^{1}

^{2}

^{1}

^{2}

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

A mapping from

It is easy to find that the weak uninorms are the most general class; that is, all the

As we all know, for a weak uninorm

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

An AMC operator

An AMC operator

An AMC operator

Clearly, if

An AMC operator

Obviously, if

An AMC operator is called an

Obviously, each uninorm and nullnorm is an

An AMC operator

If

One can easily see that

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.

If

its idempotent elements are just 0 and 1;

there exists some element

there exists some element

then

From this theorem, we have the following theorem.

Let

if there exists some

if there exists some

Before the proof, let us show the following lemma firstly.

Let

If

If for all

(1) Since for all

(2) It is similarly.

Now, let us show the proof of Theorem

Since a weak uninorm is an AMC operator. From Theorem

that is,

Similarly.

This theorem shows that there are no continuous weak uninorms with no nontrivial idempotents, except the

There are no continuous weak uninorms with just one idempotent element

Suppose that

Let

Since

Obviously, they have no nontrivial idempotent elements. From Theorem

If

If

If both of them are

which could get that

Similarly, if both of them are

In this section, we will give some examples of weak uninorms, which have no nontrivial idempotent elements. And none of them is an

The unit interval

Then,

Actually, for

Obviously,

For any

If there is some

If there are some

If there are some

The last case is that

Next, let us show that

For any

Since

As a result,

It is obvious that it is not an

Note that it is not difficult to find that

Let

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.

The unit interval

For some given

Note that, in this example, if

In

This example shows that

The following is a weak uninorm with only trivial idempotent elements:

Let

Graphical representation of

The following defined

Define a mapping

Examples

Define mappings

Graphical representation of

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).