C-sets of n-uninorms

Fuzzy Sets and Systems 160 (2009) 1 – 21 www.elsevier.com/locate/fss

C-sets of n-uninorms夡 Prabhakar Akella Department of Mathematics and Computer Science, Sri Sathya Sai University, Prasanthi Nilayam, A.P. 515134, India Received 10 April 2007; received in revised form 5 March 2008; accepted 20 April 2008 Available online 14 May 2008

Abstract The concept of n-uninorms was introduced in our earlier paper which is based on the existence of an n-neutral element for an associative, monotone non-decreasing in both variables and commutative (AMC) binary operator on [0, 1]. There it was shown that the number of subclasses of n-uninorms is the (n + 1)th Catalan number. An expression for the arbitrary member of each subclass was also given which is recursive in nature. In this paper we introduce a unique ordered set of distinct integers between 0 and n, called C-sets, for each subclass of n-uninorms. This enables us to (1) (2) (3) (4)

obtain an expression for arbitrary member of each subclass which is non-recursive in nature, identify the minimum and the maximum members in general and of idempotent members in particular in each subclass, relate C-sets of De Morgan pairs (for strict negation) of n-uninorms from different subclasses, convert theoretical results into construction procedures which are algorithmic in nature.

In process we generalize some of the existing results for uninorms in the literature. © 2008 Elsevier B.V. All rights reserved. Keywords: Strict negation; Strong negation; Uninorm; n-neutral element; n-uninorm; Idempotent n-uninorm; De Morgan triple

1. Introduction Associative, monotone non-decreasing in both variables and commutative (AMC) binary operators on [0, 1] have been extensively studied in fuzzy logic. Two particular classes of operators namely triangular norms (t-norms) and triangular conorms (t-conorms) have been thoroughly investigated from both theoretical and practical point of view in [6]. The largest t-norm is minimum and the smallest t-conorm is maximum, which makes them unsuitable for aggregation purpose. For this reason two aggregation operators based on t-norms and t-conorms have been recently introduced. One is uninorm in [9] and explored in [5] and the other is t-operator in [8]. Nullnorm was introduced in [3] in the context of Frank equation which has an annihilator and 0 acts as the neutral element for members smaller than the annihilator and 1 acts as the neutral element for all members larger than the annihilator. It was also proved there that t-operators and nullnorms are equivalent. In [1] the 0, 1 combination of nullnorm was called a 2-neutral element and the concept was extended to n-neutral element which in turn gave rise to the concept of n-uninorm. It was shown that the number of subclasses of n-uninorms is the (n + 1)th Catalan number and a recursive expression for members of each subclass of n-uninorms was obtained. 夡

This work was done with the research grant that was given by Sri Sathya Sai University to the author. E-mail address: [email protected].

0165-0114/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2008.04.011

2

P. Akella / Fuzzy Sets and Systems 160 (2009) 1 – 21

In this paper the concept of ‘C-set’ (‘Class-set’) which is a unique ordered set of distinct integers between 0 and n, for each subclass of n-uninorms has been introduced. C-set is closely related to a depth first search algorithm Ext(n, 0), which extracts the elements of the C-set for a given n-uninorm. Not every ordered set of distinct integers between 0 and n qualifies to be a C-set. The algorithm Val({At }, n, 0) checks the validity of the given set {At } to be a C-set. The significance of a C-set is that it defines within intervals all members of a subclass to which it corresponds. This representation is non-recursive in nature. This is done by an algorithm Const({Ct }n , n, 0) for a given valid C-set {Ct }n . The role of C-set is further explored. First, we identify the minimum and the maximum members of each subclass in general and idempotent ones in particular. Secondly, we establish the relation between the C-sets of a De Morgan pair (for a strict negation) of n-uninorms from different subclasses. In the process we generalize some of the results in literature on idempotent uninorms in [7] which states that a uninorm is idempotent if and only if it is locally internal (i.e., U (x, y) ∈ {x, y}) and on De Morgan triple of uninorms with a strict negation which generalizes the results in [5] where De Morgan triple of uninorms with only strong negation was considered. This paper is organized as follows. In Section 2 we give the required definitions and results. In Section 3 we introduce the concept of C-set of an n-uninorm which identifies the subclass the n-uninorm belongs to. We give an algorithm to extract the C-set of a given n-uninorm, give an algorithm to check if a given set of integers is a valid C-set and finally give an algorithm to construct an n-uninorm with the given C-set. We also give the expression of the smallest and the largest operators with a given C-set and study idempotent n-uninorms with the given C-set. In Section 4 we study the De Morgan law involving n-uninorms with a strict negation. We also explore the relation between C-sets of a De Morgan pair of n-uninorms. 2. Preliminaries 2.1. Uninorms Definition 1 (Fodor et al. [5]). A uninorm is a two-place function U : [0, 1]2 → [0, 1] which is associative, commutative and monotone non-decreasing in each place with a neutral element e ∈ [0, 1] i.e., U (e, x) = x, for all x ∈ [0, 1]. If e = 0 then U is a t-conorm and if e = 1 then U is a t-norm. Proposition 1 (Fodor et al. [5]). Let U be a uninorm with neutral element e. If e ∈ (0, 1] then the binary operator TU defined by (1) is a t-norm and if e ∈ [0, 1) then the binary operator SU defined by (2) is a t-conorm. TU (x, y) =

U (ex, ey) , e

(1)

SU (x, y) =

U (e + (1 − e)x, e + (1 − e)y) − e . 1−e

(2)

For any uninorm U, min(x, y) U (x, y)  max(x, y) when min(x, y) < e < max(x, y). Therefore, in general the values of U (x, y) when min(x, y) < e < max(x, y) are not determined. We have general representation of uninorms as follows: ⎧ x y  ⎪ , , x, y ∈ [0, e], ⎨ eT e e  (3a) U (x, y) = x−e y−e ⎪ , , x, y ∈ [e, 1], ⎩ e + (1 − e)S 1−e 1−e and by monotonicity we have, U (x, y) ∈ [min(x, y), max(x, y)],

(x, y) ∈ (e, 1] × [0, e) or (x, y) ∈ [0, e) × (e, 1],

where T and S are t-norm and t-conorm, respectively. Lemma 1 (Fodor et al. [5]). Let U be a uninorm. Then U (0, 1) ∈ {0, 1}.

(3b)

P. Akella / Fuzzy Sets and Systems 160 (2009) 1 – 21

3

A uninorm U such that U (1, 0) = 0 is called a conjunctive uninorm and if U (1, 0) = 1 it is called a disjunctive uninorm. We denote the class of conjunctive uninorms by Uc and the class of disjunctive uninorms by Ud . Based on continuity considerations, some special classes of Uc and Ud are characterized as follows. Let U ∈ Uc be a uninorm with neutral element e ∈ (0, 1]. If the function U (x, 1) is continuous except at the point x = e, then there exists a t-norm T and a t-conorm S such that ⎧ x y  ⎪ eT , , x, y ∈ [0, e], ⎪ ⎪ ⎨ e e  U (x, y) = e + (1 − e)S x − e , y − e , x, y ∈ [e, 1], (4) ⎪ ⎪ 1 − e 1 − e ⎪ ⎩ min(x, y) elsewhere. This class of operators are called as UMin . Let U ∈ Ud be a uninorm with neutral element e ∈ [0, 1). If the function U (x, 0) is continuous except at the point x = e, then there exists a t-norm T and a t-conorm S such that ⎧ x y  ⎪ eT , , x, y ∈ [0, e], ⎪ ⎪ ⎨ e e  x − e y − e U (x, y) = e + (1 − e)S (5) , , x, y ∈ [e, 1], ⎪ ⎪ 1 − e 1 − e ⎪ ⎩ max(x, y) elsewhere. This class of operators are called as UMax . Also, it was shown in [5] that a uninorm U with neutral element e satisfies U e U U e , where ⎧ x, y ∈ [0, e), ⎨ 0, U e (x, y) = max(x, y), x, y ∈ [e, 1], ⎩ min(x, y) elsewhere, ⎧ ⎨ min(x, y), x, y ∈ [0, e], x, y ∈ (e, 1], U e (x, y) = 1, ⎩ max(x, y) elsewhere.

(6)

(7)

In the framework of n-uninorms, a uninorm is a 1-uninorm with a 1-neutral element e1 = {e} and is denoted as U 1 . As class of uninorms has two subclasses based on the value of its annihilator, we have the following notations: U01 = Uc and U11 = Ud , where the subscript denotes the value of annihilator for the class. Clearly, U e (x, y) ∈ U01 and we denote 1

1

it as U 10 and U e (x, y) ∈ U11 and we denote it as U 1 . The largest operator in U01 denoted as U 0 and smallest operator in U11 denoted as U 11 are as follows: ⎧ 0 if 0 x, y < e1 , ⎪ ⎪ ⎨ max(x, y) if e1 x, y 1, 1 U 1 (x, y) = (8) 1 if x = 1 or y = 1, ⎪ ⎪ ⎩ min(x, y) otherwise. ⎧ min(x, y) ⎪ ⎪ ⎨ 1 1 U 0 (x, y) = 0 ⎪ ⎪ ⎩ max(x, y)

if 0 x, y e1 , if e1 < x, y 1, if x = 0 or y = 0, otherwise. 1

(9)

1

Clearly for any operators U01 and U11 , U 10 U01 U 0 and U 11 U11 U 1 . Lemma 2 (Mas et al. [8]). Let V to be an AMC binary operator on [0, 1] such that V (0, 0) = 0 and V (1, 1) = 1. Then V (0, 1) = k is the annihilator of V.

