t-norm is strict whenever it is strictly increasing over (0; 1) × (0; 1). The algebraic t-norm ta with ata b := ab is a typical example of a strict t-norm, whereas the Lukasiewicz t-norm tL with atL b := 0 ∨ (a + b − 1) is a typical nonstrict Archimedean t-norm. The class of all nonstrict Archimedean t-norms will be denoted by Natn. Archimedean t-norms seem to be especially important and interesting. One of reasons is that, for each continuous t, either t = ∧ or t is Archimedean or t is the ordinal sum of a family of Archimedean t-norms; the de:nition of ordinal sums can be found e.g. in [6,9]. Moreover, Archimedean t-norms do have a nice characterization theorem by AczKel-Ling [1,10]; see also [6,9,13]. Theorem 1.1. A t-norm t is Archimedean i9 there exists a strictly decreasing and continuous function g : [0; 1] → [0; ∞] such that g(1) = 0 and ∀a; b ∈ [0; 1]: atb = g−1 (g(0) ∧ (g(a) + g(b))): Moreover, t is strict i9 g(0) = ∞. The function g satisfying the conditions of Theorem 1.1 is called an (additive) generator of t. If g(0) = 1; g is said to be normed. Since generators of Archimedean t-norms are uniquely determined up to a positive factor, we can restrict ourselves to normed generators when considering t-norms from Natn. By mathematical induction, k a1 ta2 t : : : tak = g−1 g(0)∧ g(ai ) (1.1) i=1
for each Archimedean t with generator g and each system of numbers a1 ; a2 ; : : : ; ak ∈ [0; 1] with k ∈ N; in particular, a1 ta2 t : : : tak = 1 for k = 0. Hence a1 ta2 t : : : tak ¿ 0 i?
k
g(ai ) ¡ g(0)
(1.2)
i=1
and, thus, a1 ta2 t : : : tak ¿0 ⇔ a1 ; a2 ; : : : ; ak ¿0 when t is strict, which cannot be extended to t-norms from the class Natn. So, strict t-norms do not have zero divisors, whereas nonstrict Archimedean ones do have, i.e. atb = 0 is then possible for a; b¿0. In other words, that t ∈ Natn has zero divisors does mean that t has some inertia in attaining positive values: each positive argument of t is treated as zero whenever the other one is not suMciently “large”. For instance, 0:4tL 0:5 = 0 but 0:4tL 0:7¿0. Clearly, this feature of a nonstrict Archimedean t-norm can be useful and desired in many applications. Each nonincreasing function v : [0; 1] → [0; 1] with v(0) = 1 and v(1) = 0 will be called a negation. The negation v∗ such that v∗ (a) := 0 for each a¿0 is thus the smallest possible negation, whereas v∗ with v∗ (a) := 1 for each a¡1 is the largest one. Strictly decreasing and continuous negations are called strict negations. Their class will be denoted by Sng. If a strict negation is involutive (v(v(a)) = a for each a), it is said to be a strong negation. A fundamental example of a strong negation is the Lukasiewicz negation vL with vL (a) := 1 − a. Each t-norm t does induce a negation
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vt de:ned as ∀a ∈ [0; 1]: vt (a) :=
{c ∈ [0; 1]: atc = 0}:
(1.3)
One has vt = v∗ whenever t is strict. If t ∈ Natn and g is its generator, then vt is strong and, by Theorem 1.1, vt (a) = g−1 (g(0) − g(a)):
(1.4)
It is clear that v ∈ Sng has a unique :xed point a∗ ∈ (0; 1). For instance, a∗ = 0:5 for v = vL . Generally, v(a∗ ) = a∗ and ∀a = a∗ : v(a) ¡ a∗ ¡ a ⊥ a ¡ a∗ ¡ v(a); i.e. ∀a ∈ [0; 1]: a ∨ v(a) ¿ a∗ ;
(1.5)
where ⊥ denotes the inclusive disjunction connective. It is a routine task to check that ∀a ∈ [0; 1]: atvt (a) = 0
(1.6)
for t ∈ Natn. By (1.6), if a∗ is the :xed point of vt with a nonstrict Archimedean t, we have the following: a∗ tvt (a∗ ) = vt (a∗ )tvt (a∗ ) = a∗ ta∗ = 0; ∀a 6 a∗ : ata = 0; ∀a ¿ a∗ : vt (a)tvt (a) = 0:
(1.7)
That :xed point of vt is equal to g−1 (0:5g(0)), i.e. a∗ = g−1 (0:5)
(1.8)
whenever g is normed, which follows from (1.4). Consider an example in which t ∈ Natn is a Schweizer t-norm, i.e. t = tS;p for p¿0 and atS;p b := [0 ∨ (ap + bp − 1)]1=p
(1.9)
with normed generator g(a) = 1 − ap . So, g−1 (y) = (1 − y)1=p and, by (1.4), vt (a) = (1 − ap )1=p . The :xed point of vt is thus a∗ = 0:51=p . Obviously, t = tL and vt = vL for p = 1. Use t-norms, t-conorms and negations to introduce operations on fuzzy sets via pointwise de:nitions: the intersection A ∩t B of A and B induced by a t-norm t with (A ∩t B)(x) := A(x)tB(x), the sum A ∪s B induced by a t-conorm s with (A ∪s B)(x) := A(x)sB(x), and the complement Av of A induced by a negation v with Av (x) := v(A(x)): ∩ := ∩∧ ; ∪ := ∪∨ and := v with v := vL are the standard operations on fuzzy sets.
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2. Generalized cardinals Let A ∈ FFS and = |A|. At the beginning of the 1980s the following three main types of generalized cardinals were proposed: (a1)
(k) := [A]k
for k ∈ N: is then called the FGCount of A (see [22,23]). In the language of many valued logic, (k) expresses a degree to which A has at least k elements. So, this kind of does form a lower cardinal evaluation of A. (b1)
(k) := 1 − [A]k+1
with k ∈ N (see [22]). What we now get as is the FLCount of A which forms an upper cardinal evaluation of A because (k) is a degree to which A contains at most k elements. (c1)
(k) := [A]k ∧ (1 − [A]k+1 )
for k ∈ N: is then called the FECount of A (see [22,14,16]). This intersection of the lower and upper cardinal evaluations can be viewed as a “proper” generalized cardinal of A: (k) becomes a degree to which A has exactly k elements. De:nitions (a1), (b1) and (c1) remain reasonable if A is in:nite (see [15]). On the other hand, as pointed out in [19,21], they are suitable only for fuzzy sets with the standard operations. Their appropriate generalizations to fuzzy sets with triangular norm-based operations are presented below; t is an arbitrary t-norm and v denotes a negation. (a2)
(k) := [A]1 t[A]2 t : : : t[A]k
with k ∈ N. In the vector notation, the resulting generalized FGCount of A can be written as = (1; [A]1 ; [A]1 t[A]2 ; : : : ; [A]1 t[A]2 t : : : t[A]n ): We see that (k) is a degree to which A contains (at least) i elements for each i6k. The quanti:cation “for each i6k” is here realized by means of t. Clearly, collapses to the FGCount of A if t = ∧. (b2)
(k) := v([A]k+1 )tv([A]k+2 )t : : : = v([A]k+1 )tv([A]k+2 )t : : : tv([A]n )
for k ∈ N, i.e. (k) is a degree to which A contains at most i elements for each i¿k. Thus, = (v([A]1 )tv([A]2 )t : : : tv([A]n ); v([A]2 )tv([A]3 )t : : : tv([A]n ); .. . v([A]n−1 )tv([A]n ); v([A]n ); (1):
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is now called the generalized FLCount of A. t = ∧ with v = vL gives the FLCount of A. (c2)
(k) := [A]1 t[A]2 t : : : t[A]k tv([A]k+1 )t : : : tv([A]n )
with k ∈ N, i.e. = (v([A]1 )tv([A]2 )t : : : tv([A]n ); [A]1 tv([A]2 )t : : : tv([A]n ); [A]1 t[A]2 tv([A]3 )t : : : tv([A]n ); .. . [A]1 t[A]2 t : : : t[A]n−1 tv([A]n ); [A]1 t[A]2 t : : : t[A]n : This generalized FECount of A, induced by t and v, is the t-based intersection of the generalized cardinals from (a2) and (b2). It is always convex. If t = ∧ and v = vL ; collapses to the FECount. Another worth mentioning particular case is that with t = ∧ and v = v∗ . Then = (0; 0; : : : ; 0; 1; [A]m+1 ; [A]m+2 ; : : : ; [A]n ) with m zeros; i.e. becomes the generalized cardinal of A due to Dubois [3]. GFEt; v will denote the family of all generalized FECounts induced by a t-norm t and a negation v. A detailed study of generalized FGCounts, FLCounts and FECounts is placed in [19,20]. Our further discussion in this paper will be limited to generalized FECounts with an additional restriction to t ∈ Natn and v ∈ Sng. As we will see, an interesting cardinal singularity of many fuzzy sets is connected with such generalized FECounts. 3. Singular fuzzy sets Let A; B ∈ FFS; |A| = ; |B| = and ; ∈ GFEt; v with v ∈ Sng. If t is strict or t = ∧, a very natural equipotency relation∼ful:lling the requirement = ⇔ A ∼ B corresponds to generalized FECounts, namely (see [19,20]): A ∼ B ⇔ ∀k ∈ N: [A]k = [B]k ⇔ ∀t ∈ (0; 1]: |At | = |Bt | ⇔ ∀t ∈ [0; 1): |At | = |Bt |: A ∼ B thus means that A and B are identical up to the permutation of their positive membership values, including possible repetitions, which is equivalent to the usual equipotency of all the corresponding t-cut sets (sharp or not) of A and B. Consequently, two equipotent fuzzy sets do have equipotent cores and equipotent supports. Since t does not have zero divisors, we get = (0), i.e. always di?ers from the zero function. Let us stress in this context that = 1{k } whenever A collapses
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to a k-element set. So, in particular, the generalized FECount of the empty fuzzy set 1∅ is equal to 1{0} ∈ [0; 1]N . The situation is quite di?erent if t ∈ Natn. t then has zero divisors and, therefore, an equipotency relation similar to that for a strict t-norm, including ∧, cannot be constructed. The equality = may hold for A and B having di?erent cardinalities of the corresponding t-cut sets, cores and supports. What is more, = (0) becomes possible. The generalized cardinal (0) will be called the
k ¿ s ⇒ [A]k 6 a∗ ⇒ v([A]k ) ¿ [A]k :
Consequently, for each k = s, we have (k) = [A]1 t : : : t[A]k tv([A]k+1 )t : : : tv([A]n ) 6 [A]1 t : : : t[A]s tv([A]s+1 )t : : : tv([A]n ) = (s): Thesis (b) is an immediate consequence of (a). The procedure of checking whether (k) = 0 for each k can thus be reduced to verifying the single condition (s) = 0. This is important because the general form of (k) is rather complex.
