Some remarks on n -uninorms in IF-sets

: Uninorms and nullnorms are well-known monoidal and monotone operations on the unit interval. Akella [2007] proposed their genaralization to n -uninorms. Really, we get both, proper uninorms as well as proper nullnorms as special cases of 2-uninorms. Moreover, proper uninorms as well as proper nullnorms can be characterized as 2-uninorms with some special types of 2-neutral elements. In the present paper, we discuss a classiﬁcation of 2-uninorms from another point of view as it was done by Akella in 2007 and 2009. Then, we look at 2-uninorms in IF-sets and point out some differences between 2-uninorms on the unit interval and 2-uninorms in IF-sets.


Introduction
Uninorms were introduced by Yager and Rybalov [21] as a generalization of both t-norms and t-conorms (for details on t-norms and their duals, t-conorms, see, e.g., [13,17]). Since that time, researchers study properties of several distinguished families of uninorms. In [16], Karaçal and Mesiar introduced uninorms in bounded lattices. In [5], Bodjanova and Kalina constructed uninorms in bounded lattices with arbitrarily given underlying t-norm and t-conorm.
Another generalization of t-norms and t-conorms, called t-operators, was introduced by Mas et al. In [18,19], Mas et al. studied t-operators on finite chains. In 2001, Calvo et al. [6] introduced nullnorms when trying to solve Frank's functional equation [11] where one of the operations in the equation was a uninorm. Afterwards, Mas et al. [20] showed that nullnorms and t-operators coincide in the unit interval. Karaçal et al. [15] introduced nullnorms in bounded lattices.
Akella [1,2] introduced 2-and n-uninorms in the unit interval and gave a characterization of these operations. In this paper, we characterize 2-uninorms (or more general on n-uninorms) from the point of view of their two-neutral elements. Particularly, we split the system of all 2-uninorms into 9 (not necessarily disjoint) subclasses. Afterwards, we point out some differences in the structure of 2-(and n-) uninorms in IF sets.
Intuitionistic fuzzy sets (also, IF-sets), introduced by Atanassov, are a special type of latticevalued fuzzy sets, introduced by Goguen [12]. Important milestones in the theory of IF-sets, besides the monograph by Atanassov [3], are the papers by Deschrijver [7,8], and Deschrijver and Kerre [9]. In [7] Deschrijver has shown that there exist t-norms which are not representable as a pair of a t-norm and a t-conorm. In [8] the author has shown that there exist uninorms in IF-sets which are neither conjunctive nor disjunctive. In [9], Deschrijver and Kerre have shown that the theory of IF-sets is equivalent to the theory of interval-valued sets.
A further development of uninorms in IF-sets (or, equivalently, in interval-valued sets) is the paper by Kalina and Král' [14], where the authors have shown that for arbitrary pair (a, e) of incomparable elements of interval-valued sets there exists a uninorm having a as the annihilator and e as the neutral element.

Basic definitions and some known facts
An IF-set [3] can be represented as a special case of L-fuzzy set [12], where L is a bounded lattice. Membership grades of an IF-set are elements (x 1 , x 2 ) ∈ [0, 1] 2 such that x 1 + x 2 ≤ 1. The set of all IF-membership grades will be denoted by L * . For arbitrary (x 1 , x 2 ), (y 1 , y 2 ) ∈ L * the following holds Thus, the least and the greatest elements of L * are 0 = (0, 1), 1 = (1, 0), respectively. We will write these values in bold letters to distinguish them from the real numbers 0 and 1.
Following the notation introduced in [4], we will write x y if x, y ∈ L * are incomparable. For x ∈ L * we denote x = {z ∈ L * ; z x}.

Lemma 1 ([10]).
A uninorm U is a t-norm whenever its neutral element is e = 1. In that case the annihilator of U is a = 0.
U is a t-conorm whenever its neutral element is e = 0. In that case the annihilator of U is a = 1. Lemma 2 ([10]). Let U be a uninorm, e ∈ ]0, 1[ be its neutral element. Then are a t-norm and a t-conorm, respectively.
The operations T U and S U from Lemma 2 are called the underlying t-norm and the underlying t-conorm, respectively.

Definition 3 ([6]
). An associative, commutative and monotone operation V : Then a is the annihilator of V .
Similarly like for uninorm U , also for nullnorm V there exist its undrlying t-norm V T and t-conorm S T given by, respectively, be a monotone, commutative and associative operation that has a 2-neutral element {e 1 , e 2 } z .

