Fuzzy absolute value on a ring

Abstract

In this paper, fuzzy absolute value on a ring is defined. It is proved that every absolute value can induce a fuzzy absolute value. It is also proved that every fuzzy valued ring induce a fuzzy metric space and a fuzzy norm space. In addition, the notion of a Cauchy sequence and completion of a fuzzy valued ring are given and some of their properties are investigated. Finally, it is proved that every fuzzy valued ring has a completion.

Keywords

Completion fuzzy absolute value fuzzy metric space fuzzy set fuzzy valued ring

1 Introduction

In 1965, Zadeh [31] introduced the concept of fuzzy set theory, as an extension of the classical notion of set. After introducing the fuzzy sets by Zadeh, many authors have attempted to develop the classical results in this new context. In this regard, this concept has been studied and used for the topology, analysis, algebra, modelling, engineering and control theory. Today, the theory of fuzzy set applies to a wide range of issues such as communication, gaming, signal processing, modelling theory, image processing and etc. Many books have been written about fuzzy set theory and its applications, for instance, the reader is refferred to [3 , 29]. The concept of fuzzy metric spaces were initiated by Kramosil and Mich $\overset{´}{a}$ lek [15] as a generalization the concept of the statistical metric space to fuzzy situation. Since then, many of the authors have introduced the concept of fuzzy metric spaces in different ways [6 , 12]. In 1994, George and Veeramani [10] modified the concept of fuzzy metric space introduced by Kramosil and Mich $\overset{´}{a}$ lek and define a Hausdorff topology on this fuzzy metric space. This concept is mainly used in quantum physics, fuzzy optimization and pattern recognition. A large number of papers have been published in this field by different authors (for reference please see [11 , 30]). In this paper, the notation of fuzzy absolute value on a ring is introduced and it is proved that every fuzzy valued ring introduce a fuzzy metric space.

Over the long time, a significant problem was finding an appropriate definition of a fuzzy normed space. An idea of fuzzy norm on a linear space first introduced by Katsaras [14]. Since then several definitions for a fuzzy norms on a linear space were introduced by different authors [2 , 22]. In 1992, Felbin [9], defined the fuzzy normed spaces by assigning a fuzzy real number to each element of a linear space and verified some properties of finite dimensional fuzzy normed space. In 1994, Cheng and Mordeson [4] introduced another type of fuzzy norm on a linear space whose associated fuzzy metric is of Kramosil and Mich $\overset{´}{a}$ lek [15] type. In 1999, Lee et al. [18] discussed the completions of fuzzy metric spaces and fuzzy normed spaces. Bag and Samanta [2] introduced the definition of fuzzy norm which is a little different from the one defined by Cheng and Mordeson. Bag and Samanta have studied finite dimensional fuzzy normed linear spaces. Fuzzy norm space has been studied in many different aspects, all of them have tried to simplify and improve it [1 , 23]. In this article, an attempt is made to a connection between the fuzzy absolute value and fuzzy norm.

Absolute values of a field and their completions played an important role in the development of number theory in the beginning of the 20th century. In this paper, the concept of fuzziness is applied to the clasical notions of absolute value and valued ring. In this regard, a definition of fuzzy absolute value on a ring is introduced. Some properties of fuzzy absolute value, are also given in this paper. Further, the notion of Cauchy sequences in fuzzy valued ring and completion of fuzzy valued ring are introduced and then prove a very important and useful theorem, which states that every fuzzy valued ring has a completion.

2 Preliminaries

In this section, some definitions and some results of absolute valued field and fuzzy set theory which will be used in the later section are recalled.

Let K be a field. An absolute value on K is a map || : K → [0, ∞) satisfying the following axioms for all x, y ∈ K :

|x|=0 if and only if x = 0 ;

|xy| = |x||y| ;

|x + y| ≤ |x| + |y| .

Proposition 2.1. [7] The set ${| n . 1 | \leq 1 | n \in ℤ}$ is bounded if and only if satisfies the ultrametric inequality $| x + y | \leq max {| x |, | y |}$ for all x, y ∈ K .