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Theorem 3.2. (s) = 0 i9
s
i=1
g([A]i ) +
n
i=s+1
g(v([A]i ))¿1.
Proof. The thesis immediately follows from (1.2) applied to the general formula for (s) given in (c2) in Section 2. Since g and v are decreasing, Theorem 3.2 says that A is nonsingular i? the membership values [A]i with i6s are suMciently near to 1 (are suMciently larger than a∗ ) and, moreover, the [A]i ’s with s + 16i6n are suMciently smaller than the :xed point a∗ . Intuitively speaking, = (0) is thus possible only if A is “similar” to an s-element set (see also Example 3.4). This suggests a relationship between singularity=nonsingularity of a fuzzy set and its fuzziness, which will be investigated in Section 4. Measuring that fuzziness, however, we have to use a fuzziness measure constructed in a special way: a maximal fuzziness should be assigned to the fuzzy set identically equal to a∗ (see Example 3.4(iii)). Corollary 3.3. Let s¿1 be such that a∗ ¡g−1 (1=s). If [A]1 6g−1 (1=s), then A is singular. Proof. Indeed, s¿1 implies [A]1 ¿a∗ , which means that [A]1 6g−1 (1=s) is possible only if s ful:ls ∗ −1 −1 the condition s a ¡g (1=s). s The inequality [A]1 6g (1=s) then leads to g([A]1 )¿1=s and, consequently, i=1 g([A]i ) ¿ i=1 g([A]1 )¿1. By Theorem 3.2, we get (s) = 0, i.e. A is singular. Although Corollary 3.3 looks rather technical, it is very useful. For instance, let t = tL and v = vL , i.e. a∗ = 0:5 and g(a) = 1 − a. The condition a∗ ¡g−1 (1=s) is now equivalent to s¿3. The corollary says that if s¿3 and [A]1 61 − 1=s, then A is singular. A with, say, s = 10 is thus singular whenever [A]1 60:9. Example 3.4. In the light of Theorem 3.2, as already mentioned, a nonsingular fuzzy set A must be “similar” to an s-element set. For better understanding, let us consider a few related instances. (i) Again, assume t = tL and v = vL with a∗ = 0:5. Then A = 0:8=x1 + 0:2=x2 + 0:1=x3 + 0:1=x4 is nonsingular because s = 1 and (s) = 0:8tv(0:2)tv(0:1)tv(0:1)¿0. We see that A is really similar to a 1-element set. (ii) Let us use the t and v from (i), and take A = 0:95=x1 + 0:95=x2 + 0:95=x3 + 0:5=x4 + 0:4=x5 : Now s = 3 and (s) = 0:95t0:95t0:95t0:5t0:6 = 0, i.e. A is singular. Its similarity to a 3-element set seems to be low because the membership grades of x4 and x5 are too large. More precisely speaking, they are not suMciently smaller that a∗ . The fuzzy set B = 0:9=x1 + 0:9=x2 + 0:9=x3 + 0:2=x4 + 0:1=x5 ; a fuzzy subset of A, is nonsingular although sB; v = 3, too. The reason is that B is more similar to a 3-element set than A.
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(iii) Take t = tS;4 and v = vt , i.e. a∗ = 0:51=4 ≈ 0:84 (see (1.8) and (1.9)). Consider again the fuzzy set A from (ii). As previously, we have s = 3 but this time (s)¿0 and, hence, A is nonsingular. The reason is that a∗ always plays the role of a threshold point: the membership values ¿a∗ are considered to be more or less similar to the maximal membership value 1, whereas those 6a∗ are considered to be more or less similar to the minimal membership value 0. In this example, we have a∗ ≈ 0:84. The membership values of x4 and x5 are now considerably smaller than the threshold value and, consequently, A becomes more similar to a 3-element set than in (ii). It is easy to notice that each singular fuzzy set has a nonsingular proper fuzzy subset, at least the empty fuzzy set 1∅ . As mentioned in Example 3.4(ii), this is because a proper fuzzy subset of A may be more similar to an s-element set than A itself. Theorem 3.5. If A is singular and B is such that A ∩ B = 1∅ , then A ∪ B is singular, too. Proof. Assume A is singular and A ∩ B = 1∅ . Put ci := [A ∪ B]i for i¿1; r := sA∪B; v = max{i: ci ¿a∗ }; n∗ := |supp(B) and l := |supp(A ∪ B)| = n + n∗ . As previously, s := max{i: [A]i ¿a∗ }. The sequence c1 ; c2 ; : : : ; cl is a result of joining the sequences [A]1 ; [A]2 ; : : : ; [A]n and [B]1 ; [B]2 ; : : : ; [B]n∗ into one nonincreasing sequence. Hence r¿s and, by Theorem 3.2, r i=1
g(ci ) +
l i=r+1
g(v(ci )) ¿
s i=1
g([A]i ) +
n
g(v([A]i )) ¿ 1;
i=s+1
i.e. A ∪ B is singular. The property of singular fuzzy sets described in Theorem 3.5 will be called the absorption property. This property is also reOected in the results generated by the triangular norm-based extension principle ( + )(k) := {(i)t(j): i + j = k}: Indeed, we get (0) + = (0) for each ∈ GFEt; v with t ∈ Natn and v ∈ Sng. Let i:k := ik . Corollary 3.6. If a fuzzy set ai:1 =xi:1 +ai:2 =xi:2 +· · ·+ai:k =xi:k is singular for some indices 16i:1¡i:2¡ · · · ¡i:k6n with 2 6 k¡n, then A = a1 =x1 + a2 =x2 + · · · + an =xn is singular, too. Proof. An immediate consequence of Theorem 3.5 combined with the trivial fact that each fuzzy set is a sum of disjoint singletons. So, a fuzzy set is singular whenever its proper fuzzy subset resulting from a reduction of the support is singular. On the other hand, that A is singular does not generally imply the existence of a singular fuzzy subset ai:1 =xi:1 + ai:2 =xi:2 + · · · + ai:k =xi:k of A. An instance is A = 0:6=x1 + 0:6=x2 + 0:6=x3 with t = tL and v = vL . A is then singular but |0:6=xi + 0:6=xj | = (0; 0; 0:2) = (0). The suMcient condition in Corollary 3.6 for A to be singular does form a useful criterion because in practice it is easy to check if a fuzzy set with a small support having, say, 2 or 3 elements is singular. The inequality k¿2 reOects the property that a singleton cannot be singular as |a=x| = (v(a); a) for
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a ∈ (0; 1]. By the way, it is usually easy to construct a singular fuzzy set with a support of cardinality ¿3. The task of :nding a singular fuzzy set supported by a 2-element set is more diMcult and can even be unrealizable. The conditioning factor is the form of v (see Theorem 3.9 and subsequent remarks). Generally, the larger the support the easier the construction of a singular fuzzy set (cf. Corollary 3.3). A group of useful properties of singular fuzzy sets can be formulated for v = vt (see (1.3) and (1.4)). The :rst one o?ers a simpli:cation of the criterion from Theorem 3.2. We still use the assumptions and notation established just before Theorem 3.1. Theorem 3.7. If v = vt , then (s) = 0 i9
s
n
g([A]i ) −
i=1
g([A]i ) ¿ s − n + 1:
i=s+1
Proof. It follows from Theorem 3.2 and (1.4). The theorem above allows to formulate explicit singularity criteria for concrete t-norms with negations induced by them. Let us give two instances. (i) t = tS;p with p¿0 and v = vt (see (1.9)). Then (s) = 0 ⇔
s
([A]i )p −
i=1
n
([A]i )p 6 s − 1:
i=s+1
So, if t = tL and v = vL , we get (s) = 0 ⇔
s i=1
[A]i −
n
[A]i 6 s − 1:
i=s+1
(ii) t = tW; and v = vt , where tW; is the Weber t-norm with parameter ¿−1 (see [13]): atW; b := 0 ∨ ((a + b − 1 + ab)=(1 + )) and g(a) = 1 − ln(1 + a)= ln(1 + )
for 0 =
¿ −1:
Routine transformations do lead to the equivalence (s) = 0 ⇔
s i=1
s −1
(1 + [A]i ) 6 (1 + )
n
(1 + [A]i )
i=s+1
for ¿0; if −1¡ ¡0, the inequality symbol 6 should be replaced by ¿. The case omitted because tW; 0 = tL .
