# Remarks on Sugeno Integrals on Bounded Lattices

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

- (i)
- ${x}_{i}\in \mathrm{Term}\left(n\right)$ for $i=1,\cdots ,n$,
- (ii)
- if $\mathsf{p},\mathsf{q}\in \mathrm{Term}\left(n\right)$ then $(\mathsf{p}\vee \mathsf{q}),(\mathsf{p}\wedge \mathsf{q})\in \mathrm{Term}\left(n\right)$.

- (i)
- If $\mathsf{p}={x}_{i}$, then $p({a}_{1},\cdots ,{a}_{n})={a}_{i}$ for any $i=1,\cdots ,n$.
- (ii)
- If $p({a}_{1},\cdots ,{a}_{n})=a$, $q({a}_{1},\cdots ,{a}_{n})=b$ and $\mathsf{r}=\mathsf{p}\vee \mathsf{q}$, $\mathsf{t}=\mathsf{p}\wedge \mathsf{q}$, then $r({a}_{1},\cdots ,{a}_{n})=a\vee b$ and $t({a}_{1},\cdots ,{a}_{n})=a\wedge b$.

**Proposition**

**1.**

## 3. Results

- (${\mathsf{D}}_{M}^{\mathcal{K}}$)
- For any two n-ary functions $f,g\in \mathcal{K}\left(L\right)$, the property $f\left(\mathbf{y}\right)=g\left(\mathbf{y}\right)$ for all $\mathbf{y}\in {M}^{n}$ implies $f\left(\mathbf{x}\right)=g\left(\mathbf{x}\right)$ for all $\mathbf{x}\in {L}^{n}$.

**Lemma**

**1.**

**Proof.**

**Corollary**

**1.**

**Theorem**

**1.**

**Proof.**

- (i)
- If $\mathbf{b}\nleqq \mathbf{a}$, then there is $j\in \{1,\cdots ,n\}$ with ${b}_{j}=1$ and ${a}_{j}=0$. In this case $j\in {\mathbf{b}}^{-1}\left(1\right)$, which yields $\bigwedge \{{a}_{i}\mid i\in {\mathbf{b}}^{-1}\left(1\right)\}=0$. Consequently, from (3), we obtain ${G}_{\mathbf{b}}\left(\mathbf{a}\right)=f\left(\mathbf{b}\right)\wedge 0=0$.
- (ii)
- Let $\mathbf{b}=\mathbf{a}$. Then evidently $\bigwedge \{{a}_{i}\mid i\in {\mathbf{a}}^{-1}\left(1\right)\}=1$, and we obtain ${G}_{\mathbf{a}}\left(\mathbf{a}\right)=f\left(\mathbf{a}\right)\wedge 1=f\left(\mathbf{a}\right)$.
- (iii)
- Assume $\mathbf{b}<\mathbf{a}$. Since the function f is monotone and $\mathbf{b}<\mathbf{a}$, from (3), it follows that ${G}_{\mathbf{b}}\left(\mathbf{a}\right)\le f\left(\mathbf{b}\right)\le f\left(\mathbf{a}\right)$.

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

**Theorem**

**3.**

- (i)
- L is distributive.
- (ii)
- Any two distinct elements of L can be separated by a prime ideal.
- (iii)
- L fulfills the condition $\left({\mathsf{D}}_{\{0,1\}}^{\mathcal{P}}\right)$.
- (iv)
- L fulfills the condition $\left({\mathsf{D}}_{\{0,1\}}^{w\mathcal{P}}\right)$.
- (v)
- L fulfills the condition $\left({\mathsf{D}}_{\{0,1\}}^{m\mathcal{C}}\right)$.
- (vi)
- L fulfills the condition $\left({\mathsf{D}}_{\{0,1\}}^{\mathcal{C}}\right)$.

**Remark**

**2.**

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Halaš, R.; Pócs, J.; Pócsová, J.
Remarks on Sugeno Integrals on Bounded Lattices. *Mathematics* **2022**, *10*, 3078.
https://doi.org/10.3390/math10173078

**AMA Style**

Halaš R, Pócs J, Pócsová J.
Remarks on Sugeno Integrals on Bounded Lattices. *Mathematics*. 2022; 10(17):3078.
https://doi.org/10.3390/math10173078

**Chicago/Turabian Style**

Halaš, Radomír, Jozef Pócs, and Jana Pócsová.
2022. "Remarks on Sugeno Integrals on Bounded Lattices" *Mathematics* 10, no. 17: 3078.
https://doi.org/10.3390/math10173078