![MathJax Logo](/templates/jsp/_style2/_tandf/pb2/images/math-jax.gif)
ABSTRACT
In this paper, we investigated some related properties and applied the concept of fuzzy congruence relation to a pseudo-UP algebra X. Additionally, we used their level set to define the fuzzy congruence relation on X. The Cartesian product of two fuzzy congruences and the intersection of two fuzzy congruences are discussed. We also provided the connection between the collection of fuzzy pseudo-UP ideals and the collection of fuzzy congruence relations in pseudo-UP algebras. We then demonstrated that there exists an associated algebra X that is a pseudo-UP algebra for each fuzzy ideal ζ and fuzzy congruence ϕζ generated by ζ. Additionally, we created a congruence relation on a pseudo-UP algebra using the fuzzy pseudo-UP ideal and fuzzy congruence relation ϕζ produced by ζ.
1. Introduction
Georgescu and Iorgulescu (Citation2001) introduced the concept of pseudo-BCK algebra as an extension of BCK algebra. Y. B. Jun (Citation2003) introduced the pseudo-ideal of pseudo-BCK algebra. Dudek and Jun (Citation2008) introduced and analyzed the concept of pseudo-BCI algebra as a generalization of BCI-algebra. Y. Jun et al. (Citation2006) introduced the concept of pseudo-BCI ideal in pseudo-BCI algebra. Kim and Kim (Citation2007) introduced the notion of BE-algebra which is another generalization of BCK-algebra. Borzooei et al. (Citation2013) generalized the notion of BE-algebra and introduced the notion of pseudo BE algebra. Rezaei et al. (Citation2014) introduced the notion of congruence relation on pseudo BE-algebra and constructed the quotient pseudo BE algebra via this congruence relation. Prabpayak and Leerawat (Citation2009) introduced a new algebraic structure which is called KU-algebra. As a generalization of that Iampan (Citation2017) introduced a new algebraic structure which is called UP-algebra, and he studied ideal, congruence, and homomorphism of UP-algebra and investigated some related properties. Moreover Iampan derived some properties of the relation between quotient UP-algebra and isomorphism. Romano (Citation2020a) introduced the concept of pseudo-UP algebra and derived basic properties. Romano (Citation2020b) in a forthcoming article introduced the concepts of pseudo-UP ideal and pseudo-UP filter in pseudo-UP algebra. As a continuation of these papers, Romano (Citation2021) introduced the concept of homomorphism between pseudo-UP algebra. The notion of homomorphism between pseudo-UP algebra is designed in the same way as it was done in Y. B. Jun (Citation2003), Y. Jun et al. (Citation2006), and Lee and Park (Citation2009) when analyzing pseudo-BCK and pseudo-BCI algebras. Furthermore, Iampan and Romano (Citation2022) introduced the concept of congruence relation on a pseudo-UP algebra.
Zadeh (Citation1965) introduced the notion of fuzzy sets and fuzzy relation. Then many authors have studied about it. Lee (Citation2009) defined the notion of ideal in pseudo BCI-algebra. Dymek and Walendziak (Citation2012) investigated fuzzy ideals of pseudo BCK-algebra. Recently, Rezaei et al. (Citation2015) discussed on (fuzzy) congruence relation in (pseudo) CI/BE-algebras and studied some of their properties.
The notion of congruence relations are one of the important concept in algebraic structure. Motivated by all the above results, we applied, in this paper, the notion of fuzzy congruence relation on pseudo-UP algebra. Additionally, we used their level set to define the fuzzy congruence relation on X. The Cartesian product of two fuzzy congruences and the intersection of two fuzzy congruences are discussed. We also provided the connection between the collection of fuzzy pseudo-UP ideal and the collection of fuzzy congruence relation in pseudo-UP algebra. We then demonstrated that there exists an associated algebra X that is a pseudo-UP algebra for each fuzzy ideal ζ and fuzzy congruence ϕζ generated by ζ. Additionally, we created a congruence relation on a pseudo-UP algebra using the fuzzy pseudo-UP ideal and fuzzy congruence relation ϕζ produced by ζ.
2. Preliminaries
In this section, we discussed some fundamental concepts and important basic results associated with our study, like pseudo-UP algebras, pseudo-UP ideal and congruence relation of pseudo-UP algebra. Furthermore, fuzzy set, fuzzy pseudo-UP ideals are discussed. These concepts are taken from Iampan and Romano (Citation2022), Romano (Citation2020a, Citation2020b), and Zadeh (Citation1965).
