![MathJax Logo](/templates/jsp/_style2/_tandf/pb2/images/math-jax.gif)
ABSTRACT
Numerous scholars have studied fuzzy congruence relations in various algebraic structures. In this paper, we introduce a new kind of fuzzy congruence relations on a fuzzy algebraic structure (i.e. fuzzy lattice). Additionally, we provide a characterization of fuzzy congruence relations using its level sets and support. We also make a theoretical study of their basic properties analogous to those of ordinary congruence relations. Furthermore, we study the lattice of fuzzy congruence relations and also, we show that the lattice of all fuzzy congruence relations is complete, distributive and a special class of this lattice forms a modular lattice under certain conditions.
1. Introduction
The concept of a fuzzy set and fuzzy relation was first introduced by Zadeh in his pioneering paper (Zadeh, Citation1971). Since then many fuzzy algebraic structures have been developed by utilizing the concept of fuzzy relation in many directions. Starting from the earlier periods, several researchers have extensively studied fuzzy congruence relations on various crisp algebraic structures. A fuzzy congruence relation is a fuzzy equivalence relations which is fuzzy compatible with all basic operations of the algebras.
In (Murali, Citation1989, Citation1991), Murali studied fuzzy equivalence and congruence relations in universal algebra, developed some properties, and proved various results related to these relations. Furthermore, he developed certain lattice theoretic properties of fuzzy equivalence and congruence relations. Samhan (Citation1993) discussed the fuzzy congruence generated by a fuzzy relation and studied on the lattice of fuzzy congruence relations on a semigroup. Kondo (Citation2004) studied the notion of fuzzy congruence relation on a group, derived some of their fundamental properties. Also, he proved that there is an isomorphism between the set of fuzzy normal subgroups of a group and the set of fuzzy congruences on the group. More recently, Ullah in 2021 (Ullah et al., Citation2021), established the idea of fuzzy congruences on Abel-Grassmann’s group (AG-group) and studied different outcomes of fuzzy congruences on AG-groups. Similarly, numerous researchers have also examined the concept of fuzzy congruence relation with respect to different areas of algebra, including semi-group, group, ring, etc. (see Das, Citation1997; Debnath, Citation2022; Kim & Bae, Citation1997; Konwar & Debnath, Citation2018a, Citation2018b; Kuroki, Citation1992, Citation1997; Samhan, Citation1995; Tan, Citation2001), and more specifically, on different types of lattices (see Addis, Citation2022; Alaba & Mulat Addis, Citation2017; Alemayehu & Ayenew Ageze, Citation2022; Alemayehu et al., Citation2021, Citation2022; Rasuli, Citation2021).
On the other hand, the concept of fuzzy lattice was presented by (Chon, Citation2009) in 2009 as an extension of fuzzy partial order relation which is defined by (Zadeh, Citation1965). Developed some basic properties, characterized a fuzzy lattice using its level set, presented the ideas of distributive and modular fuzzy lattices, and measured several undeveloped properties of fuzzy lattices. Despite the fact that there are numerous papers about fuzzy congruence relations on algebraic structures, we were unable to find papers about fuzzy congruence’s defined on fuzzy algebraic structures. We are thus motivated to research on fuzzy congruence relation which are defined based on a fuzzy lattice.
Thus, in this work, using (Chon, Citation2009) definition of fuzzy lattices, we introduce the concept of fuzzy congruence relation, characterize its properties using its level sets and support, and finally, we prove that the lattice of all fuzzy congruence relations is a distributive lattice and special class of this lattice forms a modular lattice. We organized the remaining part of this paper as follows. In Section 2, some concepts, terminologies, notations, and important results on fuzzy relation, fuzzy posets, and fuzzy lattices are recalled. In Section 3, by the use of Chons’s definition of fuzzy lattice, a new kind of fuzzy algebraic structure, i.e. fuzzy congruence relation is introduced. Basic, fundamental properties of fuzzy congruence relation are presented. Finally, in Section 4, we investigate the lattice structure of fuzzy congruence relations on a fuzzy lattice.