4

P. Akella / Fuzzy Sets and Systems 160 (2009) 1 – 21

2.2. n-uninorms Definition 2 (Akella [1]). Let V be a binary operator on [0, 1] which is commutative. Then {e1 , e2 , . . . , en }z1 ,z2 ,...,zn−1 is called an n-neutral element of V if for 0 = z0 < z1 < · · · < zn = 1 and ei ∈ [zi−1 , zi ], i = 1, 2, . . . , n we have V (ei , x) = x for all x ∈ [zi−1 , zi ]. Definition 3 (Akella [1]). A binary operator on [0, 1], denoted by U n , is an n-uninorm if it is associative, monotone non-decreasing in each variable and commutative and has an n-neutral element {e1 , e2 , . . . , en }z1 ,z2 ,...,zn−1 . The class of all n-uninorms is denoted as Un . When n = 1, we have uninorms with two classes Uc and Ud . When n = 2 we have 2-uninorms. It was shown in [1] that there are five subclasses of 2-uninorms. There are denoted as Uz21 , 2 , U2 , U2 2 U0,z 0,1 1,z1 and U1,0 . A general representation of an arbitrary member in each of the subclasses was given. We 1 recall only two of them as they are used in this work. ⎧  x y ⎪ ⎪ , x, y ∈ [0, z1 ], , ⎪ z1 U1 ⎪ ⎪ z 1 z1  ⎪ ⎪ ⎪ x − z 1 y − z1 ⎪ ⎪ , x, y ∈ [z1 , 1], , ⎪ z1 + (1 − z1 )U2 ⎪ ⎨ 1 − z1 1 − z1 2 (x, y) ∈ [e1 , z1 ] × [z1 , e2 ] (10a) U0,1 (x, y) = z1 , ⎪ ⎪ or (x, y) ∈ [z , e ] × [e , z ], ⎪ 1 2 1 1 ⎪ ⎪ ⎪ 1, x = 1 and y e1 ⎪ ⎪ ⎪ ⎪ or y = 1 and x e1 , ⎪ ⎪ ⎩ 0, x = 0 or y = 0 and by monotonicity we have, ⎧ [z1 , max(x, y)], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ [ min(x, y), z1 ], 2 U0,1 (x, y) ∈ ⎪ ⎪ ⎪ ⎪ [ min(x, y), max(x, y)], ⎪ ⎪ ⎩

(x, y) ∈ [e1 , z1 ) × (e2 , 1) or (x, y) ∈ (e2 , 1) × [e1 , z1 ), (x, y) ∈ (0, e1 ) × (z1 , e2 ] or (x, y) ∈ (z1 , e2 ] × (0, e1 ), (x, y) ∈ (0, e1 ) × (e2 , 1] or (x, y) ∈ (e2 , 1] × (0, e1 ),

where U1 ∈ U01 and U2 ∈ U11 . ⎧  x y ⎪ ⎪ , , z1 U1 ⎪ ⎪ ⎪ z 1 z1  ⎪ ⎪ ⎪ x − z 1 y − z1 ⎪ ⎪ z1 + (1 − z1 )U2 , , ⎪ ⎪ ⎨ 1 − z1 1 − z1 2 U1,0 (x, y) = z1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1, and by monotonicity we have, ⎧ [z1 , max(x, y)], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ [ min(x, y), max(x, y)], 2 (x, y) ∈ U1,0 ⎪ ⎪ ⎪ ⎪ [ min(x, y), z1 ], ⎪ ⎪ ⎩ where U1 ∈ U01 and U2 ∈ U11 .

(10b)

x, y ∈ [0, z1 ], x, y ∈ [z1 , 1], (x, y) ∈ [e1 , z1 ] × [z1 , e2 ] or (x, y) ∈ [z1 , e2 ] × [e1 , z1 ], x = 0 and y e2 or y = 0 and x e2 , x = 1 or y = 1

(x, y) ∈ [e1 , z1 ) × (e2 , 1) or (x, y) ∈ (e2 , 1) × [e1 , z1 ), (x, y) ∈ [0, e1 ) × (e2 , 1) or (x, y) ∈ (e2 , 1) × [0, e1 ), (x, y) ∈ (0, e1 ) × (z1 , e2 ] or (x, y) ∈ (z1 , e2 ] × (0, e1 ),

(11a)

(11b)

P. Akella / Fuzzy Sets and Systems 160 (2009) 1 – 21

5

Fig. 1. General representation of U 4 .

We recall a few results from [1] which will be used in this work. Lemma 3. For an n-uninorm U n , U n (0, 1) ∈ {zi |i = 0, 1, . . . , n}. Lemma 4. For any U n , U n (zi , zj ) ∈ {zk |k = i, i + 1, . . . , j } and i j . Corollary 1. For any U n , U n (zi , zj ) is the annihilator for all x ∈ [zi , zj ] where i j . For any U n , there exists Ui ∈ U1 for i = 1, 2, . . . , n such that  x − zi−1 y − zi−1 U n (x, y) = zi−1 + (zi − zi−1 )Ui , zi − zi−1 zi − zi−1

for (x, y) ∈ [zi−1 , zi ].

(12)

This shows that U n has an ordinal sum like structure of n uninorms. For all i, j, 0 < i j < n, 0 < i we have U n (ei , zj ) = U n (zi , zj ) = U n (ej +1 , zi ) = U n (ej +1 , ei ) = zk

for some k where i k j.

(13)

We have the expression for any U n as given in Fig. 1. For the sake of illustration we take n = 4. In the figure, for example, {z1 , z2 } in a block denotes that the value of U 4 in that block is either z1 or z2 . In the rest of the region the value of U n can be specified only within an interval. 3. Construction of n-uninorms We know that the number of subclasses in Un is the (n + 1)th Catalan number, C(n + 1). Therefore, the question is how do we specify an operator U n belongs to a particular subclass in Un . When n = 1 we have seen the two subclasses are {Uc , Ud } based on the value of the annihilator namely U 1 (0, 1) and when n = 2 the subclasses are 2 , U2 , U2 , U2 }. As we can see in the case of U2 , the value of the annihilator alone is not enough, for {Uz21 , U0,z 0,1 1,z1 1,0 1 example, in all cases except Uz21 . In this section we give a methodology to identify the members of all the C(n + 1) number of subclasses in Un . For this study we take n to be fixed, 0 = z0 , 1 = zn and i, j to be non-negative integers less than n. We know that U n belongs to a particular subclass of Un when the values U n (zi , zj ) are determined for all i, j ∈ {0, 1, . . . , n} such that i  = j (as U n (zi , zi ) = zi for each i). Such a U n is said to be resolved. We show that we require no more than a set of n values to resolve a U n . We say that a given set of values of the type U n (zi , zj ) for some 0 i < j n, to be a generating set (for resolving) U n provided all the values {U n (zi , zj ) : 0 i < j n} can be computed from the given set. One obvious generating set is {U n (zi , zj ) : 0 i < j n}. We show that the minimum

6

P. Akella / Fuzzy Sets and Systems 160 (2009) 1 – 21

number of values required in a generating set of U n is n/2 (if n is even) and (n + 1)/2 (if n is odd). Generating set need not be unique for a particular subclass of U n . We give an algorithm called Ext(n, 0) which extracts an ordered generating set of no more than n values for a given U n . We call this particular generating set as C-set and denote it as {Ct }n,0 where t n. We show that C-set for a subclass of U n is unique. Next, given an ordered set of values, we examine whether this set is valid C-set (i.e., whether this is the set Ext(n, 0) extracts) for some U n . This algorithm is called Val({Ct }n,0 , n, 0). Our next algorithm is Res({Ct }n,0 , i, j ) which returns the value of U n (zi , zj ) from the given valid C-set {Ct }n,0 . Thus the algorithm can be used to compute all other values required to resolve U n by using the C-set. The last algorithm along this line is Const({Ct }n,0 , n, 0) which constructs the U n in its entirety for a given C-set, i.e., determines U n (x, y) for all x, y ∈ [0, 1]. We would like to point out that this representation of U n is non-recursive in nature unlike the ones given in [1] which are recursive in nature. Based on this algorithm we define two particular classes of operators with the same C-set. Based on these two classes we give the expressions of the smallest and the largest operators with a particular C-set. Finally, we study idempotent n-uninorms with the same C-set and show that they have an ordinal sum like structure of n idempotent uninorms. Also, we find the smallest and the largest idempotent n-uninorms in a subclass. We first give a few definitions and notations that build a base for the algorithms to follow. 3.1. C-sets of n-uninorms Based on the commutativity of U n we have the following definition. Definition 4. U n ∈ Un is said to be resolved if the values U n (zi , zj ) for all i < j n are known.

One can verify that we need to determine ni=1 i number of values to resolve a U n . Define U j,i : [zi , zj ]2 → [zi , zj ] where i < j n as U j,i (x, y) = U n (x, y)

for all x, y ∈ [zi , zj ],

(14)

i.e., U j,i is the restriction of U n to [zi , zj ]2 . By annihilator of U j,i we mean an element a ∈ [zi , zj ] such that U j,i (a, x) = a for all x ∈ [zi , zj ]. From Corollary 1 the value at U n (zi , zj ) is the annihilator of U j,i . Similar to Definition 4 we say that U j,i is resolved if the values U n (zp , zq ) for all i p < q j are determined, i.e., the annihilator of U q,p is determined for all i p < q j . We can now give the following proposition. Proposition 2. U n is resolved iff U j,i has been resolved for all i < j n. That is, the process of resolving U n is based on determining the annihilator of U j,i for all i < j n. We know from Lemma 4 that annihilator of U j,i is zk for some i k j and by monotonicity if U n (zi , zj ) = zk for some i k j then U n (zp , zq ) = zk for all j q k and i p k. Therefore, the annihilator of U q,p is also zk for all j q k and i p k. Thus in the next two stages we need to resolve U k−1,i and U j,k+1 so as to resolve U j,i . As U j,i has an ordinal sum like structure of j − i uninorms (12), this process will terminate when we reach uninorms. Thus there can be a maximum of j − i stages. As at each stage we get one value which is the value of the annihilator of U q,p for some i p < q j , we can see that we need a maximum of j − i values to resolve U j,i . We now find the smallest number of values required to resolve U j,i . For a uninorm we require one value to resolve it. Consider a U 2 . We can see from its subclasses that the smallest number of values required to resolve U 2 is 1 (in the case of Uz21 ). Proceeding this way we can see that the smallest number of values required to resolve U n for any n is one more than the smallest number of values required to resolve U n−2 . We show it by induction. Let m be the smallest number of values required to resolve U n for n > 2. Let m = n/2 when n is even and m = (n + 1)/2 when n is odd. We see that U n+1 has two more neutral elements (ei s) than U n−1 and hence U n+1 has one more zj (between the two ei s) than U n−1 . Consider n + 1 when it is even. Clearly the smallest number of values required to resolve U n+1 is one more than the smallest number of values required to resolve U n−1 . As n − 1 is also even the smallest number of values required to resolve U n+1 is (n + 1)/2. Similarly, for the case when n + 1 is odd we get (n + 2)/2. Summarizing the above discussion we have the following proposition. Proposition 3. Let m be the number of values required to resolve U j,i . Then (j − i)/2 m j − i when j − i is even and (j − i + 1)/2 m j − i when j − i is odd.