= 0 is here
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Theorem 3.8. If v = vt and [A]s+1 = [A]s+2 = a∗ , then (s) = 0. Proof. Indeed, applying (1.7) to the formula (s) = [A]1 t : : : t[A]s tv([A]s+1 )tv([A]s+2 )t : : : tv([A]n ) we get (s) = 0 whenever [A]s+1 = [A]s+2 = a∗ and v = vt . So, if A attains the value a∗ (at least) at two points, then A is singular provided that vt is used as negation. This means that nonsingularity of A implies |{x: A(x) = a∗ }|61. In particular, A with t = tS;p is singular whenever |{x: A(x) = 0:51=p }|¿2. Let us notice that Theorem 3.8 can also be derived from Theorem 3.2 combined with (1.8). As remarked earlier in this section, singletons are never singular. The following property refers to the question of constructability of a singular fuzzy set having a 2-element support. Theorem 3.9. For v = vt , the fuzzy set a=x + b=y with a; b¿0 is singular i9 a = b = a∗ . Proof. Let v = vt . Choose arbitrary a; b ∈ (0; 1] and suppose that = |A| for A = a=x + b=y with any x; y ∈ M . So, n = 2. Without loss of generality, one can assume that a¿b. By (c2) from Section 2, we have = (v(a)tv(b); atv(b); atb): (a) If a¿b¿a∗ , then s = 2 and, by Theorem 3.7, = (0) ⇔ (2) = 0 ⇔ g(a) + g(b)¿1. However (1.8) says that a∗ = g−1 (0:5), which implies g(a)6g(b)¡g(a∗ ) = 0:5 and, hence, g(a) + g(b)¡1. Consequently, A cannot be singular. (b) If a¿a∗ ¿b, then g(a)¡g(a∗ ) = 0:56g(b), i.e. g(a) − g(b)¡0. Again, A cannot be singular because s = 1 and = (0) ⇔ (1) = 0 ⇔ g(a) − g(b)¿0. (c) Finally, suppose a∗ ¿a¿b¿0. This time g(b)¿g(a)¿0:5. On the other hand, s = 0 and (0) = 0 ⇔ g(a)+g(b)61. A is thus singular only if g(a) = g(b) = 0:5, which means that a = b = a∗ . This completes the proof. If, say, t = tL and v = vL , the unique singular fuzzy set a=x + b=y is thus that with a = b = 0:5. Theorem 3.9 does not work for v = vt . For instance, a=x + b=y is always nonsingular for t = tL and v(a) := 1 − a2 . There exist in:nitely many singular fuzzy sets a=x + b=y if one uses the Lukasiewicz t-norm accompanied by the negation v(a) := 1 − a0:5 . 4. Singularity and fuzziness of a fuzzy set Let us investigate the question of singularity=nonsingularity of a fuzzy set in connection with its fuzziness. Our starting point is a consideration concerning fuzzy sets being constant on their supports. As previously, t ∈ Natn with normed generator g; v ∈ Sng and a∗ is its :xed point. Generalizing the notation 1D , by aD with a ∈ (0; 1] and a set D ⊂ M we denote a fuzzy set such that aD (x) := a if x ∈ D, else aD (x) := 0.
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Theorem 4.1. For each a ∈ (0; 1] and each nonempty ?nite D, aD is singular i9 a ∨ v(a) 6 g−1 (1=|D|): Proof. Indeed, :x an arbitrary a ∈ (0; 1] and a :nite set D = ∅ and put := |aD |. Let p := |D| and r := max{i: [aD ]i ¿a∗ }. Obviously, [aD ]i = a for 16i6p. Suppose a ∈ (a∗ ; 1]. Then r = p and, by Theorem 3.2 and (1.5), (r) = 0 ⇔
p
g(a) ¿ 1 ⇔ v(a) ¡ a 6 g−1 (1=p):
i=1
If a ∈ (0; a∗ ], then r = 0 and (r) = 0 ⇔
p
g(v(a)) ¿ 1 ⇔ a 6 v(a) 6 g−1 (1=p):
i=1
So, in each case, aD singular i? a ∨ v(a) 6 g−1 (1=p), which completes the proof. Since g−1 (1) = 0, Theorem 4.1 supplies us with an additional con:rmation that a singleton cannot be singular. Corollary 4.2. Let |D|¿2 and a ∈ (0; 1]. (a) aD is singular i9 v−1 (g−1 (1=|D|))6a6g−1 (1=|D|). (b) If aD is singular and |D|¡|E|, then aE is singular. (c) If g−1 (1=|D|)¡a∗ , then aD is nonsingular. Proof. (a) aD is singular i? a ∨ v(a)6g−1 (1=|D|), which means that a6g−1 (1=|D|) and a¿v−1 (g−1 (1=|D|)). (b) |D|¡|E| implies g−1 (1=|D|)¡g−1 (1=|E|). Thus, using again Theorem 4.1, we conclude that aE is singular if so is aD . (c) The inequality g−1 (1=|D|)¡a∗ in combination with (1.5) does lead to g−1 (1=|D|)¡a ∨ v(a), which implies that aD is nonsingular for each a ∈ (0; 1]. A group of useful conclusions can be formulated for a = a∗ . By the way, we see that the generalized FECount of aD∗ is always a function being constant on its support because v(a∗ ) = a∗ (see (c2) in Section 2). Corollary 4.3. Let |D|¿2. (a) aD∗ is singular i9 a∗ 6g−1 (1=|D|). (b) aD∗ is singular i9 |D|¿1=g(a∗ ). Proof. (a) immediately follows from Theorem 4.1. (b) is a transformed version of (a).
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Clearly, the variant (a) of Corollary 4.3 is convenient if D and t are :xed and one looks for a negation v such that aD∗ is singular=nonsingular. (b) says that if t and v and, thus, a∗ are given, then aD∗ is always singular whenever D is suMciently large. Corollary 4.4. If v = vt , then aD∗ with |D|¿2 is singular. Proof. Straightforward consequence of (1.8) and Corollary 4.3(b). Notice that the above corollary could also be derived from Theorem 3.8. Consider an example in which t = tS;p with p¿0 and v = vt . Applying Corollary 4.2(a), we conclude that aD with a ∈ (0; 1] and |D|¿2 is then singular i? (1=|D|)1=p 6 a 6 (1 − 1=|D|)1=p : Let us introduce the following partial ordering relation for A; B ∈ FFS with v ∈ Sng: A 6v B ⇔ ∀x ∈ M : A(x) 6 B(x) 6 a∗ ⊥ a∗ 6 B(x) 6 A(x): If A6v B, we say that B is fuzzier than A or, dually, A is sharper than B. We easily notice that A6v a∗supp(A) and, generally, the fuzziest fuzzy set with a given support D is aD∗ . The relation 6v is considered in a bit di?erent terminological framework in [8]; its particular case with v = vL and a∗ = 0:5 is used e.g. in [11]. Theorem 4.5. If A is singular and A6v B, then B is singular, too. Proof. Assume t ∈ Natn; v ∈ Sng and A; B ∈ FFS are such that A6v B. Let = |A| and = |B|. Suppose A is singular, i.e. (see Theorem 3.1) (s) = [A]1 t : : : t[A]s tv([A]s+1 )t : : : tv([A]n ) = 0: Let n∗ := |supp(B)| and r := max{i: [B]i ¿a∗ }. By the de:nition of 6v ; s¿r and n6n∗ . Since [B]j = a∗ for each r¡j6s, we have v([B]j ) = [B]j for such j’s. So, (r) = [B]1 t : : : t[B]r tv([B]r+1 )t : : : tv([B]s )tv([B]s+1 )t : : : tv([B]n∗ ) = [B]1 t : : : t[B]s tv([B]s+1 )t : : : tv([B]n∗ ) 6 [A]1 t : : : t[A]s tv([A]s+1 )t : : : tv([A]n∗ ) = [A]1 t : : : t[A]s tv([A]s+1 )t : : : tv([A]n ) =0 because [B]i 6[A]i for i6s; [A]i 6[B]i for s¡i6n∗ , and v([A]i ) = 1 for i¿n. Consequently, (r) = 0 and, hence, B is singular. Thus, if A6v B and B is nonsingular, then A is also nonsingular. In other words, a fuzzy set being fuzzier than a singular fuzzy set is singular, too, whereas a fuzzy set which is sharper than a
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nonsingular one is also nonsingular. Another corollary following from Theorem 4.5 is that if A is singular, then so is aD∗ with any D such that |D|¿n. So, A with n6|D| is nonsingular whenever aD∗ with a :nite D is nonsingular. The relation 6v draws our attention to fuzziness measures answering the question how fuzzy a fuzzy set is. Recollect that there are two general classes of those measures: entropy and energy type fuzziness measures. They describe deviation of a fuzzy set from a set and from a 1-element set, respectively (see e.g. [7,11] and a review in [12]). In this paper, we restrict attention to fuzziness measures understood as entropy type ones. From now on, we shall assume that the universe M is ∗ is also :nite (see the axiom (A2) below). As previously, v ∈ Sng and :nite. This guarantees that aM a∗ denotes its unique :xed point. De)nition 4.6. A mapping E : FFS → [0; ∞) is a fuzziness measure i? the following postulates are satis:ed by each A; B ∈ FFS: (A1) (A2) (A3) (A4) (A5)
E(A) = 0 i? A ∈ FCS, ∗ , E reaches its unique maximum at aM v A6 B implies E(A)6E(B), E(A) = E(Av ), E(A ∪ B) + E(A ∩ B) = E(A) + E(B).
