# 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

## References

- Trillas, E. Assaig sobre les relacions d’indistingibilitat. In Proceedings Primer Congrés Català de Lògica Matemàtica; Barcelona Institut d’Estudis Catalans: Barcelona, Spain, 1982; pp. 51–59. [Google Scholar]
- Recasens, J. Indistinguishability Operators: Modelling Fuzzy Equalities and Fuzzy Equivalence Relations; Springer: Berlin, Germany, 2010. [Google Scholar]
- Mayor, M.; Recasens, J. Preserving T-transitivity. In Artificial Intelligence Research and Development; Nebot, À., Binefa, X., López de Mántaras, R., Eds.; IOS Press: Amsterdam, The Netherlands, 2016; pp. 79–87. [Google Scholar]
- Calvo, T.; Fuster-Parra, P. Aggregation of partial T-indistinguishability operators and partial pseudo-metrics. Fuzzy Sets Syst.
**2019**, 403, 119–138. [Google Scholar] [CrossRef] - Miñana, J.-J.; Valero, O. On indistinguishability operators, fuzzy metrics and modular metrics. Axioms
**2017**, 6, 34. [Google Scholar] [CrossRef][Green Version] - Höhle, U. Fuzzy equalities and indistinguishability. In Proceedings of EUFIT’93; Eurogress Aachen: Aachen, Germany, 1993; Volume 1, pp. 358–363. [Google Scholar]
- Höhle, U. Many valued equalities and their representation. In Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms; Klement, E.P., Mesiar, R., Eds.; Elsevier: Amsterdam, The Netherlands, 2005; pp. 301–320. [Google Scholar]
- Petry, F.E.; Bosc, P. Fuzzy Databases: Principles and Applications; Kluwer Academic Publishers: Boston, MA, USA, 1996. [Google Scholar]
- Ovchinnikov, S. Representation of transitive fuzzy relations. In Aspects of Vagueness; Skala, H.J., Termini, S., Trillas, E., Eds.; Springer: Berlin/Heidelberg, Germany, 1984; pp. 105–118. [Google Scholar]
- Boixader, D.; Recasens, J. On the relationship between fuzzy subgroups and indistinguishability operators. Fuzzy Sets Syst.
**2019**, 373, 149–163. [Google Scholar] [CrossRef] - Calvo, T.; Recasens, J. On the representation of local indistinguishability operators. Fuzzy Sets Syst.
**2021**, 410, 90–108. [Google Scholar] [CrossRef] - Recasens, J. On the Relationship between Positive Definite Matrices and t-norms. Fuzzy Sets Syst.
**2021**, in press. [Google Scholar] [CrossRef] - Bejines, C.; Chasco, M.J.; Elorza, J.; Recasens, J. Preserving fuzzy subgroups and indistinguishability operators. Fuzzy Sets Syst.
**2019**, 373, 164–179. [Google Scholar] [CrossRef] - Bejines, C.; Ardanza, S.; Chasco, M.J.; Elorza, J. Aggregation of indistinguishability operators. Fuzzy Sets Syst.
**2021**, in press. [Google Scholar] [CrossRef] - Pedraza, T.; Rodríguez-López, J.; Valero, O. Aggregation of fuzzy quasi-metrics. Inform. Sci.
**2021**, in press. [Google Scholar] - Saminger, S.; Mesiar, R.; Bodenhofer, U. Domination of aggregation operators and preservation of transitivity. Int. J. Uncertain. Fuzziness-Knowl.-Based Syst.
**2002**, 10, 11–35. [Google Scholar] [CrossRef] - Klement, E.P.; Mesiar, R.; Pap, E. Triangular Norms; Kluwer: Dordrecht, The Netherlands, 2000. [Google Scholar]
- Deza, M.M.; Deza, E. Encyclopedia of Distances; Springer: Heidelberg, Germany, 2009. [Google Scholar]
- Copson, E.T. Metric Spaces; Cambridge University Press: Cambridge, UK, 1968. [Google Scholar]
- Valverde, L. On the structure of F-indistinguishability operators. Fuzzy Set. Syst.
**1985**, 17, 313–328. [Google Scholar] [CrossRef][Green Version] - De Baets, B.; Mesiar, R. Pseudo-metrics and T-equivalences. Fuzzy. Math.
**1997**, 5, 471–481. [Google Scholar] - De Baets, B.; Mesiar, R. Metrics and T-equalities. J. Math. Anal. Appl.
**1997**, 267, 531–547. [Google Scholar] [CrossRef][Green Version] - Beliakov, G.; Bustince, H.; Calvo, T. A Practical Guide to Averaging Functions, Studies in Fuzziness and Soft Computing; Springer: Berlin/Heidelberg, Germany, 2016. [Google Scholar]
- Grabisch, M.; Marichal, J.-L.; Mesiar, R.; Pap, E. Aggregation functions, Encyclopedia of Mathematics and its Applications; Cambridge University Press: Cambridge, UK, 2009; Volume 127. [Google Scholar]
- Drewniak, J.; Dudziak, U. Aggregation in classes of fuzzy relations. Stud. Math.
**2006**, 5, 33–43. [Google Scholar] - Drewniak, J.; Dudziak, U. Preservation of properties of fuzzy relations during aggregation processes. Kybernetik
**2007**, 43, 115–132. [Google Scholar] - Pradera, P.; Trillas, E.; Castiñeira, E. On the aggregation of some classes of fuzzy relations. In Technologies for Constructing Intelligent Systems 2. Tools; Bouchon-Meunier, B., Gutiérrez-Ríos, J., Magdalena, L., Yager, R.R., Eds.; Springer: Heidelberg, Germany, 2002; pp. 125–147. [Google Scholar]
- Calvo Sánchez, T.; Fuster-Parra, P.; Valero, O. The aggregation of transitive fuzzy relations revisited. Fuzzy Sets Syst.
**2021**, in press. [Google Scholar] - Pradera, P.; Trillas, E. A note on pseudo-metrics aggregation, Int. J. Gen. Syst.
**2002**, 31, 41–51. [Google Scholar]

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**MDPI and ACS Style**

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