P. Akella / Fuzzy Sets and Systems 160 (2009) 1 – 21

7

Fig. 2. Construction of U 4 with the C-set {4, 0, 1, 3}.

We see that both these bounds are attained in the case n = 2. We would like to point out that we will be working with the index of the value of an annihilator, i.e., with k if U n (zi , zj ) = zk . We give an algorithm, which extracts no more than j − i integers between i and j for a given U j,i . This particular set of j − i or less values extracted by the algorithm is called the C-set of U j,i . In a subsequent algorithm, we show that U j,i can be fully resolved by the C-set. C-set for a given U j,i is a unique set (for the subclass to which U j,i belongs) of ordered non-negative distinct integers. The algorithm is based on Lemma 4. The algorithm is called Ext(j, i) where i < j . By ind(U n (zi , zj )) we mean the index of the value at U n (zi , zj ). Initially t = 0. Ext(j,i) (i < j ) Step 1. Step 2. Step 3. Step 4. Step 5.

t := t + 1. Ct := ind(U n (zi , zj )). If Ct − i > 1 Then CALL Ext(Ct − 1, i). If j − Ct > 1 Then CALL Ext(j, Ct + 1). Return.

Therefore, to extract the C-set of U n we call Ext(n, 0). If we run the algorithm on the U 4 of Fig. 2 at the end of this section, we get the C-set {4, 0, 1, 3}. The algorithm Ext(n, 0) picks the values in depth first search manner starting from U n (0, 1) on a binary tree and picks distinct values (i.e., indices). C-set of subclasses of Un will be written as {Ct }n,0 where t n. The set of all C-sets for a given n is denoted as {Ct }n . Clearly the cardinality of {Ct }n is C(n + 1). Let {Ct }n,0 denote the cardinality of the C-set. As shown in Proposition 3, n/2 {Ct }n,0 n when n is even and (n + 1)/2 {Ct }n,0 n when n is odd. C1 is the index of the value of the annihilator of U n (i.e., U n (0, 1)), say k where 0 k n. Then, the C-set of U n is {k, {Cp }k−1,0 , {Cq }n,k+1 } where {Cp }k−1,0 and {Cq }n,k+1 denotes the C-sets of U k−1,0 and U n,k+1 respectively. Clearly, the elements of the set {Cq }n,k+1 is greater than C1 . Also, after C1 (= k) the next few elements are the C-set {Cp }k−1,0 which are all smaller than k. Thus the first element which is greater than k is the first element of the C-set {Cq }n,k+1 , i.e., the annihilator of U n,k+1 . If there is no element greater than k then k ∈ {n − 1, n} and the C-set {Cq }n,k+1 is empty. Similarly, if the second element in the C-set {Ck }n,0 is greater than k then k ∈ {0, 1} and the C-set {Cp }k−1,0 is empty. We can see that the C-set of U n is recursive in nature, i.e., C-set of U n is the annihilator of U n with the C-sets of U k−1,0 and U n,k+1 . Next, we find a method to check if a given ordered set of non-negative integers is a valid C-set or not for given i and j. That is, does there exist a U j,i for which Ext(j, i) yields this set. One algorithm to perform this check is to generate all the C-sets for a given n and match the given set of numbers with them. Clearly we must generate C(n + 1) number of C-sets. From the representations of the members of Un we see that the C-set of Uzni is {i, {Cp }i−1,0 , {Cq }n,i+1 } where {Cp }i−1,0 and {Cq }n,i+1 are C-sets of U i−1,0 and U n,i+1 , respectively. But U n,i+1 is a scaled down (n − i − 1)-uninorm in Un−i−1 and the C-set {Cq }n−i−1,0 of any operator in Un−i−1 a subset of the set {0, 1, . . . , n − i − 1}. Therefore,

8

P. Akella / Fuzzy Sets and Systems 160 (2009) 1 – 21

we can see that by adding i + 1 to the elements of the C-set of U n−i−1 we get {Cq }n,i+1 which is the C-set of U n,i+1 . Therefore, given all the C-sets of all the operators in Ui−1 and Un−i−1 we can generate all the C-sets of operators in Uzni by the formula {i, {Cp }i−1,0 , (i + 1) + {Cq }n−i−1,0 } where by (i + 1) + {Cq }n−i−1,0 we denote the addition of i + 1 to each member of {Cq }n−i−1,0 . This method is recursive. Therefore, to generate all the C-sets for a given n we must generate all the C-sets for all m < n. As the total number of C-sets increase factorially with n, this is not a feasible algorithm. Therefore, we give an algorithm to check if a given ordered set of non-negative integers is a valid C-set or not based on the construction of C-sets. This algorithm is based on the fact that a C-set for a U n is {k, {Cp }k−1,0 , {Cq }n,k+1 } as stated above. Therefore, a C-set of U n is valid then {Cp }k−1,0 and {Cq }n,k+1 are valid C-sets of U k−1,0 and U n,k+1 , respectively. The algorithm is recursive in nature. The algorithm is called Val({At }, j, i), which checks the validity of the given ordered set of non-negative integers {At } to be a C-set of some U j,i for i < j . The cases when a set of numbers is not a valid C-set are: (1) If the cardinality of the set is less than (j − i)/2 when j − i is even or less than (j − i + 1)/2 when j − i is odd or cardinality is greater than j − i. (2) If the first element is greater than 1 and less than n − 1 and there does not exist a number in the set greater than the first element. (3) For a given i < j if the elements in the set do not belong to the set {i, i + 1, . . . , j }. (4) If there is a repetition of a number in the set. The termination criteria is when we get a uninorm U i+1,i and the set is {i} or {i + 1} or a 2-uninorm U i+2,i and the set is {i + 1}. The algorithm returns a flag variable V. If V = 0 then the given set is not a valid C-set and if V = 1 then the set is a valid C-set. At any point of time in the algorithm if a condition is not satisfied then V is assigned 0 and the algorithm stops. Only when all the conditions are satisfied in all the steps of the recursion does the algorithm return V = 1. To check the validity of {Ct }n,0 we call Val({Ct }n,0 , n, 0). Val({At }, j, i) (i < j ) Step 1. Step 2. Step 3. Step 4. Step 5. Step 6. Step 7. Step 8. Step 9. Step 10. Step 11. Step 12.

Check if the elements in {At } belong to the set {i, i + 1, . . . , j }. If NO then V := 0. Exit. If j − i is even and {At } < (j − i)/2 or {At } > j − i Then V := 0. Exit. If j − i is odd and {At } < (j − i + 1)/2 or {At } > j − i Then V := 0. Exit. If i < A1 < j and {At } = j − i Then V := 0. Exit. If j − i = 1 and A1 ∈ {i, j } Then V := 1. Return. If j − i = 2 and A1 = i + 1 Then V := 1. Return. If A1 < j − i − 1 Then check if there is an element in {At } greater than A1 . If NO Then V := 0. Exit. Else check if there is an element in {At } greater than A1 . If YES Then V := 0. Exit. If A1 = i and {At } − {A1 } = ∅ Then V := 0. Exit. Else CALL Val({At } − {A1 }, j, i + 1). If A1 = j and {At } − {A1 } = ∅ Then V := 0. Exit. Else CALL Val({At } − {A1 }, j − 1, i). Let m := [The position of the first element of {At } which is larger than A1 ]. If {A2 , . . . , Am−1 }  = ∅ Then CALL Val({A2 , . . . , Am−1 }, A1 − 1, i). Else If A1 > i + 1 Then V := 0. Exit. If {At } − {A1 , . . . , Am−1 }  = ∅ Then CALL Val({At } − {A1 , . . . , Am−1 }, j, A1 + 1). Else If A1 < j − 1 Then V := 0. Exit.

From now on by a C-set we mean a valid C-set. Next, we see the algorithm to resolve a U n . Given a C-set {Ct }n,0 we give an algorithm to determine the values n U (zi , zj ) for all i < j n. The algorithm is called Res({Ct }n,0 , i, j ) where the algorithm returns the value U n (zi , zj ) from the C-set {Ct }n,0 . Res({Ct }n,0 , i, j) (i < j ) Step 1. t = 1 and m = 0. Step 2. For (m n) Repeat If i Ct+m and j Ct+m Then Return zCt+m . If i, j < Ct+m Then t := t + 1.