The number E(A) is called the fuzziness measure of A. The classical de:nition of a fuzziness measure given in [11] is a particular case of De:nition 4.6 with v = vL and a∗ = 0:5. That particular formulation is suitable only if one uses the standard complement A induced by the Lukasiewicz negation. De:nition 4.6 does form its natural generalization to complements induced by an arbitrary strict negation. Postulates (A1)–(A3) are also used in [8] for de:ning a general class of fuzziness measures; as shown therein, that class is identical to a class of fuzziness measures de:ned via distance between A and Av . One can say that if A ∈ FFS is nonsingular, then its fuzziness measure E(A) is “small” in comparison with E(a∗supp(A) ). Clearly, this is valid provided that |supp(A)|¿1=g(a∗ ). By Corollary 4.3(b), |supp(A)|¡1=g(a∗ ) implies that a∗supp(A) is nonsingular and, hence, A is nonsingular, too, although E(A) is possibly near to E(a∗supp(A) ). The fuzziness measures from De:nition 4.6 can be characterized in a convenient way by means of :nite sums of transformed membership values. Theorem 4.7. E : FFS → [0; ∞) is a fuzziness measure i9 E(A) = f(A(x)) = f(A(x)) x ∈M
x∈supp(A)
for each A ∈ FFS, where f : [0; 1] → [0; ∞) is such that (P1) f(0) = f(1) = 0 and f(a)¿0 for a ∈ (0; 1), (P2) f(a)¡f(a∗ ) for a = a∗ ,
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(P3) f is nondecreasing on [0; a∗ ] and nonincreasing on [a∗ ; 1], (P4) f(a) = f(v(a)) for a ∈ [0; 1]. Proof. (⇒) Suppose E is a fuzziness measure satisfying the axioms (A1)–(A5). By virtue of(A1) and (A5), if A ∩ B = 1∅ , then E(A ∩ B) = 0 and, hence, E(A ∪ B) = E(A)+E(B). So, E(A) = E( x∈M A(x)=x) = x∈M E(A(x)=x). Since E : FFS → [0; ∞) and it seems to be reasonable to accept an implicit assumption that E(a=x) does not depend on x, we get E(a=x) = f(a) for a ∈ [0; 1] with f : [0; 1] →[0; ∞) such that f(0) = f(1) = 0 and f(a)¿0 for a ∈ (0; 1),which follows from (A1). The function f thus satis:es (P1) and, moreover, we have E(A) = x∈M f(A(x)) = x∈supp(A) f(A(x)). (A2) implies (P2). Indeed, if f(a)¿f(a∗ ) for some a = a∗ , then E(aM ) = x∈M f(a)¿ x∈M ∗ ), which contradicts (A2). f(a∗ ) = E(aM By (A3), a=x6v b=x with a; b ∈ [0; 1] does imply E(a=x)6E(b=x), which means that f(a)6f(b). As a 6 b6a∗ or a∗ 6b6a, we conclude that f ful:ls (P3). Finally, (A4) and (A1) do lead to f(a) = E(a=x) = E(v(a)=x ∪ 1{y ∈ M : y = x} ) = E(v(a)=x) + E(1{y ∈ M : y = x} ) = E(v(a)=x) = f(v(a)) for each a ∈ [0; 1], i.e. (P4) is satis:ed. (⇐) Suppose E(A) = x∈M f(A(x)) for each A ∈ FFS with f ful:lling (P1)–(P4). Let us show that E is a fuzziness measure in the sense of De:nition 4.6. By (P1), E(A) = 0 ⇔ x∈M f(A(x)) = 0 ⇔ ∀x ∈ M : f(A(x)) = 0 ⇔ ∀x ∈ M : A(x) (A1) holds true. ∈ {0; 1}, i.e. ∗ ) whenever A = a∗ , i.e. The postulate (P2) implies E(A) = x∈M f(A(x))¡ x∈M f(a∗ ) = E(aM M (A2) is ful:lled. In the same simple way one shows that (P3) and (P4), respectively, do imply (A3)and (A4), respectively. f(b) for a; b ∈ [0; 1], we ob Since f(a ∨ b) + f(a ∧ b) = f(a) + tain x∈M f((A ∪ B)(x)) + x∈M f((A ∩ B)(x)) = x∈M f(A(x)) + x∈M f(B(x)). The valuation property (A5) is thus satis:ed, which completes the proof. The characterization formulated in Theorem 4.7 does generalize that given in [5] for the particular case of fuzziness measures with v = vL and a∗ = 0:5 (see also [2]). References [1] J. AczKel, Sur des operations de:nies pour des nombres rKeels, Bull. Soc. Math. France 76 (1949) 59–64. [2] C. Alsina, E. Trillas, Sur les measures du degrKe de Oou, Stochastica 3 (1979) 81–84. [3] D. Dubois, H. Prade, Fuzzy cardinality and the modeling of imprecise quanti:cation, Fuzzy Sets and Systems 16 (1985) 199–230. [4] K. Dyczkowski, M. Wygralak, On cardinality and singular fuzzy sets, in: B. Reusch (Ed.), Computational Intelligence—Theory and Applications, Lecture Notes in Computer Science, vol. 2206, Springer, Berlin, Heidelberg, New York, 2001, pp. 261–268. [5] B.R. Ebanks, On measures of fuzziness and their representations, J. Math. Anal. Appl. 94 (1983) 24–37. [6] S. Gottwald, Many-valued logic and fuzzy set theory, in: U. H˝ohle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets. Logic, Topology, and Measure Theory, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999, pp. 5–89. [7] S. Gottwald, E. Czogala, W. Pedrycz, Measures of fuzziness and operations with fuzzy sets, Stochastica 6 (1982) 187–205. [8] M. Higashi, G.J. Klir, On measures of fuzziness and fuzzy complements, Internat. J. Gen. Systems 8 (1982) 169–180. [9] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic Publishers, Dordrecht, Boston, London, 2000. [10] C.H. Ling, Representation of associative functions, Publ. Math. Debrecen 12 (1965) 189–212.
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[11] A. De Luca, S. Termini, On some algebraic aspects of the measures of fuzziness, in: M.M. Gupta, E. Sanchez (Eds.), Fuzzy Information and Decision Processes, North-Holland, Amsterdam, 1982, pp. 17–24. [12] N.R. Pal, J.C. Bezdek, Measuring fuzzy uncertainty, IEEE Trans. Fuzzy Systems 2 (1994) 107–118. [13] S. Weber, A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms, Fuzzy Sets and Systems 11 (1983) 115–134. [14] M. Wygralak, Fuzzy cardinals based on the generalized equality of fuzzy subsets, Fuzzy Sets and Systems 18 (1986) 143–158. [15] M. Wygralak, Vaguely De:ned Objects. Representations, Fuzzy Sets and Nonclassical Cardinality Theory, Kluwer Academic Publishers, Dordrecht, Boston, London, 1996. [16] M. Wygralak, Questions of cardinality of :nite fuzzy sets, Fuzzy Sets and Systems 102 (1999) 185–210. [17] M. Wygralak, Triangular operations, negations, and scalar cardinality of a fuzzy set, in: L.A. Zadeh, J. Kacprzyk (Eds.), Computing with Words in Information=Intelligent Systems, vol. 1, Foundations, Physica-Verlag, Heidelberg, New York, 1999, pp. 326 –341. [18] M. Wygralak, An axiomatic approach to scalar cardinalities of fuzzy sets, Fuzzy Sets and Systems 110 (2000) 175–179. [19] M. Wygralak, Fuzzy sets with triangular norms and their cardinality theory, Fuzzy Sets and Systems 124 (2001) 1–24. [20] M. Wygralak, Cardinalities of Fuzzy Sets, Physica-Verlag, Heidelberg, New York, in preparation. [21] M. Wygralak, D. Pilarski, FGCounts of fuzzy sets with triangular norms, in: R. Hampel, M. Wagenknecht, N. Chaker (Eds.), Fuzzy Control. Theory and Practice, Physica-Verlag, Heidelberg, New York, 2000, pp. 121–131. [22] L.A. Zadeh, A computational approach to fuzzy quanti:ers in natural languages, Comput. Math. Appl. 9 (1983) 149–184. [23] L.A. Zadeh, From computing with numbers to computing with words—From manipulation of measurements to manipulation of perceptions, IEEE Trans. Circuits and Systems—I: Fund. Theory Appl. 45 (1999) 105–119.
www.elsevier.com/locate/fss
On triangular norm-based generalized cardinals and singular fuzzy sets Krzysztof Dyczkowski, Maciej Wygralak ∗ Adam Mickiewicz University, Faculty of Mathematics and Computer Science, Matejki 48=49, 60-769 Pozna&n, Poland Received 6 August 2001; received in revised form 6 June 2002; accepted 14 June 2002
Abstract There are three basic types of triangular norm-based generalized cardinals of fuzzy sets, namely generalized FGCounts, FLCounts and FECounts. All of them are convex fuzzy sets of usual cardinal numbers. Our attention will be focused on generalized FECounts. If nonstrict Archimedean triangular norms are involved, generalized FECounts of many fuzzy sets become the zero function. Those singular fuzzy sets, being totally dissimilar to any set of any cardinality, and their properties are the subject of this paper. A relationship between singularity and fuzziness measures suited to fuzzy sets with complements induced by a strict negation will also be discussed. c 2002 Elsevier Science B.V. All rights reserved. Keywords: Cardinalities of fuzzy sets; Triangular norms; Singular fuzzy sets; Fuzziness measures
1. Introduction Two dominating streams are present in contemporary cardinality theory of fuzzy sets. The :rst one comprises so-called scalar approaches in which the cardinality of a fuzzy set is a single ordinary cardinal number or a nonnegative real number (see e.g. [3,11,14] for details and further references). An axiomatic theory of scalar cardinalities of fuzzy sets with triangular norms is constructed in [17,18]. Our attention in this paper will be focused on the other stream which o?ers a “fuzzy” perception of the cardinality of a fuzzy set. It is then a convex fuzzy set of usual cardinal numbers, a weighted family of usual cardinals, called a generalized cardinal number. This perception is more advanced and complicated from both the conceptual and operational viewpoints, but at that price it gives us in comparison with any scalar approach a considerably more complete and adequate cardinal ∗
Corresponding author. Tel.: +48-61-866-6615; fax: +48-61-866-2992. E-mail address: [email protected] (M. Wygralak).