2. Preliminaries
In this section, we summarize well-known definitions, and results that are required later in coming section of this paper. For contents of this section, we refer (Chon, Citation2009; Murali, Citation1989, Citation1991).
2.1. Fuzzy equivalence relation on a set
Definition 2.1.
(Zadeh, Citation1971) Let X be anon empty set. Then, any mapping is called fuzzy relation on X (or a binary fuzzy relation on X).
Definition 2.2.
(Zadeh, Citation1971) Let µ be a fuzzy relation in on X. For , the set (a crisp set)
is called (or
) subset of the fuzzy relation µ
Thus, the set µα form a nested sequence of crisp relations, with
Definition 2.3.
(Zadeh, Citation1971) Let X be a non empty set and let be a fuzzy relation on X. Then the support of a fuzzy relation µ is denoted by
is defined to be the crisp relation on X over which
for every
That is,
Definition 2.4.
Let X be a non-empty set and let µ1 and µ2 be fuzzy relations on X. Then
(1) | µ1 is said to be contained in fuzzy relation µ2 and is denoted by | ||||
(2) | µ1 and µ2 and are said to be equal denoted by µ1 = µ2 if and only if |
Definition 2.5.
(Zadeh, Citation1971) Let X be a non-empty set and let µ1 and µ2 be fuzzy relations on X. Then
(1) | The union of fuzzy relations µ1 and µ2 is denoted by for all (x, y) in X × Y | ||||
(2) | The intersection of fuzzy relations µ1 and µ2 is denoted by for all (x, y) in X × X |
Remark.
If is a family of fuzzy relations, we shall write
Definition 2.6.
(Zadeh, Citation1965) Let X be a non empty set and let π1 and π2 be fuzzy relations on X, then the composition or, more specifically, the max-min composition of π1 and π2 is a fuzzy relation on X denoted by and is defined by
for every and the sup is taken for every
Remark.
The composition of fuzzy relation is associative. That is, for any fuzzy relations on X we have,
Definition 2.7.
(Zadeh, Citation1971) Let X be a non empty set. A fuzzy relation π on X is said to be,
(a) | Reflexive:- If and only if | ||||
(b) | Symmetric:- If and only if | ||||
(c) | Anti-symmetric:- If and only if | ||||
(d) | Transitive:- If and only if |
Definition 2.8.
(Murali, Citation1989) Let X be a non empty set and π be a fuzzy relation on X. Then, π is said to be fuzzy equivalence relation on X if and only if π is Reflexive, Symmetric and Transitive fuzzy relation on set X.
Proposition 2.1.
If π1 and π2 are fuzzy relations on set X, then
Proof. The proof is clear.
Proposition 2.2.
Let X be a non-empty set and π be a fuzzy relation on X. Then, π is transitive fuzzy relation on X if and only if
Proof. See in (Murali, Citation1989) Proposition 3.1.
Notation.
Let X be a non-empty set then, the family of all fuzzy equivalence relations on set X is denoted by FEq(X)
Definition 2.9.
Let X be a crisp set . Define the fuzzy equivalence relations Δ and on set X by
(a) | for all | ||||
(b) |
|
Remark.
It is clear that Δ and are fuzzy equivalence relations and are called the diagonal and total fuzzy relations on X, respectively.
Proposition 2.3.
(Murali, Citation1991)
Let be a non-empty family of fuzzy equivalence relations on a set X. Then,
is a fuzzy equivalence relation on set X.
2.2. Fuzzy lattice
Definition 2.10.
(Zadeh, Citation1971) Let X be a non empty set and µ be a fuzzy relation on X, then µ is said to be fuzzy partial ordered relation on X if and only if µ is reflexive, anti-symmetric and transitive.
Definition 2.11.
(Chon, Citation2009) If µ is fuzzy partial ordered relation in X, then the ordered pair is called a fuzzy partially ordered set or a fuzzy poset.
Definition 2.12.
(Chon, Citation2009) If µ is fuzzy total order relation on X, then the ordered pair is called a fuzzy total ordered set or a fuzzy chain.