P. Akella / Fuzzy Sets and Systems 160 (2009) 1 – 21

9

If i, j > Ct+m Then t := [1 − The position of the first element of the C-set which is larger than Ct+m ]. m := m + 1. End Repeat Given an n the class of all n-uninorms with the same C-set is denoted as Un ({Ct }n ). Therefore, in the framework of representing the subclasses of n-uninorms in terms of the C-sets we have the following notations (simplified) of the subclasses of uninorms. U01 = U1 (0) and U11 = U1 (1). Similarly the subclasses of 2-uninorms are U2 (1), U2 (0, 1), U2 (0, 2), U2 (2, 1) and U2 (2, 0). Finally, we give an algorithm to construct a U n ∈ Un ({Ct }n ). We see that there will be regions in [0, 1]2 where the values can be determined only in an interval to which they belong to. Based on this algorithm we give two special cases of the algorithm which give two classes of n-uninorms whose values in [0, 1]2 are given exactly (i.e., single valued). The values of U n on [zi−1 , zi ]2 for all i = 1, 2, . . . , n are given by 1-uninorm Ui (12). Thus this algorithm determines the value of U n on [0, 1]2 − [zi−1 , zi ]2 for all i = 1, 2, . . . , n. The algorithm also determines the subclass of 1-uninorm to which Ui ’s belong to. The algorithm is based on fact that if U n (zi , zj ) = zk where i k j then U n (zp , zq ) = zk for all j q k and i p k. Thus for the given C-set of U j,i we have U n (zi , zj ) = zC1 , U n (zp , zC1 ) = zC1 for all i p C1 and U n (zC1 , zq ) = zC1 for all j q C1 . If C1 > i then U n (zi , zC1 −1 ) = C2 we must determine the values in (zC1 −1 , zC1 ) × (zi , zC1 −1 ). Similarly if C1 < j we need to determine the values in (zC1 , zC1 +1 ) × (zC1 +1 , zj ). The algorithm is called as Const({At }, j, i) which constructs U j,i with respect to the given C-set {At }j,i . The algorithm is recursive in nature. Step 1 and Step 2 are termination cases which cannot be handled in a general way as that of Step 2 and U 2 which are given in (10a), (10b), 4. These two steps give the representation of U j,i similar to the cases of U0,1 1,0 n,0 n (11a) and (11b). We call Const({Ct }n , n, 0) for U (i.e., U ). The algorithm has an internal variable B. Const({At }, j, i) (i < j )(n2) 2 (10a) and (10b) */ Step 1. /* This step is similar to the case U0,1 If j − i = 2 and {At } = {i, j } Then,

U n (zi , zj ) := zi and U n (zi+1 , zj ) := zj , Ui+1 ∈ U01 and Uj ∈ U11 , U n (x, zj ) = zj , x ∈ [ei+1 , zi+1 ), U n (zj , y) = zj , y ∈ [ei+1 , zi+1 ), U n (x, y) = zi+1 , (x, y) ∈ [ei+1 , zi+1 ) × (zi+1 , ej ] or (x, y) ∈ (zi+1 , ej ] × [ei+1 , zi+1 ), ⎧ ⎪ ⎪ [zi+1 , max(x, y)], ⎪ ⎪ ⎪ ⎪ ⎨ [ min(x, y), zi+1 ], n U (x, y) ∈ ⎪ ⎪ ⎪ ⎪ [ min(x, y), max(x, y)], ⎪ ⎪ ⎩

(x, y) ∈ [ei+1 , zi+1 ) × (ej , zj ) or (x, y) ∈ (ej , zj ) × [ei+1 , zi+1 ) (x, y) ∈ (zi , ei+1 ) × (zi+1 , ej ] or (x, y) ∈ (zi+1 , ej ] × (zi , ei+1 ) (x, y) ∈ (zi , ei+1 ) × (ej , zj ] or (x, y) ∈ (ej , zj ] × (zi , ei+1 ).

Return. 2 (11a) and (11b) */ Step 2. /* This step is similar to the case U1,0 If j − i = 2 and {At } = {j, i} Then, U n (zi , zj ) := zj and U n (zi , zi+1 ) := zi , Ui+1 ∈ U01 and Uj ∈ U11 , U n (x, zi ) = zi , x ∈ (zi+1 , ej ], U n (zi , y) = zi , y ∈ (zi+1 , ej ], U n (x, y) = zi+1 , (x, y) ∈ [ei+1 , zi+1 ) × (zi+1 , ej ] or (x, y) ∈ (zi+1 , ej ] × [ei+1 , zi+1 ),

10

P. Akella / Fuzzy Sets and Systems 160 (2009) 1 – 21

⎧ [zi+1 , max(x, y)], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ [ min(x, y), zi+1 ], U n (x, y) ∈ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [ min(x, y), max(x, y)], ⎩

(x, y) ∈ [ei+1 , zi+1 ) × (ej , zj ) or (x, y) ∈ (ej , zj ) × [ei+1 , zi+1 ) (x, y) ∈ (zi , ei+1 ) × (zi+1 , ej ] or (x, y) ∈ (zi+1 , ej ] × (zi , ei+1 ) (x, y) ∈ [zi , ei+1 ) × (ej , zj ) or (x, y) ∈ (ej , zj ) × [zi , ei+1 ).

Return. /* In the following steps if any of the subscripts are not valid for e or z (e.g., if the subscript is less than 0 or greater than n) then those steps are not performed*/ Step 3. U n (zi , zj ) := zA1 . If A1 > i Then UA1 ∈ U11 . If A1 < j Then UA1 +1 ∈ U01 . Let m := [The position of the first element in {At } which is larger than A1 ]. If there is no element in {At } which is larger than A1 Then m := {At } + 1. Step 4. /* The bounds of the intervals follow from monotonicity and commutativity of U n . Note that this step is not executed when the algorithm is first called, i.e., j = n and i = 0. */ If(1) j + 1 n OR i − 1 0 Then /* (13) */ U n (x, y) = zA1 , (x, y) ∈ [ei , zA1 ] × [zj , ej +1 ] or (x, y) ∈ [zj , ej +1 ] × [ei , zA1 ], U n (x, y) = zA1 , (x, y) ∈ [ei , zi ] × [zA1 , zj ] or (x, y) ∈ [zA1 , zj ] × [ei , zi ]. If(2) B = 0 Then U n (x, y) ∈ [min(x, y), max(x, y)], (x, y) ∈ (zi−1 , ei ) × (ej +1 , zj +1 ] or (x, y) ∈ (ej +1 , zj +1 ] × (zi−1 , ei ), End(If(2)) If(3) B = 1 Then U n (x, y) ∈ [min(x, y), max(x, y)], (x, y) ∈ [zi−1 , ei ) × (ej +1 , zj +1 ) or (x, y) ∈ (ej +1 , zj +1 ) × [zi−1 , ei ), End(If(3))

⎧ [zA1 , max(x, y)], (x, y) ∈ [ei , zA1 ] × (ej +1 , zj +1 ) ⎪ ⎪ ⎨ or (x, y) ∈ (ej +1 , zj +1 ) × [ei , zA1 ], n U (x, y) ∈ [ min(x, y), z ], (x, y) ∈ (zi−1 , ei ) × [zA1 , ej +1 ] ⎪ A 1 ⎪ ⎩ or (x, y) ∈ [zA1 , ej +1 ] × (zi−1 , ei ).

If i > 0 and A1 − i = 1 Call Const({i − 1, A1 }, A1 , i − 1). If j < n and j − A1 = 1 Then Const({j + 1, A1 }, j + 1, A1 ). End(If(1)). Step 5. If m > 2 and {A2 , . . . , Am−1 }  = ∅ Then B := 0 Call Const({A2 , . . . , Am−1 }, A1−1,i ). If {At } − {A1 , . . . , Am−1 }  = ∅ Then B := 1 Call Const({At } − {A1 , . . . , Am−1 }, j, A1 + 1). Return. If we run the algorithm on the C-set {4, 0, 1, 3} when j = 4, i = 0, the construction of the U 4 is as shown in Fig. 2. 3.2. Smallest and largest operators in each subclass of n-uninorms In this section we give a non-recursive expression of the smallest and the largest n-uninorms with the given C-set. We define two classes of n-uninorms with a given C-set {Ct }n which are well defined (i.e., single valued) on [0, 1]2 . This follows as a result of two special cases of the Const algorithm called Const-S({Ct }n , n, 0) and Const-L({Ct }n , n, 0) whose steps are same as Const({Ct }n , n, 0) with the following differences: (1) The points in [0, 1]2 for whom bounds are given as intervals in Const({Ct }n , n, 0) are assigned the lower bound in Const-S({Ct }n , n, 0).

P. Akella / Fuzzy Sets and Systems 160 (2009) 1 – 21

11

(2) The points in [0, 1]2 for whom bounds are given as intervals in Const({Ct }n , n, 0) are assigned the upper bound in Const-L({Ct }n , n, 0). The classes of n-uninorms which are arrived at as a result of Const-S({Ct }n , n, 0) and Const-L({Ct }n , n, 0) are denoted as USn ({Ct }n ) and ULn ({Ct }n ), respectively. Such operators can be shown to be associative. We give a brief insight into the proof of associativity of a U n ∈ USn ({Ct }n ). When x, y, z ∈ [zi−1 , zi ] associativity of U n follows from associativity of the underlying n uninorms. There are many cases to be considered to show the associativity of U n . We illustrate only one case as others can be shown in a similar way. Consider x, y ∈ (zi−1 , ei ) and y ∈ (zj , ej +1 ] where 0 < i j n − 1. Then U n (x, y) ∈ [zi−1 , min(x, y)] ⊂ [zi−1 , ei ]. From Step 4 of Const-S algorithm we have U n (x, z) = min(x, z) = x. We can easily see that U n (U n (x, y), z)) = U n (x, y) and U n (x, U n (y, z)) = U n (x, y). Thus associativity holds. We conclude this section by giving a non-recursive expression of the smallest and the largest n-uninorms with the given C-set. For the uninorms Ui ’s, denoted as U i and U i , are the smallest and largest uninorms, as given in Eqs. (6)–(9), in the corresponding subclasses as resolved in the algorithm respectively, we have the following theorem. n

