Further properties of the fuzzy complete a-fuzzy normed algebra

In this paper further properties of the fuzzy complete a-fuzzy normed algebra have been introduced. Then we found the relation between the maximal ideals of fuzzy complete a-fuzzy normed algebra and the associated multiplicative linear function space. In this direction we proved that if ℓ is character on Z then ker ℓ is a maximal ideal in Z. After this we introduce the notion structure of the a-fuzzy normed algebra then we prove that the structure, st(Z) is ω ∗ -fuzzy closed subset of fb(Z, ℂ ) when (Z, 𝑛 𝑍 , ⊛ , ⊙ ) is a commutative fuzzy complete a-fuzzy normed algebra with identity e


Introduction
In this paper we continue the previous study of fuzzy complete a-fuzzy normed algebra.We prove here the other important properties of fuzzy complete a-fuzzy normed algebra.The organization of this paper is as follows: we divided this research into four sections; the introduction will be in section one, after that in section two important properties of fuzzy length space and a-fuzzy normed space are recalled.Furthermore, basic important properties of afuzzy normed algebra that will be needed later can be found in section three.Moreover, further properties of fuzzy complete a-fuzzy normed algebra have been proved as a main results in the same section.Finally, in section four we highlight the conclusion for this research.

Preliminaries about a-fuzzy normed algebra Definition 2.1 [1]:
Let ⊛:I × I →I be a binary operation function, then it is said to be continuous t-conorm (or simply t-conorm) if it satisfies the following conditions: ) whenever p ≥ q and z ≥ w.For all p, q, z, w ∈ I = [0, 1].

Definition 2.5 [3]:
If (ℂ,  ℂ , ⊛ ) is a-fuzzy length space, Z is a vector space over ℂ, and ⊛ is a t-conorm and   : Z→I is a fuzzy set.Then   is a-fuzzy norm on Z if For all z, y ∈ Z. Then (Z,   , ⊛) is a-fuzzy normed space (or simply a-FNS).

Definition 2.7 [3]:
If (  ) is a sequence in Z, then (  ) is fuzzy converges to the limit z as k → ∞, if for all  ∈ (0,1), we can find N ∈ ℕ when   (z k −z) < , for all k≥N, if (Z,   , ⊛) is a-FNS.If (  ) is a fuzzy converges to z we write lim k→∞ z k =z, or z k →z, or lim k→∞   (z k − z)=0.

Definition 2.8 [3]:
If (  ) is a sequence in Z, then (  ) is fuzzy Cauchy sequence in Z if for all  ∈ (0, 1), we can find N ∈ ℕ when   (z k − z m ) < s, for all k, m ≥N, if (Z,   , ⊛) is a-FNS.

Definition 2.9 [3]:
If for any (  ) fuzzy Cauchy in Z, there is z∈Z such that z k →z, then the a-FNS (Z,   , ⊛) is a fuzzy complete

Theorem 2.11 [4]:
When (Z, n Z , ⊛) and (W, n W , ⊛) are two a-FNS.Then the operator H : Z→W is a fuzzy continuous at z∈Z if and only if whenever (z k ) is a fuzzy converges to z∈Z then (H(z k )) is a fuzzy converges to H(z) ∈ W.

Theorem 2.14 [4]:
The space afb(Z, Y) is a fuzzy complete if Y is a fuzzy complete when (Z,   , ⊛) and (Y,   , ⊛) are two a-FNS.

Definition 2.18 [4]:
={z+D: z∈Z} is a -space with the operations ; (v+D) +(z+D) = (v+z) +D and (z+D) = (z)+D.If Z is a vector space over the field  and D is a closed subspace of Z.

Definition 2.19 [5]:
Define a-fuzzy norm for the quotient space When (Z,   , ⊛) be a-FNS and D⊂Z is a fuzzy closed in Z.

Theorem 2.22 [5]:
, q, ⊛) is a fuzzy complete then (Z,   , ⊚) is a fuzzy complete when (Z,   , ⊛) be a-FNS and D⊂Z is a fuzzy closed in Z.

Theorem 2.23 [5]:
, q, ⊚) is a fuzzy complete when (Z,   , ⊛) is a-FNS and D⊂Z is a fuzzy closed in Z.

Definition 2.24 [1]:
Let ⊙:I × I →I be a binary operation function then ⊚ is said to be continuous t-norm (or simply t-norm) if it satisfies the following conditions: ) whenever p ≤ q and z ≥ w.For all p, q, z, w ∈ I = [0, 1].

Theorem 2.32 [6]:
An a-FNA (Z, n Z , ⊛, ⊙) without identity can be embedded into a-FNA, Z e having the identity , also Z is considered as an ideal in Z e .

Theorem 2.34 [6]:
Every a-FNA can be embedded as a closed subalgebra of afb(Z, Z).

Definition 2.41 [6]:
Let ={  :j∈J} be a family of subsets of a space Z.The family  is centered if for any finite number of sets  1 ,  2 , …,  ∈  we have ∩ =1    ≠ ∅.

Definition 2.42 [6]:
Let Z be a non-empty set.A collection T of a subset of Z is said to be a fuzzy topology on Z if (i)Z ∈ T and  ∈ T; (iii)If {A j : jJ}∈ T then ∪ jJ A j ∈ T. Then (Z, T) is called a fuzzy topological space.

Theorem 2.43:
If Z is a fuzzy topological space then the following statement are equivalent: (1) Z is a fuzzy compact; (2) For any centred family  of a fuzzy closed subset of Z we have ∩ ∈ ≠ ∅.