Remark.
(Mezzomo et al., Citation2015) Let is fuzzy poset and
. If
and
, then
.
Theorem 2.4.
(Chon, Citation2009) Let be a collection of fuzzy partial order relations in a set X, then
is a fuzzy poset.
Remark.
It is easy to see that for fuzzy partial order relations µ1 and µ2 on a set X, and
are not necessarily a fuzzy poset.
Example 2.13.
Let and define the fuzzy relations µ1 and µ2 on X by the fuzzy relational matrices given in Table and Table respectively.
Then it is easily checked that, and
are fuzzy posets. And also, as given in the table (Table ) since,
and
. Then,
is not antisymmetric. Hence,
is not fuzzy partial ordering relation. Therefore,
is not a fuzzy poset.
Similarly, as given in the , since, and
. Then,
is not antisymmetric. Hence,
is not fuzzy partial ordering relation. Therefore,
is not a fuzzy poset.
Table 1. Relational matrix of the fuzzy relation µ1 on X
Table 2. Relational matrix of the fuzzy relational µ2 on X
Table 3. Relational matrix of the fuzzy relation on X
Table 4. Relational matrix of the fuzzy relation on X
The following Theorems show the characterization of fuzzy relations in terms of their level subsets.
Theorem 2.5.
(Mezzomo et al., Citation2015) Let µ be a fuzzy relation in X. Then µ is a fuzzy partial order relation if and only if every α-cut, µα is a partial order relation in X for all α such that .
Theorem 2.6.
(Mezzomo et al., Citation2015) Let µ be a fuzzy relation in X and let be a support of the fuzzy relation µ defined on X. If µ is a fuzzy partial order relation on X. Then,
is a partial order relation on X.
Definition 2.14.
(Chon, Citation2009) Let be a fuzzy poset and let
.
(1) | An element | ||||
(2) | An upper bound u0 for B is said to be the least upper bound (lub) of B if and only if | ||||
(3) | An element | ||||
(4) | A lower bound v0 for B is said to be the greatest lower bound (glb) of B if and only if |
Notation.
We denote the least upper bound of the set if it exists by
and denote the greatest lower bound of the set
if it exists by
.
Definition 2.15.
(Chon, Citation2009) Let be a fuzzy poset. Then,
is said to be a fuzzy lattice if and only if
and
exist for all
.
Remark.
Let be a fuzzy poset and let
. Then, the least upper bound and greatest lower bound of Y if it exists is unique.
Theorem 2.7.
(Chon, Citation2009) (Properties of Fuzzy Lattice)
Let be a fuzzy lattice and let
. Then,
(i) |
| ||||
(ii) |
| ||||
(iii) | If | ||||
(iv) | If | ||||
(v) |
|
Theorem 2.8.
(Chon, Citation2009) Let be a fuzzy relation in X and let
be
sub set of the fuzzy relation µ defined on X. If
is a lattice for all
, then
is a fuzzy lattice.
Remark.
The converse of the above Theorem is not necessarily true.
But we can find a lattice from a fuzzy lattice by specifying the α-cut as follow
Theorem 2.9.
(Chon, Citation2009) Let be a fuzzy relation on X and let
be
sub set of the fuzzy relation µ defined on X. If
is a fuzzy lattice, then
is a lattice for some
.
Proposition 2.10.
(Mezzomo et al., Citation2015) Let be a fuzzy lattice and let
, then
and
coincides with
and
respectively in
.
Theorem 2.11.
(Chon, Citation2009) Let µ be a fuzzy relation in X and let be a support of the fuzzy relation µ defined on X. If
is a fuzzy lattice, then
is a lattice on X.
3. Fuzzy congruence relation on fuzzy lattice
In this section, we introduce the notion of fuzzy congruence relations on a fuzzy lattice and discuss basic properties related to its level sets.
Remark.
Throughout the rest of this paper stands for a fuzzy lattice
.
Definition 3.1.