Theorem 1. For a given n, U n ({Ct }n ) and U ({Ct }n ) are the smallest and largest operators in Un ({Ct }n ), respectively, n if and only if U n ({Ct }n ) ∈ USn ({Ct }n ) with Ui = U i and U ({Ct }n ) ∈ ULn ({Ct }n ) with Ui = U i for all i = 1, 2, . . . , n. 3.3. Idempotent n-uninorms In this section we turn our attention to idempotent n-uninorms. A binary operator V on [0, 1] is idempotent operator if V (x, x) = x for all x ∈ [0, 1]. In [7] idempotent uninorms have been completely characterized where it has been shown that a uninorm is idempotent if and only if it is locally internal, i.e., U (x, y) ∈ {x, y} for all x, y ∈ [0, 1]. As a consequence T = Min and S = Max. We prove the following theorem. Theorem 2. Let U n be an n-uninorm. U n is idempotent if and only if U n (x, y) ∈ {x, y, {zk }} for all x, y ∈ [0, 1], x zk y and Ti = Min and Si = Max, i = 1, 2, . . . , n. Proof. Let U n be an idempotent n-uninorm. From (12) when x, y ∈ [zi−1 , zi ], i = 1, 2, . . . , n the result follows from the case of uninorms. Therefore, Ti = Min and Si = Max, i = 1, 2, . . . , n. From (13) we know that for (x, y) ∈ [ei , zi ] × [zj , ej +1 ] or (x, y) ∈ [zj , ej +1 ] × [ei , zi ], 0 < i j n − 1 we have U n (x, y) = zk where min(x, y) zk  max(x, y). U n (x, 0) = U n (0, x) = zk for some k and for all x ∈ [zi , ei+1 ] and zk zi and U n (x, 1) = U n (1, x) = zk for some k and for all x ∈ [ei , zi ] and zk zi , i = 1, 2, . . . , n − 1. Consider (x, y) ∈ [zi−1 , ei ) × (ej , zj ] where 0 < i < j n. By monotonicity we have x = U n (ei , x) U n (x, y)  n U (ej , y) = y. We have the following subcases: (1) Let zi−1 U n (x, y)ei . As Ti = Min we have U n (x, U n (x, y)) = min(x, U n (x, y)) = x. But U n (x, U n (x, y)) = U n (U n (x, x), y) = U n (x, y). Therefore, U n (x, y) = x. (2) Let ek < U n (x, y)zk where i k < j n. U n (y, U n (x, y)) U n (zk , U n (x, y)) = zk . But U n (x, y) = U n (x, U n (y, y)) = U n (y, U n (x, y))zk which implies U n (x, y) = zk . (3) Let zk U n (x, y)ek+1 where i k < j n. U n (x, U n (x, y)) U n (zk , U n (x, y)) = zk . But U n (x, y) = U n (U n (x, x)), y) = U n (x, U n (x, y))zk which implies U n (x, y) = zk . (4) Let zj U n (x, y)ej . As Sj = Max we have U n (y, U n (x, y)) = max(y, U n (x, y)) = y. But U n (y, U n (x, y)) = U n (x, U n (y, y)) = U n (x, y). Therefore, U n (x, y) = y. By commutativity the above results hold for (x, y) ∈ (ej , zj ] × [zi−1 , ei ) where 0 < i < j n. Consider (x, y) ∈ (zi−1 , ei )×(zj , ej +1 ] where 0 < i j n−1. By monotonicity we have U n (x, y) U n (zj , y) = zj and U n (x, y)U n (x, ei ) = x. We have the following subcases: (1) Let zi−1 U n (x, y)ei . U n (x, y) = x follows as in above subcase (1). (2) Let ek < U n (x, y)zk where i k j n. From (13), U n (y, U n (x, y)) zk . Also, U n (y, U n (x, y)) = U n (x, y). Therefore, U n (x, y) = zk .

12

P. Akella / Fuzzy Sets and Systems 160 (2009) 1 – 21

(3) Let zk−1 U n (x, y)ek where i + 1k j n. But by monotonicity we have U n (x, U n (x, y)) U n (zk−1 , U n (x, y))U n (zk−1 , ek ) = zk−1 . As U n (x, U n (x, y)) = U n (x, y), we have U n (x, y) = zk−1 . By commutativity the above results hold for (x, y) ∈ (zj , ej +1 ] × (zi−1 , ei ) where 0 < i j n − 1. Finally, consider (x, y) ∈ [ei , zi ] × (ej , zj ) i < j n. By monotonicity we have U n (x, y) U n (ej , y) = y and n U (x, y)U n (x, zi ) = zi . The proof of its subcases follow as in the above two cases. Conversely, the given condition implies U n (x, x) = x for all x ∈ [0, 1]. Hence the result.  Corollary 2 (Martin et al. [7]). Let U 1 be a uninorm. U 1 is idempotent if and only if U 1 is locally internal. Therefore, idempotent n-uninorms have an ordinal sum like structure of n idempotent uninorms. n ({C } ). From the condition of Theorem The class of idempotent n-uninorms having a C-set {Ct }n is denoted as Uid t n n 2 we see that the smallest and the largest idempotent n-uninorms in Uid ({Ct }n ) belong to USn ({Ct }n ) and ULn ({Ct }n ), respectively. 1 (0) denoted as U 1 (0) and U 1 (0), The expressions of the smallest and the largest idempotent uninorms in Uid id id respectively, are given as follows:  max(x, y), x, y ∈ [e, 1], 1 U id (0)(x, y) = (15) min(x, y) elsewhere, ⎧ ⎨ min(x, y), x, y ∈ [0, e], 1 x = 0 or y = 0, U id (0)(x, y) = 0, (16) ⎩ max(x, y) elsewhere. 1

1 (1) denoted as U 1 (1) and U (1), respecThe expressions of the smallest and the largest idempotent uninorms in Uid id id tively, are given as follows: ⎧ ⎨ max(x, y), x, y ∈ [e, 1], x = 1 or y = 1, U 1id (1)(x, y) = 1, (17) ⎩ min(x, y) elsewhere.  min(x, y), x, y ∈ [0, e], 1 U id (1)(x, y) = (18) max(x, y) elsewhere.

We have the following theorem. n

Theorem 3. For a given n, U nid ({Ct }n ) and U id ({Ct }n ) are the smallest and the largest idempotent operators in n ({C } ), respectively, if and only if U n ({C } ) ∈ Un ({C } ) with U = U 1 of the subclass as resolved in Uid t n t n t n i id id S n

1

Const({Ct }n , n, 0) and U id ({Ct }n ) ∈ ULn ({Ct }n ) with Ui = U id of the subclass as resolved in Const({Ct }n , n, 0) for all i = 1, 2, . . . , n. Therefore, the smallest and the largest idempotent n-uninorms with the given C-set {Ct }n are like ordinal sums of n idempotent uninorms as defined in (15)–(18) depending on the subclasses of Ui ’s as resolved in Const({Ct }n , n, 0). 4. De Morgan triple involving n-uninorms The De Morgan law when T is a t-norm, S is a t-conorm and m is a strict negation can be expressed in two ways [4]: T (x, y) = m−1 (S(m(x), m(y))) for all x, y ∈ [0, 1],

(19)

S(x, y) = m(T (m−1 (x), m−1 (y))) for all x, y ∈ [0, 1].

(20)

One can easily verify that given T, S and m, (19) is true if and only if (20) is true. (T , S, m) is called as a De Morgan triple if (19) (or (20)) is satisfied.

P. Akella / Fuzzy Sets and Systems 160 (2009) 1 – 21

13

In [5] De Morgan triple involving two uninorms and a strong negation m was studied. Let U1 and U2 be two uninorms with neutral elements e1 ∈ (0, 1) and e2 ∈ (0, 1), respectively, and let m be a strong negation. The triple (U1 , U2 , m) is called a De Morgan triple if we have U1 (x, y) = m(U2 (m(x), m(y))) for all x, y ∈ [0, 1]. The study with strong negations is valid because of the following two results. Lemma 5. Given any e1 , e1 ∈ (0, 1), there exists a strong negation m such that m(e1 ) = e1 . Proposition 4 (Fodor et al. [5]). Given any e1 , e1 ∈ (0, 1), there exist uninorms U1 , U2 with neutral elements e1 and e1 , respectively, and a strong negation m such that (U1 , U2 , m) is a De Morgan triple. Thus Proposition 4 shows that given any two points e1 , e1 ∈ (0, 1) we can find two uninorms U1 , U2 with neutral elements e1 , e1 , respectively, and a strong negation m such that (U1 , U2 , m) is a De Morgan triple. But such a result cannot be stated for n-uninorms with n > 1. We prove it by showing that there does not exist a result in the lines of Lemma 5 for points e1 , e2 , e1 , e2 ∈ (0, 1). Elaborating further, given any points e1 , e2 , e1 , e2 ∈ (0, 1) such that e1 e2 and e1 e2 there does not exist a strong negation m such that m(e1 ) = e1 and m(e2 ) = e2 . We show this by giving an example. Let e1 = 1/4, e2 = 1/2, e1 = 3/4 and e2 = 1/4. There does not exist a strong negation m such that m(e1 ) = e1 and m(e2 ) = e2 as m(1/4)  = m−1 (1/4). But we can always find a strict negation m such that m(e1 ) = e1 and m(e2 ) = e2 for any points e1 , e2 , e1 , e2 ∈ (0, 1) such that e1 e2 and e1 e2 . Therefore, we first study De Morgan triple involving uninorms and strict negation and show that the study in [5] is a special case of our study. We then generalize this study to n-uninorms with a strict negation m and find the conditions for m to be strong. 4.1. De Morgan triple of uninorms with a strict negation In this section we study the De Morgan triple involving uninorms with respect to a strict negation m and also find the conditions for m to be a strong negation. Let U1 and U2 be two uninorms with neutral elements e1 ∈ (0, 1) and e2 ∈ (0, 1), respectively, and let m be a strict negation. Definition 5. The triple (U1 , U2 , m) is called a De Morgan triple if we have U1 (x, y) = m−1 (U2 (m(x), m(y)))

(21)

for all x, y ∈ [0, 1]. One can easily verify that (21) holds if and only if the following holds: U2 (x, y) = m(U1 (m−1 (x), m−1 (y)))

(22)

for all x, y ∈ [0, 1]. Let Ti , Si be the underlying t-norms and t-conorms of uninorms Ui for i = 1, 2 and let m be a strict negation. Lemma 6. Let (U1 , U2 , m) be a De Morgan triple then, (1) m(e1 ) = e2 . (2) There exist two strict negations m1 , m2 such that (T1 , S2 , m1 ) and (T2 , S1 , m2 ) are De Morgan triples. Proof. (1) Substituting x = e1 and y = m−1 (e2 ) in (21) we get m(e1 ) = e2 .