c 2002 Elsevier Science B.V. All rights reserved. 0165-0114/02/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 3 3 2 - 9
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information. Generalized cardinals will be denoted by lowercase letters ; ; ; : : : from the beginning of the Greek alphabet. If does express the cardinality of a fuzzy set A : M → [0; 1], we write |A| = and we say that the cardinality of A is equal to . Reviews of generalized cardinals of fuzzy sets with the standard operations ∧ := min and ∨ := max can be found in [3,14–16]; as usual, := stands for “equals by de:nition”. A theory of generalized cardinals for fuzzy sets with triangular norms is constructed and developed in [19–21]. It o?ers three basic classes of such cardinals, namely generalized (or triangular norm-based) FGCounts, FLCounts and FECounts. Their short presentation is placed in Section 2. Our attention in the next sections will be restricted to generalized FECounts. If one uses them and nonstrict Archimedean triangular norms are involved, many fuzzy sets appear to be singular: their generalized FECounts become the zero function. This singularity of a fuzzy set thus means its total dissimilarity to any set of any cardinality. The phenomenon of singularity is the main subject of this paper. It will be investigated in Sections 3 and 4. The :rst formulation of the problem and some initial results can be found in [4,19]. Our further discussion requires more notation and terminology. First of all, it will be restricted to :nite fuzzy sets with triangular norm-based operations. The reason in the context of generalized cardinals is the lack of fully satisfactory extensions of triangular norms di?ering from ∧ to in:nitely, including uncountably, many arguments (cf. [9]). This restriction to :nite fuzzy sets is easy to accept because just they play the central role in applications. So, from now on, the phrase “fuzzy set” will mean “:nite fuzzy set” if not emphasized otherwise. The universe M is however assumed to be of a quite arbitrary cardinality, except for the last part of Section 4 in which M is :nite. The family of all :nite crisp sets in M will be denoted by FCS. FFS symbolizes the family of all (:nite) fuzzy sets in M . Their generalized cardinals are just some convex fuzzy sets in the set N of all nonnegative integers. The following convenient vector notation will be used for those cardinals: = (a0 ; a1 ; : : : ; ak ; (a))
with k ∈ N;
which means that (i) = ai for i6k and (i) = a for i¿k. Moreover, (a0 ; a1 ; : : : ; ak ) := (a0 ; a1 ; : : : ; ak ; (0)): So, = (0; 0:3; 0:6; 0:4; 0:1) is the notation for such that (0) = 0; (1) = 0:3; (2) = 0:6; (3) = 0:4; (4) = 0:1 and (i) = 0 for i¿4. Throughout, let A ∈ FFS with n := |supp(A)| and m := |core(A)|. Generally, single capitals in italic will denote fuzzy sets in M , whereas single bold italicized capitals will be symbols of sets. The characteristic function of D ⊂ M is denoted by 1D . Let At := {x ∈ M : A(x)¿t} with t ∈ (0; 1] (t-cut t set of A), A := {x ∈ M : A(x)¿t} with t ∈ [0; 1) (sharp t-cut set of A), and [A]i := {t: |At |¿i} with i ∈ N. We see that [A]0 = 1 and [A]i = 0 for i¿n, whereas each [A]i with 0¡i6n is the ith element in the nonincreasingly ordered sequence of all positive membership values A(x), including their possible repetitions. The de:nition of [A]i remains reasonable if A is in:nite and=or i is a trans:nite cardinal. Let us recollect a few de:nitions and facts from the theory of triangular norms which will be useful in the next sections. An exhaustive presentation of that theory can be found in [6,9]. A binary operation t in the closed unit interval [0; 1] is called a triangular norm (t-norm, in short) if t is commutative, associative, nondecreasing in the :rst and, hence, in each argument, and has 1 as unit. A binary operation s in [0; 1] which ful:ls the :rst three properties and has 0 as unit is
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213
said to be a triangular conorm (t-conorm). A continuous t-norm t is called Archimedean if ata¡a for each a ∈ (0; 1). One says that an Archimedean t-norm is strict whenever it is strictly increasing over (0; 1) × (0; 1). The algebraic t-norm ta with ata b := ab is a typical example of a strict t-norm, whereas the Lukasiewicz t-norm tL with atL b := 0 ∨ (a + b − 1) is a typical nonstrict Archimedean t-norm. The class of all nonstrict Archimedean t-norms will be denoted by Natn. Archimedean t-norms seem to be especially important and interesting. One of reasons is that, for each continuous t, either t = ∧ or t is Archimedean or t is the ordinal sum of a family of Archimedean t-norms; the de:nition of ordinal sums can be found e.g. in [6,9]. Moreover, Archimedean t-norms do have a nice characterization theorem by AczKel-Ling [1,10]; see also [6,9,13]. Theorem 1.1. A t-norm t is Archimedean i9 there exists a strictly decreasing and continuous function g : [0; 1] → [0; ∞] such that g(1) = 0 and ∀a; b ∈ [0; 1]: atb = g−1 (g(0) ∧ (g(a) + g(b))): Moreover, t is strict i9 g(0) = ∞. The function g satisfying the conditions of Theorem 1.1 is called an (additive) generator of t. If g(0) = 1; g is said to be normed. Since generators of Archimedean t-norms are uniquely determined up to a positive factor, we can restrict ourselves to normed generators when considering t-norms from Natn. By mathematical induction, k a1 ta2 t : : : tak = g−1 g(0)∧ g(ai ) (1.1) i=1
for each Archimedean t with generator g and each system of numbers a1 ; a2 ; : : : ; ak ∈ [0; 1] with k ∈ N; in particular, a1 ta2 t : : : tak = 1 for k = 0. Hence a1 ta2 t : : : tak ¿ 0 i?
k
g(ai ) ¡ g(0)
(1.2)
i=1
and, thus, a1 ta2 t : : : tak ¿0 ⇔ a1 ; a2 ; : : : ; ak ¿0 when t is strict, which cannot be extended to t-norms from the class Natn. So, strict t-norms do not have zero divisors, whereas nonstrict Archimedean ones do have, i.e. atb = 0 is then possible for a; b¿0. In other words, that t ∈ Natn has zero divisors does mean that t has some inertia in attaining positive values: each positive argument of t is treated as zero whenever the other one is not suMciently “large”. For instance, 0:4tL 0:5 = 0 but 0:4tL 0:7¿0. Clearly, this feature of a nonstrict Archimedean t-norm can be useful and desired in many applications. Each nonincreasing function v : [0; 1] → [0; 1] with v(0) = 1 and v(1) = 0 will be called a negation. The negation v∗ such that v∗ (a) := 0 for each a¿0 is thus the smallest possible negation, whereas v∗ with v∗ (a) := 1 for each a¡1 is the largest one. Strictly decreasing and continuous negations are called strict negations. Their class will be denoted by Sng. If a strict negation is involutive (v(v(a)) = a for each a), it is said to be a strong negation. A fundamental example of a strong negation is the Lukasiewicz negation vL with vL (a) := 1 − a. Each t-norm t does induce a negation
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vt de:ned as ∀a ∈ [0; 1]: vt (a) :=
{c ∈ [0; 1]: atc = 0}:
(1.3)
One has vt = v∗ whenever t is strict. If t ∈ Natn and g is its generator, then vt is strong and, by Theorem 1.1, vt (a) = g−1 (g(0) − g(a)):
(1.4)
It is clear that v ∈ Sng has a unique :xed point a∗ ∈ (0; 1). For instance, a∗ = 0:5 for v = vL . Generally, v(a∗ ) = a∗ and ∀a = a∗ : v(a) ¡ a∗ ¡ a ⊥ a ¡ a∗ ¡ v(a); i.e. ∀a ∈ [0; 1]: a ∨ v(a) ¿ a∗ ;
(1.5)
where ⊥ denotes the inclusive disjunction connective. It is a routine task to check that ∀a ∈ [0; 1]: atvt (a) = 0
(1.6)
for t ∈ Natn. By (1.6), if a∗ is the :xed point of vt with a nonstrict Archimedean t, we have the following: a∗ tvt (a∗ ) = vt (a∗ )tvt (a∗ ) = a∗ ta∗ = 0; ∀a 6 a∗ : ata = 0; ∀a ¿ a∗ : vt (a)tvt (a) = 0:
(1.7)
That :xed point of vt is equal to g−1 (0:5g(0)), i.e. a∗ = g−1 (0:5)
(1.8)
whenever g is normed, which follows from (1.4). Consider an example in which t ∈ Natn is a Schweizer t-norm, i.e. t = tS;p for p¿0 and atS;p b := [0 ∨ (ap + bp − 1)]1=p
(1.9)
with normed generator g(a) = 1 − ap . So, g−1 (y) = (1 − y)1=p and, by (1.4), vt (a) = (1 − ap )1=p . The :xed point of vt is thus a∗ = 0:51=p . Obviously, t = tL and vt = vL for p = 1. Use t-norms, t-conorms and negations to introduce operations on fuzzy sets via pointwise de:nitions: the intersection A ∩t B of A and B induced by a t-norm t with (A ∩t B)(x) := A(x)tB(x), the sum A ∪s B induced by a t-conorm s with (A ∪s B)(x) := A(x)sB(x), and the complement Av of A induced by a negation v with Av (x) := v(A(x)): ∩ := ∩∧ ; ∪ := ∪∨ and := v with v := vL are the standard operations on fuzzy sets.