Definition 2.1.
A pseudo-UP algebra is an algebra of type
which satisfies the following axioms: for any
1) | |||||
2) | |||||
3) | |||||
4) |
Lemma 2.2.
In a pseudo-UP algebra X the following holds, for each ,
1) | |||||
2) |
| ||||
3) |
Proposition 2.3.
In a pseudo-UP algebra X the following holds: for any
1) |
| ||||
2) |
|
Theorem 2.4.
In a pseudo-UP algebra X the following holds: for any
1) | |||||
2) | |||||
3) | |||||
4) |
| ||||
5) |
Definition 2.5.
A pseudo-UP ideal of X is a nonempty subset J of a pseudo-UP algebra X that has the following: for each .
1) | |||||
2) |
| ||||
3) |
The following theorem describes the distinguishing characteristics of these substructures.
Theorem 2.6.
Let J be a pseudo-UP ideal in a pseudo-UP algebra, . Then:
1) | |||||
2) | |||||
3) |
| ||||
4) |
Definition 2.7.
Let be a pseudo-UP algebra and ϕ be an equivalence relation on the set X.
i) | ϕ is a congruence of type 1 on X if the following holds: for all | ||||
ii) | ϕ is a congruence of type 2 on X if the following holds |
Definition 2.8.
A fuzzy subset ζ of a pseudo-UP algebra of X is called fuzzy pseudo-UP ideal of X if and only if it fulfill the following axioms: for any
1) | |||||
2) |
| ||||
3) |
Theorem 2.9.
Let ζ be a fuzzy pseudo-UP ideal of a pseudo-UP algebra of X, then it satisfies the following assertions: for each
1) | |||||
2) | |||||
3) | |||||
4) |
Definition 2.10.
Let ζ and µ be fuzzy subsets of X. Then
1) |
More generally, if | ||||
2) |
More generally, if |
3. Fuzzy congruence relations on pseudo-up algebra
In this section, we study the basic properties of a fuzzy congruence relation on a pseudo-UP algebra. Let X be a pseudo-UP algebra. A fuzzy relation ϕ on X is a mapping .
Definition 3.1.
A fuzzy relation ϕ on X is called a fuzzy congruence relation on a pseudo-UP algebra of X, if it satisfies the following axioms: for any
1) | |||||
2) | |||||
3) | |||||
4) |
| ||||
5) |
|
Example 3.2.
Let with two binary operations
and
defined by the following Cayley table.
Then is a pseudo-UP algebra. Consider a fuzzy relation ϕ from X × X to
with
,
and
. It is easily checked that ϕ a fuzzy congruence relation on X.
Proposition 3.3.
If ϕ is a fuzzy congruence relation on a pseudo-UP algebra of X, then, for all
i) | |||||
ii) |
| ||||
iii) | if ϕ satisfies Definition 2.1 |
Proof.
i) | We have | ||||
ii) | By Definition 2.1 (5), we have | ||||
iii) | Suppose Conversely, assume |
Theorem 3.4.
A fuzzy equivalence relation ϕ of X is a fuzzy congruence relation pseudo-UP algebra of X if and only if and
.
Proof.
Suppose that a fuzzy equivalence relation ϕ is a fuzzy congruence on X. Then for all .
And,
Therefore, and
.
Conversely, assume ϕ is a fuzzy equivalence relation on X that satisfies and
.
Now, and
Similarly,
and
Therefore, ϕ is a fuzzy congruence relation on a pseudo-UP algebra of X.
Theorem 3.5.
If ϕ and θ are fuzzy congruence relations on X. Then is a fuzzy congruence relation on X × X.
Proof.
1) | Let | ||||
2) | Let | ||||
3) | Let | ||||
4) | Let Similarly, | ||||
5) | By similarly way, we have |
Definition 3.6.
Let be fuzzy congruence relations of X. Define the composition
by:
Definition 3.7.
Let ϕ be a fuzzy congruence relation on X and . Then, the level congruence relation
of ϕ and strong level congruence
of ϕ are defined as the follows:
Example 3.8.
