A Study of Axiom of Choice for Fs-Sets

ABSTRACT In this paper, based upon Fs-set theory [Yogesara V, Srinivas G, Rath B. A theory of Fs-sets, Fs-complements and Fs-de Morgan laws. IJARCS. 2013;4(10)], we define Fs-Cartesian product of given family Fs-subsets of give Fs-set and we prove Axiom of choice for Fs-sets and we study the validity of converse of the Axiom of choice for Fs-sets.


Introduction
Ever since Zadeh [1] introduced the notion of fuzzy sets in his pioneering work, several mathematicians studied numerous aspects of fuzzy sets.
Murthy [2] introduced f-set in order to prove Axiom of choice for fuzzy sets which is not true for L-fuzzy sets introduced by Goguen [3]. The collection of all f-subsets of given f-set with his definition f-complement [4] could not form a compete Boolean algebra also for any f-subset B = (B,B,L B ) and for any b ∈ B the complement of Bb − denoted by(Bb) c is not discussed in the f-set theory interdicted by Neog and Sut [5,6]. Recently many researchers put their efforts in order to prove collection of all fuzzy subsets of given fuzzy set is Boolean algebra under suitable operations and it seems among them the efforts of Neog and Sut [5,6] and Mamoni [7] are most successful. The definition of fuzzy set given by Neog is based on the definition of fuzzy set given by Baruah [8]. Particularly in the definition of membership function of Neog and Sut [5] namely, μ 1 (x)−μ 2 (x), −μ 2 (x) will not be in the real interval [0,1]. To eliminate those lacunae Vaddiparthi Yogeswara, G. Srinivas and Biswajit Rath introduced the concept of Fs-set and developed the theory of Fs-sets in order to prove collection of all Fs-subsets of given Fs-set is a complete Boolean algebra under Fs-unions, Fs-intersections and Fs-complements. The Fs-sets they introduced contain Boolean valued membership functions. They are successful in their efforts in proving that result with some conditions. In papers [9] and [10] Vaddiparthi Yogeswara, Biswajit Rath and S. V. G. Reddy introduced the concept of Fs-Function between two Fs-subsets of given Fs-set and defined an image of an Fs-subset under a given Fs-function. Also they studied the properties of images under various kinds of Fs-functions.
In this paper, we introduced the concept of Fs-Cartesian product of given family of Fssubsets of give Fs-set and prove Axiom of choice for Fs-sets also we study the validity of converse of the Axiom of choice for Fs-sets For smooth reading of paper, the theory of Fs-sets and Fs-functions in brief is dealt with in first two sections. We denote the largest element of a complete Boolean algebra L A [1.1] by M A or 1. We denote Fs-union and crisp set union by same symbol ∪ and similarly Fs-intersection and crisp set intersection by the same symbol ∩ [11]. For all lattice theoretic properties and Boolean algebraic properties one can refer Szśz [12], Garret [13], Steven and Paul [14], James [15] and Thomas [16].

Fs-Set
Definition 2.1: Let U be a universal set, A 1 ⊆U and let A ⊆ U be non-empty. A four tuple For some L X , such that L X = L A a four tuple X =(X 1 , X,X(μ 1X 1 , μ 2X ), L X ) is not an Fs-set if, and only if (a) X X 1 or (6) μ 1X 1 x μ 2X x , for some x ∈ X∩X 1 Here onwards, any object of this type is called an Fs-empty set of first kind and we accept that it is an Fs-subset of B for any B⊆A . Definition 2.5: Let B 1 =(B 11 , B 1 , B 1 (μ 1B 11 , μ 2B 1 ), L B 1 ) and B 2 =(B 12 , B 2 , B 2 (μ 1B 12 , μ 2B 2 ), L B 2 ) be a pair of Fs-sets. We say that B 1 and B 2 are equal, denoted by B 1 = B 2 if, and only if

Remark 2.3:
We can easily observed that 3(a) and 3(b) not equivalent statements.

Proposition 2.8: B ∪C is an Fs-subset of A .
Definition 2.9: Fs-intersection for a given pair of Fs-subsets of A : be a pair of Fs-subsets of A satisfying the following conditions: which is the Fs-empty set of first kind.

Proposition 2.10: For any Fs-subsets B, C and D of
, the following associative laws are true:

Definition of Fs-union is as follows 2.12:
Case (1): For I = ,define Fs-union of (B i ) i∈I , denoted by i∈I B i as i∈I B i = A , which is the Fs-empty set Case (2): Define for I = , Fs-union of (B i ) i∈I denoted by i∈I B i as follow where,

Definition 2.13 (Fs-intersection):
Case (1): For I = , we define Fs-intersection of (B i ) i∈I , denoted by i∈I B i as i∈I B i =A Case (2): Suppose i∈I B 1i ⊇ i∈I B i and i∈I μ 1B 1i |( i∈I B i )≥ i∈I μ 2B i Then, we define Fs-intersection of (B i ) i∈I , denoted by i∈I B i as follows  ( if, and only if (using the diagrams shown in Figure 1).
is denoted by f Proposition 3.2:

Definition 3.3:
Increasing Fs-function f is said to be an increasing Fs-function, and denoted by f i if ,and only if (using Figure 1)

Proposition 3.9:
• B = C • f , provided f is Fs-preserving function

Definition 3.10 (Composition of two Fs-function):
Given two Fs-functions f : B→C and g:C →D.We denote composition of g and f as g•f and define as (g•f)= (g 1 , g, )

Proposition 3.12: The class of all Fs-sets as objects together with morphism sets Fs-functions under the partial operation denoted by • is called composition between Fs-functions whenever it exists is a category denoted by Fs-SET.
Here

Fs-cartesian product
Definition 4.1: Let (A i ) i∈I be a non-empty family of non-empty Fs-sets. Define Fs-Cartesian Product of (A i ) i∈I , denoted by i∈I A i as follows.
is a non-degenerating complete Boolean algebra and A i a i =0 for at least one a i ∈ A i Here i∈I A i =X = X 1 , X,X μ 1X 1 , μ 2X , L X , where X 1 = i∈I A 1i such that i∈I A 1i , (P 1i ) i∈I is the product of (A 1i ) i∈I in SET, the category of sets with usual maps between crisp sets. X = i∈I A i such that i∈I A i , (P i ) i∈I is the product of (A i ) i∈I in SET, the category of sets with usual maps between crisp sets. L X = i∈I L A i such that i∈I L A i ,(π i ) i∈I is the product of L A i i∈I in CBOO, the category of complete Boolean algebras with complete homomorphism between complete Boolean algebra.
is an Fs-set The Fs-function (P 1i , P i , π i ) : X −→ A i are Fs-projections In particular i∈I A i =X =A I where A i = A , ∀i ∈ I.

Definition 4.2 (Fs-Cartesian Product of non-empty family of non-empty Fs-subsets of A ):
Let (B i ) i∈I be a non-empty family of non-empty Fs-subset of A . Define Fs-Cartesian Product of (B i ) i∈I , denoted by i∈I B i as follows. Here i∈I B i =C = C 1 , C,C μ 1C 1 , μ 2C , L C , where C 1 = i∈I B 1i such that i∈I B 1i , (P 1i ) i∈I is the product of (B 1i ) i∈I in SET, the category of sets with usual maps. C = i∈I B i such that i∈I B i , (P i ) i∈I is the product of (B i ) i∈I in SET, the category of sets with usual maps. L C = i∈I L B i such that i∈I B i , (π i ) i∈I is the product of L B i i∈I in CBOO, the category of complete Boolean algebras.
i∈I is an Fs-subset of A I Figure 2. (P 1i , P i , π i ) : C −→B i .
(P 1i , P i , π i ) : C −→B i is an Fs-function is a preserving function, where P 1i | C = P i . I.e following diagrams shown in Figure 2 are commutative.
Hence π i : i∈I L B i −→ L B i is a complete homomorphism.

Axiom of choice for Fs-Set
Theorem 4.3: Let (B i ) i∈I be a non-empty family of non-empty Fs-subsets of A , then Fs-Cartesian Product of (B i ) i∈I , namely i∈I B i is a non-empty Fs-subset, that is, i∈I B i = A Proof: Also observed that i∈I L B i is non-degenerate because (a i ) i∈I < (b i ) i∈I where a i = 0 and b i = 1 for each i ∈ I · · · (3) Hence from (1) , (2) and (3)  Case (I): B 1i 0 B i 0 ⇒ i∈I B 1i 0 i∈I B i 0 (∵ i∈I B 1i ⊇ i∈I B i if, and only if B 1i ⊇B i for each i ∈ I ) Hence i∈I B i = A Fs-empty set of first is contradiction. Case (II): μ 1B 1i 0 b i 0 ≯μ 2B i 0 b i 0 for some b i 0 ∈ B i 0 ⊆B 1i 0 for any i 0 ∈ I ⇒(μ 1B 1i | B i ) i∈I ≯(μ 2B i ) i∈I Hence i∈I B i = A is contradiction.
From case (I) and (II) we get B i are NON-Fs-empty set of first kind for each i ∈ I Example 4.1: A non-empty Fs-Cartesian product of X of Fs-subset such that one of the member in X is Fs-empty set of second kind Let B= B 1 , B,B μ 1B 1 , μ 2B , L B and C = C 1 , C,C μ 1C 1 , μ 2C , L C are Fs-subsets of an Fs-set A = A 1 , A,A μ 1A 1 , μ 2A , L A where A 1 =B 1 =C 1 = B = C = A, μ 1B 1 a = 1,μ 2B a = 0 and μ 1C 1 a =μ 2C a = 1, ∀a ∈ A and L B =L C = {0, 1} . Observe that C is an Fs-empty set of second kind.