Some Examples of Weak Uninorms

and Applied Analysis 3 Since λ, 0, and 1 are idempotent elements and R is continuous, from Theorem 2.8 in [14] (or Definition 3.1 in [9]), R and R are continuous weak uninorms. Obviously, they have no nontrivial idempotent elements. FromTheorem 8, they are t-norms or t-conorms. (1) If R is a t-normand R is a t-conorm, then λ is the neutral element of R; that is, R is a uninorm. (2) If R is a t-conorm and R is a t-norm, then λ is an absorbing element λ ∈ [0, 1], and for all x ∈ [0, λ], V(x, 0) = x and for all x ∈ [λ, 1], V(x, 1) = x; that is, R is a nullnorm. (3) If both of them are t-norms, let us show that R is a t-norm, a special uninorm. It just needs to show for allx ∈ [0, λ], R (x, 1) = R (R (x, λ) , 1) = R (x, R (λ, 1)) = R (x, λ) = x, (5) which could get that R and R are t-norms. (4) Similarly, if both of themare t-conorms, R is also a tconorm, a special uninorm. 3. Weak Uninorms with No Nontrivial Idempotent Elements In this section, wewill give some examples of weak uninorms, which have no nontrivial idempotent elements. And none of them is an n-uninorm; that is, all the examples in this section are nontrivial weak uninorms. Example 11. The unit interval [0, 1] is divided into infinitely many sections as (1 − (1/2), 1 − (1/2)], with n = 1, 2, . . . and 1. Let a n = 1 − (1/2 n−1 ); define a mapping R 1 as follows:

It is easy to find that the weak uninorms are the most general class; that is, all the -norms, the -conorms, the (-)uninorms, and the nullnorms are weak uninorms.Conversely, it is not valid; that is, a weak uninorm could be none of the others [12].
As we all know, for a weak uninorm , its idempotent elements are the points  subject to (, ) = .The elements 0 and 1 are the trivial idempotent elements of all the weak uninorms.All the common examples of the nontrivial weak uninorms are with infinite idempotents.Then, the following problem arises.
Problem 1 (see [14]).Is there a nontrivial weak uninorm with no more than one nontrivial idempotent elements?This problem can be divided into two parts: continuous weak uninorms and discontinuous ones.In this paper, we will give answers to this problem separately.
The content will be arranged as follows: in Section 2, some basic definitions will be given, and it will be proved that there is no nontrivial continuous weak uninorms with none or one nontrivial idempotent element.In Section 3, some examples of weak uninorms with nontrivial idempotent elements are given.These examples give positive answers to the problem above.Section 4 also shows examples of weak uninorms, which have one or more idempotent elements.Section 5 gives a conclusion of this paper.
Obviously, each uninorm and nullnorm is an -uninorm.And the converse is not valid.Examples could be found in [10].
One can easily see that -uninorms are weak uninorms.However, a weak uninorm may not be an -uninorm and thus neither a nullnorm nor a uninorm.Examples are in [12,13].
As a result, the problem in the introduction arises.And now, let us give an answer to it: there are no nontrivial continuous weak uninorms, but there exist discontinuous ones.
From this theorem, we have the following theorem.
Before the proof, let us show the following lemma firstly.

Lemma 9.
Let  be a continuous AMC operator with no idempotent elements except 0 and 1.
This theorem shows that there are no continuous weak uninorms with no nontrivial idempotents, except the norms and the -conorms.For weak uninorms with just one nontrivial idempotent element, we have a similar result.Theorem 10.There are no continuous weak uninorms with just one idempotent element  ∈ (0, 1), except the uninorms and the nullnorms.
Proof.Suppose that  is a continuous weak uninorm, with just one nontrivial idempotent element .Let's show that it is either a uninorm or a nullnorm.
Obviously, they have no nontrivial idempotent elements.From Theorem 8, they are -norms or -conorms.
(1) If   is a -norm and   is a -conorm, then  is the neutral element of ; that is,  is a uninorm.

Weak Uninorms with No Nontrivial Idempotent Elements
In this section, we will give some examples of weak uninorms, which have no nontrivial idempotent elements.And none of them is an -uninorm; that is, all the examples in this section are nontrivial weak uninorms.
Proof.Obviously,  1 is monotone and commutative.Let's show it is associative.
For any , ,  ∈ [0, 1], if one of them is 0 or 1, then it is trivial.
Next, let us show that  1 is a weak uninorm with no idempotent elements, except 0 and 1.
As a result,  1 is a weak uninorm with idempotent elements no more than 0 and 1.

Theorem 12. Let 𝑅 be an AMC operator on
is the ordinal sum of the semigroups ((  ,   )) ∈ , in which  is an infinite set, each   is in the form (, ], and each   is Archimedean, then  is a weak uninorm with no idempotent elements, except 0 and 1.
Next, let us construct some more examples of weak uninorms.In these examples, if the ordinal sums are replaced as in this theorem, then they are still weak uninorms with no idempotent elements.

(12)
This example shows that   could not be replaced by any Archimedean -norm.
Example 16.The following is a weak uninorm with only trivial idempotent elements: The demarcation point of  4 is 0.5.
Example 17.Let  5 be defined by Then,  5 is a weak uninorm with nontrivial idempotent elements.See Figure 1.

Examples of Weak Uninorms with One or More Nontrivial Idempotent Elements
Example 18.The following defined  6 and  7 are weak uninorms, with just one nontrivial idempotent element 0.5: Example 19.Define a mapping  8 by Then,  8 is a weak uninorm with idempotent elements 0, 0.5, and 1.
Examples 18 and 19 are constructed by  1 .The next example is not in this case.
Example 20.Define mappings  9 and  10 (see Figure 2) as follows:   10 (, ) Then  9 is a weak uninorm with two nontrivial idempotent elements 0.3 and 0.6;  10 is a weak uninorm with just one nontrivial idempotent element 0.6.

Conclusion
In this paper, it is proved that there are no nontrivial continuous weak uninorms with none or one idempotent element.Moreover, some nontrivial examples of weak uninorms are given.These examples are with no more than two nontrivial idempotent elements, which is a positive answer to the question in [14].

( 2 )
It is similarly.Now, let us show the proof of Theorem 8.