Let be a fuzzy lattice as defined by (Chon, Citation2009) and let π be a fuzzy relation on L. Then, π is said to be
(i) | Fuzzy join compatible if | ||||
(ii) | Fuzzy meet compatible if | ||||
(iii) | Fuzzy compatible if it is both fuzzy join and fuzzy meet compatible. |
Definition 3.2.
Let . Then, π is called fuzzy join congruence (fuzzy meet congruence, fuzzy congruence) relation on fuzzy lattice L if it is fuzzy join compatible (meet compatible, compatible) on fuzzy lattice L.
Notation.
Let be a fuzzy lattice then, the family of all fuzzy congruence relation on fuzzy lattice L is denoted by FCon(L).
Theorem 3.1
Let be a fuzzy lattice and
. Then, π is fuzzy compatible if and only if
and
for all
.
Proof. Suppose π is fuzzy compatible. Let , then
Similarly,
Conversely, suppose the condition and to show π is fuzzy compatible. Let , then
Similarly, . Therefore, π is fuzzy compatible.
Based on the definition of fuzzy congruence relation and Theorem (2.11) we have the following result,
Theorem 3.2
Let be a fuzzy lattice and π be a fuzzy relation on L. Then
if and only if
for all
whenever
.
Proof. Suppose
and let
and assume
(i) | Let | ||||
(ii) | Let | ||||
(iii) | Let Therefore, | ||||
(iv) | Let |
Conversely, suppose
for all
.
(i) | Let | ||||
(ii) | Let | ||||
(iii) | Let Since | ||||
(iv) | Let |
Hence, π is compatible and therefore, .
Definition 3.3.
Let be a fuzzy lattice and a and b be pair of elements of L such that
. We define the interval
on L by
Proposition 3.3.
Let be a fuzzy lattice and
. Then, the following are equivalent for all
(i) | |||||
(ii) | |||||
(iii) | If |
Proof. (i ii): Suppose
. Since
, then
. Similarly,
. Thus by transitivity on π we have
.
(ii iii): Suppose
. Let
such that
. Then,
and
. Hence,
and
and hence
and
. Then, by transitivity of π we have
.
(iii i): Suppose, if
such that
, then
. Since
, then
.
Definition 3.4.
Let be a fuzzy lattice and let
. Then, π1 and π2 are said to be permutable fuzzy congruences if
.
Definition 3.5.
If every pair of elements of FCon(L) are permutable then the fuzzy lattice L is called fuzzy congruence permutable.
Lemma 3.4.
If π1 and π2 are compatible fuzzy congruences on . Then,
is also fuzzy compatible.
Proof. Let . Then,
Similarly, . Therefore,
is fuzzy compatible.
Proposition 3.5.
Let . Then, π1 and π2 are permutable fuzzy congruences if and only if
.
Proof. Suppose π1 and π2 are permutable fuzzy congruences.
(i) | Let Thus, | ||||
(ii) | Let Therefore, | ||||
(iii) | Since Thus, | ||||
(iv) | Fuzzy compatibility is clear by Lemma (3.4) |
Therefore, .
Conversely, suppose and let
, then
Therefore, π1 and π2 permutable fuzzy congruences.
Lemma 3.6.
Let be fuzzy lattice and let
. Then,
Proof. Suppose
(i) | Since | ||||
(ii) | Let | ||||
(iii) | Let Hence, | ||||
(iv) | Let Similarly, |
Remark.
However the union of two fuzzy congruence relations may not be a fuzzy congruence relation
Example 3.6.
Let and define a fuzzy relation µ1 on X by the fuzzy relational matrix given in
Thus, from figure it is easy to verify that is fuzzy lattice.
Similarly consider the following fuzzy congruence relations π1 and π2 on L1 whose represntations given in Table and Table respectively
Table 5. Representation of the fuzzy congruence relations π1 and π2 on fuzzy lattice L1 by relational matrices
Then, the relational matrix corresponds to the fuzzy relation is given in Table ,
Table 6. Representation of fuzzy congruence relations on L1 by relational matrix
Now since, . Then,
is not a transitive and hence it is not a fuzzy congruence relation on L1.