Characterization and classes of 2-uninorms and a generalization to n-uninorms
Let us consider proper uninorms and proper nullnorms as 1-uninorms. Then we adopt the the following definition: Definition 8. Let F n be an n-uninorm for n > 1. We say that F n is a proper n-uninorm if F n is not an (n − 1)-uninorm.
For a proper 2-uninorm F , the operations U 1 and U 2 given by equality (1), will be called the lower and the upper underlying uninorm, respectively.
Let us a look at 2-neutral elements. For a given 0 < z < 1, there are 9 possibilities how to set a 2-neutral element {e 1 , e 2 } z . Namely,

1.
As a corollary to Lemma 1 we get the following Let us check all 9 possibilities of setting a 2-neutral element.
Proof. We will prove only the t-norm case. As the first step, let us prove that F (0, 1) = 0. Since {z, 1} z is the 2-neutral element of F , we have by associativity Monotonicity of F implies that 0 is the annihilator of F . As the second step, we prove that 1 is the neutral element of F . Since we know that 1 is the partial neutral element of F in the interval [z, 1]. Let x ∈ [0, z].
The proof is completed. Lemma 6. Let F be a 2-uninorm whose 2-neutral element is {z} z ({e 1 , z} z , {z, e 2 } z ) for 0 < z < 1 and 0 < e 1 < z, z < e 2 < 1. Then F is a uninorm whose neutral element is z (e 1 and the underlying t-conorm S is the ordinal sum of two t-conorms S = ( S 1 , e 1 , z , S 2 , z, 1 ), e 2 and the underlying t-norm T is the ordinal sum of two t-norms T = ( T 1 , 0, z , T 2 , z, e 2 )).
Proof. In the case that {z} z , we have that z is a partial neutral element in the interval [0, z] as well as in the interval [z, 1], i.e., z is the neutral element of F . Hence, directly by Definition 1 we get that F is a uninorm with the neutral element z.
In the case that {e 1 , z} z is the 2-neutral element of F , we get applying Lemma 5 to the interval [e 1 , 1] that S = ( S 1 , e 1 , z , S 2 , z, 1 ) is a t-conorm which is the underlying operation of F . The rest of the proof is due to Definition 1.
Dually we could prove the case when {z, e 2 } z is the 2-neutral element of F .
Proof. The fact that F is a nullnorm with the annihilator z is directly due to Definition 3.
The remaining three cases lead to proper 2-uninorms.
Then F is a proper 2-uninorm.
We omit the proof of this lemma since the assertion is obvious.
Lemma 9. Let F be a 2-uninorm whose 2-neutral element is {e, 1} z for 0 < e < z < 1. Then F is a proper 2-uninorm whose upper underlying uninorm is reduced to a t-norm.
Proof. The fact that the upper underlying uninorm is reduced to a t-norm is due to Lemma 5. The rest of the proof is obvious.
Lemma 10. Let F be a 2-uninorm whose 2-neutral element is {0, e} z for 0 < z < e < 1. Then F is a proper 2-uninorm whose lower underlying uninorm is reduced to a t-conorm.
To prove item 1), it is enough to realize that, for n ≥ 2, if there were no i such that z i−1 < e i < z i , the operation F would have diagonal blocks either (T 1 , S 1 , T 2 , S 2 , . . . ) or (S 1 , T 1 , S 2 , T 2 , . . . ), where T 1 , T 2 are t-norms, and S 1 , S 2 are t-conorms. In each of these two cases the n-neutral element could be reduced to the (n − 1)-neutral element, since in the first case e 1 = e 2 and in the second case e 2 = e 3 either {e 2 , . . . , e n } (z 2 ,...,e i ,z i ,...,z n−1 ) or {e 1 , . . . , e n } (z 1 ,...,e i ,z i ,...,z n−1 ) , respectively. This proves the item 1) for n ≥ 3. For n = 2 the statement is due to Lemmas 8, 9 and 10. Item 2) is a direct consequence of item 1).

2-uninorms on IF-sets
is a bounded lattice with incomparable elements. The incomparability of some elements will be crucial in our considerations.
On the other hand, since x = ∅, we can definẽ x otherwise. ( Hence,T ∧ is not a t-norm, but {x, 1} x is a 2-neutral element ofT ∧ . This means thatT ∧ is a proper 2-uninorm.
and for z 1 ∈ [x, 1] and z 2 ∈ x , and for z 2 ∈ [x, 1] and z 1 ∈ x , The operationŨ is restricted to [0, x] ∪ [x, 1], ifŨ has no neutral element on the whole L * , i.e., it is not a uninorm. On the other hand, {x} x is a 2-neutral element, henceŨ is a proper 2-uninorm.
x otherwise. (4) V is a nullnorm whose annihilator is x. In this case, if we are looking for a modificationṼ of V in such a way thatṼ is reduced to [0, x] ∪ [x, 1], butṼ is not a nullnorm, we will not succeed. Really, we have that V (1, 0) = x and hence alsoṼ (1, 0) = x and this implies that x is the annihilator ofṼ .
Remark 1. Dually to the operationT ∧ introduced by (2), we can define on L * an operatioñ S ∨ starting from the t-conorm S ∨ (z 1 , z 2 ) = z 1 ∨ z 2 and an element x / ∈ {0, 1}. This means that, unlike the situation with the operations in the unit interval, an arbitrary form of the 2-neutral element, except of the case when {0, 1} x is the 2-neutral element, may lead to proper 2-uninorms.
As a corollary to the above considerations in Examples 1 -3, we get the following proposition.

Conclusions
In this paper, we have discussed 2-uninorms in the unit interval and in the L * lattice of IF-membership grades. We have shown that there are substantial differences between 2-uninorms in the unit interval and 2-uninorms in the L * lattice. The results on 2-uninorms we have generalized to n-uninorms.