Some Examples of Weak Uninorms
It is proved that, except for the uninorms and the nullnorms, there are no continuous weak uninorms who have no more than one nontrivial idempotent element. And some examples of discontinuous weak uninorms are shown. All of these examples are not -uninorms, thus not uninorms or nullnorms.
A mapping fromtois called an AMC operator [1, 2], if it is associative, monotone nondecreasing in both variables, and commutative. The most famous AMC operator in fuzzy mathematics is the-norms [3–5] and the-conorms. In recent years, the weak forms of the-norms interest the logisticians, such as the uninorms [6, 7], the nullnorms (-operators) [8, 9], the -uninorms [10, 11], and the weak uninorms [12, 13].
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 .
As we all know, for a weak uninorm, its idempotent elements are the pointssubject 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 ). 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.
2. Continuous Weak Uninorms
Clearly, ifthenis a-norm and ifthenis a-conorm .
Obviously, if, then the nullnorm is a-norm; and if, then it is a-conorm.
Obviously, each uninorm and nullnorm is an-uninorm. And the converse is not valid. Examples could be found in .
Ifin particular is a fixed valuefor all, thenis called the neutral element of. And in this case,comes to be uninorm . If there is some element, subject to for all ,, and for all , , it is a nullnorm .
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.
Theorem 7 (Theorem 7 in ). Ifis an AMC operator on that satisfies the following conditions: (i)its idempotent elements are just 0 and 1;(ii)there exists some elementwith;(iii)there exists some elementwith;
thenis not continuous, andis a discontinuous point, withthe demarcation point.
From this theorem, we have the following theorem.
Theorem 8. Letbe a continuous weak uninorm with no nontrivial idempotents. Then, we have the following results: (1)if there exists some subject to , thenis a-norm;(2)if there exists some subject to , thenis a-conorm.
Before the proof, let us show the following lemma firstly.
Lemma 9. Letbe a continuous AMC operator with no idempotent elements except 0 and 1. (1)Iffor all , then for any, there exists some natural number, subject to.(2)If for all , then for any, there exists some natural number, subject to.
Proof. (1) Since for all , for any,
Thus, the limitexists, denoted by. From the continuity of, we could know that
This means that is an idempotent element of. Because,, that is, for any, the limit ofis 0. Thus, for any , there exists some natural number, subject to.
(2) It is similarly.
Now, let us show the proof of Theorem 8.
Proof. Since a weak uninorm is an AMC operator. From Theorem 7, we could know that the squares of the elements inare either all strictly smaller than themselves or all strictly bigger than themselves; that is, (1)for all . Let’s show that is a-norm. From the definition of weak uninorms, for any, there is some, subject to. If, from Lemma 9, there exists some, subject to. Thus, , contradiction. Therefore,; that is, for any,. From the monotonicity of, we could know that and from the idempotence of 1, we have. As a result, that is,is a-norm.(2)Similarly.
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, except the uninorms and the nullnorms.
Proof. Suppose thatis a continuous weak uninorm, with just one nontrivial idempotent element. Let’s show that it is either a uninorm or a nullnorm.
Let, and, for all . Obviously,andare continuous AMC operators. Let’s show that they are weak uninorms, that is, for each element,exists.
Since, 0, and 1 are idempotent elements andis continuous, from Theorem 2.8 in  (or Definition 3.1 in ),andare continuous weak uninorms.
Obviously, they have no nontrivial idempotent elements. From Theorem 8, they are-norms or-conorms. (1)Ifis a-norm andis a-conorm, thenis the neutral element of; that is,is a uninorm.(2)Ifis a-conorm andis a-norm, thenis an absorbing element, and for all ,and for all ,; that is,is a nullnorm.(3)If both of them are-norms, let us show that is a-norm, a special uninorm. It just needs to show for all , which could get that andare-norms.(4)Similarly, if both of them are-conorms,is also a-conorm, a special uninorm.
3. 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.
Example 11. The unit intervalis divided into infinitely many sections as, withand 1. Let; define a mappingas follows:
Then, is a weak uninorm with no nontrivial idempotent elements; that is, its idempotent elements are just 0 and 1. But it is not an-uninorm and thus neither a uninorm nor a nullnorm.
Actually, for,. For convenience, it is shortly rewritten as In this formula, there is a symbol. It means an injection of the operator, that is, for all ,, instead of . Similar for the following examples.
Proof. Obviously,is monotone and commutative. Let’s show it is associative.
For any, if one of them is 0 or 1, then it is trivial.
If there is some, subject to, from the associativity of,is associative.
If there are some, subject toand, then we haveand. Therefore,
If there are some, subject toand, then, and. Thus,. Therefore,
The last case is that , and , with different , , . In this case, the result is always equal to the smallest element; that is, the associative law is valid. Now, the associativity ofhas already been proved.
Next, let us show that is a weak uninorm with no idempotent elements, except 0 and 1.
For any, there is some, subject to. Let, then. Together withand,is a weak uninorm.
Since, for all , we have, for all .
As a result,is a weak uninorm with idempotent elements no more than 0 and 1.
It is obvious that it is not an-uninorm, thus, neither a uninorm nor a nullnorm.
Note that it is not difficult to find that, in whichis the restriction of the weak uninormon the square, is the ordinal sum  of the semigroups. Butitself is not. Actually, similar to the proof of Example 11, we can obtain the following property.
Theorem 12. Letbe an AMC operator onwith. Ifis the ordinal sum of the semigroups, in whichis an infinite set, eachis in the form, and eachis Archimedean, thenis 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.
Example 13. The unit intervalis divided into infinitely many sections as, withand 0. Then, the following definedis a weak uninorm with no non trivial idempotent elements: that is, for,
is the dual of. Thus, it is a weak uninorm.
Example 14. For some given, define a mappingas follows:
in which. Then, is a weak uninorm with idempotent elements 0 and 1 only.
Note that, in this example, if, the associativity will not be valid; that is,will no longer be a weak uninorm.
Example 15. In, ifis replaced by, denoted as , then it will no longer be a weak uninorm. Since the associativity is not valid,
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 ofis 0.5.
Example 17. Letbe defined by Then, is a weak uninorm with nontrivial idempotent elements. See Figure 1.
4. Examples of Weak Uninorms with One or More Nontrivial Idempotent Elements
Example 18. The following definedandare weak uninorms, with just one nontrivial idempotent element 0.5:
Example 19. Define a mappingby Then, is a weak uninorm with idempotent elements, and 1.
Example 20. Define mappingsand(see Figure 2) as follows: Thenis a weak uninorm with two nontrivial idempotent elements 0.3 and 0.6;is a weak uninorm with just one nontrivial idempotent element 0.6.
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 .
This project is supported by the Tianyuan special funds of the National Natural Science Foundation of China (Grant no. 11226265), Promotive research fund for excellent young and middle-aged scientists of Shandong Province (Grant no. 2012BSB01159), and Foundation of the Education Department Henan Province (no. 13A110552).
E. P. Klement, R. Mesiar, and E. Pap, Triangular Norms, Kluwer Academic, Dodrecht, The Netherlands, 2000.View at: MathSciNet