4. Lattice of fuzzy congruence relation
In this section, we discuss the lattice structure of the fuzzy congruence relations on the fuzzy lattice which is defined above.
Lemma 4.1.
Let be a fuzzy lattice. Then,
is complete lattice where
is a relation defined by
if and only if
for all
and
.
Proof. It is clear that is a poset with Δ and
are the least and greatest elements of
. Now let
be any collection of fuzzy congruences on fuzzy lattice L and suppose
for
. Then,
(i) | Since | ||||
(ii) | Let | ||||
(iii) | Let Hence, π is transitive. | ||||
(iv) | Let Similarly, |
Theorem 4.2
Let be a fuzzy lattice and let
. Then,
(a) | |||||
(b) | That is, for every |
Proof. Let
(a) | Clearly | ||||
(b) | Let |
(i) | Let | ||||
(ii) | Let which shows θ is symmetric. | ||||
(iii) | To show θ is transitive it is sufficient to show that Thus, | ||||
(iv) | Let Similarly, | ||||
(v) | Finally to show that θ is least upper bound of This shows θ is the least upper bound of Therefore, |
Proposition 4.3.
Let such that
. Then
Proof. Let such that
. Then, for
Therefore, and similarly
. Thus,
is upper bound of
. Now let,
be upper bound of
. Then for
Then, . Which shows
is least upper bound of
. Therefore,
.
By combining Propositions (3.5) and (4.3) we have the following result:
Corollary 4.4.
Let . If π1 and π2 are permutable fuzzy congruences, then
Theorem 4.5
Let be a fuzzy lattice and
. Then, the following are equivalent.
(i) | |||||
(ii) | |||||
(iii) |
Proof. : Suppose
. Then
and also ,
: Suppose
. Now,
: Suppose
. Then,
Therefore,
Definition 4.1.
A fuzzy lattice is said to be fuzzy congruence modular if Fcon(L) is modular lattice. That is, if
such that
, then
Similarly,
Definition 4.2.
A fuzzy lattice is said to be fuzzy congruence distributive if Fcon(L) is distributive lattice. That is, if
then,
.
Theorem 4.6
If a fuzzy lattice is fuzzy congruence permutable then
is modular lattice
Proof. Let and suppose
. It is clear that
. To show
it is enough to show
. Again, since Fcon(L) is fuzzy permutable we need to show
. Now, let
, then
Thus, . Therefore,
is modular lattice.
Proposition 4.7.
If a fuzzy lattice is fuzzy congruence permutable then
is distributive lattice.
Proof. Let . To show that
. Since L is fuzzy congruence permutable then
for all
. Thus to show Fcon(L) is distributive it is enough to show that
. Now let
. Then,
Hence, . Therefore, Fcon(L) is distributive.
In the next theorem, we will prove the above theorem without the condition is fuzzy congruence permutable.
Theorem 4.8
If L is fuzzy lattice. Then, Fcon(L) is distributive.
Proof. Let . To show that
. Since, it is clear that
then it is enough to show that
.
Now let . Then,
This shows . Therefore, Fcon(L) is distributive lattice.
5. Discussion
As explained earlier in the introduction, a lot of authors have attempted to study fuzzy congruence relation on different algebraic structures. So, it is a natural question to extend these concept to the case of fuzzy congruence relation on fuzzy algebraic structures. Hence, the current study presented a novel approach to define fuzzy congruence relation on fuzzy lattice. The earlier publications, on the other hand, focused solely on fuzzy congruence relation on crisp algebraic structures. Additionally, our study showed that there is connection between a fuzzy congruence relation on fuzzy lattice with the congruence relation on a level set. Also, as expected we were able to show that the fuzzy congruence relation we defined here preserve the main properties that the classical congruence relation holds. Furthermore, in this study, we studied the lattice theoretic structure of the fuzzy congruence relations on the fuzzy lattice. Because the concept distributive fuzzy lattice was not thoroughly studied in literature, we are unable to explore the relationship between fuzzy congruence relation and an ideal (or fuzzy ideal) of fuzzy lattice. Because of this, we plan to expand the theory of distributive fuzzy lattice in more depth soon and explore the connection between fuzzy ideals and fuzzy congruences on a distributive fuzzy lattice, just as we found in the classical lattice theory.