Consider the pseudo-UP algebra given in Example 3.2. Then
i) | if t = 0.4, then | ||||
ii) | if t = 0.6, then | ||||
iii) | if t = 1, then | ||||
iv) | if t = 0.4, then | ||||
v) | if t = 0.6, then | ||||
vi) | if t = 1, then |
Theorem 3.9.
Let and ϕ2 be fuzzy congruence relations of X. Then
i) |
| ||||
ii) | For any |
Proof.
i) | Let Conversely, let ɛ > 0 be given and let From And | ||||
ii) | Suppose |
Therefore, .
Theorem 3.10.
Let be non empty and for any
. A fuzzy relation ϕ on X is a fuzzy congruence relation on X if and only if
is a congruence relation on X.
Proof.
Suppose that ϕ is fuzzy congruence relation on X. We need to prove that is a congruence relation on X. Let
be such that
. Let
such that
. Then
. Since ϕ is fuzzy congruence relation, then we have
. Which implies that
. Thus
is reflexive. Let
. Then
. This implies that
. Hence,
is symmetric. Let
. Then
and
. Now,
implies that
. Hence
is transitive. Let
then
and
implies that
and
belongs to
. Similarly, Let
then
and
implies that
and
belongs to
. Therefore,
is a congruence relation on pseudo-UP algebra of X.
Conversely, assume a level subset is a congruence relation on X. We need to show that ϕ is a fuzzy congruence relation on X. Let
be such that
. Let
such that
. Then
and take
. Since
is a congruence relation on X, we have
such that
. In the same way, we have
. Hence,
. Let
then
and
. Because
is a congruence relation on X. Thus
. In the same way, we have
. Hence,
. Let
, then
and
and take
. Since
is a congruence relation, we have
. Then
. Let
then
. Since,
is a congruence relation on X, we have
and
. Then
implies that
and
implies that
Similarly,
implies that
and
implies that
. Therefore, ϕ is a fuzzy congruence relation on pseudo-UP algebra of X.
Proposition 3.11.
A relation ϕ of X is a congruence relation of X if and only if its characteristic function χϕ is a fuzzy congruence relation on X.
Proof.
Suppose a relation ϕ of X is a congruence relation on X. We need to show that χϕ is a fuzzy congruence relation on X. For ,
, then
, since
because ϕ is a congruence relation on X. Let
, then
, since ϕ is a congruence relation, we have
, then
. So
If
then
otherwise
which is a contradiction. This implies that
. Therefore, in any case
. Let
. Since ϕ is a congruence relation X, then
. Now,
. Suppose
, then
. Now,
. Finally, let
and
, then
. Since ϕ is a congruence relation such that
. Then
and
. Similarly,
and
. Therefore, χϕ is a fuzzy congruence relation on X.
Conversely, assume χϕ is a fuzzy congruence relation on X. We want to prove that ϕ is a congruence relation on X. Let , then
. Thus
. Let
, then
Thus
For any
then
. Thus
Let
, then
and
. Hence,
and
belongs to ϕ. Similarly,
and
. Hence,
and
belongs to ϕ. Therefore, ϕ is a congruence relation on X.
Theorem 3.12.
If is a family of fuzzy congruence relation on X. Then
is a fuzzy congruence relation of X.
Proof.
Suppose . Then
. For
,
. For any
,
Finally,
and
. Similarly,
and
Therefore,
is a fuzzy congruence relation on X.
Remark 3.13.
The union of any two fuzzy congruence relation on X is not necessarily a fuzzy congruence relation on X.
Example 3.14.
Let be a set with binary operations
and
by the following Cayley table.
Then is a pseudo-UP algebra. Define the fuzzy relations ϕ and θ as follows:
Then ϕ and θ are fuzzy congruence relation on X. It is easily checked that is not a fuzzy congruence relation on X. Because Definition 2.1 (3) is not valid.
Theorem 3.15.
Let f be an endomorphism of X. If ϕ is fuzzy congruence relation of X, ϕf is defined by is also a fuzzy congruence relation on X.
Proof.
For any . If
. Then,
1) | |||||
2) | |||||
3) | |||||
4) |
| ||||
5) |
|
Definition 3.16.
Let ϕ be a fuzzy congruence relation in X and . Define the fuzzy set
which is defined by
for any
is called a fuzzy congruence relation class containing x.
Theorem 3.17.