(23)

14

P. Akella / Fuzzy Sets and Systems 160 (2009) 1 – 21

(2) Proof is similar to Proposition 4.1 in [8]. Consider x, y ∈ [0, e1 ]. Then (21) implies,    x y m(x) − e2 m(y) − e2 m e1 T1 = e2 + (1 − e2 )S2 . , , e 1 e1 1 − e2 1 − e2 Define m1 (x) =

m(e1 x) − e2 1 − e2

for all x ∈ [0, 1].

(24)

One can easily verify that m1 is a strict negation. Therefore, we have      y x y x m1 T1 = S2 m1 , m1 . , e1 e 1 e1 e1 Let u = x/e1 , v = y/e1 for all x, y ∈ [0, e1 ]. Clearly u, v ∈ [0, 1] and we see that m1 (T1 (u, v)) = S2 (m1 (u), m1 (v)) holds for all u, v ∈ [0, 1] and hence (T1 , S2 , m1 ) is a De Morgan triple. Similarly we can show for the case x, y ∈ [e1 , 1] that (T2 , S1 , m2 ) is a De Morgan triple where m2 is defined as m2 (x) =

m−1 (e2 x) − e1 1 − e1

for all x ∈ [0, 1].



(25)

Theorem 4. Suppose U1 ∈ UMin with neutral element e1 ∈ (0, 1) and U2 ∈ UMax with neutral element e2 ∈ (0, 1) then there exists a strict negation m such that (U1 , U2 , m) is a De Morgan triple iff there exist two strict negations m1 , m2 such that (T1 , S2 , m1 ) and (T2 , S1 , m2 ) are De Morgan triples. Proof. The necessity part follows from Lemma 6. Proof of sufficiency is similar to Theorem 4.4 in [8]. Suppose that there exist two strict negations m1 , m2 such that (T1 , S2 , m1 ) and (T2 , S1 , m2 ) are De Morgan triples. Define a function m : [0, 1] → [0, 1] as ⎧  x ⎪ ⎪ , x e1 , ⎨ e2 + (1 − e2 )m1  e1 m(x) = (26) x − e1 ⎪ ⎪ , x e . ⎩ e2 m−1 1 2 1 − e1 One can easily verify that m is a strict negation with m(e1 ) = e2 . • If x, y e1 then m(x), m(y) e2 and therefore (21) follows from (T1 , S2 , m1 ) being a De Morgan triple. • If x, y e1 then m(x), m(y) e2 and therefore (21) follows from (T2 , S1 , m2 ) being a De Morgan triple. • If x e1 y then m(x)e2 m(y) and (21) holds since m(min(x, y)) = max(m(x), m(y)) is true for any strict negation m.  Next, we see the conditions under which m is a strong negation. Theorem 5. Let m be defined as in (26) where m1 and m2 are strict negations. Then m is a strong negation if and only if the following equations hold:   x x e2 + (1 − e2 )m1 = e1 + (1 − e1 )m2 for x e1 , e1 e2   x − e1 x e2 m−1 = e for e1 x e2 , + (1 − e )m 1 1 2 2 1 − e1 e2   x − e1 −1 x − e2 e2 m−1 = e for x e2 m 1 1 2 1 − e1 1 − e2 when e1 e2 . If e1 e2 then the same equations hold with the indices reversed.

P. Akella / Fuzzy Sets and Systems 160 (2009) 1 – 21

15

Proof. Let e1 e2 . If let m be a strong negation then m = m−1 . From the definition of m we can see that m−1 is given by ⎧  x ⎪ ⎪ , x e2 , ⎨ e1 + (1 − e1 )m2 e 2  m(x) = (27) x − e2 ⎪ ⎪ , x e . ⎩ e1 m−1 2 1 1 − e2 One can easily see that if m = m−1 then the above equations hold and also if e1 e2 then the same equations hold with the indices reversed. Proof of the sufficient part, i.e., to show that m is involutive, can be seen in Theorem 4.4 in [8] where three cases are distinguished. When x e1 , e1 x e2 and e2 x. One can easily verify that m(m(x)) = x holds when m is substituted from the given equations for each of the three cases.  4.2. De Morgan triple of n-uninorms with a strict negation In this section we study the De Morgan triple involving n-uninorms with respect to a strict negation m and also find the conditions for m to be a strong negation. The results in this section are a generalization of the results in Section 4.1. n be two n-uninorms for a given n and let m be a strict negation. Let U n and U n , m) is called a De Morgan triple if we have Definition 6. The triple (U n , U n (m(x), m(y))) U n (x, y) = m−1 (U

(28)

for all x, y ∈ [0, 1]. One can easily verify that (28) holds if and only if the following holds: n (x, y) = m(U n (m−1 (x), m−1 (y))) U

(29)

for all x, y ∈ [0, 1]. In this work U n is an n-uninorm with n-neutral element {e1 , e2 , . . . , en }z1 ,z2 ,...,zn−1 where zi−1 < ei < zi and Ti ’s, n is another Si ’s and Ui ’s are its underlying t-norms, t-conorms and uninorms, respectively, for i = 1, 2, . . . , n and U          n-uninorm with n-neutral element{e1 , e2 , . . . , en }z1 ,z2 ,...,zn−1 where zi−1 < ei < zi and Ti ’s, Si ’s and Ui ’s are its underlying t-norms, t-conorms and uninorms, respectively, for i = 1, 2, . . . , n. n , m) be a De Morgan triple then, Lemma 7. Let (U n , U  , m(ei ) = e(n+1)−i

(30)

 m(zj ) = zn−j

(31)

for i = 1, 2, . . . , n and j = 0, 1, . . . , n.  holds when j = 0, n for any negation m. We prove the above equations by induction. Proof. Clearly m(zj ) = zn−j  . We first show that m(e1 ) = en and m(z1 ) = zn−1 ∗  Let m(e1 ) = e > en . As m is strict and continuous there exist e1 < x < y < z1 such that en < m(y) < m(x) < e∗ . n (m(x), m(y))  U n (m(x), en ) = m(x). As (U n , U n , m) is a De Morgan Clearly U n (x, y)U n (e1 , y) = y and U n n n n (m(x), m(y)) which triple, we have m(U (x, y)) = U (m(x), m(y)). But we have m(U (x, y)) m(y) < m(x)  U ∗  contradicts the above equality. Similarly one can verify for the case m(e1 ) = e < en .  . Clearly, z∗ < en as en = m(e1 ) and e1 < z1 . As m is strict and continuous there exist z1 < Let m(z1 ) = z∗ > zn−1  n (m(x), m(y))  U n (m(y), x < y < e2 such that zn−1 < m(y) < m(x) < z∗ . Clearly U n (x, y) U n (e2 , x) = x and U n , m) is a De Morgan triple, we have m(U n (x, y)) = U n (m(x), m(y)). But we have en ) = m(y). As (U n , U n n m(U (x, y))m(x) > m(y)  U (m(x), m(y)) which contradicts the above equality. Similarly one can verify for  the case m(z1 ) = z∗ < zn−1 .

16

P. Akella / Fuzzy Sets and Systems 160 (2009) 1 – 21    Assume that m(ek ) = en+1−k and m(zk ) = zn−k where 1 < k < n. We show that m(ek+1 ) = en−k and m(zk+1 ) =

 . zn−k−1

   Let m(ek+1 ) = e∗ > en−k . Clearly e∗ < zn−k as zn−k = m(zk ) and zk < ek+1 . As m is strict and continuous there  exist ek+1 < x < y < zk+1 such that en−k < m(y) < m(x) < e∗ . The argument is similar to the case m(e1 ) = e∗ > en .   . As m is continuous, strict and m(zk ) = zn−k there exist zk < x < y < ek+1 such that Let m(ek+1 ) = e∗ < en−k  ∗ e < m(y) < m(x) < en−k . The argument is similar to the case m(e1 ) = e∗ < en .  If k = n − 1 the m(zn ) = m(1) = 0 = z0 . Therefore, let k < n − 1. Argument for m(zk+1 ) = zn−k−1 is similar to  the case m(z1 ) = zn−1 . 

n , m) be a De Morgan triple. Then there exist 2n strict negations mi , m for i = 1, 2, . . . , n Lemma 8. Let (U n , U i   such that (Ti , Sn+1−i , mi ) and (Tn+1−i , Si , mi ) are De Morgan triples for all i = 1, 2, . . . , n. Proof. Consider x, y ∈ [zi−1 , ei ] for i = 1, 2, . . . , n. Then (28) together with (12) implies  

x − zi−1 y − zi−1 , m zi−1 + (ei − zi−1 )Ti ei − zi−1 ei − zi−1     m(x) − e m(y) − e n+1−i n+1−i     = en+1−i . + (zn+1−i − en+1−i )Sn+1−i ,     zn+1−i − en+1−i zn+1−i − en+1−i Define mi (x) =

 m(zi−1 + (ei − zi−1 )x) − en+1−i

for all x ∈ [0, 1].

  zn+1−i − en+1−i

(32)

One can easily verify that mi is a strict negation for all i = 1, 2, . . . , n. Let u = (x − zi−1 )/(ei − zi−1 ) and v = (y − zi−1 )/(ei − zi−1 ) where x, y ∈ [zi−1 , ei ]. Clearly u, v ∈ [0, 1]. By definition of mi we see that  mi (Ti (u, v)) = Sn+1−i (mi (u), mi (v)).  , mi ) is a De Morgan triple for all i = 1, 2, . . . , n. Hence (Ti , Sn+1−i  , Si , mi ) is a De Morgan triple Similarly one can show for the case x, y ∈ [ei , zi ] for all i = 1, 2, . . . , n that (Tn+1−i  for all i = 1, 2, . . . , n where mi is defined as

mi (x) =

   )x) − e m−1 (zn−i + (en+1−i − zn−i i

zi − e i

for all x ∈ [0, 1].