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2. Generalized cardinals Let A ∈ FFS and = |A|. At the beginning of the 1980s the following three main types of generalized cardinals were proposed: (a1)
(k) := [A]k
for k ∈ N: is then called the FGCount of A (see [22,23]). In the language of many valued logic, (k) expresses a degree to which A has at least k elements. So, this kind of does form a lower cardinal evaluation of A. (b1)
(k) := 1 − [A]k+1
with k ∈ N (see [22]). What we now get as is the FLCount of A which forms an upper cardinal evaluation of A because (k) is a degree to which A contains at most k elements. (c1)
(k) := [A]k ∧ (1 − [A]k+1 )
for k ∈ N: is then called the FECount of A (see [22,14,16]). This intersection of the lower and upper cardinal evaluations can be viewed as a “proper” generalized cardinal of A: (k) becomes a degree to which A has exactly k elements. De:nitions (a1), (b1) and (c1) remain reasonable if A is in:nite (see [15]). On the other hand, as pointed out in [19,21], they are suitable only for fuzzy sets with the standard operations. Their appropriate generalizations to fuzzy sets with triangular norm-based operations are presented below; t is an arbitrary t-norm and v denotes a negation. (a2)
(k) := [A]1 t[A]2 t : : : t[A]k
with k ∈ N. In the vector notation, the resulting generalized FGCount of A can be written as = (1; [A]1 ; [A]1 t[A]2 ; : : : ; [A]1 t[A]2 t : : : t[A]n ): We see that (k) is a degree to which A contains (at least) i elements for each i6k. The quanti:cation “for each i6k” is here realized by means of t. Clearly, collapses to the FGCount of A if t = ∧. (b2)
(k) := v([A]k+1 )tv([A]k+2 )t : : : = v([A]k+1 )tv([A]k+2 )t : : : tv([A]n )
for k ∈ N, i.e. (k) is a degree to which A contains at most i elements for each i¿k. Thus, = (v([A]1 )tv([A]2 )t : : : tv([A]n ); v([A]2 )tv([A]3 )t : : : tv([A]n ); .. . v([A]n−1 )tv([A]n ); v([A]n ); (1):
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is now called the generalized FLCount of A. t = ∧ with v = vL gives the FLCount of A. (c2)
(k) := [A]1 t[A]2 t : : : t[A]k tv([A]k+1 )t : : : tv([A]n )
with k ∈ N, i.e. = (v([A]1 )tv([A]2 )t : : : tv([A]n ); [A]1 tv([A]2 )t : : : tv([A]n ); [A]1 t[A]2 tv([A]3 )t : : : tv([A]n ); .. . [A]1 t[A]2 t : : : t[A]n−1 tv([A]n ); [A]1 t[A]2 t : : : t[A]n : This generalized FECount of A, induced by t and v, is the t-based intersection of the generalized cardinals from (a2) and (b2). It is always convex. If t = ∧ and v = vL ; collapses to the FECount. Another worth mentioning particular case is that with t = ∧ and v = v∗ . Then = (0; 0; : : : ; 0; 1; [A]m+1 ; [A]m+2 ; : : : ; [A]n ) with m zeros; i.e. becomes the generalized cardinal of A due to Dubois [3]. GFEt; v will denote the family of all generalized FECounts induced by a t-norm t and a negation v. A detailed study of generalized FGCounts, FLCounts and FECounts is placed in [19,20]. Our further discussion in this paper will be limited to generalized FECounts with an additional restriction to t ∈ Natn and v ∈ Sng. As we will see, an interesting cardinal singularity of many fuzzy sets is connected with such generalized FECounts. 3. Singular fuzzy sets Let A; B ∈ FFS; |A| = ; |B| = and ; ∈ GFEt; v with v ∈ Sng. If t is strict or t = ∧, a very natural equipotency relation∼ful:lling the requirement = ⇔ A ∼ B corresponds to generalized FECounts, namely (see [19,20]): A ∼ B ⇔ ∀k ∈ N: [A]k = [B]k ⇔ ∀t ∈ (0; 1]: |At | = |Bt | ⇔ ∀t ∈ [0; 1): |At | = |Bt |: A ∼ B thus means that A and B are identical up to the permutation of their positive membership values, including possible repetitions, which is equivalent to the usual equipotency of all the corresponding t-cut sets (sharp or not) of A and B. Consequently, two equipotent fuzzy sets do have equipotent cores and equipotent supports. Since t does not have zero divisors, we get = (0), i.e. always di?ers from the zero function. Let us stress in this context that = 1{k } whenever A collapses
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to a k-element set. So, in particular, the generalized FECount of the empty fuzzy set 1∅ is equal to 1{0} ∈ [0; 1]N . The situation is quite di?erent if t ∈ Natn. t then has zero divisors and, therefore, an equipotency relation similar to that for a strict t-norm, including ∧, cannot be constructed. The equality = may hold for A and B having di?erent cardinalities of the corresponding t-cut sets, cores and supports. What is more, = (0) becomes possible. The generalized cardinal (0) will be called the
k ¿ s ⇒ [A]k 6 a∗ ⇒ v([A]k ) ¿ [A]k :
Consequently, for each k = s, we have (k) = [A]1 t : : : t[A]k tv([A]k+1 )t : : : tv([A]n ) 6 [A]1 t : : : t[A]s tv([A]s+1 )t : : : tv([A]n ) = (s): Thesis (b) is an immediate consequence of (a). The procedure of checking whether (k) = 0 for each k can thus be reduced to verifying the single condition (s) = 0. This is important because the general form of (k) is rather complex.
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Theorem 3.2. (s) = 0 i9
s
i=1
g([A]i ) +
n
i=s+1
g(v([A]i ))¿1.
Proof. The thesis immediately follows from (1.2) applied to the general formula for (s) given in (c2) in Section 2. Since g and v are decreasing, Theorem 3.2 says that A is nonsingular i? the membership values [A]i with i6s are suMciently near to 1 (are suMciently larger than a∗ ) and, moreover, the [A]i ’s with s + 16i6n are suMciently smaller than the :xed point a∗ . Intuitively speaking, = (0) is thus possible only if A is “similar” to an s-element set (see also Example 3.4). This suggests a relationship between singularity=nonsingularity of a fuzzy set and its fuzziness, which will be investigated in Section 4. Measuring that fuzziness, however, we have to use a fuzziness measure constructed in a special way: a maximal fuzziness should be assigned to the fuzzy set identically equal to a∗ (see Example 3.4(iii)). Corollary 3.3. Let s¿1 be such that a∗ ¡g−1 (1=s). If [A]1 6g−1 (1=s), then A is singular. Proof. Indeed, s¿1 implies [A]1 ¿a∗ , which means that [A]1 6g−1 (1=s) is possible only if s ful:ls ∗ −1 −1 the condition s a ¡g (1=s). s The inequality [A]1 6g (1=s) then leads to g([A]1 )¿1=s and, consequently, i=1 g([A]i ) ¿ i=1 g([A]1 )¿1. By Theorem 3.2, we get (s) = 0, i.e. A is singular. Although Corollary 3.3 looks rather technical, it is very useful. For instance, let t = tL and v = vL , i.e. a∗ = 0:5 and g(a) = 1 − a. The condition a∗ ¡g−1 (1=s) is now equivalent to s¿3. The corollary says that if s¿3 and [A]1 61 − 1=s, then A is singular. A with, say, s = 10 is thus singular whenever [A]1 60:9. Example 3.4. In the light of Theorem 3.2, as already mentioned, a nonsingular fuzzy set A must be “similar” to an s-element set. For better understanding, let us consider a few related instances. (i) Again, assume t = tL and v = vL with a∗ = 0:5. Then A = 0:8=x1 + 0:2=x2 + 0:1=x3 + 0:1=x4 is nonsingular because s = 1 and (s) = 0:8tv(0:2)tv(0:1)tv(0:1)¿0. We see that A is really similar to a 1-element set. (ii) Let us use the t and v from (i), and take A = 0:95=x1 + 0:95=x2 + 0:95=x3 + 0:5=x4 + 0:4=x5 : Now s = 3 and (s) = 0:95t0:95t0:95t0:5t0:6 = 0, i.e. A is singular. Its similarity to a 3-element set seems to be low because the membership grades of x4 and x5 are too large. More precisely speaking, they are not suMciently smaller that a∗ . The fuzzy set B = 0:9=x1 + 0:9=x2 + 0:9=x3 + 0:2=x4 + 0:1=x5 ; a fuzzy subset of A, is nonsingular although sB; v = 3, too. The reason is that B is more similar to a 3-element set than A.
