# Andness Directedness for t-Norms and t-Conorms

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Fuzzy Operators: t-Norms and t-Conorms

- $\top (a,b)=\top (b,a)$ (commutativity)
- $\top (a,b)\le \top (c,d)$ when $a\le c$ and $b\le d$ (monotonicity)
- $\top (a,\top (b,c\left)\right)=\top (\top (a,b),c)$ (associativity)
- $\top (a,1)=a$ (identity)

- $\perp (a,b)=\perp (b,a)$ (commutativity);
- $\perp (a,b)\le \perp (c,d)$ when $a\le c$ and $b\le d$ (monotonicity);
- $\perp (a,\perp (b,c\left)\right)=\perp (\perp (a,b),c)$ (associativity);
- $\perp (a,0)=a$ (identity).

#### 2.2. Andness and Orness

## 3. Andness Directedness for t-Norms

**Establish the relationship between the parameter and andness.**Compute the andness of ${\top}_{p}$ for p in $range\left(p\right)$. As we cannot consider the full range of p, we will select a finite subset of this range large enough to have a good approximation. We denote this finite subset of the range by ${R}_{p}$. This process results in pairs ${\left\{(p,andness\left({\top}_{p}\right))\right\}}_{p\in {R}_{p}}$. As andness is monotonic with respect to p, we can interpolate (e.g., linearly) the relationship between p and the andness easily.Figure 1 displays the andness for the t-norms (left) and t-conorms (right) discussed in Section 2. Therefore, we include ${T}_{D}$, ${T}_{SS1}$, ${T}_{SS2}$, ${T}_{SS3}$, and ${T}_{Y}$ as t-norms, and ${S}_{D}$, ${S}_{SS1}$, ${S}_{SS2}$, ${S}_{SS3}$, ${S}_{Y}$ as t-conorms. It is worth noting that in the figure only four lines are visible because the lines for ${T}_{D}$ (Dombi’s t-norm) and ${T}_{Y}$ are indistinguishable. In Figure 1, we consider positive parameters $\lambda $, p, and w for all t-norms and t-conorms. We have discussed before that the parameter for both t-norm and t-conorm ${T}_{SS1}$ and ${S}_{SS1}$ can take any value except zero. For this reason, we represent the andness of these function independently in Figure 2.**Establish the parameter given the andness.**Given the andness level $\alpha $ and the pairs ${\left\{({p}_{i},{\alpha}_{i})\right\}}_{i\in \{1,\cdots ,|{R}_{p}\left|\right\}}$ check if there is a value ${\alpha}_{i}=\alpha $. If this is the case, select ${p}_{i}$. Otherwise, find a pair of values ${\alpha}_{i}$ and ${\alpha}_{i+1}$ that satisfy ${\alpha}_{i}\le \alpha \le {\alpha}_{i+1}$. Then, first, compute $r=(\alpha -{\alpha}_{i})/({\alpha}_{i+1}-{\alpha}_{i})$ and, second, compute $p={p}_{i}\ast (1-r)+{p}_{i+1}\ast r$. For example, given ${\alpha}_{p}=\alpha \left(p\right)=\mathit{andness}\left({\top}_{p}\right)$ for a given t-norm ⊤, we compute ${\alpha}_{p}^{-1}\left(\alpha \right)$, or, in other words, we approximate the parameter p as the inverse of the andness for the given andness $\alpha $.We will use $\lambda \left(\alpha \right)$, $p\left(\alpha \right)$, and $w\left(\alpha \right)$ to denote this process. For example, for $\alpha =1.5$, we obtain the following 5 parameters. For ${T}_{D}$, $\lambda =0.4143849$; for ${T}_{SS1}$, $p=0.9999819$; ${T}_{SS2}$, $p=0.4080617$; ${T}_{SS3}$$p=0.6143033$ and for ${T}_{Y}$, $w=1.0000$.

#### 3.1. Differences between t-Norms for a Given Andness

#### 3.2. Differences between t-Norms and High Hyperconjunction for a Given Andness

#### 3.3. Analysis

## 4. Conclusions and Future Directions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Andness of t-norms ${T}_{D}$, ${T}_{SS1}$, ${T}_{SS2}$, ${T}_{SS3}$, ${T}_{Y}$ (

**left**) and t-conorms ${S}_{D}$, ${S}_{SS1}$, ${S}_{SS2}$, ${S}_{SS3}$, ${S}_{Y}$ (

**right**) in terms of their parameters ($\lambda $, p, or w).

**Figure 2.**Andness of t-norm ${T}_{SS1}$ (

**left**) and t-conorm ${S}_{SS1}$ (

**right**) given its parameter p.

**Figure 3.**Given andness $\alpha $, maximum difference between pairs of t-norms ${T}_{D}$, ${T}_{SS1}$, ${T}_{SS2}$, ${T}_{SS3}$, ${T}_{Y}$ (

**left**), and difference between high hyperconjunction $hC$ and the same t-norms (

**right**). The parameters of the t-norms ($\lambda $, p, or w) have been selected so that the t-norm has precisely andness $\alpha $ (i.e., ($\lambda =\lambda \left(\alpha \right)$, $p=p\left(\alpha \right)$, and $w=w\left(\alpha \right)$).

**Figure 4.**Given andness $\alpha $, difference between pairs of t-norms ${T}_{D}$, ${T}_{SS1}$, ${T}_{SS2}$, ${T}_{SS3}$, ${T}_{Y}$. Their parameters ($\lambda $, p, or w) have been selected so that the t-norm has precisely andness $\alpha $ (i.e., ($\lambda =\lambda \left(\alpha \right)$, $p=p\left(\alpha \right)$, and $w=w\left(\alpha \right)$).

t-norm | ${\mathit{T}}_{\mathit{m}\mathit{n}}$ | ${\mathit{T}}_{\mathit{a}\mathit{p}}$ | ${\mathit{T}}_{\mathit{b}\mathit{d}}$ | ${\mathit{T}}_{\mathit{d}\mathit{r}}$ |

andness | 1.000 | 1.250 | 1.500 | 2.000 |

t-conorm | ${S}_{mx}$ | ${S}_{as}$ | ${S}_{bs}$ | ${S}_{dr}$ |

andness | 0.000 | −0.250 | −0.500 | −1.000 |

Case | ${\mathit{T}}_{\mathit{D}}$ | ${\mathit{T}}_{\mathbf{SS}1}$ | ${\mathit{T}}_{\mathbf{SS}2}$ | ${\mathit{T}}_{\mathbf{SS}3}$ | ${\mathit{T}}_{\mathit{Y}}$ |
---|---|---|---|---|---|

(0.8, 0.8) | 0.3566074 | 0.5738743 | 0.4089883 | 0.4146452 | 0.5387788 |

(0.5, 0.5) | 0.1217016 | 0 | 0.1120718 | 0.06492248 | 0 |

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Torra, V.
Andness Directedness for t-Norms and t-Conorms. *Mathematics* **2022**, *10*, 1598.
https://doi.org/10.3390/math10091598

**AMA Style**

Torra V.
Andness Directedness for t-Norms and t-Conorms. *Mathematics*. 2022; 10(9):1598.
https://doi.org/10.3390/math10091598

**Chicago/Turabian Style**

Torra, Vicenç.
2022. "Andness Directedness for t-Norms and t-Conorms" *Mathematics* 10, no. 9: 1598.
https://doi.org/10.3390/math10091598