If ϕ is a fuzzy congruence relation on X. Then ϕ0 is a fuzzy pseudo-UP ideal of X.
Proof.
Suppose ϕ is a fuzzy congruence relation on X. We need to show that ϕ0 is a fuzzy pseudo-UP ideals of X. From Definition 3.16 and by Proposition 3.3(i), we have
which implies that
. Next for any
be such that
and
. From
Which implies that
Also, since ϕ is a fuzzy congruence relation on X, we have and from
, we have
implies that
, by transitive. Which implies that
. Hence,
.
Similarly, for any be such that
and
. From
Which implies that
Also, since ϕ is a fuzzy congruence relation on X, we have and from
, we have
implies that
, by transitive. Which implies that
. Thus,
.
Therefore, ϕ0 is a fuzzy pseudo-UP ideal of X.
4. Fuzzy congruence relations induced by fuzzy pseudo-UP ideal
Let ζ be a fuzzy pseudo-UP ideals of X, let us define the fuzzy relation ϕζ on X define as follows:
=min
Then we have the following results.
Theorem 4.1.
The fuzzy relation ϕζ is a fuzzy congruence relation on X.
Proof.
Assume that ζ is a fuzzy pseudo-UP ideals of X. Define a fuzzy relation on X by:
. Then for all
, we have,
1) | Similarly, | ||||
2) | Similarly, | ||||
3) | For Similarly, | ||||
4) | Let
By similar argument, | ||||
5) |
|
Therefore, by ϕζ is a fuzzy congruence relation on pseudo-UP algebra of X.
From this, a fuzzy congruence relation ϕζ we say that it is generated by fuzzy pseudo-UP ideal ζ of X.
Theorem 4.2.
Let ζ be a fuzzy pseudo-UP ideals of X. Then, there is a fuzzy congruence relation ϕ in X such that .
Proof.
By Theorem 3.17 and 4.1 it is easily checked that for each fuzzy pseudo-UP ideal ζ of X and fuzzy congruence relation ϕ of X.
Let ζ be a fuzzy pseudo-UP ideal of ζx denote the fuzzy congruence class of x induced by ζ in X for every
.
Proposition 4.3.
Let ζ be a fuzzy pseudo-UP ideal of X. Then if and only if
.
Proof.
Let , for
. We have
, for any
. Since
for any
. It follows that
. Since X is pseudo-UP algebra such that
.
Similarly, . Since X is pseudo-UP algebra such that
.
Conversely, let By Theorem 2.9, we have
, since X is a pseudo-UP algebra, then we have
. Which implies that
implies
, since
. So,
, since ζ is fuzzy pseudo-UP ideal of X, i.e
. Thus
…….
. Again by Theorem 2.9
, since X is a pseudo-UP algebra then we have
. Which implies that
implies
, since
. Thus
……..
. From
and
we have
. Similarly, we have
. Now
. Therefore,
.
Theorem 4.4.
Let ζ be a fuzzy pseudo-UP ideal of X and ϕζ be a fuzzy congruence relation on X. Then is a pseudo-UP algebra of X.
Proof.
For every , we define
and
. Let
, we have
, since X is pseudo-UP algebra. Similarly,
, since X is pseudo-UP algebra. Let
and
.
It follows that and
. This means,
and
and
and
. That is
. Thus
. Similarly, Let
and
.
It follows that and
. This means,
and
and
and
. That is
. Thus
. Let
, then
and
. Therefore,
is a pseudo-UP algebra of X.
5. Conclusion
In this paper, we introduced the notion fuzzy congruence relation on pseudo-UP algebra, considering the concept of fuzzy congruence relation in certain algebraic structures and investigated some associated properties. We characterized the fuzzy congruence relation on X by their level set. The intersection of two fuzzy congruence and the Cartesian product of two fuzzy congruence are discussed. We provided the relationship between the set of fuzzy pseudo-UP ideal and the set of fuzzy congruence relation in pseudo-UP algebra. Then, we show that for each fuzzy ideal ζ and fuzzy congruence ϕζ generated by ζ, there is an associated algebra that is a pseudo-UP algebra. Moreover, using fuzzy pseudo-UP ideal and fuzzy congruence relation ϕζ generated by ζ, we constructed a congruence relation on a pseudo-UP algebra. Since congruence relations are interesting and important subjects in fuzzy logic, we hope that we helped to open new fields to anyone that is interested to studying of these concepts in pseudo-UP algebra.
