An Independence Property for General Information

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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 [(p_{1}) - (p_{5})] 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 [(P_{1}) - (P_{5})] is

(2)

where h is any continuous, strictly increasing function with and

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 [11] . The inequality (P_{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

(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 [(p_{1}) − (p_{4})] [5] [6] . Therefore, we are looking for functions (4) solutions of the system [(P_{1}) − (P_{5})]. We have again the similar result:

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

References

[1] Kampé de Fériet, J. and Forte, B. (1967) Information et Probabilité. Comptes Rendus de l’Académie des Sciences Paris, 265, 110-114, 142-146, 350-353.

[2] Forte, B. (1969) Measures of Information: The General Axiomatic Theory. RAIRO Informatique Théorique et Applications, 63-90.

[3] Kampé de Feriét, J. (1970) Mesures de l'information fornie par un evénement. Colloque International du Centre National de la Recherche Scientifique, 186, 191-221.

[4] Benvenuti, P., Vivona, D. and Divari, M. (1990) A General Information for Fuzzy Sets. Uncertainty in Knowledge Bases, Lectures Notes in Computer Sciences, 521, 307-316. http://dx.doi.org/10.1007/BFb0028117

[5] Zadeh, L.A. (1965) Fuzzy Sets. Information and Control, 8, 338-353.
http://dx.doi.org/10.1016/S0019-9958(65)90241-X

[6] Klir, G.J. and Folger, T.A. (1988) Fuzzy Sets, Uncertainty and Information. Prentice-Hall International Editions, Englewood Cliffs.

[7] Halmos, P.R. (1965) Measure Theory. Van Nostrand Company, Princeton.

[8] Benvenuti, P. (2004) L’opera scientifica. Roma, Ed.Univ.La Sapienza, Italia.

[9] Aczél, J. (1966) Lectures on Functional Equations and Their Applications. Academic Press, New York.

[10] Forte, B. (1970) Functional Equations in Generalized Information Theory. In: Applications of Functional Equations and Inequalities to Information Theory, Ed. Cremonese, Roma-Italy, 113-140.

[11] Ling, C.-H. (1995) Representation of Associative Functions. Publicationes Mathematicae, 12, 189-212.