Received 25 December 2015; accepted 19 February 2016; published 22 February 2016

1. Introduction

Since 1967-69, J. Kampé de Ferét and B. Forte have introduced, by axiomatic way, new information measures without probability [1] - [3] ; later, in analogous way, with P. Benvenuti we have defined information measures without probability or fuzzy measure [4] for fuzzy sets [5] [6] . This form of information measure is again called general information.

In Information Theory an important role has played by an independence property with respect to a given information measures J applied to crisp sets [7] . These sets are called J-independent (i.e. independent each other with the respect to J) [8] .

For this reason we will propose a generalization of J-independence property.

The paper develops in the following way: in Section 2 we recall some preliminaires; in Section 3 the generalization of J-indepedence is proposed; the result is extended to fuzzy sets in Section 4. Section 5 is devoted to the conclusion.

2. Preliminaires

J. Kampé de Ferét and B. Forte gave the following definition [1] [2] :

Definition 2.1 Measure of general information J for crisp sets is a mapping

such that:

(i)

(ii)

(iii)

If the couple satisfies the (iii), we say that and are J-independent, i.e. independent each other with respect to information J.

3. A Generalization of the J-Independence Property

In this paragraph we are going to present a generalization of the J-independence property.

We propose the following:

Definition 3.1 Given a general information J, let and be two crisp sets in C such that We say that and are J-idependent each other if there exists a continuous function such that

(1)

We shall characterize the function, taking into account the properties of the intersection for every:

Putting the properties [(p₁) - (p₅)] have translated in the fol- lowing system of functional equations and inequalities [9] [10] :

We can give the following

Proposition 3.2 A class of solutions of the system [(P₁) - (P₅)] is

(2)

where h is any continuous, strictly increasing function with and

Proof. The class of functions (2) satisfy the equations [(P₁)-(P₃)] and the inequality (P₄) by appling the Ling Theorem about the representation of a function which is monotone, commutative, associative with neutral element [11] . The inequality (P₅) is a consequence of the monotonicity of h.

So, from (2), we have

Proposition 3.3 The generalization of the J-independence property for crisp sets is

(3)

where h is any continuous, strictly increasing function with and

Remark When h is linear, the generalization (3) coincide with the property (iii).

4. Extension to Fuzzy Setting

In this paragraph, we are considering the extension of J-independence property at fuzzy setting.

Definition 4.1 Measure of general information in fuzzy setting is a mapping such that:

(i')

(ii')

(iii')

If the couple satisfies the (iii'), we say that and are J'-independent, i.e. independent each other with respect to information.

Also in fuzzy setting, we generalize the (iii'), setting

(4)

The properties of the intersection between fuzzy sets are the similar to the [(p₁) − (p₄)] [5] [6] . Therefore, we are looking for functions (4) solutions of the system [(P₁) − (P₅)]. We have again the similar result:

Proposition 4.2 A class of solution of the system [(P₁) − (P₅)] is

(5)

where k is any continuous, strictly increasing function with and

From (5), we get

Proposition 4.3 A generalization of the J'-independence property between two fuzzy set is

(6)

where k is any continuous, strictly increasing function with and

Proof. The proof is similar to that given for crisp sets.

Remark. When k is linear, the generalization (6) coincide with the property (iii').

5. Conclusions

In this paper we have proposed a genralization of J-independence property between crisp sets:

where h is any continuous, strictly increasing function with and

Therefore, we have extended the result to fuzzy setting:

where k is any continuous, strictly increasing function with and