Journal of Mathematics

Journal of Mathematics / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 5562828 | https://doi.org/10.1155/2021/5562828

Ying Wang, Abid Mahboob, Abdul Rauf Khan, Zafar Ullah, Zahid Karim, Mamoona Karim, "On -Multiplier on Almost Distributive Lattices", Journal of Mathematics, vol. 2021, Article ID 5562828, 4 pages, 2021. https://doi.org/10.1155/2021/5562828

On -Multiplier on Almost Distributive Lattices

Academic Editor: Efthymios G. Tsionas
Received11 Feb 2021
Revised09 May 2021
Accepted20 May 2021
Published02 Jun 2021

Abstract

In this paper, we initiate the concept of -multiplier on almost distributive lattices. We prove some useful results by using the notion of -multiplier and generalize the idea of multiplier on almost distributive lattices.

1. Introduction

A lattice is an advanced abstract structure that has been studied in abstract algebra during last few decades. Birkhoff introduced the concept of lattice theory in 1940 [1]. Lattice is generalization of Boolean and fuzzy algebras. Latter on Gratzer and Schmidt worked together and showed their interest in the development of lattice theory [2]. In 1955, Helgason introduced the concept of multiplier in Banach Algebra [3]. The idea of multiplier in lattice was given by Larsen [4] in 1971, and Cornish extended this concept of multiplier in distributive lattice [5].

In 1981, the idea of ADLs was initiated by Swamy and Rao [6]. An almost distributive lattice satisfies all the properties of distributive lattice except commutativity of and and right distributivity of over . Recently, Kim has introduced the idea of a multiplier in ADLs [7] and discussed some fundamental properties of this notion. For detailed study of the subject, we refer to readers [810].

Now, we have generalized certain properties of -multiplier. The notion of -multiplier for an almost distributive lattice is introduced, and some related properties are investigated. Moreover, we introduced principle -multiplier and isotone -multiplier on almost distributive lattices.

2. Preliminaries

Definition 1. (see [6]). An algebra is said to be an almost distributive lattice if it satisfies the following:(i)(ii)(iii)(iv)(v)(vi)(vii)(viii), , ,

Lemma 1. (see [6]). Let be an almost distributive lattice. For any , , we have(i)(ii)(iii)(iv)(v)(vi)(vii)

Definition 2. (see [6]). For any , , we say that if or equivalently, .

Lemma 2. (see [6]). Let be an almost distributive lattice. For any , , , , then the following identities hold:(i) and (ii) whenever (iii)(iv) and

Definition 3. (see [6]). Let be a lattice, and 0 is known as a zero element of a lattice if .

Lemma 3. (see [6]). Let be an almost distributive lattice. If has 0, then for any , , the following identities hold:(i) and (ii)(iii) if and only if

Definition 4. (see [6]). Let be a nonempty subset of which is called an ideal of if and whenever , and .
If is an ideal of and , , then if and only if .

Lemma 4. (see [6]). For any , , we have(i)(ii)(iii)

Definition 5. (see [7]). Let be an almost distributive lattice and be two self maps. We define by .

Definition 6. (see [7]). Let and be two almost distributive lattices. Then, is also an with respect to the pointwise operation given by ,

Definition 7. (see [7]). Let be an almost distributive lattice and be a multiplier of . Define a set by .

Definition 8. (see [7]). Let be an almost distributive lattice. For any , define , where is a principle multiplier induced by .

3. -Multiplier on Almost Distributive Lattices

Definition 9. Let be an almost distributive lattice. A function is called -multiplier if , , where is a mapping on .

Example 1. Let be an almost distributive lattice with . A function defined by is called zero -multiplier.

Proof. Let be an almost distributive lattice with ; then, we have to prove that is a zero -multiplier.
Let , . Hence, is a zero -multiplier.

Lemma 5. Let be -multiplier of . If is homomorphism, then following conditions hold:(i)(ii),

Proof. (i)Since , it implies that .(ii)Let , . We have to show that .. It implies .

Definition 10. Let be an almost distributive lattice. A function is defined by , where is a homomorphism, then is multiplier of , and such multiplier of is called a principle multiplier of .

