# Some Remarks on Strong Fuzzy Metrics and Strong Fuzzy Approximating Metrics with Applications in Word Combinatorics

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## Abstract

**:**

## 1. Introduction

#### 1.1. Discrepancy of Ordinary Metrics for the Problems of Word Combinatorics

## 2. Preliminaries

#### 2.1. t-Norms

**Definition**

**1.**

- (1tn)
- ∗ is monotone: $\alpha \le \beta \Rightarrow \alpha \ast \gamma \le \beta \ast \gamma $ for all $\alpha ,\beta ,\gamma \in [0,1]$;
- (2tn)
- ∗ is commutative: $\alpha \ast \beta =\beta \ast \alpha $ for all $\alpha ,\beta \in [0,1]$;
- (3tn)
- ∗ is associative: $(\alpha \ast \beta )\ast \gamma =\alpha \ast (\beta \ast \gamma )$ for all $\alpha ,\beta ,\gamma \in [0,1]$;
- (4tn)
- $\alpha \ast 1=\alpha $ for all $\alpha \in [0,1]$.

**Example**

**1.**

- The minimum t-norm is defined by $\alpha \ast \beta :=\alpha \wedge \beta $ where ∧ denotes the operation of taking minimum in [0, 1].
- The product t-norm is defined by $\alpha \ast \beta :=\alpha \xb7\beta $ where · is the product.
- The Łukasiewicz t-norm $L=[0,1]$ is defined by $\alpha {\ast}_{L}\beta :=max(\alpha +\beta -1,0).$
- The Drastic t-norm is defined by$${T}_{D}(\alpha ,\beta )=\left\{\begin{array}{cc}\hfill \alpha ,\hfill & \phantom{\rule{4.pt}{0ex}}\mathit{if}\phantom{\rule{4.pt}{0ex}}\beta =1,\hfill \\ \beta ,\hfill & \phantom{\rule{4.pt}{0ex}}\mathit{if}\phantom{\rule{4.pt}{0ex}}\alpha =1,\hfill \\ 0,\hfill & \phantom{\rule{4.pt}{0ex}}\mathit{otherwise}\phantom{\rule{4.pt}{0ex}}\hfill \end{array}\right.$$
- The nilpotent minimum t-norm is defined by$${T}_{nM}(\alpha ,\beta )=\left\{\begin{array}{cc}min\{\alpha ,\beta \},\hfill & \phantom{\rule{4.pt}{0ex}}\mathit{if}\phantom{\rule{4.pt}{0ex}}\alpha +\beta >1,\hfill \\ 0,\hfill & \phantom{\rule{4.pt}{0ex}}\mathit{otherwise}.\phantom{\rule{4.pt}{0ex}}\hfill \end{array}\right.$$

**Remark**

**1.**

#### 2.2. Fuzzy (Pseudo)Metrics

**Definition**

**2**

- (0FKM)
- $M(x,y,0)=0$ for all $x,y\in X$;
- (1FKM)
- $M(x,y,t)=1\phantom{\rule{4pt}{0ex}}forall\phantom{\rule{4pt}{0ex}}twheneverx=y$;
- (2FKM)
- $M(x,y,t)=M(y,x,t)$ for all $x,y\in X$, for all $t\in {\mathbb{R}}^{+}$;
- (3FKM)
- $M(x,z,t+s)\ge M(x,y,t)\ast M(y,z,s)$ for all $x,y\in X$, for all $t\in {\mathbb{R}}^{+}$;
- (4FKM)
- $M(x,y,-):{\mathbb{R}}^{+}\to [0,1]$ is left continuous for all $x,y\in X$.

- (1FKM)
- $M(x,y,t)=1\phantom{\rule{4pt}{0ex}}forall\phantom{\rule{4pt}{0ex}}tifandonlyifx=y$;

**Definition**

**3**

- (3${}^{s}$FKM)
- $M(x,z,t)\ge M(x,y,t)\ast M(y,z,t)$ for all $x,y,z\in X$ and for all $t\in {\mathbb{R}}^{+}$.
- (4${}^{s}$FKM)
- $M(x,y,-):{\mathbb{R}}^{+}\to [0,1]$ is left continuous and increasing, (i.e., $t<s\u27f9M(x,y,t)\le M(x,y,s)$ for all $x,y\in X.$).