If an absolute value satisfies ultrametric inequality, it is called non-archimedean; otherwise it is called archimedean.

A typical example of an archimedean absolute value is $| x | = {\begin{matrix} x & x \geq 0 \\ - x & x \leq 0 \end{matrix}$ for all $x \in ℝ$ . This absolute value is called usual absolute value on $ℝ$ .

Next, the most basic examples of non-archimedean absolute values are considered. For every rational prime p, the p-adic absolute value ||_p on $ℚ$ is defined by |0|_p = 0 and $| p^{ν} \frac{m}{n} |_{p} = \frac{1}{e^{ν}},$ where e is the base of the natural logarithms, $ν \in ℤ$ and $n, m \in ℤ \ {0}$ are not divisible by p .

Let X be a non-empty set. A fuzzy subset A in X is characterized by a membership function μ_A : X → [0, 1].

Definition 2.2. [25] A binary operation ∗ : [0, 1] × [0, 1] → [0, 1] is called tringular norm (t-norm) if it satisfies the following conditions:

∗ is associative and commutative;

∗ is continuous;

a ∗ 1 = a for all a ∈ [0, 1];

a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d, with a, b, c, d ∈ [0, 1] .

There are four main examples of continuous t-norms for all a, b ∈ [0, 1] :

a ∗ ₁b = min {a, b} .

a ∗ ₂b = ab .

a ∗ ₃b = max {a + b - 1, 0} .

$a *_{4} b = \frac{ab}{max {a, b, λ}}$ for λ ∈ (0, 1) .

Definition 2.3. [15] The 3-tuple (X, M, ∗) is said to be a fuzzy metric space if X is an arbitrary set, ∗ is a continuous t-norm and M is a fuzzy set on X² × [0, ∞) such that for all x, y, z ∈ X:

M (x, y, 0) =0 ;

M (x, y, t) =1 for all t > 0 if and only if x = y ;

M (x, y, t) = M (y, x, t) for all t> 0 ;

M (x, y, t) ∗ M (y, z, s) ≤ M (x, z, t + s) , for all t, s≥ 0 ;

M (x, y, t) is left continuous for all t > 0.

Definition 2.4. [2] Let U be a linear space over a field K and ∗ be a continuous t-norm. A fuzzy subset N of $U \times ℝ$ is called a fuzzy norm on U if the following condition, are satisfied for all x, y ∈ U and c ∈ K:

N (x, t) =0, for all $t \in ℝ$ with t≤ 0 ;

N (x, t) =1, for all t > 0 if and only if x = 0 ;

$N (cx, t) = N (x, \frac{t}{| c |}), \forall t \in ℝ, t > 0$ and c≠ 0 ;

N (x+ y, t + s) ≥ N (x, t) ∗ N (y, s) ;

N (x, .) is left continuous and lim N (x, t) =1 as t → ∞ .

The triple (U, N, ∗) is said to be a fuzzy normed space.

3 Fuzzy absolute value

In this section, first the notion of fuzzy absolute value is introduced and it is expressed some fundamental properties of fuzzy absolute value. Next, the concept of Cauchy sequences and completion of fuzzy valued ring are introduced and obtain some interesting results.

Definition 3.1. Let R be a ring. A fuzzy subset ρ of R × [0, ∞) is called a fuzzy absolute value of R when it satisfies the following conditions for all x, y ∈ R:

ρ (x, t) =0 if and only if t = 0;

ρ (x, t) =1, for all t > 0 if and only if x = 0;

ρ (x + y, t + s) ≥ min {ρ (x, t) , ρ (y, s)};

ρ (xy, ts) ≥ ρ (x, t) ρ (y, s);

ρ (- x, t) = ρ (x, t) .

The pair (R, ρ) , where ρ is a fuzzy absolute value of R, is called a fuzzy absolute valued ring, or simply a fuzzy valued ring.