Proof: (2)⟹(1)
Let ={  :j∈J} be a fuzzy open cover of Z.We need to show that  has a finite subcover.For j∈J, define   =Z−   this gives a family =={  :j∈J} of fuzzy closed sets in Z.We have   and so Z is a fuzzy compact since {  : j=1,2, …, k} is a finite subcover of .

Fuzzy Tychonoff Theorem 2.44:
If {  : j∈J} is a family of fuzzy topological spaces and   is a fuzzy compact ∀ j∈J, then the product space Π j∈J   is a fuzzy compact.

Proof:
Let Z=Π j∈J   where   is a fuzzy compact ∀ j∈J.Let  be a centred family of fuzzy closed subset of Z.We will show that there exists z = (  ), j∈J ∈Z such that z ∈ ∩ ∈ A. Let D denote the set consisting of all centred families ℱ[ not necessarily fuzzy closed] of subset of Z such that  ⊆ ℱ.The set D is partially ordered set by ⊆ .
We will show that every chain in D has an upper bound.Indeed, if {ℱ  : j∈J} is A chain in D then take ℱ= ∪ j∈J ℱ  .Since ℱ is centred family and ℱ  ⊆ ℱ for all j∈J thus ℱ is an upper bound of {ℱ  : j∈J}.Now by Zorn ' s Lemma we obtain that the set D contains a maximal element ℳ.We will show that there exists z ∈Z such that z∈ ∩ ∈ℳ  ̅ .Since  ⊆ ℳ and  contains of fuzzy closed sets we have, ∩ ∈ℳ  ̅ ⊆ ∩ ∈ A. Therefore, it will follow that z∈ ∩ ∈ A and ∩ ∈ A ≠ ∅.
Construction of the element z proceed as follows.For j∈J let   :Z→   be the projection onto the jth coordinate.Now for each j∈J the family {  () ̅̅̅̅̅̅̅̅ :  ∈ ℳ} is centred family of fuzzy closed subsets of   , so by the fuzzy compactness of   there exists   ∈   such that   ∈ ∩ ∈ℳ   () ̅̅̅̅̅̅̅̅ .We set z = (  ), j∈J.
In order to see that z∈ ∩ ∈ℳ  ̅ notice that ℳ the following property: This means that for any  ∈ ℳ we have z∈  ̅ , and hence z ∈ ∩ ∈ℳ  ̅ .

3.Further properties of fuzzy complete a-fuzzy normed algebra Definition 3.1:
An ideal  in an algebra (Z, +, .) is maximal if  ⊂Z (that is  ≠Z), and if there is an ideal  with  ⊂  then  =Z.

Proposition 3.2:
Every maximal ideal  in Z where (Z,   ,⊛, ⊙) is fuzzy complete a-fuzzy normed algebra with an identity e, is fuzzy closed.

Proof:
If  be a maximal ideal in Z, then  must does not contains any invertible element, otherwise =Z.This implies that  ⊆Z− (Z).But (Z) is fuzzy open so Z− (Z) is fuzzy closed, hence  ⊆  ̅ ⊆ Z− (Z).As special case,  ≠ Z. Since  ⊆  ̅ so  ̅ = but  is maximal ideal.Hence  is a fuzzy closed.

Proof:
The case when =0 then it is fuzzy continuous.Let  ≠0 and Z has an identity e.Now ∀ u∈Z, (u)= (u.e)= (u).(e), and so (e)=1.If u∈Z with (u)≠0, then b=u− (u).e∈ ker  and so b is not invertible [or 1= (b −1 )=(b).( −1 ) which is not correct ].Therefore, (u) ∈   (u) and this implies that  ℂ [(u)] ≤   (u).This inequality stall true when (u)=0 and hence  is fuzzy continuous on Z. [ if (z k ) be a sequence in Z converge to z∈Z that is lim If Z does not have an identity, we consider   instead.Define  ′ :   → ℂ by:  ′ [(u, )] = (u) +  for all (u, ) ∈   .Then  ′ is a homomorphism is clear and therefore by the first part of the prove,  ′ is fuzzy continuous on   .specially, its restriction to Z in   is fuzzy continuous i.e.,  is a fuzzy continuous.

Definition 3.4:
A homomorphism :Z→ ℂ where (Z,   ,⊛, ⊙) is a fuzzy complete a-FNA is called a character.Character is fuzzy continuous by Proposition 3.3.

Theorem 3.5:
If  is a character on Z, then ker  is a maximal ideal in Z, and every maximal ideal has this form for some unique character, when (Z,   ,⊛, ⊙) is a commutative fuzzy complete a-FNA with identity e.

Proof:
If :Z→ ℂ is a character and =ker ℓ, it is clear that  ≠Z because  =0.If z ∉  then for any u ∈ Z is represented by u=z ] ∈ ker =, we see that Z= ℂz +  and therefore  is a maximal ideal.This implies that  is fuzzy closed and hence   is fuzzy Proposition 3.10: If (Z,   , ⊛, ⊙) is a fuzzy complete a-FNA having an identity e then ω * −fuzzy topology on afb(Z, ℂ) is a Hausdorff space.

Conclusions
In [6] we proved some properties of fuzzy complete a-fuzzy normed algebra.In this paper we recall the definition of a-fuzzy normed algebra in order to prove other properties of fuzzy complete a-fuzzy normed algebra.
and D is a fuzzy closed ideal in Z. Also Z D has an identity if Z has an identity.As well as the identity of Z D has a fuzzy norm equal to 1.