(33)

1 n , m) be a De Morgan triple. Then there exist n strict negations m1 such that (Ui , U  Corollary 3. Let (U n , U i n+1−i , mi ) and are De Morgan triples for all i = 1, 2, . . . , n.

Proof. Consider x, y ∈ [zi−1 , zi ] for i = 1, 2, . . . , n. Then (28) implies  

x − zi−1 y − zi−1 , m zi−1 + (zi − zi−1 )Ui zi − zi−1 zi − zi−1     m(x) − z m(y) − z n−i n−i     = zn−i . + (zn+1−i − zn−i )Un+1−i ,     zn+1−i − zn−i zn+1−i − zn−i Define m1i (x) =

 m(zi−1 + (zi − zi−1 )x) − zn−i   zn+1−i − zn−i

for all x ∈ [0, 1].

(34)

One can easily verify that m1i is a strict negation for all i = 1, 2, . . . , n. Let u = (x − zi−1 )/(zi − zi−1 ) and v = (y − zi−1 )/(zi − zi−1 ) where x, y ∈ [zi−1 , zi ]. Clearly u, v ∈ [0, 1]. By definition of m1i we see that  (m1i (u), m1i (v)). m1i (Ui (u, v)) = Un+1−i  Hence (Ui , Un+1−i , m1i ) is a De Morgan triple for all i = 1, 2, . . . , n. 

P. Akella / Fuzzy Sets and Systems 160 (2009) 1 – 21

17

Moreover     ⎧ 

    − zn−i en+1−i en+1−i − zn−i ⎪ x(zi − zi−1 ) ei − zi−1 ⎪ ⎪ mi , x ∈ 0, , + 1−  ⎪    ⎪ − zn−i zn+1−i − zn−i ei − zi−1 zi − zi−1 ⎨ zn+1−i 1 mi (x) =   

 ⎪   ⎪ en+1−i − zn−i (zi − zi−1 )x − ei + zi−1 ei − zi−1 ⎪  −1 ⎪ ⎪ (mi ) , x∈ ,1 ⎩ z  zi − e i zi − zi−1 n+1−i − zn−i for all i = 1, 2, . . . , n. Thus any analysis done with the 2n strict negations mi and mi (for t-norms and t-conorms) will hold for the n strict negations m1i (for uninorms) for all i = 1, 2, . . . , n and vice versa. Consider U n ({Ct }n ). Let {Ct1 }n = {C11 , {Cp1 }C 1 −1,0 , {Cq1 }n,C 1 +1 }, where {Cp1 }C 1 −1,0 and {Cq1 }n,C 1 +1 are the C-sets 1

1

of U C1 −1,0 and U n,C1 +1 , respectively. We define m-dual of {Ct1 }n as follows. 1

1

1

1

Definition 7. m({Ct1 }n ) = {n − C11 , m({Cq1 }n,C 1 +1 ), m({Cp1 }C 1 −1,0 )}. 1

1

Thus m({Ct1 }n ) is defined recursively. n ({Ct2 }n ), m) be a De Morgan triple then Lemma 9. Let (U n ({Ct1 }n ), U {Ct2 }n ≡ m({Ct1 }n ) = {n − C11 , m({Cq1 }n,C 1 +1 ), m({Cp1 }C 1 −1,0 )}. 1

1

Proof. Case 1: 1 < C11 < n − 1. Let {Ct1 }n = {C11 , C21 , . . . , Cs1 } and let Cr1 be the first element in the C-set which is larger 1 , C 1 , C 1 , . . . , C 1 } and {C 1 , . . . , C 1 } = {C 1 } than C11 (1 < r < n). Thus {Ct1 }n = {C11 , C21 , . . . , Cr−1 r s p C 1 −1,0 and r+1 2 r−1 1

1 , . . . , C 1 } = {C 1 } 2 2 2 {Cr1 , Cr+1 s q n,C11 +1 . Let Cr  be the first element in {Ct }n which is larger than C1 . As the elements n (m(0), m(1)) = m(U n (0, 1)) = m(z 1 ) = z 1 where the last equality of the C-set are indices of zi ’s, z 2 = U C1

comes from (31). Thus C12 = n − C11 . Also, z z 2 Cr 

n (z = U

C12 +1

, 1) =

m(U n (z

C22

C11 −1 , 0))

n (z = U

= m(zC 1 ) = 2

C1

n−C1

 , 0) = m(U n (zC 1 +1 , 1)) = m(zCr1 ) = zn−C 1 and

C12 −1 z 1 . n−C2

1

r

Therefore, C22 = n − Cr1 and Cr2 = n − C21 .

Proceeding this way and due to the recursive nature of C-sets we get the result. Case 2: If C1 = 0 or 1, then {Ct1 }n = {C11 , {Cq1 }n,C 1 +1 }. One can verify that m({Ct1 }n ) = {n − C11 , m({Cq1 }n,C 1 +1 )}. 1

1

Case 3: If C1 = n or n − 1, then {Ct1 }n = {C11 , {Cp1 }C 1 −1,0 }. One can verify that m({Ct1 }n ) = {n − C11 , m({Cp1 }C 1 −1,0 )}. 1

1



From the definition of m-dual of a C-set and Corollary 3 we have the following necessary condition for (U n ({Ct1 }n ), n ({Ct2 }n ), m) to be a De Morgan triple. U n ({Ct2 }n ), m) be a De Morgan triple. Then m({Ct1 }n ) = {Ct2 }n and there exists n strict Lemma 10. Let (U n ({Ct1 }n ), U  , m1i ) are De Morgan triples for all i = 1, 2, . . . , n. negations m1i such that (Ui , Un+1−i n ({Ct2 }n ). As they are constructed using an algorithm we cannot have a general representaConsider U n ({Ct1 }n ) and U n 1 n ({Ct2 }n ) with general C-sets {Ct1 }n and {Ct2 }n . This is the limitation of the method tion of n-uninorms U ({Ct }n ) and U n ({Ct2 }n ), m) to be a of constructing an n-uninorm using C-sets. Thus we show a sufficient condition for (U n ({Ct1 }n ), U 1 2 De Morgan triple by taking particular examples of C-sets. Let {Ct }n = {0, 1, . . . , n−1} and {Ct }n = {n, n−1, . . . , 1}.

18

P. Akella / Fuzzy Sets and Systems 160 (2009) 1 – 21

Then any USn ∈ USn (0, 1, . . . , n − 1) has the following expression: ⎧  x − zi−1 y − zi−1 ⎪ ⎪ , , zi−1 + (zi − zi−1 )Ui ⎪ ⎪ ⎪ zi − zi−1 zi − zi−1 ⎪ ⎨ min(x, y), USn (x, y) = ⎪ ⎪ ⎪ ⎪ zi , ⎪ ⎪ ⎩

(x, y) ∈ [zi−1 , zi ], (x, y) ∈ [zi−1 , ei ) × [zi , 1] or (x, y) ∈ [zi , 1] × [zi−1 , ei ), (x, y) ∈ [ei , zi ] × [zi , 1] or (x, y) ∈ [zi , 1] × [ei , zi ],

(35)

where z0 = 0, i = 1, 2, . . . , n and Ui ∈ UMin and any ULn ∈ ULn (n, n − 1, . . . , 1) has the following expression: ⎧  x − zi−1 y − zi−1 ⎪  ⎪ , (x, y) ∈ [zi−1 , zi ], , ⎪ zi−1 + (zi − zi−1 )Ui ⎪ ⎪ zi − zi−1 zi − zi−1 ⎪ ⎨ (x, y) ∈ [0, zi−1 ] × [zi−1 , ei ] zi−1 , ULn (x, y) = (36) or (x, y) ∈ [zi−1 , ei ] × [0, zi−1 ], ⎪ ⎪ ⎪ ⎪ max(x, y), (x, y) ∈ [0, zi−1 ] × (ei , zi ] ⎪ ⎪ ⎩ or (x, y) ∈ (ei , zi ] × [0, zi−1 ], where z0 = 0 i = 1, 2, . . . , n, and Ui ∈ UMax . USn and ULn are defined as in Section 3.2. Now we are in a position to prove a result analogue to Theorem 4 for n-uninorms. n ∈ Un ({Ct2 }n ). Theorem 6. Let {Ct1 }n = {0, 1, . . . , n − 1}, {Ct2 }n = {n, n − 1, . . . , 1}, U n ∈ USn ({Ct1 }n ) and U L n n , m) is a De Morgan triple iff there exist 2n strict negations Then there exists a strict negation m such that (U , U   mi , mi for i = 1, 2, . . . , n such that (Ti , Sn+1−i , mi ) and (Tn+1−i , Si , mi ) are De Morgan triples where Ti and Si are  the underlying t-norms and t-conorms, respectively, of Ui and Ti and Si are the underlying t-norms and t-conorms, respectively, of Ui for all i = 1, 2, . . . , n and m({Ct1 }n ) = {Ct2 }n . Proof. One can easily verify that m({0, 1, . . . , n − 1}) = {n, n − 1, . . . , 1}. The rest of the proof of necessary part follows from Lemma 8.   Suppose there exist 2n strict negations mi , mi for i = 1, 2, . . . , n such that (Ti , Sn+1−i , mi ) and (Tn+1−i , Si , mi ) are De Morgan triples for all i = 1, 2, . . . , n. Define m : [0, 1] → [0, 1] as ⎧  x − zi−1 ⎪    ⎪ , x ∈ [zi−1 , ei ], ⎨ en+1−i + (zn+1−i − en+1−i )mi  ei − zi−1 m(x) = (37) ⎪   )(m )−1 x − ei  ⎪ , x ∈ [ei , zi ] + (en+1−i − zn−i ⎩ zn−i i zi − e i   and m(zj ) = zn−j for for all i = 1, 2, . . . , n. One can easily verify that m is a strict negation with m(ei ) = e(n+1)−i i = 1, 2, . . . , n and j = 0, 1, . . . , n.    • If x, y ∈ [zi−1 , ei ] then m(x), m(y) ∈ [en+1−i , zn+1−i ] and therefore (28) holds as (Ti , Sn+1−i , mi ) is a De Morgan triple for all i = 1, 2, . . . , n.  , e   • If x, y ∈ [ei , zi ] then m(x), m(y) ∈ [zn−i n+1−i ] and therefore (28) holds as (Tn+1−i , Si , mi ) is a De Morgan triple for all i = 1, 2, . . . , n.   • If zi−1 x < ei y 1 then 0 m(y)en+1−i < m(x) zn+1−i . From (35) and (36) we see that

n (m(x), m(y)) ⇔ m(min(x, y)) = max(m(x), m(y)), m(U n (x, y)) = U which is true from the definition of m.   • If ei x zi y 1 then 0 m(y) zn−i m(x) en+1−i . From (35) and (36) we see that n (m(x), m(y)) ⇔ m(zi ) = z , m(U n (x, y)) = U n−i which is true from the definition of m. n , m) is a De Morgan triple.  Hence (U n , U