**Remark**

**2.**

**Definition**

**4**

- (0FGV)
- $M(x,y,t)>0$ for all $x,y\in X$ and all $s,t\in {\mathbb{R}}_{0}^{+}$;
- (1FGV)
- $M(x,y,t)=1\phantom{\rule{4pt}{0ex}}whenever\phantom{\rule{4pt}{0ex}}x=y$;
- (2FGV)
- $M(x,y,t)=M(y,x,t)$ for all $x,y\in X$ and all $s,t\in {\mathbb{R}}_{0}^{+}$;
- (3FGV)
- $M(x,z,t+s)\ge M(x,y,t)\ast M(y,z,s)$ for all $x,y,z\in X$ and for all $s,t\in {\mathbb{R}}_{0}^{+}$;
- (4FGV)
- $M(x,y,-):{\mathbb{R}}^{+}\to [0,1]$ is continuous for all $x,y\in X$.

- (1′FGV)
- $M(x,y,t)=1\phantom{\rule{4pt}{0ex}}forall\phantom{\rule{4pt}{0ex}}tifandonlyifx=y$;

**Definition**

**5**

- (3${}^{s}$FGV)
- $M(x,z,t)\ge M(x,y,t)\ast M(y,z,t)$ for all $x,y,z\in X$, for all $t\in {\mathbb{R}}_{0}^{+}$.
- (4${}^{s}$FGV)
- $M(x,y,-):{\mathbb{R}}_{0}^{+}\to [0,1]$ is continuous and increasing (that is $t<s\u27f9M(x,y,t)\le M(x,y,s)\forall x,y\in X.$)

**Remark**

**3.**

## 3. Strongness of Standard Fuzzy Pseudometrics

**Definition**

**6.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**2.**

**Theorem**

**3.**

**Proof.**

- If $a\ne 0$ and $b\ne 0$, then left side of inequality is equal to 0
- If $a=0$ and $b\ne 0$, then $c\le b$ and $\frac{t}{t+c}\ge \frac{t}{t+b}$. We similarly reason if $b=0$ and $a\ne 0$
- If $a=b=0$, then $c=0$ and $1\ge 1$.

**Example**

**2.**

**Example**

**3.**

**Proof.**

#### Strongness of Standard Fuzzy k-Pseudometrics

**Definition**

**7**

- (1Mk)
- $d(x,y)=0\u27fax=y$;
- (2Mk)
- $d(x,y)=d(y,x)\phantom{\rule{4pt}{0ex}}\forall x,y\in X$;
- (3Mk)
- $d(x,z)\le k\xb7\left(d\right(x,y)+d(y,z\left)\right)\phantom{\rule{4pt}{0ex}}\forall x,y,z\in X$.

**Example**

**4.**

**Definition**

**8**

- (0FKMk)
- $M(x,y,0)=0$ for all $x,y\in X$;
- (1FKMk)
- $M(x,y,t)=1\phantom{\rule{4pt}{0ex}}forall\phantom{\rule{4pt}{0ex}}twheneverx=y$;
- (2FKMk)
- $M(x,y,t)=M(y,x,t)$ for all $x,y\in X$, for all $t\in {\mathbb{R}}^{+}$;
- (3FKMk)
- $M(x,z,k(t+s\left)\right)\ge M(x,y,t)\ast M(y,z,s)$ for all $x,y\in X$, for all $t\in {\mathbb{R}}^{+}$;
- (4FKMk)
- $M(x,y,-):{\mathbb{R}}^{+}\to [0,1]$ is left continuous for all $x,y\in X$.

- (3${}^{s}$FKMk)
- $M(x,z,kt)\ge M(x,y,t)\ast M(y,z,t)$;
- (4${}^{s}$FKMk)
- $M(x,y,-):{\mathbb{R}}^{+}\to [0,1]$ is left continuous and increasing for all $x,y\in X$.

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

## 4. Topological and Lattice Structure of Some Families of Strong Fuzzy Metric Spaces

#### 4.1. Some Remarks on t-Norms That Ensure Strongness of Standard Fuzzy Metrics

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

#### 4.2. Some Remarks about the Set of Strong Fuzzy Metrics for a Fixed t-Norm

**Theorem**

**9.**

**Proof.**

**Remark**

**4.**

**Theorem**

**10.**

**Proof.**

**Theorem**

**11.**

**Proof.**

- If $x=z$, then $M(x,x)=1\ge T\left(M(x,y),M(y,z)\right)$
- If $x\ne z$, then either $x\ne y$ or $y\ne z$. Therefore either $M(x,y)=0$ or $M(y,z)=0$. So $M(x,z)=0\ge T\left(M(x,y),M(y,z)\right)=0$.