Disclosure statement
No potential conflict of interest was reported by the authors.
Additional information
Funding
Notes on contributors
![](/cms/asset/ffec7b50-7568-4228-83be-baeb90a6d80e/oama_a_2235843_ilg0001.jpg)
Alachew Amaneh Mechdesro
Alachew Amaneh Mechderso received the MSc degree in mathematics from Ambo University, in 2018. He is currently a lecturer at Department of Mathematics, Kebri Dehar University, Ethiopia, and also a PhD student at Department of Mathematics, Bahir Dar University, Ethiopia. His research interests include Boolean algebra, fuzzy set theory and soft set theory. Alachew Amaneh Mechderso is a PhD scholar at the Department of Mathematics, Science College, Bahir Dar University, Ethiopia. His research interest includes the development of a Fuzzy pseudo-UP Algebra.
References
- Borzooei, R., Saeid, A., Rezaei, A., Radfar, A., & Ameri, R. (2013). On pseudo BE-algebras. Discussiones Mathematicae-General Algebra and Applications, 33(1), 95–108. https://doi.org/10.7151/dmgaa.1205
- Dudek, W. A., & Jun, Y. B. (2008). Pseudo-BCI algebras. East Asian Mathematical Journal, 24(2), 187–190.
- Dymek, G., & Walendziak, A. (2012). Fuzzy ideals of pseudo-BCK algebras. Demonstratio Mathematica, 45(1), 1–15. https://doi.org/10.1515/dema-2013-0346
- Georgescu, G., & Iorgulescu, A. (2001). Pseudo-BCK Algebras: An Extension of BCK Algebras. In C. S. Calude, M. J. Dinneen, & S. Sburlan (Eds.), Combinatorics (pp. 97–114). Springer-Verlag London.
- Iampan, A. (2017). A new branch of the logical algebra: UP-algebras. Journal of Algebra and Related Topics, 5(1), 35–54.
- Iampan, A., & Romano, D. A. (2022). Congruence relations on pseudo-UP algebras. Computer Science, 17(1), 1–9.
- Jun, Y. B. (2003). Pseudo-ideals of pseudo-BCK algebras. Scientiae Mathematicae Japonicae Online, 8, 87–91.
- Jun, Y., Kim, H., & Neggers, J. (2006). On pseudo-BCI ideals of pseudo-BCl algebras. Matematički vesnik, 58(1–2), 39–46. https://doi.org/10.1515/dema-2006-0204
- Kim, H. S., & Kim, Y. H. (2007). On BE-algebras. Scientiae Mathematicae Japonicae, 66(1), 113–116.
- Lee, K. J. (2009). Fuzzy ideals of pseudo BCI-algebras. Journal of Applied Mathematics & Informatics, 27(3–4), 795–807.
- Lee, K. J., & Park, C. H. (2009). Some ideals of pseudo BCI-algebras. Journal of Applied Mathematics & Informatics, 27(1–2), 217–231.
- Prabpayak, C., & Leerawat, U. (2009). On ideals and congruences in KU-algebras. Science Magna, 5(1), 54–57.
- Rezaei, A., Saeid, A. B., Radfar, A., & Borzooei, R. A. (2014). Congruence relations on pseudo BE–algebras. Annals of the University of Craiova-Mathematics and Computer Science Series, 41(2), 166–176.
- Rezaei, A., Said, A. B., & Daneshpayeh, R. (2015). Some results on fuzzy congruence relations in pseudo BE-algebras. Proceedings of the 2015 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE) (pp. 1–4). IEEE.
- Romano, D. A. (2020a). Pseudo-UP algebras, an introduction. Bulletin of International Mathematical Virtual Institute, 10(2), 349–355.
- Romano, D. A. (2020b). Pseudo-UP ideals and pseudo-UP filters in pseudo-UP algebras. Mathematical Sciences and Applications E-Notes, 8(1), 155–158. https://doi.org/10.36753/mathenot.630110
- Romano, D. A. (2021). Homomorphisms of Pseudo-UP algebras. Bulletin of International Mathematical Virtual Institute, 11(1), 47–53.
- Zadeh, L. A. (1965). Fuzzy sets. Information & Control, 8(3), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X