Some Results about Weak UP-algebras

Chaudhry, Muhammad Anwar; Fahad, Asfand; Qureshi, Muhammad Imran; Riasat, Urwa

doi:https://doi.org/10.1155/2022/1206804

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Abstract Introduction Conclusion Abbreviations Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 1206804 | https://doi.org/10.1155/2022/1206804

Some Results about Weak UP-algebras

Muhammad Anwar Chaudhry,¹Asfand Fahad,²Muhammad Imran Qureshi,³and Urwa Riasat¹

Academic Editor: Musavarah Sarwar

Received28 Apr 2022

Accepted17 Aug 2022

Published16 Sept 2022

Abstract

We introduce a new class of algebras, weak UP-algebras, briefly written as WUP-algebras, as a wider class than the classes of UP-algebras and KU-algebras. We investigate fundamental aspects including maximal elements, branches and the subalgebra consisting of maximal elements of a WUP-algebra. We also study regular congruences on a WUP-algebra as well as the corresponding quotient WUP-algebras. We prove that the congruence generated from the branches of a WUP-algebra is regular and the corresponding quotient algebra is isomorphic to the WUP-algebra , the subalgebra of consisting of its maximal elements.

1. Introduction

A few abstract algebras, such as BCK-algebras and BCI-algebras, have their roots in implicational logic, see [1, 2]. Many facets of daily life have benefited substantially from recent advances in artificial intelligence. Logical systems are the fundamental artificial intelligence techniques that aid in decision-making. The evolution and introduction of BCK-algebras and BCI-algebras are driven by the essential axioms of the implicational calculus, see [1, 2]. Taking into account the important linkages between these algebras and related logics, translation techniques have been created to link the theorems and formulae of a logic and the corresponding algebra. As a result, those involved in the fields of artificial intelligence, logical systems, and algebraic structures continue to be interested in the study of abstract algebras (and their generalizations) that have been inspired by logical systems. As a result, the BCH-algebra class, a new class of algebras, has been introduced [3]. Recently, another algebra class, known as gBCH-algebra class, has been introduced and studied in [4, 5]. The significant feature of this new class is its ability to cover all other mentioned classes. In the context of this class, further study through introduction of generalized -algebras and isomorphism theorems in generalized -algebras has been developed [6]. For more on such other algebras such as SU-algebra [7], BCC algebra [8] and the significance of these algebras we refer to [9–11]. Although the immediate applications played vital role in getting attention to the above mentioned algebras, however, the development of theoretical aspects of these algebras can not be ignored. Particularly, when they provide different (or a generalized) framework. Consequently, the study on these algebras continued and recently the notion of a KU-algebra [12] was introduced and studied further in [13]. Later, a generalized class of KU-algebras, known as UP-algebras [14] was defined and investigated. The range of the notions studied for these algebras is very vast, such as: the ideals and congruences [12], cubic KU-ideals [13], -fuzzy KU- Ideals [15], interval-valued fuzzy KU ideals and -fuzzy KU-ideals [16, 17], anti fuzzy and intuitionistic fuzzy KU-ideals [18, 19] in KU-algebras. In UP-algebras, the fuzzy sets and the generalized fuzzy sets were introduced and studied in [20, 21]. Moreover, the derivations of UP-algebras have been studied in [22, 23]. In 2019, the isomorphism theorems for UP-algebras were proved [24] and the results were extended later on. For more on the recent related study/applications of these algebras in multiple directions, we refer [25, 26] and references therein. Keeping the above motivation in view, we introduce a general class of algebras, known as weak UP-algebras (WUP-algebras) in such a way that the classes of KU-algebras and UP-algebras become the subclasses of the class of WUP-algebras. Further, we give necessary examples to show that the UP-algebra (consequently) class is a proper sub-class of the WUP-algebra class. Consequently, all investigations of this paper are valid for the KU-algebra class and the UP-algebra class. In particular, we study some fundamental aspects including maximal elements, branches and the subalgebra consisting of maximal elements of a WUP-algebra. We also define regular congruences on a WUP-algebra and study the corresponding quotient algebras. We prove that the congruence generated from the maximal elements of a WUP-algebra is regular and the corresponding quotient algebra is isomorphic to , the subalgebra consisting of maximal elements of .