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(iii) Take t = tS;4 and v = vt , i.e. a∗ = 0:51=4 ≈ 0:84 (see (1.8) and (1.9)). Consider again the fuzzy set A from (ii). As previously, we have s = 3 but this time (s)¿0 and, hence, A is nonsingular. The reason is that a∗ always plays the role of a threshold point: the membership values ¿a∗ are considered to be more or less similar to the maximal membership value 1, whereas those 6a∗ are considered to be more or less similar to the minimal membership value 0. In this example, we have a∗ ≈ 0:84. The membership values of x4 and x5 are now considerably smaller than the threshold value and, consequently, A becomes more similar to a 3-element set than in (ii). It is easy to notice that each singular fuzzy set has a nonsingular proper fuzzy subset, at least the empty fuzzy set 1∅ . As mentioned in Example 3.4(ii), this is because a proper fuzzy subset of A may be more similar to an s-element set than A itself. Theorem 3.5. If A is singular and B is such that A ∩ B = 1∅ , then A ∪ B is singular, too. Proof. Assume A is singular and A ∩ B = 1∅ . Put ci := [A ∪ B]i for i¿1; r := sA∪B; v = max{i: ci ¿a∗ }; n∗ := |supp(B) and l := |supp(A ∪ B)| = n + n∗ . As previously, s := max{i: [A]i ¿a∗ }. The sequence c1 ; c2 ; : : : ; cl is a result of joining the sequences [A]1 ; [A]2 ; : : : ; [A]n and [B]1 ; [B]2 ; : : : ; [B]n∗ into one nonincreasing sequence. Hence r¿s and, by Theorem 3.2, r i=1
g(ci ) +
l i=r+1
g(v(ci )) ¿
s i=1
g([A]i ) +
n
g(v([A]i )) ¿ 1;
i=s+1
i.e. A ∪ B is singular. The property of singular fuzzy sets described in Theorem 3.5 will be called the absorption property. This property is also reOected in the results generated by the triangular norm-based extension principle ( + )(k) := {(i)t(j): i + j = k}: Indeed, we get (0) + = (0) for each ∈ GFEt; v with t ∈ Natn and v ∈ Sng. Let i:k := ik . Corollary 3.6. If a fuzzy set ai:1 =xi:1 +ai:2 =xi:2 +· · ·+ai:k =xi:k is singular for some indices 16i:1¡i:2¡ · · · ¡i:k6n with 2 6 k¡n, then A = a1 =x1 + a2 =x2 + · · · + an =xn is singular, too. Proof. An immediate consequence of Theorem 3.5 combined with the trivial fact that each fuzzy set is a sum of disjoint singletons. So, a fuzzy set is singular whenever its proper fuzzy subset resulting from a reduction of the support is singular. On the other hand, that A is singular does not generally imply the existence of a singular fuzzy subset ai:1 =xi:1 + ai:2 =xi:2 + · · · + ai:k =xi:k of A. An instance is A = 0:6=x1 + 0:6=x2 + 0:6=x3 with t = tL and v = vL . A is then singular but |0:6=xi + 0:6=xj | = (0; 0; 0:2) = (0). The suMcient condition in Corollary 3.6 for A to be singular does form a useful criterion because in practice it is easy to check if a fuzzy set with a small support having, say, 2 or 3 elements is singular. The inequality k¿2 reOects the property that a singleton cannot be singular as |a=x| = (v(a); a) for
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a ∈ (0; 1]. By the way, it is usually easy to construct a singular fuzzy set with a support of cardinality ¿3. The task of :nding a singular fuzzy set supported by a 2-element set is more diMcult and can even be unrealizable. The conditioning factor is the form of v (see Theorem 3.9 and subsequent remarks). Generally, the larger the support the easier the construction of a singular fuzzy set (cf. Corollary 3.3). A group of useful properties of singular fuzzy sets can be formulated for v = vt (see (1.3) and (1.4)). The :rst one o?ers a simpli:cation of the criterion from Theorem 3.2. We still use the assumptions and notation established just before Theorem 3.1. Theorem 3.7. If v = vt , then (s) = 0 i9
s
n
g([A]i ) −
i=1
g([A]i ) ¿ s − n + 1:
i=s+1
Proof. It follows from Theorem 3.2 and (1.4). The theorem above allows to formulate explicit singularity criteria for concrete t-norms with negations induced by them. Let us give two instances. (i) t = tS;p with p¿0 and v = vt (see (1.9)). Then (s) = 0 ⇔
s
([A]i )p −
i=1
n
([A]i )p 6 s − 1:
i=s+1
So, if t = tL and v = vL , we get (s) = 0 ⇔
s i=1
[A]i −
n
[A]i 6 s − 1:
i=s+1
(ii) t = tW; and v = vt , where tW; is the Weber t-norm with parameter ¿−1 (see [13]): atW; b := 0 ∨ ((a + b − 1 + ab)=(1 + )) and g(a) = 1 − ln(1 + a)= ln(1 + )
for 0 =
¿ −1:
Routine transformations do lead to the equivalence (s) = 0 ⇔
s i=1
s −1
(1 + [A]i ) 6 (1 + )
n
(1 + [A]i )
i=s+1
for ¿0; if −1¡ ¡0, the inequality symbol 6 should be replaced by ¿. The case omitted because tW; 0 = tL .
= 0 is here
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Theorem 3.8. If v = vt and [A]s+1 = [A]s+2 = a∗ , then (s) = 0. Proof. Indeed, applying (1.7) to the formula (s) = [A]1 t : : : t[A]s tv([A]s+1 )tv([A]s+2 )t : : : tv([A]n ) we get (s) = 0 whenever [A]s+1 = [A]s+2 = a∗ and v = vt . So, if A attains the value a∗ (at least) at two points, then A is singular provided that vt is used as negation. This means that nonsingularity of A implies |{x: A(x) = a∗ }|61. In particular, A with t = tS;p is singular whenever |{x: A(x) = 0:51=p }|¿2. Let us notice that Theorem 3.8 can also be derived from Theorem 3.2 combined with (1.8). As remarked earlier in this section, singletons are never singular. The following property refers to the question of constructability of a singular fuzzy set having a 2-element support. Theorem 3.9. For v = vt , the fuzzy set a=x + b=y with a; b¿0 is singular i9 a = b = a∗ . Proof. Let v = vt . Choose arbitrary a; b ∈ (0; 1] and suppose that = |A| for A = a=x + b=y with any x; y ∈ M . So, n = 2. Without loss of generality, one can assume that a¿b. By (c2) from Section 2, we have = (v(a)tv(b); atv(b); atb): (a) If a¿b¿a∗ , then s = 2 and, by Theorem 3.7, = (0) ⇔ (2) = 0 ⇔ g(a) + g(b)¿1. However (1.8) says that a∗ = g−1 (0:5), which implies g(a)6g(b)¡g(a∗ ) = 0:5 and, hence, g(a) + g(b)¡1. Consequently, A cannot be singular. (b) If a¿a∗ ¿b, then g(a)¡g(a∗ ) = 0:56g(b), i.e. g(a) − g(b)¡0. Again, A cannot be singular because s = 1 and = (0) ⇔ (1) = 0 ⇔ g(a) − g(b)¿0. (c) Finally, suppose a∗ ¿a¿b¿0. This time g(b)¿g(a)¿0:5. On the other hand, s = 0 and (0) = 0 ⇔ g(a)+g(b)61. A is thus singular only if g(a) = g(b) = 0:5, which means that a = b = a∗ . This completes the proof. If, say, t = tL and v = vL , the unique singular fuzzy set a=x + b=y is thus that with a = b = 0:5. Theorem 3.9 does not work for v = vt . For instance, a=x + b=y is always nonsingular for t = tL and v(a) := 1 − a2 . There exist in:nitely many singular fuzzy sets a=x + b=y if one uses the Lukasiewicz t-norm accompanied by the negation v(a) := 1 − a0:5 . 4. Singularity and fuzziness of a fuzzy set Let us investigate the question of singularity=nonsingularity of a fuzzy set in connection with its fuzziness. Our starting point is a consideration concerning fuzzy sets being constant on their supports. As previously, t ∈ Natn with normed generator g; v ∈ Sng and a∗ is its :xed point. Generalizing the notation 1D , by aD with a ∈ (0; 1] and a set D ⊂ M we denote a fuzzy set such that aD (x) := a if x ∈ D, else aD (x) := 0.
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Theorem 4.1. For each a ∈ (0; 1] and each nonempty ?nite D, aD is singular i9 a ∨ v(a) 6 g−1 (1=|D|): Proof. Indeed, :x an arbitrary a ∈ (0; 1] and a :nite set D = ∅ and put := |aD |. Let p := |D| and r := max{i: [aD ]i ¿a∗ }. Obviously, [aD ]i = a for 16i6p. Suppose a ∈ (a∗ ; 1]. Then r = p and, by Theorem 3.2 and (1.5), (r) = 0 ⇔
p
g(a) ¿ 1 ⇔ v(a) ¡ a 6 g−1 (1=p):
i=1
If a ∈ (0; a∗ ], then r = 0 and (r) = 0 ⇔
p
g(v(a)) ¿ 1 ⇔ a 6 v(a) 6 g−1 (1=p):
i=1
So, in each case, aD singular i? a ∨ v(a) 6 g−1 (1=p), which completes the proof. Since g−1 (1) = 0, Theorem 4.1 supplies us with an additional con:rmation that a singleton cannot be singular. Corollary 4.2. Let |D|¿2 and a ∈ (0; 1]. (a) aD is singular i9 v−1 (g−1 (1=|D|))6a6g−1 (1=|D|). (b) If aD is singular and |D|¡|E|, then aE is singular. (c) If g−1 (1=|D|)¡a∗ , then aD is nonsingular. Proof. (a) aD is singular i? a ∨ v(a)6g−1 (1=|D|), which means that a6g−1 (1=|D|) and a¿v−1 (g−1 (1=|D|)). (b) |D|¡|E| implies g−1 (1=|D|)¡g−1 (1=|E|). Thus, using again Theorem 4.1, we conclude that aE is singular if so is aD . (c) The inequality g−1 (1=|D|)¡a∗ in combination with (1.5) does lead to g−1 (1=|D|)¡a ∨ v(a), which implies that aD is nonsingular for each a ∈ (0; 1]. A group of useful conclusions can be formulated for a = a∗ . By the way, we see that the generalized FECount of aD∗ is always a function being constant on its support because v(a∗ ) = a∗ (see (c2) in Section 2). Corollary 4.3. Let |D|¿2. (a) aD∗ is singular i9 a∗ 6g−1 (1=|D|). (b) aD∗ is singular i9 |D|¿1=g(a∗ ). Proof. (a) immediately follows from Theorem 4.1. (b) is a transformed version of (a).
