On some methods of study of states on interval valued fuzzy sets

: In this paper the state on interval valued fuzzy sets is studied. Two methods are considered: a representation of a state by a Kolmogorov probability and an embedding to an MV -algebra. The Butnariu–Klement representation theorem for interval valued fuzzy sets as a relation between probability measure and state is presented.


Introduction
Intuitionistic fuzzy sets were introduced by Prof. Krassimir Atanassov in 1983 [1]. A similar model was introduced by Prof. Lotfi Zadeh in 1975, namely, interval valued fuzzy sets [10]. Although the motivations as well as the applications were different, actually they are isomorphic. Here we will present the isomorphism and show that these two models can be studied from the M V -algebra theory point of view.
Let us start with the definition of the basic terms. Consider a measurable space (Ω, S) with a σ-algebra S. Let J be the family of all measurable functions f : Ω → [0, 1]. To define the state on J we need two binary operations. In this paper we will use the Łukasiewicz operations: These binary operations play the same role as the union and the intersection in the set theory. A state on J is a mapping m J : J → [0, 1] satisfying the following conditions [5]: Since J contains the functions f : Ω → [0, 1], then J represents the family of fuzzy sets. In the fuzzy set theory, f is called membership function. Now, let us look at the more general structures -intuitionistic fuzzy sets, shortly IF -sets, and interval valued fuzzy sets, shortly IVsets. The Similarly as in the fuzzy set theory, the function µ A is called the membership function. The second function, ν A is called the non-membership function. Denote by F the family of all IF -sets such that µ A , ν A are S-measurable. On the set F we could define the Łukasiewicz operations and the ordering by the following way By this ordering for each A ∈ F it holds that (0 Ω , 1 Ω ) A (1 Ω , 0 Ω ). The family J can be considered as a subset of F if for f ∈ J we put A = (f, 1 Ω − f ). Then µ A + ν A = 1 Ω .
By an IV -set we shall consider the pair C = (π C , ρ C ) such that π C : Ω → [0, 1], ρ C : Ω → [0, 1] and π C ≤ ρ C . If (Ω, S) is a measurable space, then by K we shall denote the family of all IV -sets with the measurable functions π C , ρ C : (Ω, S) → [0, 1]. Then for any C, D ∈ K we could define Proposition 1. The systems (F, , ⊕, ) and (K, , , ) are isomorphism by the mapping ψ : On the other hand, Similarly the equality can be obtain.
Also the states on F and K are isomorphism. For the first recall the definitions of these structures.

State representation theorem
The isomorphism between F and K presented in Proposition 1 and Proposition 4 permits us to translate results holding in F to K. As an illustration, we shall present here the state representation theorem. This theorem points on the relation between probability measure and state. In [2] there was proved the Butnariu-Klement theorem: This theorem has been generalized also for Atanassov IF-sets. The generalization of the Butnariu-Klement theorem has the form: This theorem is called Representation theorem for probabilities on IF-states and it was proved in [3]. The Butnariu-Klement theorem can be obtained as a consequence if ν A = 1 Ω − µ A . Now we shall use the isomorphism between F and K to build representation theorem for IV-states.
Theorem 7. Let k be a state defined on K. Then there exists a probability measure P : S → [0, 1] and α ∈ [0, 1] such that for any C = (π C , ρ C ) ∈ K it holds Proof. Put m = k • ψ : F → [0, 1] and use the results of Theorem 6. Then This completes the proof.

States and M V -algebras
Similarly as in the case of F also K can be embedded into the suitable M V -algebra ( [7]). For the first recall the definition of an M V -algebra.
Definition 8. By an -group G we consider an algebraic system (G, +, ≤) such that We shall use the form of M V -algebra which is generated by fuzzy sets ( [9]).
The element u is supposed to be a strong unit of G.
In the paper [6] there was proved that an IF-set could be embedded into the M V -algebra M by using following construction. Let us take M V -algebra M such that G = R 2 . Then the addition of two elements could be defined as This addition represents the addition of two vectors with a fixed point (0, 1). For any point From this reason, we can define the functionm : M → [0, 1] by the formulā This function represents the state on M generated by a state defined on F. Proof. Let (G, +, ≤) be the -group where G consists of all mappings C : (π C , ρ C ) → R 2 where π C , ρ C are S-measurable functions. Define and Then the element (0 Ω , 0 Ω ) is the neutral element, i.e., Consider the element (1 Ω , 1 Ω ) and define M as the structure Then for any C, D ∈ M it holds C D = [(π C , ρ C ) + (π D , ρ D )] ∧ (1 Ω , 1 Ω ) = ((π C + π D ) ∧ 1 Ω ), (ρ C + ρ D ) ∧ 1 Ω ), We have obtained an M V -algebra M such that K ⊂ M. Letm : M → [0, 1] be a state on M.
This completes the proof.

Conclusion
Representation theorems as well as assertions about isomorphism of some mathematical structures belong to the most useful mathematical means. In this paper, we used the representation of IF -states by Kolmogorov probability measures and the isomorphism between the Atanassov IF -sets and the Zadeh IV -sets. An important point is the possibility to embed fuzzy families to multivalued algebras. Namely, some useful results of M V -algebra measure theory can be applied to some fuzzy structures. In this way, we have two ways to obtain fuzzy measure constructions. The first is in using set probability theory. The second in using the M V -algebra probability theory.