2. Weak UP-Algebras (WUP-Algebras)

In this section, we introduce the notion of a Weak UP-algebra and prove that the classes of KU-algebras and UP-algebras are proper subclasses of the class of WUP-algebras. Before proceeding further, we recall the notions of a KU-algebra [12] and an UP-algebra [14].

By an algebra of type , we mean a non-empty set together with a binary operation “.” and a distinguished element 0 satisfying some specified axioms. Further, for , we shall write and will mean “.” between the bracketed expressions.

Defintion 1. (KU-algebra). An algebra of type is called a KU-algebra if it satisfies the following conditions for any :
,
,
,
xy = 0 and yx = 0 imply x = y.

Defintion 2. (UP-algebra). An algebra of type is called a UP-algebra if it satisfies the following conditions for any :
,
,
,
xy = 0 and yx = 0 imply x = y.
It is shown in [14] that every KU-algebra is an UP-algebra. Moreover, it is described in [14] that the algebra of the following Example 1 is an UP-algebra but not a KU-algebra. Consequently, the class of UP-algebras is wider than the class of KU-algebras.

Example 1. (see [14]). Letand the binary operation defined as (see Table1):
We now give our notion of a weak UP-algebra (WUP-algebra).

Defintion 3. (Weak UP-algebra). An algebra of type is called a weak UP-algebra (WUP-algebra) if it satisfies the following conditions for any :
,
,
implies ,
xy = 0 and yx = 0 imply x = y.

Defintion 4. (Subalgebra of a WUP-algebra). A subset of a WUP-algebra is called a subalgebra of if is closed under the operation of .
Now, we prove the following proposition.

Proposition 1. Every UP-algebra is a WUP-algebra.

Proof. It is sufficient to prove WUP-3. Let be such that . The condition UP-3 gives and hence becomes . Consequently, by UP-2 . On the other hand, and WUP-3 gives . Thus, . Hence is a WUP-algebra.
On a WUP-algebra , we define the binary relation by: if and only if . The following example shows that the converse of Proposition 1 is not true, in general.

Example 2. Let and the binary operation be defined as in the following table: (see Table 2):
Routine calculations show that is a WUP-algebra but not a UP-algebra because .

Example 3. Let and the binary operation be defined as in the following table: (see Table 3):
Routine calculations show that is a WUP-algebra but not a UP-algebra because .

Remark 1. From Proposition 1 and Examples 2 and 3, we deduce that every UP-algebra is a WUP-algebra but converse is not true. Thus the class of WUP-algebras contains both classes of KU-algebras and UP-algebras. Consequently, we feel that the investigation of the properties of WUP-algebras will add some important results to the mathematical literature.
Now, we prove some important properties of WUP-algebras.

Theorem 1. For a WUP-algebra and for any in , we have:(1),(2) and imply ,(3) implies ,(4) implies ,(5).

Proof. The proofs of the parts of this theorem are exactly the same as in Proposition 1.7, parts (page 39 of [14]). So, we only prove (5). For this, consider (by WUP-1).

Remark 2. (1)In Theorem 1 (5), if we substitute for , then we get , that is, which gives for all .(2)In an UP-algebra, , so Theorem 1 (5) gives or , which yields proposition of [14].(3)In the sequel, we denote the element of a WUP-algebra by .(4)From Theorem 1, we conclude that the relation on a WUP-algebra is reflexive as well as transitive.

Defintion 5. (UP-Part). Let be a WUP-algebra. The UP-part of , denoted by , is defined as .

Example 4. In WUP-algebras of Examples 2-3, their UP-parts are and , respectively.

Defintion 6. (Maximal Element). Let be a WUP-algebra. The element corresponding to is called a maximal element of . We also call it the maximal element corresponding to and denote it by . That is .
We denote the set of all maximal elements of a WUP-algebra by . That is, .
Now, we define the notion of a branch of a WUP-algebra.

Defintion 7. (Branch of a WUP-algebra). Let be a maximal element of a WUP-algebra . Then the branch of determined by , denoted by , is the subset = of .