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Clearly, the variant (a) of Corollary 4.3 is convenient if D and t are :xed and one looks for a negation v such that aD∗ is singular=nonsingular. (b) says that if t and v and, thus, a∗ are given, then aD∗ is always singular whenever D is suMciently large. Corollary 4.4. If v = vt , then aD∗ with |D|¿2 is singular. Proof. Straightforward consequence of (1.8) and Corollary 4.3(b). Notice that the above corollary could also be derived from Theorem 3.8. Consider an example in which t = tS;p with p¿0 and v = vt . Applying Corollary 4.2(a), we conclude that aD with a ∈ (0; 1] and |D|¿2 is then singular i? (1=|D|)1=p 6 a 6 (1 − 1=|D|)1=p : Let us introduce the following partial ordering relation for A; B ∈ FFS with v ∈ Sng: A 6v B ⇔ ∀x ∈ M : A(x) 6 B(x) 6 a∗ ⊥ a∗ 6 B(x) 6 A(x): If A6v B, we say that B is fuzzier than A or, dually, A is sharper than B. We easily notice that A6v a∗supp(A) and, generally, the fuzziest fuzzy set with a given support D is aD∗ . The relation 6v is considered in a bit di?erent terminological framework in [8]; its particular case with v = vL and a∗ = 0:5 is used e.g. in [11]. Theorem 4.5. If A is singular and A6v B, then B is singular, too. Proof. Assume t ∈ Natn; v ∈ Sng and A; B ∈ FFS are such that A6v B. Let = |A| and = |B|. Suppose A is singular, i.e. (see Theorem 3.1) (s) = [A]1 t : : : t[A]s tv([A]s+1 )t : : : tv([A]n ) = 0: Let n∗ := |supp(B)| and r := max{i: [B]i ¿a∗ }. By the de:nition of 6v ; s¿r and n6n∗ . Since [B]j = a∗ for each r¡j6s, we have v([B]j ) = [B]j for such j’s. So, (r) = [B]1 t : : : t[B]r tv([B]r+1 )t : : : tv([B]s )tv([B]s+1 )t : : : tv([B]n∗ ) = [B]1 t : : : t[B]s tv([B]s+1 )t : : : tv([B]n∗ ) 6 [A]1 t : : : t[A]s tv([A]s+1 )t : : : tv([A]n∗ ) = [A]1 t : : : t[A]s tv([A]s+1 )t : : : tv([A]n ) =0 because [B]i 6[A]i for i6s; [A]i 6[B]i for s¡i6n∗ , and v([A]i ) = 1 for i¿n. Consequently, (r) = 0 and, hence, B is singular. Thus, if A6v B and B is nonsingular, then A is also nonsingular. In other words, a fuzzy set being fuzzier than a singular fuzzy set is singular, too, whereas a fuzzy set which is sharper than a
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nonsingular one is also nonsingular. Another corollary following from Theorem 4.5 is that if A is singular, then so is aD∗ with any D such that |D|¿n. So, A with n6|D| is nonsingular whenever aD∗ with a :nite D is nonsingular. The relation 6v draws our attention to fuzziness measures answering the question how fuzzy a fuzzy set is. Recollect that there are two general classes of those measures: entropy and energy type fuzziness measures. They describe deviation of a fuzzy set from a set and from a 1-element set, respectively (see e.g. [7,11] and a review in [12]). In this paper, we restrict attention to fuzziness measures understood as entropy type ones. From now on, we shall assume that the universe M is ∗ is also :nite (see the axiom (A2) below). As previously, v ∈ Sng and :nite. This guarantees that aM a∗ denotes its unique :xed point. De)nition 4.6. A mapping E : FFS → [0; ∞) is a fuzziness measure i? the following postulates are satis:ed by each A; B ∈ FFS: (A1) (A2) (A3) (A4) (A5)
E(A) = 0 i? A ∈ FCS, ∗ , E reaches its unique maximum at aM v A6 B implies E(A)6E(B), E(A) = E(Av ), E(A ∪ B) + E(A ∩ B) = E(A) + E(B).
The number E(A) is called the fuzziness measure of A. The classical de:nition of a fuzziness measure given in [11] is a particular case of De:nition 4.6 with v = vL and a∗ = 0:5. That particular formulation is suitable only if one uses the standard complement A induced by the Lukasiewicz negation. De:nition 4.6 does form its natural generalization to complements induced by an arbitrary strict negation. Postulates (A1)–(A3) are also used in [8] for de:ning a general class of fuzziness measures; as shown therein, that class is identical to a class of fuzziness measures de:ned via distance between A and Av . One can say that if A ∈ FFS is nonsingular, then its fuzziness measure E(A) is “small” in comparison with E(a∗supp(A) ). Clearly, this is valid provided that |supp(A)|¿1=g(a∗ ). By Corollary 4.3(b), |supp(A)|¡1=g(a∗ ) implies that a∗supp(A) is nonsingular and, hence, A is nonsingular, too, although E(A) is possibly near to E(a∗supp(A) ). The fuzziness measures from De:nition 4.6 can be characterized in a convenient way by means of :nite sums of transformed membership values. Theorem 4.7. E : FFS → [0; ∞) is a fuzziness measure i9 E(A) = f(A(x)) = f(A(x)) x ∈M
x∈supp(A)
for each A ∈ FFS, where f : [0; 1] → [0; ∞) is such that (P1) f(0) = f(1) = 0 and f(a)¿0 for a ∈ (0; 1), (P2) f(a)¡f(a∗ ) for a = a∗ ,
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(P3) f is nondecreasing on [0; a∗ ] and nonincreasing on [a∗ ; 1], (P4) f(a) = f(v(a)) for a ∈ [0; 1]. Proof. (⇒) Suppose E is a fuzziness measure satisfying the axioms (A1)–(A5). By virtue of(A1) and (A5), if A ∩ B = 1∅ , then E(A ∩ B) = 0 and, hence, E(A ∪ B) = E(A)+E(B). So, E(A) = E( x∈M A(x)=x) = x∈M E(A(x)=x). Since E : FFS → [0; ∞) and it seems to be reasonable to accept an implicit assumption that E(a=x) does not depend on x, we get E(a=x) = f(a) for a ∈ [0; 1] with f : [0; 1] →[0; ∞) such that f(0) = f(1) = 0 and f(a)¿0 for a ∈ (0; 1),which follows from (A1). The function f thus satis:es (P1) and, moreover, we have E(A) = x∈M f(A(x)) = x∈supp(A) f(A(x)). (A2) implies (P2). Indeed, if f(a)¿f(a∗ ) for some a = a∗ , then E(aM ) = x∈M f(a)¿ x∈M ∗ ), which contradicts (A2). f(a∗ ) = E(aM By (A3), a=x6v b=x with a; b ∈ [0; 1] does imply E(a=x)6E(b=x), which means that f(a)6f(b). As a 6 b6a∗ or a∗ 6b6a, we conclude that f ful:ls (P3). Finally, (A4) and (A1) do lead to f(a) = E(a=x) = E(v(a)=x ∪ 1{y ∈ M : y = x} ) = E(v(a)=x) + E(1{y ∈ M : y = x} ) = E(v(a)=x) = f(v(a)) for each a ∈ [0; 1], i.e. (P4) is satis:ed. (⇐) Suppose E(A) = x∈M f(A(x)) for each A ∈ FFS with f ful:lling (P1)–(P4). Let us show that E is a fuzziness measure in the sense of De:nition 4.6. By (P1), E(A) = 0 ⇔ x∈M f(A(x)) = 0 ⇔ ∀x ∈ M : f(A(x)) = 0 ⇔ ∀x ∈ M : A(x) (A1) holds true. ∈ {0; 1}, i.e. ∗ ) whenever A = a∗ , i.e. The postulate (P2) implies E(A) = x∈M f(A(x))¡ x∈M f(a∗ ) = E(aM M (A2) is ful:lled. In the same simple way one shows that (P3) and (P4), respectively, do imply (A3)and (A4), respectively. f(b) for a; b ∈ [0; 1], we ob Since f(a ∨ b) + f(a ∧ b) = f(a) + tain x∈M f((A ∪ B)(x)) + x∈M f((A ∩ B)(x)) = x∈M f(A(x)) + x∈M f(B(x)). The valuation property (A5) is thus satis:ed, which completes the proof. The characterization formulated in Theorem 4.7 does generalize that given in [5] for the particular case of fuzziness measures with v = vL and a∗ = 0:5 (see also [2]). References [1] J. AczKel, Sur des operations de:nies pour des nombres rKeels, Bull. Soc. Math. France 76 (1949) 59–64. [2] C. Alsina, E. Trillas, Sur les measures du degrKe de Oou, Stochastica 3 (1979) 81–84. [3] D. Dubois, H. Prade, Fuzzy cardinality and the modeling of imprecise quanti:cation, Fuzzy Sets and Systems 16 (1985) 199–230. [4] K. Dyczkowski, M. Wygralak, On cardinality and singular fuzzy sets, in: B. Reusch (Ed.), Computational Intelligence—Theory and Applications, Lecture Notes in Computer Science, vol. 2206, Springer, Berlin, Heidelberg, New York, 2001, pp. 261–268. [5] B.R. Ebanks, On measures of fuzziness and their representations, J. Math. Anal. Appl. 94 (1983) 24–37. [6] S. Gottwald, Many-valued logic and fuzzy set theory, in: U. H˝ohle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets. Logic, Topology, and Measure Theory, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999, pp. 5–89. [7] S. Gottwald, E. Czogala, W. Pedrycz, Measures of fuzziness and operations with fuzzy sets, Stochastica 6 (1982) 187–205. [8] M. Higashi, G.J. Klir, On measures of fuzziness and fuzzy complements, Internat. J. Gen. Systems 8 (1982) 169–180. [9] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic Publishers, Dordrecht, Boston, London, 2000. [10] C.H. Ling, Representation of associative functions, Publ. Math. Debrecen 12 (1965) 189–212.
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