# Aggregation of Indistinguishability Fuzzy Relations Revisited

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## Abstract

**:**

## 1. Introduction

- (E1)
- $E(x,x)=1$;
- (E2)
- $E(x,y)=E(y,x)$;
- (E3)
- $T\left(E\right(x,y),E(y,z\left)\right)\le E(x,z)$.

- (d1)
- $d(x,x)=0$,
- (d2)
- $d(x,y)=d(y,x)$,
- (d3)
- $d(x,z)\le d(x,y)+d(y,z)$.

**Theorem**

**1.**

- (1)
- F aggregates $\mathcal{T}$-equivalences into a T-equivalence.
- (2)
- The function $H:{\prod}_{i=1}^{n}{[0,{f}_{{T}_{i}}\left(0\right)]}^{n}\to [0,{f}_{T}\left(0\right)]$ aggregates every collection ${\left\{{d}_{i}\right\}}_{i=1}^{n}$ of ${\left({f}_{{T}_{i}}\left(0\right)\right)}_{i=1}^{n}$-bounded pseudo-metrics into a ${f}_{T}\left(0\right)$-bounded pseudo-metric, where $H={f}_{T}\circ F\circ ({f}_{{T}_{1}}^{-1}\times \dots \times {f}_{{T}_{n}}^{-1})$.

**Corollary**

**1.**

- (1)
- F aggregates $\mathcal{T}$-equivalences into a T-equivalence.
- (2)
- The function $H:{[0,+\infty ]}^{n}\to [0,\infty ]$ aggregates every collection ${\left\{{d}_{i}\right\}}_{i=1}^{n}$ of extended pseudo-metrics into an extended pseudo-metric, where $H={f}_{T}\circ F\circ ({f}_{{T}_{1}}^{-1}\times \dots \times {f}_{{T}_{n}}^{-1})$.

**Theorem**

**2.**

- (1)
- F aggregates $\mathcal{T}$-equivalences into a T-equivalence.
- (2)
- F holds the following conditions:
- (2.1)
- $F\left({1}_{n}\right)=1$, where ${1}_{n}\in {[0,1]}^{n}$ with ${1}_{n}=(1,\dots ,1)$.
- (2.2)
- F transforms n-dimensional $\mathcal{T}$-triangular triplets into a one-dimensional T-triangular triplet.

**Theorem**

**3.**

- (1)
- F aggregates $\mathcal{T}$-equalities into a T-equality.
- (2)
- F holds the following conditions:
- (2.1)
- $F\left({1}_{n}\right)=1$, where ${1}_{n}\in {[0,1]}^{n}$ with ${1}_{n}=(1,\dots ,1)$.
- (2.2)
- Let $a\in {[0,1]}^{n}$. If $F\left(a\right)=1$, then there exists $i\in \{1,\dots ,n\}$ such that ${a}_{i}=1$.
- (2.3)
- If $a,b,c\in {[0,1[}^{n}$ such that $(a,b,c)$ is a n-dimensional $\mathcal{T}$-triangular triplet, then, $\left(F\right(a),F(b),F(c\left)\right)$ is a one-dimensional T-triangular triplet.

## 2. Aggregation of $\mathcal{T}$-Equivalences

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Theorem**

**4.**

- (1)
- F aggregates $\mathcal{T}$-equivalences into a T-equivalence.
- (2)
- The function G: ${[0,+\infty ]}^{n}\to [0,{f}_{T}\left(0\right)]$ transforms n-dimensional triangular triplets in ${[0,\infty ]}^{n}$ into a one-dimensional triangular triplet in $[0,{f}_{T}\left(0\right)]$ and $G(0,\dots ,0)=0$, where $G={f}_{T}\circ F\circ ({f}_{{T}_{1}}^{(-1)}\times \dots \times {f}_{{T}_{n}}^{(-1)})$.
- (3)
- The function G: ${[0,+\infty ]}^{n}\to [0,{f}_{T}\left(0\right)]$ aggregates every collection ${\left\{{d}_{i}\right\}}_{i=1}^{n}$ of extended pseudo-metrics into a ${f}_{T}\left(0\right)$-bounded pseudo-metric, where $G={f}_{T}\circ F\circ ({f}_{{T}_{1}}^{(-1)}\times \dots \times {f}_{{T}_{n}}^{(-1)})$.
- (4)
- The function H: ${\prod}_{i=1}^{n}[0,{f}_{{T}_{i}}\left(0\right)]\to [0,{f}_{T}\left(0\right)]$ transforms n-dimensional triangular triplets in ${\prod}_{i=1}^{n}[0,{f}_{{T}_{i}}\left(0\right)]$ into a one-dimensional triangular triplet in $[0,{f}_{T}\left(0\right)]$ and $H(0\dots ,0)=0$, where $H={f}_{T}\circ F\circ ({f}_{{T}_{1}}^{-1}\times \dots \times {f}_{{T}_{n}}^{-1})$.

**Proof.**

**Theorem**

**5.**

- (1)
- F aggregates $\mathcal{T}$-equivalences into a T-equivalence.
- (2)
- The function $G:{[0,+\infty ]}^{n}\to [0,+\infty ]$ transforms n-dimensional triangular triplets into a one-dimensional triangular triplets in $[0,+\infty ]$ and $G(0,\dots ,0)=0$, where $G=t\circ F\circ ({f}_{{T}_{1}}^{-1}\times \dots \times {f}_{{T}_{n}}^{-1})$.
- (3)
- The function $G:{[0,+\infty ]}^{n}\to [0,+\infty ]$ aggregates every collection ${\left({d}_{i}\right)}_{i=1}^{n}$ of extended pseudo-metrics into an extended pseudo-metric, where $G=t\circ F\circ ({f}_{{T}_{1}}^{-1}\times \dots \times {f}_{{T}_{n}}^{-1})$.

