# Fuzzy Analogues of Sets and Functions Can Be Uniquely Determined from the Corresponding Ordered Category: A Theorem

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## Abstract

**:**

## 1. Introduction

**Category theory is one of the main tools of modern mathematics.**

- $\mathrm{Ob}$ is the set whose elements are called objects,
- $\mathrm{Mor}$ is a set whose elements are called morphisms,
- $:\mathrm{Mor}\to \mathrm{Ob}\times \mathrm{Ob}$ is a mapping that assigns, to each morphism $f\in \mathrm{Mor}$ a pair of objects $(a,b)\in \mathrm{Ob}\times \mathrm{Ob}$; this is denoted by $f:a\to b$; the object a is called f’s domain, and b is called f’s range;
- $\mathrm{id}$ is a mapping that assigns, to each object $a\in \mathrm{Ob}$, a morphism ${\mathrm{id}}_{a}:a\to a$; and
- $\circ $ is a mapping that assigns, to each pair of morphisms $f:a\to b$ and $g:b\to c$ for which the range of f is equal to the domain of g, a new morphism $g\circ f:a\to c$ so that for every $f:a\to b$, we have ${\mathrm{id}}_{b}\circ f=f\circ {\mathrm{id}}_{a}=f$.

- Set theory is naturally described in terms of a category Set in which objects are sets and morphisms are functions.
- Topology is described in terms of a category Top in which objects are topological spaces and morphisms are continuous mappings.
- Linear algebra is naturally described in terms of a category Lin, in which objects are linear spaces, and morphisms are linear mappings, etc.

**What happens in the fuzzy case?**

**Need for an ordered category.**

**Formulation of the problem.**

**What we do in this paper.**

## 2. Results

**Towards a precise formulation of the problem.**It is easy to see that if we have a 1-1 mapping $\pi :U\to U$ of the Universe of discourse U onto itself (i.e., a bijection), then the corresponding transformation $R(x,y)\to R\left(\pi \right(x),\pi (y\left)\right)$ is an automorphism of the corresponding category in the sense that it preserves the identity, composition, and order.

**Definition**

**1.**

**Definition**

**2.**

- the only object is the set U,
- morphisms are fuzzy relations, i.e., mappings $R:U\times U\to [0,1]$,
- the morphism $\mathrm{id}$ is defined as the mapping for which $\mathrm{id}(x,x)=1$ and $\mathrm{id}(x,y)=0$ for $x\ne y$,
- the composition of morphisms is defined by the formula$$(g\circ f)(x,z)=\underset{y}{\mathrm{max}}\mathrm{min}(f(x,y),g(y,z)),$$
- the order between the morphisms is the component-wise order: $f\le g$ means that $f(x,y)\le g(x,y)$ for all x and y.

**Definition**

**3.**

- for all f, a, and b, we have $f:a\to b$ if and only if $G\left(f\right):F\left(a\right)\to F\left(b\right)$;
- for all f and g, we have $G(f\circ g)=G\left(f\right)\circ G\left(g\right)$,
- for all a, we have $G\left({\mathrm{id}}_{a}\right)={\mathrm{id}}_{F\left(a\right)}$, and
- for all f and g, we have $f\le g$ if and only if $G\left(f\right)\le G\left(g\right)$.

**Proposition**

**1.**

**Theorem**

**1.**

## 3. Proofs

#### 3.1. Proof of the Proposition

#### 3.2. Proof of the Theorem

- $g(x,y)=f(x,y)>0$ and $g({x}^{\u2033},{y}^{\u2033})=0$ for all pairs $({x}^{\u2033},{y}^{\u2033})\ne (x,y)$, and
- ${g}^{\prime}(x,y)=f({x}^{\prime},{y}^{\prime})>0$ and $g({x}^{\u2033},{y}^{\u2033})=0$ for all pairs $({x}^{\u2033},{y}^{\u2033})\ne ({x}^{\prime},{y}^{\prime})$.

- $f(x,x)>0$ for this element x and
- $f({x}^{\prime},{y}^{\prime})$ for all pairs $({x}^{\prime},{y}^{\prime})\ne (x,x)$.

- ${f}_{x,v}(x,x)=v$ and
- ${f}_{x,v}({x}^{\prime},{y}^{\prime})=0$ for all pairs $({x}^{\prime},{y}^{\prime})\ne (x,x)$.

- ${f}_{x\to y}(x,y)=w$ and ${f}_{x\to y}({x}^{\prime},{y}^{\prime})=0$ for all other pairs $({x}^{\prime},{y}^{\prime})\ne (x,y)$, and, similarly,
- ${f}_{y\to x}(y,x)=w$ and ${f}_{y\to x}({x}^{\prime},{y}^{\prime})=0$ for all other pairs $({x}^{\prime},{y}^{\prime})\ne (y,x)$.

- $c(x,{y}^{\prime})=f(x,{y}^{\prime})$ for all ${y}^{\prime}$ and
- $c({x}^{\prime},{y}^{\prime})=0$ for all ${y}^{\prime}$ and for all ${x}^{\prime}\ne a$.

- ${c}^{\prime}(x,y)=f(x,y)$, and
- ${c}^{\prime}({x}^{\prime},{y}^{\prime})=0$ for all other pairs $({x}^{\prime},{y}^{\prime})\ne (x,y)$.

- ${f}_{y\to x}(y,x)=v$, and
- ${f}_{y\to x}({x}^{\prime},{y}^{\prime})=0$ for all pairs $({x}^{\prime},{y}^{\prime})\ne (y,x)$.

## 4. Conclusions

- elements of the original universe of discourse (modulo a 1-1 permutation), and
- fuzzy degrees (modulo a 1-1 monotonic mapping from the interval $[0,1]$ onto itself).

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Servin, C.; Muela, G.D.; Kreinovich, V.
Fuzzy Analogues of Sets and Functions Can Be Uniquely Determined from the Corresponding Ordered Category: A Theorem. *Axioms* **2018**, *7*, 8.
https://doi.org/10.3390/axioms7010008

**AMA Style**

Servin C, Muela GD, Kreinovich V.
Fuzzy Analogues of Sets and Functions Can Be Uniquely Determined from the Corresponding Ordered Category: A Theorem. *Axioms*. 2018; 7(1):8.
https://doi.org/10.3390/axioms7010008

**Chicago/Turabian Style**

Servin, Christian, Gerardo D. Muela, and Vladik Kreinovich.
2018. "Fuzzy Analogues of Sets and Functions Can Be Uniquely Determined from the Corresponding Ordered Category: A Theorem" *Axioms* 7, no. 1: 8.
https://doi.org/10.3390/axioms7010008