Remark 3. We note that

Example 5. In Example 2, the elements are maximal elements and the branches are and , whereas in Example 3, are maximal elements and the branches are ,, and.
In the following theorem, we prove a sufficient condition for two elements and of a WUP-algebra to have same corresponding maximal elements.

Theorem 2. Let be a WUP-algebra. If are such that , then .

Proof. Since , so by applying Theorem 1 (4) twice we get . That is, , which gives . Now, by WUP-3, we get .

Proposition 2. Let be a WUP-algebra. If , then .

Proof. Let , we consider , by WUP-1. Thus , which by WUP-3 gives .

Theorem 3. Let be a WUP-algebra. If then and is a subalgebra of .

Proof. Let . Then there exist such that and . By using Theorem 1 (3) and (4), we get and . Using Theorem 2, we get and . Thus . Since and , therefore . Now, by Theorem 2 and Proposition 2, we get . Hence . Since , so . Thus is a subalgebra of .

Theorem 4. Let be a WUP-algebra. Then for all , if and only if and belong to the same branch of .

Proof. Let . Then and . So by Theorem 1 (3), we get . Thus , which implies . Conversely, let and . Thus . By Theorem 2, we get . This gives , which by WUP-3 gives . So belong to the same branch of .

Proposition 3. Let be a WUP-algebra. Let , . Then

Proof. Clearly, . Also . Since is a subalgebra of , so implies . Hence, gives □

Proposition 4. Let be a WUP-algebra. Then its UPP-part is a subalgebra of .

Proof. Let . So by above proposition, we get . Hence is a subalgebra of .
In the next theorem, we show that the collection of branches produced by all the maximal elements in a WUP-algebra partitions :

Theorem 5. Let be a WUP-algebra. Let be the collection of all maximal elements of . Then(a).(b) for any and .

Proof. (a)Obviously is a non-empty subset of because gives . Consequently, . Also for each , there is such that , so . Hence . Combining, we get .(b)Let and . Let . So, and . That is and . Now, WUP-1 gives . So, . That is, . So, . Now, WUP-3 give . That is, . Again, WUP-3 gives , a contradiction. Thus, .

Remark 4. (1)Every WUP-algebra is the union of disjoint branches of , determined by its maximal elements.(2)Every belongs to a unique branch determined by its corresponding maximal element . That is, for every , there is a unique maximal element such that .

3. Regular Congruences on WUP-Algebras

In this section, we discuss regular congruences on WUP-algebras and the corresponding quotient WUP-algebras.

Defintion 8. Let be a WUP-algebra. A relation on is called a congruence on if for any :, the following conditions hold:(1),(2) implies ,(3), imply (4), imply .

Defintion 9. (Regular Congruence). A congruence on a WUP-algebra is called a regular congruence if implies , .

Example 6. Let be a WUP-algebra. We define a relation by: , if and only if . Then the relation is a regular congruence on .
Verification.(1)Let , since therefore .(2)Let and , therefore . So . Thus .(3)Let , and . Then and . Thus and hence .(4)Let with and . Hence and . Now, consider . So, . Thus is a congruence on . We, now show that it is a regular congruence.(5)Let such that . Thus , which gives . Now, by WUP-3, we get . This gives . Hence is a regular congruence of .

Defintion 10. Let be a WUP-algebra and let be a congruence on . Let . We define:and call , the class determined by . Then the setis called the quotient set of with respect to the congruence .

Theorem 6. Let be a WUP-algebra and a regular congruence on . Let be the quotient set of . Let the binary operation in be defined as:for all . Then is a WUP-algebra.

Proof. We first show that the operation defined by (1) is well defined. Let and and . By property of equivalence relation , we have and . Since is a congruence so . Thus . That is .
Now, we verify WUP-1, WUP-2, WUP-3 and WUP-4.(1)We, now consider .(2)Let , so .(3)Let . Then . So . That is, . Since is a regular congruence, so . Hence . So . Thus, implies .(4)Let and . Thus and . This implies . Since is a regular congruence on , so . Thus, .

Remark 5. We know that, for a WUP = algebra , the relation defined on by: if and only if , that is, if and only if belong to the same branch of , is a regular congruence on . Thus from Theorem 6, we have the following corollary.

