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
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