One of the algebraic structures that are most extensively used and applied in the fuzzy mathematical theory is certainly the fuzzy congruence structure. Fuzzy congruence theory has several applications that are quite similar to those of the classical congruence theory, including those in physics, business, computer science, coding theory, and other fields. More specifically, congruences and their classes on the ring of integers are important tools in the design of round-robin tournaments (Childs, Citation1995), fuzzy congruence relations and their classes are important structures in the study of knowledge representation systems (De Baets & Kerre, Citation1994), congruence arithmetic has many applications in the foundation of modern cryptography in public-key encryption, secret sharing, wireless authentication, and many other applications for data security (Debnath & Mohiuddine, Citation2021; Nair & Mordeson, Citation2001; Schroeder, Citation2008; Yan, Citation2002).
6. Conclusions
In this paper, we have introduced and studied a new kind of fuzzy congruence relation on a fuzzy lattice, gave an equivalent definition for a fuzzy congruence relation on fuzzy lattice, investigated some of their properties, and also characterized a fuzzy congruence relation using its level set. Moreover, we constructed a lattice of fuzzy congruence relations, and characterize the join and meet of fuzzy congruence relation in the lattice. Finally, we have shown that the lattice of fuzzy congruence relation is complete, distributive lattice and also under certain conditions the lattice becomes a modular lattice.
Acknowledgements
The authors express their gratitude to the anonymous reviewers for providing valuable feedback and suggestions that significantly enhanced the quality of the paper.
Disclosure statement
No potential conflict of interest was reported by the authors.
Additional information
Notes on contributors
Sileshe Gone Korma
Sileshe Gone Korma is a PhD scholar at the Department of Mathematics, College of Natural Sciences, Arba Minch University, Ethiopia. His research interest includes fuzzy algebra.
References
- Addis, G. M. (2022). L-fuzzy congruence classes in universal algebras. International Journal of Intelligent Systems, 37(1), 386–423. https://doi.org/10.1002/int.22631
- Alaba, B. A., & Mulat Addis, G. (2017). Fuzzy congruence relations on almost distributive lattices. Annals of Fuzzy Mathematics and Informatics, 14(3), 315–330. https://doi.org/10.30948/afmi.2017.14.3.315
- Alemayehu, T. G., Abeje Engidaw, D., Mulat Addis, G., & Al-Shami, T. (2022). Kernel l-ideals and l-congruence on a subclass of Ockham algebras. Journal of Mathematics, 2022, 1–9. https://doi.org/10.1155/2022/7668044
- Alemayehu, T. G., Abeje Engidaw, D., Mulat Addis, G., & Papadopoulos, B. (2021). L-fuzzy congruences and l-fuzzy kernel ideals in Ockham algebras. Journal of Mathematics, 2021, 1–12. https://doi.org/10.1155/2021/6644443
- Alemayehu, T. G., & Ayenew Ageze, Z. (2022). Fuzzy ideals and fuzzy congruences of distributive demi-p-algebras. Journal of Discrete Mathematical Sciences and Cryptography, 1–11. https://doi.org/10.1080/09720529.2021.2012889
- Childs, L. N. (1995). Applications of congruences. In Childs, L. N. (Ed.), A concrete introduction to higher algebra, undergraduate texts in mathematics (2 ed.). Springer New York. https://doi.org/10.1007/978-1-4419-8702-0_7
- Chon, I. (2009). Fuzzy partial order relations and fuzzy lattices. Korean Journal of Mathematics, 17(4), 361–374.
- Das, P. (1997). Lattice of fuzzy congruences in inverse semigroups. Fuzzy Sets and Systems, 91(3), 399–408. https://doi.org/10.1016/S0165-0114(96)00133-9
- De Baets, B., & Kerre, E. (1994). In Hawkes, P. W. (Ed.), Advances in Electronics and Electron Physics (Vol. 89, pp. 255–324). https://doi.org/10.1016/S0065-2539(08)60075-X0
- Debnath, P. (2022). Some results on Cesàro summability in intuitionistic fuzzy n-normed linear spaces. Sahand Communications in Mathematical Analysis, 19(1), 77–87.
