International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(1): 48-58
Published online March 25, 2022
https://doi.org/10.5391/IJFIS.2022.22.1.48
© The Korean Institute of Intelligent Systems
Emel A?ıcı
Department of Software Engineering, Faculty of Technology, Karadeniz Technical University, Trabzon, Turkey
Correspondence to :
Emel A?ıcı (emelkalin@hotmail.com)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
In recent years, construction methods for triangular norms (t-norms) and triangular conorms (t-conorms) on bounded lattices have been studied extensively. This paper presents the continued study of the construction of t-norms and t-conorms on bounded lattices. We introduce new methods for constructing t-norms and t-conorms on an arbitrary bounded lattice. Furthermore, we provide illustrative examples for clarity. Subsequently, we demonstrate how our new construction methods differ from certain existing methods for the construction of t-norms and t-conorms on bounded lattices. Finally, we reveal that new construction methods can be generalized by induction to a modified ordinal sum for t-norms and t-conorms on an arbitrary bounded lattice.
Keywords: t-norm, t-conorm, Bounded lattice
In 1960, Schweizer and Sklar [1] presented
The above authors presented a new method for constructing the
Moreover, when considering the existence of
In this paper, we introduce a new ordinal sum construction of
A lattice is a partially ordered set (
Let (
Given a bounded lattice (
Similarly, [
Let (
Let (
In this section, we present a new means of constructing
Consider (
is an ordinal sum (<
Consider (
is an ordinal sum (<
In Formula (
Consider the bounded lattice (
We can observe that the function
Let (
It can easily be observed that
We prove that if
1.1
1.1.1.
1.1.2.
1.1.3.
1.2.
1.2.1.
1.2.2.
1.2.3.
1.3.
Because
1.4.
1.4.1.
1.4.2.
1.4.3.
Thus, it must be the case that
2.1
2.1.1.
2.1.2.
2.1.3.
2.1
2.1.1.
2.1.2.
2.1.3.
Therefore, it must be the case that
3.1
3.1.1.
3.1.2.
3.2.
As
3.3
3.3.1.
3.3.2.
If at least one of
1.1
1.1.1.
1.1.2.
1.1.3.
1.2.
1.2.1.
1.2.2.
1.2.3.
1.3.
1.3.1.
1.3.2.
It holds that
2.1
2.1.1.
2.1.2.
2.1.3.
2.2.
2.2.1.
2.2.2.
2.2.3.
2.3.
2.3.1.
2.3.2.
Then,
3.1
3.1.1.
3.1.2.
3.1.3.
3.2.
3.2.1.
3.2.2.
3.2.3.
3.3.
3.3.1.
Thus, by assumption,
3.3.2.
If we use
Consider the bounded lattice (
Thus, the function
Let (
Consider the bounded lattice (
Thus, the function
Let (
Consider the bounded lattice (
Therefore, the function
We may question whether the
Consider the bounded lattice (
Consider the bounded lattice (
Moreover, consider any
In Example 6, if we take the
Consider the bounded lattice (
Furthermore, consider any
We present the same arguments for
Let (
In Theorem 3, if we use
In this section, based on the proposed approaches for constructing
Let (
Let (
In this study, we have investigated and introduced new construction methods for building
Table 1. The function
0 | 1 | ||||||
---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | ||||
0 | |||||||
0 | |||||||
1 | 0 | 1 |
Table 2. The
0 | 1 | |||||||
---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
1 | 0 | 1 |
Table 3. The function
0 | 1 | |||||||
---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
1 | 0 | 1 |
Table 4. The function
0 | 1 | |||||||
---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
1 | 0 | 1 |
Table 5. The
0 | 1 | |||||||
---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
1 | 0 | 1 |
Table 6. The
0 | 1 | |||||||
---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
1 | 0 | 1 |
Table 7. The
0 | 1 | ||||||
---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | |||
0 | 0 | 0 | 0 | 0 | |||
0 | 0 | 0 | 0 | 0 | |||
0 | 0 | 0 | 0 | 0 | |||
0 | |||||||
1 | 0 | 1 |
Table 8. The
0 | 1 | ||||||
---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | |||||
0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | ||||
0 | |||||||
1 | 0 | 1 |
Table 9. The
0 | 1 | |||||
---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | |||
0 | 0 | 0 | 0 | |||
0 | 0 | 0 | 0 | |||
0 | 0 | 0 | ||||
1 | 0 | 1 |
Table 10. The
0 | 1 | |||||
---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | ||||
1 | 0 | 1 |
E-mail: emelkalin@hotmail.com
International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(1): 48-58
Published online March 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.1.48
Copyright © The Korean Institute of Intelligent Systems.