Example 1. Consider an arbitrary fixed number α ∈ (0, 1) . Then the following map is a fuzzy absolute value. $ρ (x, t) = {\begin{matrix} 0 & t = 0 \\ 1 & t > 0, x = 0 \\ α & otherwise \end{matrix}$

Example 2. Let ||_p be the p-adic absolute value on $ℚ .$ It is not difficult to see that the following map is a fuzzy absolute value. $ρ (x, t) = {\begin{matrix} 0 & t = 0 \\ 1 & t > 0, x = 0 \\ | x |_{p} & | x |_{p} < 1 \\ \frac{1}{| x |_{p}} & | x |_{p} > 1 \end{matrix}$

In the following example it is shown that every absolute value induces a fuzzy absolute value.

Example 3. If || is an absolute value of ring R, then the mapping $ρ (x, t) = \frac{t}{t + | x |}$ is a fuzzy absolute value. In this case, it is said that the fuzzy absolute value ρ is depending on the absolute value || . We shall show that $ρ (x, t) = \frac{t}{t + | x |}$ is a fuzzy absolute value.

From the definition ρ (x, t) =0, if and only if t = 0 .

Let ρ (x, t) =1 for all t ∈ (0, ∞) . Hence $\frac{t}{t + | x |} = 1,$ therefore t = t + |x| . So |x|=0 and according to the definition of absolute value x = 0 . Conversely, if x = 0, then $ρ (x, t) = \frac{t}{t} = 1,$ for all t ∈ (0, ∞) .

For all s, t∈ [0, ∞) , x, y ∈ R ; if s = t = 0 then in this case the relation is obvious, we consider the case s> 0, t > 0 ; without loss of generality assume that min {ρ (x, t) , ρ (y, s)} = ρ (y, s) . Now $\begin{matrix} ρ (x + y, t + s) - ρ (y, s) \\ = \frac{t + s}{t + s + | x + y |} - \frac{s}{s + | y |} \\ = \frac{(t + s) | y | - s | x + y |}{(t + s + | x + y |) (s + | y |)} \\ \geq \frac{t | y | - s | x |}{(t + s + | x + y |) (s + | y |)} \\ > 0 . \end{matrix}$ Hence ρ (x + y, t + s) ≥ min {ρ (x, t) , ρ (y, s)} .

The following inequality holds for all x, y ∈ R and t, s ∈ [0, ∞) $\begin{matrix} ρ (xy, ts) \\ = \frac{ts}{ts + | xy |} \\ \geq \frac{ts}{(t + | x |) (s + | y |)} \\ = ρ (x, t) ρ (y, s) . \end{matrix}$

ρ (- x, t) = ρ (x, t) , since |x| = | - x| .

Lemma 3.2. Let || be a non-archimedean absolute value on ring R, then the depending fuzzy absolute value ρ satisfies the following inequality: $ρ (x + y, t) \geq min {ρ (x, t), ρ (y, t)} .$

Proof. Since || is a non-archimedean absolute value on ring R, then |x + y| ≤ max {|x|, |y|} , from this we have the following inequality: $\begin{matrix} ρ (x + y, t) & = & \frac{t}{t + | x + y |} \\ \geq & \frac{t}{(t + max {| x |, | y |}} \\ \geq & min {\frac{t}{t + | x |}, \frac{t}{t + | y |}} \\ = & min {ρ (x, t), ρ (y, t)} . □ \end{matrix}$

Proposition 3.3. Let || be an absolute value on ring R, then the fuzzy absolute value ρ depending on ||, satisfies the following inequality: $ρ (x + y, t) \leq max {ρ (x, t), ρ (y, t)} .$

Proof. Since || is an absolute value on ring R, then |x + y| ≥ min {|x|, |y|}, hence $\frac{1}{| x + y |} \leq \frac{1}{min {| x |, | y |}} .$ So $ρ (x + y, t) = \frac{t}{t + | x + y |} \leq max {\frac{t}{t + | x |}, \frac{t}{t + | y |}} = max {ρ (x, t), ρ (y, t)} .$ □

Theorem 3.4. Let || be a non-archimedean absolute value on ring R, and ρ be a fuzzy absolute value depending on || . If ρ (x, t) ≤ ρ (y, t) , then ρ (x + y, t) = ρ (x, t) .