**Remark**

**5.**

**Proof.**

## 5. Fuzzy Approximating Metrics and Strong Fuzzy Approximating Metrics

**Definition**

**9.**

- (0FAKM)
- $M(x,y,0)=0\phantom{\rule{4pt}{0ex}}\forall x,y\in X$;
- (1FAKM)
- $M(x,x,t)\ge M(x,y,t)\phantom{\rule{4pt}{0ex}}\forall x,y\in X$;
- (2FAKM)
- If $x,y\in X$ then ${lim}_{t\to \infty}M(x,y,t)=1$ whenever $x=y$;
- (3FAKM)
- $M(x,y,t)=M(y,x,t)\phantom{\rule{4pt}{0ex}}\forall x,y\in X,\phantom{\rule{4pt}{0ex}}\forall t\in {\mathbb{R}}^{+}$
- (4FAKM)
- $M(x,z,t+s)\ge M(x,y,t)\ast M(y,z,s)\forall x,y,z\in X,\phantom{\rule{4pt}{0ex}}\forall t,s\in {\mathbb{R}}^{+}$
- (5FAKM)
- $M(x,y,-):{\mathbb{R}}^{+}\to [0,1]$ is lower semicontinuous for all $x,y\in X.$

**Definition**

**10.**

- (${4}^{s}$FAKM )
- $M(x,z,t)\ge M(x,y,t)\ast M(y,z,t)\forall x,y,z\in X,\phantom{\rule{4pt}{0ex}}\forall t\in {\mathbb{R}}^{+}$
- (${5}^{s}$FAKM)
- $M(x,y,-):{\mathbb{R}}^{+}\to [0,1]$ is lower semicontinuous and increasing for all $x,y\in X.$

**Remark**

**6.**

**Remark**

**7.**

## 6. Some Examples of Application of Strong Fuzzy Approximating Metrics in Words Combinatorics

**Theorem**

**12.**

**Proof.**

- If $c\ge t$ then we have$$0\ge \frac{t-a-b-100}{t+100}\iff t\le a+b+100,$$
- If $c<t$ then we have$$\frac{t-c}{t+100}\ge \frac{t-a-b-100}{t+100}\iff c\le a+b+100,$$

**Corollary**

**3.**

**Example**

**5.**

**Proof.**

**Proposition**

**1.**

**Proposition**

**2.**

**Theorem**

**13.**

**Example**

**6.**

**Theorem**

**14.**

**Corollary**

**4.**

**Remark**

**8.**

**Remark**

**9.**

## 7. Conclusions

- To consider strong fuzzy pseudometric spaces and strong fuzzy approximating metrics as categories, In particular, investigate products, coproducts, and other operations in these categories. To study interrelations between these categories.
- As an important problem to be investigated in our future work, we consider the study of interrelations between our fuzzy approximating (in particular strong) metrics with partial and especially fuzzy partial metrics. An attentive reader probably will notice some similarity between our approximating metrics on one side and partial and especially fuzzy partial metric on the other. Partial metrics were introduced in 1994 by Matthews [30] and now are the focus of interest for some mathematicians and theoretical computer scientists (see, e.g., the survey [31]). Based on the concept of a partial metric, V. Gregori, J-J. Minana, and D. Miravet [32] introduced the concept of a fuzzy partial metric. Many researchers working in theoretical computer science showed serious interest in partial metrics, and recently also in fuzzy partial metrics in view of their perspectives of the use in domain theory and some other areas of theoretical computer science. An attentive reader of our paper will probably notice its certain common features with partial and fuzzy partial metrics, and this is not a surprise, since the idea of both approaches when applied to evaluation of two infinite strings is that the result will not be achieved immediately or at some step, but in the process of comparing these strings. On the other hand, we apply essentially different approaches to realize this evaluation. It is one of our principal goals for future work to investigate the relations, in particular, on the categorical level, between these theories.
- We illustrated the opportunities provided by strong fuzzy approximating metrics by some examples and comments in Section 6. We view this material only as the first step in the developing methods for the study of the problems of words combinatorics. This work will be continued in particular in the next work (in preparation) where fuzzy approximating metrics based on different t-norms and parameters will be used and the obtained results will be analyzed for a series of numerical examples.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Bēts, R.; Šostak, A.
Some Remarks on Strong Fuzzy Metrics and Strong Fuzzy Approximating Metrics with Applications in Word Combinatorics. *Mathematics* **2022**, *10*, 738.
https://doi.org/10.3390/math10050738

**AMA Style**

Bēts R, Šostak A.
Some Remarks on Strong Fuzzy Metrics and Strong Fuzzy Approximating Metrics with Applications in Word Combinatorics. *Mathematics*. 2022; 10(5):738.
https://doi.org/10.3390/math10050738

**Chicago/Turabian Style**

Bēts, Raivis, and Alexander Šostak.
2022. "Some Remarks on Strong Fuzzy Metrics and Strong Fuzzy Approximating Metrics with Applications in Word Combinatorics" *Mathematics* 10, no. 5: 738.
https://doi.org/10.3390/math10050738