- Debnath, P., & Mohiuddine, S. A. (2021). Soft computing techniques in engineering, health, mathematical and social sciences. CRC Press.
- Kim, J. P., & Bae, D. R. (1997). Fuzzy congruences in groups. Fuzzy Sets and Systems, 85(1), 115–120. https://doi.org/10.1016/0165-0114(95)00334-7
- Kondo, M. (2004). Fuzzy congruences on groups. Quasigroups and Related Systems, 11(1), 59–70.
- Konwar, N., & Debnath, P. (2018a). Some new contractive conditions and related fixed point theorems in intuitionistic fuzzy n-Banach spaces. Journal of Intelligent & Fuzzy Systems, 34(1), 361–372. https://doi.org/10.3233/JIFS-171372
- Konwar, N., & Debnath, P. (2018b). Intuitionistic fuzzy n-normed algebra and continuous product. Proyecciones (Antofagasta), 37(1), 68–83. https://doi.org/10.4067/S0716-09172018000100068
- Kuroki, N. (1992). Fuzzy congruences and fuzzy normal subgroups. Information Sciences, 60(3), 247–259. https://doi.org/10.1016/0020-0255(92)90013-X
- Kuroki, N. (1997). Fuzzy congruences on inverse semigroups. Fuzzy Sets and Systems, 87(3), 335–340. https://doi.org/10.1016/0165-0114(95)00377-0
- Mezzomo, I., Bedregal, B. C., & Santiago, R. H. (2015). Types of fuzzy ideals in fuzzy lattices. Journal of Intelligent & Fuzzy Systems, 28(2), 929–945. https://doi.org/10.3233/IFS-141374
- Murali, V. (1989). Fuzzy equivalence relations. Fuzzy Sets and Systems, 30(2), 155–163. https://doi.org/10.1016/0165-0114(89)90077-8
- Murali, V. (1991). Fuzzy congruence relations. Fuzzy Sets and Systems, 41(3), 359–369. https://doi.org/10.1016/0165-0114(91)90138-G
- Nair, P. S., & Mordeson, J. N. (2001). Fuzzy mathematics,: An introduction for engineers and scientists. Springer-Verlag.
- Rasuli, R. (2021). Fuzzy congruence on product lattices under t-norms. Journal of Information and Optimization Sciences, 42(2), 333–343. https://doi.org/10.1080/02522667.2019.1664383
- Samhan, M. A. (1993). Fuzzy congruences on semigroups. Information Sciences, 74(1–2), 165–175. https://doi.org/10.1016/0020-0255(93)90132-6
- Samhan, M. A. (1995). Fuzzy quotient algebras and fuzzy factor congruences. Fuzzy Sets and Systems, 73(2), 269–277. https://doi.org/10.1016/0165-0114(94)00300-V
- Schroeder, M. (2008). Number theory in science and communication: With applications in cryptography, physics, digital information, computing, and self-similarity. Springer Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85298-8
- Tan, Y. (2001). Fuzzy congruence on regular semigroup. Fuzzy Sets and Systems, 117(3), 447–453. https://doi.org/10.1016/S0165-0114(98)00275-9
- Ullah, A., Khan, A., Ahmadian, A., Senu, N., Karaaslan, F., & Ahmad, I. (2021). Fuzzy congruences on ag-group. AIMS Mathematics, 6(2), 1754–1768. https://doi.org/10.3934/math.2021105
- Yan, S. Y. Number theory for computing. Springer Science & Business Media, 2002.
- Zadeh, L. A. (1965). Fuzzy sets. Information & Control, 8(2), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X
- Zadeh, L. A. (1971). Similarity relations and fuzzy orderings. Information Sciences, 3(2), 177–200. https://doi.org/10.1016/S0020-0255(71)80005-1