Corollary 1. Let be a WUP-algebra. Let be the regular congruence on defined by: , if and only if . Then is a WUP-algebra with defined by (1).

Defintion 11. Let , be WUP-algebras. A mapping is called:(1)a homomorphism if for all .(2)an isomorphism if is a homomorphism and a bijection.(3)the WUP-algebras and are isomorphic if there is an isomorphism .

Theorem 7. Let be a WUP-algebra. Let be the regular congruence on defined by: , if and only if . Let be the corresponding quotient algebra. Let be the subalgebra of consisting of maximal elements of . Then is isomorphic to .

Proof. We define by for all .(a)Firstly, we show that is well-defined. Let , so . That is belong to the same branch. So . Hence, .(b)Now, we show that is homomorphism. Let . Now, . So, is a homomorphism.(c)To show that is onto, let . Then there exist such that . Hence is onto.(d)Now, we show that is one-one. Let be such that . So, . That is and belong to the same branch of . Hence . So, . Therefore, is one-one.Summing up we get, is isomorphic to .

Example 7. We consider the WUP-algebra of Example 3. We note that . We define the congruence on by: and if and only if belong to the same branch of . SoThusWe define a mapping by . It is easy to verify that is an isomorphism. So is isomorphic to .

4. Conclusion

We have introduced a new class of algebras, known as Weak UP-algebras. This class is so general that it contains the class of KU-algebras as well as the class of UP-algebras. We have studied some basic properties, maximal elements, branches, regular congruences, quotient algebras of this class of algebras. We have also established an isomorphism between and , where is the regular congruence defined on with the help of its maximal elements and is the subalgebra of consisting of maximal elements of .

5. Future Work

It will be interesting to explore other aspects of this wider class of WUP-algebras. In particular, proper commutative WUP-algebras, positive implicative WUP-algebras and implicative WUP-algebras, their properties and interconnection between them. The study of ideals, filters and different types of graphs on all above mentioned classes of algebras will also be interesting.

Abbreviations

WUP-algebras:	Interval-valued neutrosophic set
UPP:	UP-Part

Data Availability

No additional data set is used to support the study.

Conflicts of Interest

The authors hereby declare that there is no conflicts of interest regarding the publication of this paper.

Acknowledgments

All the authors are thankful to their respective institutes for providing the excellent research facilities.

