Let be two metric spaces. For a mapping , for all , if f satisfies,
where denote the metrics in the spaces , then f is called an isometry. It means that for some fixed number , assume that f preserves distance p; i.e., for all in X, if , we can get . Then we say p is a conservative distance for the mapping f. Whether there exists a single conservative distance for some f such that f is an isometry from X to Y, is the basic issue of conservative distances. It is called the Aleksandrov problem.
Theorem 1.1. (  ) Let be two real normed linear spaces (or NLS) with , and Y is strictly convex, assume that a fixed real number and that a fixed integer . Finally, if is a mapping satisfies
for all . Then f is an affine isometry. we can call Benz’s theorem.
We can see some results about the Aleksandrov problem in different spaces in  -  . A natural question is that: Whether the Aleksandrov problem can be proved in non-Archimedean 2-fuzzy 2-normed spaces under some conditions. So in this article, we will give the definition of non-Archimedean 2-fuzzy 2-normed spaces according to     , then by applying the Benz’s theorem to fix the value of p and N to solve problems.
If a function from a field K to satisfies
for all , then the field K is called a non-Archimedean field.
We can know , for all from the above definition. An example of a non-Archimedean valuation (or NAV) is the function taking and others into 1.
In 1897, Hence in  found that p-adic numbers play a vital role in the complex analysis, the norm derived from p-adic numbers is the non-Archimedean norm, the analysis of the non-Archimedean has important applications in physics.
Definition 1.2. Let X be a vector space and dim . A function is called non-Archimedean 2-norm, if and only if it satisfies
(T1) , iff are linearly dependent;
for all . Then is called non-Archimedean 2-normed space over the field K.
Definition 1.3. An NAV in a linear space X over a field K. A function is said to be a non-Archimedean fuzzy norm on X, if and only if for all and ,
(F1) with ,
(F2) iff for all ,
(F3) , for and ,
(F5) is a nondecreasing function of and .
Then is known as a non-Archimedean fuzzy normed space (or F-NANS).
Theorem 1.4. Let be an F-NANS. Assume the condition that:
(F6) for all .
Define . We call these α-norms on X or the fuzzy norm on X.
Proof: 1) Let , it implies that , then for all , , , so ;
Conversely, assume that , by (F2), for all , then for all , so .
2) By (F3), if , then
Let , then
If , then
3) We have
Example 1.5. Let be a non-Archimedean normed space. Define
for all , Then is a F-NANS.
Definition 1.6. Let Z be any non-empty set and be the set of all fuzzy sets on Z. For and , define
Definition 1.7. A non-Archimedean fuzzy linear space over the field K, we define the addition and scalar multiplication operation of X as following: , , if for every , we have a related non-negative real numebr, is the fuzzy norm of in such that
(T4) for all .
for every , then we say that X is an F-NANS.
Definition 1.8. Let X be a non-empty non-Archimedean field set, be the set of all fuzzy sets on X. If , then . Clearly, , so is a bounded function. Let , then is a non-Archimedean linear space over the field K and the addition, scalar multiplication are defined as follows
If for every , there is a related non-negative real number called the norm of f in such that for all
(T1) iff . For
(T2) . For
(T3) . For
Then the linear space is a non-Archimedean normed space.
Definition 1.9. (  ) A 2-fuzzy set on X is a fuzzy set on .
Definition 1.10. A NAV in a linear space over a field K. If a function is a non-Archimedean 2-fuzzy 2-norm on X (or a fuzzy 2-norm on ), iff for all , ,
(F1) for ;
(F2) iff are linearly dependent for all ;
(F4) , for and ;
(F6) is a nondecreasing function of R and ;
Then is called a non-Archimedean fuzzy 2-normed space (or FNA-2) or is a non-Archimedean 2-fuzzy 2-normed space.
Theorem 1.11. Let be an FNA-2. Suppose the condition that:
(F7) for all and are linearly dependent.
Define . We call these α-2-norms on or the 2-fuzzy 2-norm on X.
Proof: It is similar to the proof of Theorem 1.4.
2. Main Result
From now on, if we have no other explanation, let , . ,
Definition 2.1. Let be two FNA-2 and a mapping . If for all and , we have
then is called 2-isometry.
Definition 2.2. For a mapping and
1) If , then , we say satisfies the area one preserving property (AOPP).
2) If , then , we say satisfies the area n for each n (AnPP).
Definition 2.3. We say a mapping preserves collinear, if mutually disjoint elements of , then exist some real number t we have
Next, we denote .
Lemma 2.4. Let and be two FNA-2. If , a mapping satisfies and AOPP, then we can get where .
Proof: 1) Firstly, we prove that f preserves collinear. We assume that , according to , we get
then and are linearly dependent. So we obtain that preserves collinear.
2) Secondly, we prove that when , we can get .
Let , then , so
according to , we have
Since f preserves collinear, so there exists a real number s such that
So, we get
This contradicts with .
Lemma 2.5. Let and be two FNA-2. If a mapping satisfies AOPP and preserves collinear, then
1) is an injective;
2) if , then and with .
Proof: 1) We prove is injective. Let , since dim , there exists an element such that are linearly independent. Hence .
Let , then , and satisfies AOPP, so
we can see . So the mapping is injective.
2) Let mutually disjoint elements of and , so . Since is injective and preserves collinear, there exist such that
Since dim , there exist an element such that . Let , then and
Since , we get and
According to the mapping is injective, so , and
Let , so we have
So is additive.
From the lemma 2.4, we know that if , then satisfies 2-isometry.
so and is linearly dependent i.e. .
Next we assume ,
Thus , if , then , but
so . It contradicts with . Thus .
Lemma 2.6. Let and be FNA-2. If , a mapping satisfies and AOPP, then we can get for all , we can get .
Proof: From lemma 2.4, we know preserves collinear.
For any , there exist two numbers such that .
By lemma 2.5, we have
Lemma 2.7. Let and be two FNA-2. If a mapping satisfies AOPP and for all with , then satisfies AnPP.
Proof: Let and . Let
We know preserves collinear. So there exist a number such that
Then we have . By lemma 2.5, , so
In the same way, we can get
Theorem 2.8. Let and be two FNA-2. If a mapping satisfies AOPP and for all with , then is 2-isometry.
Proof: Since lemma 2.4, we just need to prove that with .
We can assume that when , for all , we have . and there exist a number such that
Let , then
Since preserves collinear, there exist a number such that
which is contradiction, so
Therefore, we get with . Hence
for all .
 Chu, H.Y., Park, C.G. and Park, W.G. (2004) The Aleksandrow Problem in Linear 2-Normed Spaces. Journal of Mathematical Analysis and Applications, 289, 666-672.
 Wang, D.P., Liu, Y.B. and Song, M.M. (2012) The Aleksandrov Problem on Non-Archimedean Normed Spaces. Arab Journal of Mathematical Science, 18, 135-140.
 Park, C. and Alaca, C. (2013) Mazur-Ulam Theorem under Weaker Conditions in the Framework of 2-Fuzzy 2-Normed Linear Spaces. Journal of Inequalities and Applications, 2018, 78.