Proof. By refer to the Lemma 3.2 and Proposition 3.3, ρ (x, t) ≤ ρ (x + y, t) ≤ ρ (y, t) . On the other hand we have ρ (x, t) = ρ ((x + y) - y, t) ≥ min {ρ (x + y, t) , ρ (y, t)} = ρ (x + y, t) . Hence ρ (x + y, t) = ρ (x, t) . □

The t-norm ∗ is considered as follows a ∗ b = min {a, b} . In the following, we show that every fuzzy valued ring induces a fuzzy metric space.

Theorem 3.5. Let (R, ρ) be a fuzzy valued ring, let M : R² × [0, ∞) → [0, 1] be the mapping defined by M (x, y, t) = ρ (x - y, t) for all x, y ∈ R . Then M is a fuzzy metric on R .

Proof. The properties (1) , (2) , (3) , (5) are immediate from the definition of fuzzy absolute value. Now, it is shown that M (x, y, t) ∗ M (y, z, s) ≤ M (x, z, t + s) , for all x, y, z ∈ R and t, s ≥ 0 : $\begin{matrix} M (x, z, t + s) & = & ρ (x - z, t + s) \\ = & ρ (x - y + y - z, t + s) \\ \geq & min {ρ (x - y, t), ρ (y - z, s)} \\ = & ρ (x - y, t) * ρ (y - z, s) \\ = & M (x, y, t) * M (y, z, s) . □ \end{matrix}$

Note that the inverse of the above theorem does not hold, this means that any fuzzy metric does not produce a fuzzy absolute value. For instance consider the following example.

Example 4. Let $X = ℝ .$ Define a ∗ b = min {a, b} and $M (x, y, t) = (e^{\frac{| x - y |}{t}})^{- 1}$ for all x, y ∈ X and t ∈ (0, ∞) . Then (X, M, ∗) is a fuzzy metric space. But the mapping $ρ (x, t) = (e^{\frac{| x |}{t}})^{- 1}$ is not an fuzzy absolute value (where e is the base of the natural logarithms).

In the following, an attempt is made to a connection between the fuzzy norm and fuzzy absolute value. In this regard, by considering a fuzzy absolute value depending on the absolute value, a fuzzy normed space is produced.

Theorem 3.6. A vector space V of dimension n over the fuzzy valued ring (R, ρ) is considered, where ρ is depending on the absolute value || . Let {v₁, ⋯ , v_n} be any basis of V. Now, the mapping $N : V \times ℝ \to [0, 1]$ is defined as follows. If $x = \sum_{i = 1}^{n} x_{i} v_{i}$ (with x_i ∈ R). Let $N (x, t) = {\begin{matrix} 0 & t \leq 0 \\ max {ρ (x_{i}, t) | 1 \leq i \leq n} & t > 0 \end{matrix}$ Then the pair (V, N) is a fuzzy normed space.

Proof. Proof is straightforward. □

Now, the notion of Cauchy sequences in fuzzy valued ring is introduced and prove some basic properties and relations.

Let (R, ρ) be a fuzzy valued ring and (x_n) be a sequence in R . Then (x_n) to be convergent if there exist x ∈ R such that $lim_{n \to \infty} ρ (x_{n} - x, t) = 1,$ for all t > 0, i.e. for each 0 < ɛ < 1 and t > 0, there exists $N \in ℕ$ such that ρ (x_n - x, t) >1 - ɛ for all n ≥ N . In that case x is called the limit of the sequence (x_n) and is denoted by lim x_n . A subset A of fuzzy valued ring (R, ρ) is dense if for every x ∈ R there exists sequence (x_n) ∈ A such that $lim_{n \to \infty} ρ (x_{n} - x, t) = 1 .$ A sequence (x_n) in a fuzzy absolute valued ring (R, ρ) is a Cauchy sequence if for any 0 < ɛ < 1 and t > 0 there exists $N \in ℕ$ such that ρ (x_n - x_m, t) >1 - ɛ for all n, m ≥ N . The fuzzy valued ring (R, ρ) is said to be complete when every Cauchy sequence in R has a limit in R .

