Characterization on Fuzzy Soft Ordered Banach Algebra

: In this paper, we deﬁne fuzzy soft ordered Banach algebra with fuzzy soft alfebra cone, and introduce the character on fuzzy soft ordered Banach algebra in both cases real and complex. Also, we deduce some of its basic properties and we deﬁne a new concept which is a maximal fuzzy soft algebra cone and showing that the set of all fuzzy character is isomorphism to the set of all maximal fuzzy soft algebra cone. We prove that the set of all real FS x − characters is convex and extreme point, we applied Gelfand-Mazur theorem on fuzzy soft Banach algebra, we showed that FS x − character (the set of all complex continuous FS x − character) is fuzzy soft ordered Banach algebra. Also, any FS x -OBA with inverse -closed FS x -algebra cone ˘ C and a non-zero element in ˘ A has inverse we have it is an isomorphism to Banach space Ch ( ˘ C ) .


Introduction
Lotif, Z. [5] introduced fuzzy sets in his famous paper. Which have found wide's applications in many fields including: computer science, economics, control engineering, robotics, etc. Bekir and Bur [1] introduced the notion of fuzzy topological vector space, later the concept of fuzzy norm was introduced by Reza and Ranjit [10] and define the fuzzy Banach spaces and it's quotient. Sadeqi and Amiripour [4] in 2007 gave a definition of fuzzy Banach algebra. Dmitir [3] initiated the theory of soft sets. Pabitr,... etc [9] initiated the theory of fuzzy soft sets, when the introduced a concept parametrized family of sets which applied by him in many fields including basic notions of soft normed space. P.k. Maji and A.R. Roy [8] defined basic notions of soft normed spaces. Reza and Ranjit [10] studied fuzzy normed linear spaces and later Sujoy, Pinaki and Syamal [11] found the concet of soft normed space. Tangaraj and Nirmal [13] introduced new notion yn fuzzy soft bormed spaces. Thakur and S. K. Samanta [14] introduced the definition of soft Banach algebra and studied some of its properties, and preceded that they introduce a new concept of convergence sequence of soft elements and they arrived to that finiteness of parameter sets is not necessary in many cases. Tudor and Ftavius [15] gave sturdiness fuzzy normed space. Zdzis law [17] intrudes the concept of rough sets. Boushra [2] formulated the concept of real character in ordered Banach algebra. Sonja [12] solved some spectral problems in ordered Banach spaces.
In this paper we will combine the concepts of functional analysis with other algebraic concepts in fuzzy soft theory by creating the construction of the space of fuzzy soft ordered Banach algebra with study and proof of many properties in this space. We define fuzzy soft ordered Banach algebra in both cases real and complex. Also, we deduce some of the basic properties and we define a new concept which is a maximal fuzzy soft algebra cone and showing the set of all fuzzy soft character is isomorphism to the set of all maximal fuzzy soft algebra cone. We prove that the set of all real F S x −character is convex 2 B. Hussein and S. Fahim and extreme point, we applied Gelfand-Mazur theorem on fuzzy soft Banach algebra, we showed that F S x −character (the set of all complex continuous F S x −character) is fuzzy soft ordered Banach algebra. Any F S x −OBA with inverse closed F S x −algebra cone C and a non-zero element in A has inverse, we have it is an isomorphism to Banach space Ch(C). The outline of this paper, section 2 we introduce the historical introduction and we will introduce many concepts which we need like fuzzy soft set, fuzzy soft normed space and many properties which we need. In section 3, we will define new many definitions like F S x −algebra cone, F S x −ordered Banach algebra, F S x −Maximal algebra one and define ordered isomorphism between two F S x −ordered Banach algebras. In section 4, we introduce many of results as proved that the set of real character in F S x −ordered Banach algebra is convex and extreme point and proved that the set of all complex F S x −characterization in F S x −ordered Banach algebra Ch(Ȃ) is isomorphism with F S x −Maximal algebra cone. Finally we proved that there is a F S x −character between (Ȃ,C) and C • (Ch(Ȃ)) = {k : k : Ch(C) → C, k continuous }. In section 5, we clear the conclusion of our work.

Preliminaries
Definition 2.1 (6). Let B is a universe and E be a set of parameter. Let P (B) is the power set of B and X be a non-empty set of E. A pair (S, X) is called soft set over B, where S is a mapping given by S : X → P (B). That is, a soft set over B is a parametrized family of a subsets of the universe B. For e belong to X, f (e) may be considered the set of e−approximate elements of the soft set (S, X). (8). Let (V, +, .) is a vector space over a field F and E be a set of parameter.V is said to be soft vector space over F ifV (β) is a vector space of V , for all β ∈ E.     (15). Let {x n } be a sequence of soft elements of a soft normed space (V , . ) such that {x n } is said to be a Cauchy sequence if for every∈≥0, there is k ∈ N such that:

Definition 2.2
Definition 2.9 (6). Let X be a non-empty set and I = [0, 1] and I X : X −→ I such that I X = {B : B is a function from X into I}. A set B ⊂ X is a fuzzy set in X such that B : X ←→ I, that is B ∈ I X and let b ∈ B and I X is denote to the set of all fuzzy sets of X. A fuzzy set X in X is characteristic by a membership function and it is connect with every point b ∈ B by the real number is the grad membership function for the fuzzy set B. Definition 2.10 (11). Let V be a vector space and * is a continuous t−norm and B is a fuzzy set on V × (0, ∞), then (V, B, * ) is said to be fuzzy normed space if satisfies the following conditions for every a, b ∈ V and t, k ∈ (0, ∞): (16). (V, B, * ) is said to be fuzzy Banach algebra space if satisfies:

Definition 2.11
Definition 2.12 (17). Let B be a linear algebra if ab = ba for all a, b ∈ V we say that V is a commutative linear algebra.  (17). We called a linear algebra V is unital, if it has a unit element e such that e.a = a.e = a for all a ∈ V . Definition 2.15 (2). Let B is a universe and E be a set of parameters and I x is denote to the set of all fuzzy sets of X, a pair (IS x , E) is called a fuzzy soft set over X. (10).   (10). Let two fuzzy soft sets (F S x , E) , (GS x , E) over a common universe X, we say that (F S x , E) is a fuzzy soft subset of (GS x , E) if:

Definition 2.17
Definition 2.19 (10). Two soft sets (F, A) and (G, B) over a common universe X are said to be fuzzy soft equal if (F, A) is fuzzy soft subset of (G, B) and (G, B) is a fuzzy soft subset of (F, B).  Then (V , N, .) is a fuzzy soft normed space and ifV is complete said that it is fuzzy soft Banach space. (7). If V is a linear space and K ⊆ X, such K is convex and x ∈ K, then x is extreme point, that mean if x = ta + (1 − t)b, such that a, b ∈ K and 0 < t < 1 then a = b. And we denote to the set of extreme point by e (K) .

Fuzzy Soft Ordered Banach Algebras
In this section, we define many basic definitions like, fuzzy soft algebra cone, fuzzy soft ordered Banach algebra, and maximal fuzzy soft ordered Banach algebra.

Definition 3.1 (Fuzzy Soft Linear Algebra). LetV be a vector space with respect to F = C or R.V is called fuzzy soft linear algebra, if there exists an operation which is called multiplication satisfying that for allȃ,b,c∈V and
(iii)(ȃ +b) z = (ȃ z +b z), andȃ b +z =ȃ b +ȃ z.

Definition 3.2 (Fuzzy Soft Banach Algebra). Let F S x (V ) be a fuzzy soft on a vector spaceV , which is called Fuzzy Soft Banach Algebra if satisfies that: (i) Fuzzy soft Banach space.
(ii) Fuzzy soft algebra.

Definition 3.4 (Fuzzy Soft Ordered Banach Algebras). LetȂ be a complex ordered Banach algebra with identity generated byC , and F A the set of all fuzzy soft sets of ordered Banach algebra A,Ȃ = (F A , E)
is called fuzzy soft ordered Banach algebra F S x − OBA ifȂ is ordered by a relation˘ such that for everyȃ,b,c∈Ȃ and β ≥ 0:  (1)f is Linear that isf (ȃ +b) =f (ȃ) +f (b) and βf (ȃ) =f (βȃ).

So ifȂ is ordered by an
( (3)f (e) = 1.   LetC be a F S x − algebra cone we say that it is inverse-closed F S x −algebra cone ifȃǫȂ andȃ is invertible then,ȃ −1∈C .

Main Results
In this section, we introduced many theorem and properties about fuzzy soft ordered Banach algebra like, convex and extreme point, and character on fuzzy soft ordered Banach algebra.
Proof. Let L:Ȃ −→ C o (Ch(C)) defined by L(ȃ) = k(fȃ) such thatk(fȃ) =f (ȃ) for allȃ∈(Ȃ,C) such thatf is every F S x −character from closed algebraC into C. Letȃ,b∈C, α ∈ C.  Proof. Letφ:(Ȃ,C) −→ Ch C define byφ(ȃ) =fȃ such thatfȃ(x) =ȃ.x for allx∈C, for allȃ∈Ȃ and To prove that Ch C is Banach space. First, we prove that Ch C is a closed subspace of the set of all continuous functions on (Ȃ,C): Letf 1 ,f 2∈ Ch C . To show that: . For allȃ,b∈Ȃ and β ∈ C. ( Then Ch C is subspace Now we will prove that Ch C is closed. We have to prove that Ch C = Ch C . LetfǫCh C .
Then, there exists a sequence f ı in Ch C such that lim ı→∞fı =f .
To provefǫCh C .
Thenf ∈ Ch C , we obtain Ch C ⊆ Ch C , since Ch C ⊆ Ch C so Ch C is closed.
Then Ch C is Banach space.
Hence (Ȃ,C) is an isomorphism to Banach space Ch C .

Conclusion
In fuzzy soft Banach space, we needed define soft Banach algebra [14] and fuzzy Banach space [10], in our paper, we define a new concepts called fuzzy soft algebra cone, fuzzy soft ordered Banach algebra and maximal fuzzy soft algebra cone, we then proved many properties about real and complex F S x − character like the convexity and the extreme point to the F S x − character in real numbers and shown that there is isomorphism from the set of all complex F S x − character to the set of all maximal algebra cone, more than we arrived to that the fuzzy soft ordered Banach algebra is a F S x − character with the set of all continuous F S x − characterization which is a set of functions from fuzzy soft algebra cone in to the set of complex numbers.