# On Triangular Multisets and Triangular Fuzzy Multisets

## Abstract

**:**

## 1. Introduction

#### Structure of the Paper

## 2. Discrete Triangular Norms and Conorms

**Definition**

**1.**

- 1.
**Boundary condition**$a\ast 1=a$and$a\ast 0=0$.- 2.
**Monotonicity**$b\le c$implies$a\ast b\le a\ast c$.- 3.
**Commutativity**$a\ast b=b\ast a$.- 4
**Associativity**$a\ast (b\ast c)=(a\ast b)\ast c$.

**Definition**

**2.**

- 1.
**Boundary condition**$a\u26050=a$and$a\u26051=1$.- 2.
**Monotonicity**$b\le c$implies$a\u2605b\le a\u2605c$.- 3.
**Commutativity**$a\u2605b=b\u2605a$.- 4.
**Associativity**$a\u2605(b\u2605c)=(a\u2605b)\u2605c$.

**Proposition**

**1.**

**Example**

**1.**

- 1.
- $gcd(m,n)=gcd(n,m)$;
- 2.
- $gcd\left(gcd(m,n),k\right)=gcd\left(m,gcd(n,k)\right)$;
- 3.
- If$m|{m}^{\prime}$and$n|{n}^{\prime}$, then$gcd(m,n)|gcd({m}^{\prime},{n}^{\prime})$; and
- 4.
- $gcd(m,+\infty )=m$.

**Example**

**2.**

## 3. Triangular Multisets and Triangular Fuzzy Multisets

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Proposition**

**2.**

- 1.
- Commutative: $A\uplus B=B\uplus B$;
- 2.
- Associative: $(A\uplus B)\uplus C=A\uplus (B\uplus C)$;
- 3.
- There exists a multiset, the null multiset ∅, such that $A\uplus \varnothing =A$.

**Definition**

**6.**

**Definition**

**7.**

## 4. Using Triangular Multi-Fuzzy Sets in MCDM

**Definition**

**8.**

## 5. Conclusions

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Syropoulos, A.
On Triangular Multisets and Triangular Fuzzy Multisets. *Mathematics* **2022**, *10*, 726.
https://doi.org/10.3390/math10050726

**AMA Style**

Syropoulos A.
On Triangular Multisets and Triangular Fuzzy Multisets. *Mathematics*. 2022; 10(5):726.
https://doi.org/10.3390/math10050726

**Chicago/Turabian Style**

Syropoulos, Apostolos.
2022. "On Triangular Multisets and Triangular Fuzzy Multisets" *Mathematics* 10, no. 5: 726.
https://doi.org/10.3390/math10050726