References

Y. Imai and K. Iseki, “On axiom systems of propositional calculi, XlV,” Proceedings of the Japan Academy, vol. 2, pp. 19–22, 1966.
View at: Google Scholar
K. Iseki, “An algebra related with a Propositional Calculus,” Proceedings of the Japan Academy Series A: Mathematical Sciences, vol. 42, no. 1, pp. 26–29, 1966.
View at: Publisher Site | Google Scholar
Q. P. Hu and X. Li, “On BCH-algebras,” Mathematics seminar notes, vol. 11, pp. 313–320, 1983.
View at: Google Scholar
M. A. Chaudhry, A. Fahad, and M. Arshad, “Some results about generalized BCH-algebras,” International Journal of Algebra, vol. 11, no. 5, pp. 231–246, 2017.
View at: Publisher Site | Google Scholar
M. Anwar Chaudhry, A. Fahad, Y. Rao, M. Imran Qureshi, and S. Gulzar, “Branchwise solid generalized BCH-algebras,” AIMS Mathematics, vol. 5, no. 3, pp. 2424–2432, 2020.
View at: Publisher Site | Google Scholar
M. A. Chaudhry, M. I. Qureshi, A. Fahad, and M. S. Bashir, “Isomorphism theorems in generalized -algebras,” Journal of Prime Research in Mathematics, vol. 17, no. 2, pp. 149–158, 2021.
View at: Google Scholar
S. Keawrahun and U. Leerawat, “On isomorphisms of SU-algebra,” Scientia Magna, vol. 7, no. 2, pp. 39–44, 2011.
View at: Google Scholar
W. A. Dudek, “On BCC-algebras,” Logique et. Analyse, vol. 129-130, pp. 103–111, 1990.
View at: Google Scholar
W. M. Bunder, “BCK and related algebras and their corresponding logics,” The Journal of non-classical logic, vol. 7, pp. 15–24, 1984.
View at: Google Scholar
W. A. Dudek, “Algebras inspired by logics,” Matematychni Studii, vol. 14, pp. 3–18, 2000.
View at: Google Scholar
W. A. Dudek, “Algebraic characterization of the equivalential calculus,” Rivista di Matematica Pura ed Applicata, vol. 17, pp. 107–112, 1996.
View at: Google Scholar
C. Prabpayak and U. Leerawat, “On ideals and congruences in KU-algebras,” Scientia Magna, vol. 5, no. 1, pp. 54–57, 2009.
View at: Google Scholar
M. Akram, N. Yaqoob, and M. Gulistan, “Cubic KU-algebras,” International Journal of Pure and Applied Mathematics, vol. 89, no. 5, pp. 659–665, 2013.
View at: Google Scholar
A. Iampan, “A new branch of logical algebra: UP-algebras,” Journal of Algebra and Related Topics, vol. 5, no. 1, pp. 35–54, 2017.
View at: Google Scholar
M. Gulistan, M. Shahzad, and S. Ahmed, “On -fuzzy KU-ideals of KU-algebras,” Afrika Matematika, vol. 26, no. 3-4, pp. 651–661, 2015.
View at: Publisher Site | Google Scholar
M. Akram, N. Yaqoob, and J. Kavikumar, “Interval-valued -fuzzy KU-ideals of KU-algebras,” Afrika Matematika, vol. 92, no. 3, pp. 335–349, 2014.
View at: Google Scholar
S. M. Mostafa, M. A. A. Naby, and O. R. Elgendy, “Interval-valued fuzzy KUideals in KU-algebras,” International Mathematical Forum, vol. 6, no. 64, pp. 3151–3159, 2011.
View at: Google Scholar
S. M. Mostafa, M. A. Abdel Naby, and M. M. M. Yousef, “Anti-fuzzy KU-ideals of KU-algebras,” International Journal of Algebra and Statistics, vol. 1, no. 1, pp. 92–99, 2012.
View at: Publisher Site | Google Scholar
S. M. Mostafa, M. A. A. Naby, and O. R. Elgendy, “Intuitionistic fuzzy KU-ideals in KU-algebras,” International Journal of Mathematical Sciences & Applications, vol. 1, no. 3, pp. 1379–1384, 2011.
View at: Google Scholar
N. Dokhamdang, A. Kesorn, and A. Iampan, “Generalized fuzzy sets in UP-algebras,” Annals of Fuzzy Mathematics and Informatics, vol. 16, no. 2, pp. 171–190, 2018.
View at: Publisher Site | Google Scholar
J. Somjanta, N. Thuekaew, P. Kumpeangkeaw, and A. Iampan, “Fuzzy sets in UP-algebras,” Annals of Fuzzy Mathematics and Informatics, vol. 12, pp. 739–756, 2016.
View at: Google Scholar
A. Iampan, “Derivation of UP-algebras by means of UP-endomorphisms,” Journal of Algebraic Structures and their Applications, vol. 3, no. 2, pp. 1–20, 2016.
View at: Google Scholar
K. Sawika, R. Intasan, A. Kaewwasri, and A. Iampan, “Derivation of UP-algebras,” Korean Journal of Mathematics, vol. 24, no. 3, pp. 345–367, 2016.
View at: Publisher Site | Google Scholar
A. Iampan, “The UP-Isomorphism theorems for UP-algebras,” Discussiones Mathematicae General Algebra and Applications, vol. 39, no. 1, pp. 113–123, 2019.
View at: Publisher Site | Google Scholar
S. Yuphaphin, P. Kankaew, N. Lapo, R. Chinram, and A. Iampan, “Picture fuzzy sets in UP-algebras by means of a special type,” The Journal of Mathematics and Computer Science, vol. 25, no. 01, pp. 37–72, 2021.
View at: Publisher Site | Google Scholar
N. Lapo, S. Yuphaphin, P. Kankaew, R. Chinram, and A. Iampan, “Interval-valued picture fuzzy sets in UP-algebras by means of a special type,” Afrika Matematika, vol. 33, pp. 55–43, 2022.
View at: Publisher Site | Google Scholar

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