Definition 3.7. A sequence (x_n) in a fuzzy absolute valued ring (R, ρ) is said to be F-bounded if and only if there exist x ∈ R, t > 0 and 0 < r < 1 such that ρ (x_n - x, t) > 1 - r .

Theorem 3.8. Let (R, ρ) be a fuzzy valued ring. Then every Cauchy sequence in (R, ρ) is F-bounded.

Proof. Let (x_n) be a Cauchy sequence in (R, ρ) . By definition $\begin{matrix} \forall ɛ > 0, t > 0, \exists N \in ℕ : \\ ρ (x_{n} - x_{m}, t) > 1 - ɛ, \forall n, m \geq N . \end{matrix}$ Particularly, setting $ɛ = \frac{1}{2}$ $\exists N_{1} : ρ (x_{n} - x_{m}, t) > 1 - \frac{1}{2}, \forall m, n \geq N_{1} .$ Note that since N₁ ≥ N₁, this means that: $ρ (x_{n} - x_{N_{1}}, t) > \frac{1}{2}, \forall n \geq N_{1} .$ To show that (x_n) is F-bounded, it is must be shown that there exists x ∈ R, t > 0 and 0 < r < 1 such that ρ (x_n - x, t) >1 - r for all x_n ∈ (x_n) . Let K = min {ρ (x_{N
₁} - x₁, t) , ρ (x_{N
₁} - x₂, t) , ⋯ , ρ (x_{N
₁} - x_N₁-1, t)} . Each x_n for n ≥ N₁ satisfies $ρ (x_{N_{1}} - x_{n}, t) \geq \frac{1}{2},$ by choice of N₁ as mentioned above. Each x_n for n < N₁ satisfies ρ (x_{N
₁} - x_n, t) ≤ K by choice of K . Now $\begin{matrix} ρ (x_{n} - x_{N_{1}}, 3 t) \\ \geq ρ (x_{n} - x_{i}, t) ρ (x_{i} - x_{j}, t) ρ (x_{j} - x_{N_{1}}, t) \\ \geq (1 - \frac{1}{2}) (1 - \frac{1}{2}) K . \end{matrix}$ Taking x = x_{N
₁}, t′ = 3t and $(1 - \frac{1}{2}) (1 - \frac{1}{2}) K > 1 - r, 0 < r < 1 .$ It is shown that each x_n satisfies ρ (x_n - x, t′) >1 - r . So (x_n) is F-bounded. □

Consider the fuzzy valued ring $(ℝ, ρ)$ in Example 3. According to the above theorem, the following example is expressed.

Example 5. Let $x_{n} = \frac{1}{n},$ for $n \in ℕ .$ Then (x_n) is a Cauchy sequence in fuzzy valued ring $(ℝ, ρ)$ and hence (x_n) is F-bounded.

Proposition 3.9. If R₀ is a subring of R, and if ρ is a fuzzy absolute value of R, then its restriction ρ₀ to R₀ is a fuzzy absolute value of R₀ .

Proof. Since ρ is a fuzzy absolute value and ρ₀ is a restriction of ρ to R₀ the conditions of Definition 3 for ρ₀ are satisfied and ρ₀ is a fuzzy absolute value. □

Definition 3.10. A fuzzy valued ring $(\hat{R}, \hat{ρ})$ is said to be a completion of the fuzzy valued ring (R, ρ) when the following properties are satisfied:

R is a subring of $\hat{R}, ρ$ is restriction of $\hat{ρ}$ to R ;

R is dense in $\hat{R}$ ;

$(\hat{R}, \hat{ρ})$ is a complete fuzzy valued ring.

Example 6. Consider a usual absolute value || on $ℝ$ and let $\hat{ρ} (x, t) = \frac{t}{t + | x |}$ be a fuzzy absolute value on $ℝ .$ Then a fuzzy valued ring $(ℝ, \hat{ρ})$ is a completion of the fuzzy valued ring $(ℚ, ρ),$ which ρ is restriction of $\hat{ρ}$ to $ℚ .$

In the following, a very important issue in the fuzzy absolute valued ring is mentioned, which states that any fuzzy valued ring has a completion.