Emel A?ıcı
Department of Software Engineering, Faculty of Technology, Karadeniz Technical University, Trabzon, Turkey
Correspondence to:Emel A?ıcı (emelkalin@hotmail.com)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
In recent years, construction methods for triangular norms (t-norms) and triangular conorms (t-conorms) on bounded lattices have been studied extensively. This paper presents the continued study of the construction of t-norms and t-conorms on bounded lattices. We introduce new methods for constructing t-norms and t-conorms on an arbitrary bounded lattice. Furthermore, we provide illustrative examples for clarity. Subsequently, we demonstrate how our new construction methods differ from certain existing methods for the construction of t-norms and t-conorms on bounded lattices. Finally, we reveal that new construction methods can be generalized by induction to a modified ordinal sum for t-norms and t-conorms on an arbitrary bounded lattice.
Keywords: t-norm, t-conorm, Bounded lattice
In 1960, Schweizer and Sklar [1] presented
The above authors presented a new method for constructing the
Moreover, when considering the existence of
In this paper, we introduce a new ordinal sum construction of
A lattice is a partially ordered set (
Let (
Given a bounded lattice (
Similarly, [
Let (
Let (
In this section, we present a new means of constructing
Consider (
is an ordinal sum (<
Consider (
is an ordinal sum (<
In Formula (
Consider the bounded lattice (
We can observe that the function
Let (
It can easily be observed that
We prove that if
1.1
1.1.1.
1.1.2.
1.1.3.
1.2.
1.2.1.
1.2.2.
1.2.3.
1.3.
Because
1.4.
1.4.1.
1.4.2.
1.4.3.
Thus, it must be the case that
2.1
2.1.1.
2.1.2.
2.1.3.
2.1
2.1.1.
2.1.2.
2.1.3.
Therefore, it must be the case that
3.1
3.1.1.
3.1.2.
3.2.
As
3.3
3.3.1.
3.3.2.
If at least one of
1.1
1.1.1.
1.1.2.
1.1.3.
1.2.
1.2.1.
1.2.2.
1.2.3.
1.3.
1.3.1.
1.3.2.
It holds that
2.1
2.1.1.
2.1.2.
2.1.3.
2.2.
2.2.1.
2.2.2.
2.2.3.
2.3.
2.3.1.
2.3.2.
Then,
3.1
3.1.1.
3.1.2.
3.1.3.
3.2.
3.2.1.
3.2.2.
3.2.3.
3.3.
3.3.1.
Thus, by assumption,
3.3.2.
If we use
Consider the bounded lattice (
Thus, the function
Let (
Consider the bounded lattice (
Thus, the function
Let (
Consider the bounded lattice (
Therefore, the function
We may question whether the
Consider the bounded lattice (
Consider the bounded lattice (
Moreover, consider any
In Example 6, if we take the
Consider the bounded lattice (
Furthermore, consider any
We present the same arguments for
Let (
In Theorem 3, if we use
In this section, based on the proposed approaches for constructing
Let (
Let (
In this study, we have investigated and introduced new construction methods for building
The lattice
The lattice
The lattice
The lattice
The lattice
The lattice
Table 1 . The function
0 | 1 | ||||||
---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | ||||
0 | |||||||
0 | |||||||
1 | 0 | 1 |
Table 2 . The
0 | 1 | |||||||
---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
1 | 0 | 1 |
Table 3 . The function
0 | 1 | |||||||
---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
1 | 0 | 1 |
Table 4 . The function
0 | 1 | |||||||
---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
1 | 0 | 1 |
Table 5 . The
0 | 1 | |||||||
---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
1 | 0 | 1 |
Table 6 . The
0 | 1 | |||||||
---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | |||||
1 | 0 | 1 |
Table 7 . The
0 | 1 | ||||||
---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | |||
0 | 0 | 0 | 0 | 0 | |||
0 | 0 | 0 | 0 | 0 | |||
0 | 0 | 0 | 0 | 0 | |||
0 | |||||||
1 | 0 | 1 |
Table 8 . The
0 | 1 | ||||||
---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | |||||
0 | 0 | 0 | |||||
0 | 0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | ||||
0 | |||||||
1 | 0 | 1 |
Table 9 . The
0 | 1 | |||||
---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | |||
0 | 0 | 0 | 0 | |||
0 | 0 | 0 | 0 | |||
0 | 0 | 0 | ||||
1 | 0 | 1 |
Table 10 . The
0 | 1 | |||||
---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | ||||
0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | ||||
1 | 0 | 1 |
The lattice
The lattice
The lattice
The lattice
The lattice
The lattice