## 3. Aggregation of $\mathcal{T}$-Equalities

**Theorem**

**6.**

- (1)
- F aggregates $\mathcal{T}$-equalities into a T-equality.
- (2)
- The function $G:{[0,+\infty ]}^{n}\to [0,{f}_{T}\left(0\right)]$, where $G={f}_{T}\circ F\circ ({f}_{{T}_{1}}^{(-1)}\times \dots \times {f}_{{T}_{n}}^{(-1)})$, fulfills the following conditions:
- (2.1)
- $G(0,\dots ,0)=0$;
- (2.2)
- Let $a\in {[0,+\infty ]}^{n}$. If $G\left(a\right)=0$, then there exists $i\in \{1,\dots ,n\}$ such that ${a}_{i}=0$;
- (2.3)
- G transforms n-dimensional triangular triplets in ${[0,+\infty ]}^{n}$ into a one-dimensional triangular triplet in $[0,{f}_{T}\left(0\right)]$.

- (3)
- The function G: ${[0,+\infty ]}^{n}\to [0,{f}_{T}\left(0\right)]$ aggregates every collection ${\left\{{d}_{i}\right\}}_{i=1}^{n}$ of extended metrics into a ${f}_{T}\left(0\right)$-bounded metric, where $G={f}_{T}\circ F\circ ({f}_{{T}_{1}}^{(-1)}\times \dots \times {f}_{{T}_{n}}^{(-1)})$.
- (4)
- The function H: ${\prod}_{i=1}^{n}[0,{f}_{{T}_{i}}\left(0\right)]\to [0,{f}_{T}\left(0\right)]$ aggregates every collection ${\left\{{d}_{i}\right\}}_{i=1}^{n}$ of ${\left({f}_{{T}_{i}}\left(0\right)\right)}_{i=1}^{n}$-bounded metrics into a ${f}_{T}\left(0\right)$-metric, where $H={f}_{T}\circ F\circ ({f}_{{T}_{1}}^{-1}\times \dots \times {f}_{{T}_{n}}^{-1})$.
- (5)
- The function H: ${\prod}_{i=1}^{n}[0,{f}_{{T}_{i}}\left(0\right)]\to [0,{f}_{T}\left(0\right)]$, where $H={f}_{T}\circ F\circ ({f}_{{T}_{1}}^{-1}\times \dots \times {f}_{{T}_{n}}^{-1})$, fulfills the following conditions:
- (5.1)
- $H(0,\dots ,0)=0$;
- (5.2)
- Let $a\in {\prod}_{i=1}^{n}[0,{f}_{{T}_{i}}\left(0\right)]$. If $H\left(a\right)=0$, then there exists $i\in \{1,\dots ,n\}$ such that ${a}_{i}=0$;
- (5.3)
- H transforms n-dimensional triangular triplets in ${\prod}_{i=1}^{n}[0,{f}_{{T}_{i}}\left(0\right)]$ into a one-dimensional triangular triplet in $[0,{f}_{T}\left(0\right)]$;

**Proof.**

**Corollary**

**2.**

- (1)
- F aggregates $\mathcal{T}$-equalities into a T-equality.
- (2)
- The function G: ${[0,+\infty ]}^{n}\to [0,+\infty ]$, where $G={f}_{T}\circ F\circ ({f}_{{T}_{1}}^{-1}\times \dots \times {f}_{{T}_{n}}^{-1})$, fulfills the following conditions:
- (2.1)
- $G(0,\dots ,0)=0$;
- (2.2)
- Let $a\in {[0,+\infty ]}^{n}$. If $G\left(a\right)=0$, then there exists $i\in \{1,\dots ,n\}$ such that ${a}_{i}=0$;
- (2.3)
- G transforms n-dimensional positive triangular triplets in ${[0,+\infty ]}^{n}$ into a one-dimensional positive triangular triplet in $[0,+\infty ]$.

- (3)
- The function $G:{[0,+\infty ]}^{n}\to [0,+\infty ]$ aggregates every collection ${\left\{{d}_{i}\right\}}_{i=1}^{n}$ of extended metrics into an extended metric, where $G={f}_{T}\circ F\circ ({f}_{{T}_{1}}^{-1}\times \dots \times {f}_{{T}_{n}}^{-1})$.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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González-Hedström, J.-D.-D.; Miñana, J.-J.; Valero, O.
Aggregation of Indistinguishability Fuzzy Relations Revisited. *Mathematics* **2021**, *9*, 1441.
https://doi.org/10.3390/math9121441

**AMA Style**

González-Hedström J-D-D, Miñana J-J, Valero O.
Aggregation of Indistinguishability Fuzzy Relations Revisited. *Mathematics*. 2021; 9(12):1441.
https://doi.org/10.3390/math9121441

**Chicago/Turabian Style**

González-Hedström, Juan-De-Dios, Juan-José Miñana, and Oscar Valero.
2021. "Aggregation of Indistinguishability Fuzzy Relations Revisited" *Mathematics* 9, no. 12: 1441.
https://doi.org/10.3390/math9121441