Theorem 3.11. Every fuzzy valued ring (R, ρ) has a completion $(\hat{R}, \hat{ρ})$ .

Proof. Let A be the set of all Cauchy sequence $(a_{n})_{n \in ℕ}$ of elements of R, with respect to ρ . Consider an addition and multiplication on A as follows: $(a_{n}) + (b_{n}) = (a_{n} + b_{n}), (a_{n}) (b_{n}) = (a_{n} b_{n}) .$ It is shown that the set A is closed with respect to these operations. (a_n + b_n) is a Cauchy sequence since $\begin{matrix} ρ (a_{n} + b_{n} - a_{m} - b_{m}, t) \\ = ρ (a_{n} + b_{n} - a_{m} - b_{m}, t - s + s) \\ \geq min {ρ (a_{n} - a_{m}, t - s), ρ (b_{n} - b_{m}, s)} \\ \geq 1 - ɛ . \end{matrix}$ (a_nb_n) is also a Cauchy sequence, for $\begin{matrix} ρ (a_{n} b_{n} - a_{m} b_{m}, t) \\ = ρ (a_{n} (b_{n} - b_{m}) + b_{m} (a_{n} - a_{m}), t - s + s) \\ \geq min {ρ (a_{n} (b_{n} - b_{m}), t - s), \\ ρ (b_{m} (a_{n} - a_{m}), s)} \\ \geq min {(1 - r_{1}) ρ (b_{n} - b_{m}, t - s), \\ (1 - r_{2}) ρ (a_{n} - a_{m}, s)}, \end{matrix}$ where ρ (a_n, t) ≥1 - r₁ for all $n \in ℕ, t > 0$ and ρ (b_n, t) ≥1 - r₂ for all $n \in ℕ, t > 0 .$ A set A with these operations is a commutative ring with identity element $1 = (1)_{n \in ℕ} .$ The set $I = {(a_{n})_{n \in ℕ} | lim_{n \to \infty} ρ (a_{n}, t) = 1, \forall t > 0}$ is an ideal of A . Indeed, the sum of two sequences in I lies in I . If (c_n) is a Cauchy sequence and if (a_n) ∈ I, then $ρ (c_{n} a_{n}, t) \geq ρ (c_{n}, t - s) ρ (a_{n}, s) \geq (1 - r) ρ (a_{n}, s),$ where ρ (a_n, s) ≥1 - r for all $n \in ℕ, s > 0 .$ Therefore (c_n) (a_n) ∈ I and its follows that the set I is an ideal of ring A, and $\hat{R} = \frac{A}{I}$ is a ring. Now define $\hat{ρ} : \hat{R} \times [0, \infty) \to [0, 1]$ by $\hat{ρ} (a_{n} + I, t) = lim_{n \to \infty} ρ (a_{n}, t),$ it must be shown that it is well defined. Let $a_{n} + I, b_{n} + I \in \hat{R}$ , then $\begin{matrix} \hat{ρ} (a_{n} + I, t) & = & M (a_{n} + b_{n} + I, b_{n} + I, t) \\ \geq M (a_{n} + b_{n} + I, a_{n} + b_{n}, t) \\ * M (a_{n} + b_{n}, b_{n}, t) \\ * M (b_{n}, b_{n} + I, t) . \end{matrix}$ Hence $\begin{matrix} lim_{n \to \infty} \hat{ρ} (a_{n} + I, t) \\ = lim_{n \to \infty} M (a_{n} + b_{n} + I, b_{n} + I, t) \\ \geq lim_{n \to \infty} M (a_{n} + b_{n} + I, a_{n} + b_{n}, t) \\ * lim_{n \to \infty} M (a_{n} + b_{n}, b_{n}, t) \\ * lim_{n \to \infty} M (b_{n}, b_{n} + I, t), \end{matrix}$ $\Rightarrow lim_{n \to \infty} \hat{ρ} (a_{n} + I, t) \geq lim_{n \to \infty} ρ (a_{n}, t) .$ In a similar way it is obtained that $lim_{n \to \infty} ρ (a_{n}, t) \geq lim_{n \to \infty} \hat{ρ} (a_{n} + I, t) .$ Now, it is proved that $\hat{ρ}$ is a fuzzy absolute value on $\hat{R} .$

$\hat{ρ} (a_{n} + I, t) = 0$ if and only if $lim_{n \to \infty} ρ (a_{n}, t) = 0,$ if and only if t = 0 .