Definition 11. Let be an almost distributive lattice and be -multiplier on . For any , define , where is a principle -multiplier induced by .

Lemma 6. Let be an almost distributive lattice. A function is defined by , where is a homomorphism, then is -multiplier of , and such -multiplier of is called a principle -multiplier of .

Proof. Let , , , and be an -multiplier; then, we have to prove that is an -multiplier.. This implies that is an -multiplier.

Definition 12. Let be an almost distributive lattice and be multiplier on , where is a mapping on . If for implies , then is an isotone multiplier.

Proposition 1. Let be an almost distributive lattice. If is an increasing homomorphism, then , and is an isotone -multiplier of .

Proof. Let be an and , with such that , then we have . It implies . Hence, is an isotone -multiplier.

Lemma 7. Let be an almost distributive lattice and be an -multiplier of and be an increasing homomorphism on . If and , then .

Proof. Let , for . Since is an increasing homomorphism, soBy using equation (3), we have since is an increasing homomorphism. It implies that .

Theorem 1. Let be an almost distributive lattice and be an -multiplier of and be an increasing homomorphism on . Then, is an isotone -multiplier.

Proof. Suppose , . By using Lemma 5 (i), we haveSince , therefore, we have . By equations (4) and (5), we have . It implies . Hence, is an -multiplier.

Proposition 2. Let be an almost distributive lattice, be an -multiplier of , and be homomorphism on . Then, , , .

Proof. Let , and be an -multiplier of ; then, we have to show that . By Definition 1, we have . By Definition 1, we have . It implies that .

Proposition 3. Let be an almost distributive lattice and and be two -multipliers of . Then, is also an -multiplier of .

Proof. Let be an and and be -multiplier such thatLet , , , and be -multipliers of . Now, by Definition 5, we have . By Definitions 1 and 5, we have which along with equation (6) implies that (. Hence, is called -multiplier of .

Proposition 4. Let and be two almost distributive lattices with 0. A function defined by and is a homomorphism. Then, is -multiplier of with pointwise operation.

Proof. Let and be two with 0. We define a mapping byThen, we have to show that is an -multiplier with pointwise operation such thatLet . By Definition 6, . By using equation (7) and Definition 1, we have . By Definition 6 and equation (7), we have . This implies that is an -multiplier with pointwise operation on .

Theorem 2. Let be an almost distributive lattice and be the set of all -multipliers of . Then, under binary operations and is an almost distributive lattice, where for any , , .

Proof. Let , . Then, by equation (9), we have (. This implies that is an -multiplier. Let , , and along with equation (9), we have . This implies that () is an -multiplier. Hence, (, , ) is closed under , . Hence, (, , ) is an .

Theorem 3. Let be an almost distributive lattice and be the set of all -multiplier on . Then, set of all principal -multiplier is distributive lattice with the following operation and  for all , .

Proof. Let , . Then, . For some , it implies that . Also . For any , it implies . Hence, (, , ) is closed, and so is sub almost distributive lattice. Moreover, for any , . Thus, . Hence, is a distributive lattice.

4. Conclusion

In this paper, we have generalized the idea of multiplier to -multiplier in almost distributive lattices and investigated some properties of ADLs. We have also explored some results by using the notion of principal -multiplier and isotone -multiplier. This generalized concept played a vital role in exploring different properties of almost distributive lattices.

Data Availability

The data used to support this study are included within this paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Ying Wang analyzed the results, drafted the final version of the paper, and arranged funding for this paper. Abdul Rauf Khan and Zafar Ullah proved the results. Zahid Karim and Abid Mahmoob approved the results and supervised this work. Mamoona Karim wrote the first version of the paper.

Acknowledgments

This work was supported by the National Key R&D Program of China (no. 2018YFB1005104), the Guangzhou Academician and Expert Workstation (no. 20200115-9), and Key Disciplines of Guizhou Province-Computer Science and Technology (no. ZDXK [2018]007). This work is supported by the Innovation Projects of Universities in Guangdong Province (No. 2020KTSCX215).

References

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Copyright © 2021 Ying Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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