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The aim of this paper was a generalization of independence property proposed by J. Kampé de Feriét and B. Forte in Information Theory without probability, called
*general information*. Therefore, its application to fuzzy sets has been presented.

Since 1967-69, J. Kampé de Ferét and B. Forte have introduced, by axiomatic way, new information measures without probability [

In Information Theory an important role has played by an independence property with respect to a given information measures J applied to crisp sets [

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.

Let

J. Kampé de Ferét and B. Forte gave the following definition [

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

such that

If the couple

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

We shall characterize the function

Putting _{1}) - (p_{5})] have translated in the fol- lowing system of functional equations and inequalities [

We can give the following

Proposition 3.2 A class of solutions of the system [(P_{1}) - (P_{5})] is

where h is any continuous, strictly increasing function

Proof. The class of functions (2) satisfy the equations [(P_{1})-(P_{3})] and the inequality (P_{4}) by appling the Ling Theorem about the representation of a function which is monotone, commutative, associative with neutral element [_{5}) 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

where h is any continuous, strictly increasing function

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

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

Let

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

If the couple

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

The properties of the intersection between fuzzy sets are the similar to the [(p_{1}) − (p_{4})] [_{1}) − (P_{5})]. We have again the similar result:

Proposition 4.2 A class of solution of the system [(P_{1}) − (P_{5})] is

where k is any continuous, strictly increasing function

From (5), we get

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

where k is any continuous, strictly increasing function

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

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

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

where h is any continuous, strictly increasing function

Therefore, we have extended the result to fuzzy setting:

where k is any continuous, strictly increasing function

DorettaVivona,MariaDivari, (2016) An Independence Property for General Information. Natural Science,08,66-69. doi: 10.4236/ns.2016.82008