Obviously, $\hat{ρ} (a_{n} + I, t) = 1$ for all $t \in ℝ, t > 0$ and a_n ∈ I . Moreover if $\hat{ρ} (a_{n} + I, t) = 1$ so $lim_{n \to \infty} ρ (a_{n}, t) = 1,$ which implies that a_n ∈ I .

For each $a_{n} + I, b_{n} + I \in \hat{R},$ it is obtained $\begin{matrix} \hat{ρ} (a_{n} + b_{n} + I, t + s) \\ = lim_{n \to \infty} \hat{ρ} (a_{n} + b_{n}, t + s) \\ \geq min {lim_{n \to \infty} ρ (a_{n}, t), lim_{n \to \infty} ρ (b_{n}, s)} \\ = min {\hat{ρ} (a_{n} + I, t), \hat{ρ} (b_{n} + I, s)} . \end{matrix}$

For each $a_{n} + I, b_{n} + I \in \hat{R},$ it is obtained $\begin{matrix} \hat{ρ} (a_{n} b_{n} + I, ts) \\ = lim_{n \to \infty} ρ (a_{n} b_{n}, ts) \\ \geq lim_{n \to \infty} ρ (a_{n}, t) lim_{n \to \infty} ρ (b_{n}, s) \\ = \hat{ρ} (a_{n} + I, t) \hat{ρ} (b_{n} + I, s) . \end{matrix}$

$\hat{ρ} (- a_{n} + I, t) = \hat{ρ} (a_{n} + I, t) .$

Now let a ∈ R . The map

f : R \to \hat{R}

defined by f (a) = (a_n) + I, where a_n = a for every n, is an isomorphism of R onto the set f (R) . Next, it is shown that f (R) is dense in

\hat{R} .

Let

α = (a_{n}) + I \in \hat{R},

and 0 < ɛ < 1. Take

N \in ℕ

such that ρ (a_n - a_m, t) >1 - ɛ for all n, m > N . Then

\hat{ρ} (α - f (a_{n})) = lim_{m \to \infty} ρ (a_{n} - a_{m}, t) > 1 - ɛ,

for every n > N . Hence

lim_{n \to \infty} f (a_{n}) = a_{n} + M .

So f (R) is dense in

\hat{R} .

Now, let (α_n) be an arbitrary Cauchy sequence of

\hat{R} .

As f (R) is dense in

\hat{R},

it is founded a sequence

(a_{n}^{'}) = f (a_{n})

of elements of f (R) such that

\hat{ρ} (a_{n}^{'} - α_{n}, t) > 1 - \frac{1}{n}, n = 1, 2, \dots .

Since

\begin{matrix} \hat{ρ} (a_{m}^{'} - a_{n}^{'}, t) \\ = \hat{ρ} (a_{m}^{'} - α_{m} + α_{m} - a_{n}^{'}, t - s + s) \\ \geq min {\hat{ρ} (a_{m}^{'} - α_{m}, t - s), \hat{ρ} (α_{m} - a_{n}^{'}, s)} \end{matrix}

(a_{n}^{'})

is a Cauchy sequence of elements of f (R) . Moreover, for

α = a_{n} + I \in \hat{R},

it follows from

lim_{n \to \infty} f (a_{n}) = α

that

\begin{matrix} \hat{ρ} (α_{n} - α, t) \\ = \hat{ρ} (α_{n} - f (a_{n}) + f (a_{n}) - α, t - s + s) \\ \geq min {\hat{ρ} (α_{n} - f (a_{n}), t - s), \hat{ρ} (f (a_{n}) - α, s)} \\ > 1 - ɛ \end{matrix}

for 0 < ɛ < 1 and all n sufficiently large. Consequently,

(α_{n})_{n \in ℕ}

converges to α, and so

\hat{R}

is complete. □

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