P. Akella / Fuzzy Sets and Systems 160 (2009) 1 – 21

19

 Lemma 11. Let ei , zi , ei , zi ∈ [0, 1] such that zi−1 < ei < zi and zi−1 < ei < zi for i = 1, 2, . . . , n where     and m(zj ) = zn−j for z0 = z0 = 0 and zn = zn = 1 then there exists a strict negation m such that m(ei ) = e(n+1)−i i = 1, 2, . . . , n and j = 0, 1, . . . , n.

Proof. Let mi and mi be any 2n strict negations then m constructed as in (37) is a strict negation satisfying the given conditions.  Because of the following theorem the study of the De Morgan triple of n-uninorms involving a strict negation assumes significance.  Theorem 7. Let ei , zi , ei , zi ∈ [0, 1] such that zi−1 < ei < zi and zi−1 < ei < zi for i = 1, 2, . . . , n where   z0 = z0 = 0 and zn = zn = 1 then there exist a strict negation m and n-uninorms U n with neutral element n with neutral element {e , e , . . . , en }z ,z ,...,z such that (U n , U n , m) is a De {e1 , e2 , . . . , en }z1 ,z2 ,...,zn−1 and U 1 2 n−1 1 2 Morgan triple.

Proof. Let mi and mi be any 2n strict negations then m constructed as in (37) is a strict negation satisfying m(ei ) =   e(n+1)−i and m(zj ) = zn−j for i = 1, 2, . . . , n and j = 0, 1, . . . , n. Let Si and Si be arbitrary t-conorms and let Ti  and Ti be t-norms for all i = 1, 2, . . . , n, such that  Ti (x, y) = m−1 i (Sn+1−i (mi (x), mi (y))),

 Tn+1−i (x, y) = (mi )−1 (Si (mi (x), mi (y))).

Let Ui be a uninorm with Ti and Si as its underlying t-norms and t-conorms, respectively, and Ui be a uninorm with Ti n as in (36). Then by Theorem and Si as its underlying t-norms and t-conorms, respectively. Define U n as in (35) and U n , m) is a De Morgan triple.  6 (U n , U We complete the study by finding the conditions for m to be strong. Given x ∈ [zi−1 , ei ] or x ∈ [ei , zi ] for some i. Then two cases are possible, i.e., x ∈ [zj −1 , ej ] or x ∈ [ej , zj ] for some j. Consider the following equations which follow from (37). For x ∈ [zi−1 , ei ]: (1) If x ∈ [zj −1 , ej ] for some j then   en+1−i +(zn+1−i

 − en+1−i )mi



x − zi−1 ei − zi−1



 =en+1−j +(zn+1−j −en+1−j )mn+1−j

(1) If x ∈ [ej , zj ] for some j then    + (zn+1−i − en+1−i )mi en+1−i



x − zi−1 ei − zi−1





x−zj −1

.

ej −zj −1 

= zn−j + (en+1−j − zn−j )m−1 n+1−j

(38a)



x − ej

zj − ej

(38b)

.

For x ∈ [ei , zi ]: (1) If x ∈ [zj −1 , ej ] for some j then  zn−i

 + (en+1−i

 − zn−i )(mi )−1



x − ei zi − e i



=

en+1−j +(zn+1−j −en+1−j )mn+1−j

(2) If x ∈ [ej , zj ] for some j then    zn−i + (en+1−i − zn−i )(mi )−1



x − ei zi − e i



 = zn−j + (en+1−j − zn−j )m−1 n+1−j

x−zj −1

ej −zj −1

x − ej

zj − ej

 .

(39a)

 .

(39b)

20

P. Akella / Fuzzy Sets and Systems 160 (2009) 1 – 21

Theorem 8. The function m defined in (37) where mi and mi are strict negations, is a strong negation iff Eqs. (38a), (38b), (39a) and (39b) hold for all i = 1, 2, . . . , n. Proof. Let m as defined in (37) be a strong negation. From (37) we can see that m−1 is given by   ⎧  x − zi−1 ⎪ ⎪   , e ], ⎪ en+1−i + (zn+1−i − en+1−i )mn+1−i , x ∈ [zi−1 ⎨ i  ei − zi−1 −1 m (x) =   ⎪ x − ei ⎪ ⎪ x ∈ [ei , zi ] ⎩ zn−i + (en+1−i − zn−i )m−1 n+1−i z − e , i i

(42)

for all i = 1, 2, . . . , n. As m is strong, m = m−1 and hence the equations follow. Conversely, let Eqs. (38a), (38b), (39a) and (39b) hold for all i = 1, 2, . . . , n and j = 1, 2, . . . , n. We need to show   + (zn+1−i − that m(m(x)) = x for all x ∈ [0, 1]. Let x ∈ [zi−1 , ei ] for any i = 1, 2, . . . , n. Then m(x) = en+1−i   x−zi−1  en+1−i )mi ei −zi−1 . We have two cases If m(x) ∈ [zj −1 , ej ] for some j then m(m(x)) =

 en+1−j

 + (zn+1−j

 − en+1−j )mj



m(x) − zj −1 ej − zj −1

.

  , zn+1−i ]. Therefore, from (38b) we have But we have m(x) ∈ [en+1−i    m(x) − en+1−i −1 m(m(x)) = zi−1 + (ei − zi−1 )mi   zn+1−i − en+1−i  ⎛ ⎞ x − zi−1     en+1−i − en+1−i + (zn+1−i − en+1−i )mi ⎜ ⎟ ei − zi−1 ⎜ ⎟ = zi−1 + (ei − zi−1 )m−1 i ⎝   ⎠ zn+1−i − en+1−i

= x. If m(x) ∈ [ej , zj ] for some j then m(m(x)) =

 zn−j

 + (en+1−j

 − zn+1−j )(mj )−1



m(x) − ej zj − e j

.

  , zn+1−i ]. Therefore, from (39b) and arguing as above we have m(m(x)) = x. Hence m But we have m(x) ∈ [en+1−i is a strong negation. The argument for the case when x ∈ [ei , zi ] is similar. 

5. Concluding remarks In [2] De Baets has classified uninorms (1-uninorm in the language of this paper) into three main classes: UMin and UMax , idempotent and representable. In this paper the first class has been generalized to USn and ULn . In [7], characterization of idempotent uninorms in terms of a decreasing function g from [0, 1] to [0, 1] has been obtained. Such a representation of n-uninorms is not yet known. Representable uninorms have been defined in [5]. Such a representation is not possible for n-uninorms as it was shown in [8] that such a representation does not exist for t-operators (2-uninorms). In this paper we have introduced the concept of C-set which enabled us to identify each subclass of n-uninorms uniquely. It further enabled us to give a procedure to construct an arbitrary member of each subclass of n-uninorms. Due to this procedure we were able to give a methodology to construct the smallest and the largest members in general and of idempotent ones in particular for each subclass of n-uninorms. It also gave strength to the study of the De Morgan triple of n-uninorms with a strict negation as we were able to relate the C-sets of De Morgan pairs of n-uninorms.

P. Akella / Fuzzy Sets and Systems 160 (2009) 1 – 21

21

Acknowledgements The author dedicates this work to the Chancellor of his university, Bhagawan Sri Sathya Sai Baba. The author is deeply grateful to Prof. C. Jagan Mohan Rao for his guidance. The author would like to thank the anonymous reviewers for their valuable suggestions and insight. References [1] P. Akella, Structure of n-uninorms, Fuzzy Sets and Systems 158 (2007) 1631–1651. [2] B. De Baets, Uninorms: the known classes, in: D. Ruan, H.A. Abderrahim, P. D’hondt, E. Kerre (Eds.), Fuzzy Logic and Intelligent Technologies for Nuclear Science and Industry, Proc. 3rd Int. FLINS Workshop, Antwerp, Belgium, World Scientific Publishing, Singapore, 1998, pp. 21–28. [3] T. Calvo, B. De Baets, J. Fodor, The functional equations of Frank and Alsina for uninorms and nullnorms, Fuzzy Sets and Systems 120 (2001) 385–394. [4] J.C. Fodor, M. Roubens, Fuzzy Preference Modeling and Multicriteria Decision Support, Kluwer, Dordrecht, 1994. [5] J.C. Fodor, R. Yager, A. Rybalov, Structure of uninorms, Internat. J. Uncertainty, Fuzziness Knowledge-Based Systems 5 (1997) 411–427. [6] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer, Dordrecht, 2000. [7] J. Martin, G. Mayor, J. Torrens, On locally internal monotonic operations, Fuzzy Sets and Systems 137 (2003) 27–42. [8] M. Mas, G. Mayor, J. Torrens, t-Operators, Internat. J. Uncertainty, Fuzziness Knowledge-Based Systems 7 (1999) 31–50. [9] R. Yager, A. Rybalov, Uninorm aggregation operators, Fuzzy Sets and Systems 80 (1996) 111–120.