Characterizations of Left H-Clifford Semirings by Their H-Ideals

Anjum, Rukhshanda; Mufti, Zeeshan Saleem; Alghazzawi, Dilshad; Al-Zidi, Nasser M.; Ullah Khalid, Muhammad Rizwan

doi:https://doi.org/10.1155/2022/3498344

Mathematical Problems in Engineering

On this page

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

Special Issue

Computational Intelligence and Fuzzy Modeling for Sustainable Decision-Making in Engineering Processes 2022

View this Special Issue

Research Article | Open Access

Volume 2022 | Article ID 3498344 | https://doi.org/10.1155/2022/3498344

Characterizations of Left H-Clifford Semirings by Their H-Ideals

Rukhshanda Anjum,¹Zeeshan Saleem Mufti,¹Dilshad Alghazzawi,²Nasser M. Al-Zidi,³and Muhammad Rizwan Ullah Khalid¹

Academic Editor: Dragan Pamučar

Received16 Jul 2022

Revised05 Aug 2022

Accepted17 Aug 2022

Published16 Sept 2022

Abstract

The main aim of this research is to introduce Left Clifford Semi-rings. Using some basic properties of regular semi-rings we shall investigate several properties of Left Clifford semi-rings and their characterizations. We will also establish that a semi-group Q will be a Left Clifford Semi-group iff the semi-group P (Q) of all subsets of Q is a Left Clifford Semiring. Also, this research will investigate the distributive congruences of regular semi-rings.

1. Introduction

Principal thought of semi-group and monoid is given by Wallis [1]. Wallis communicated that a set which fulfills the associative law under some binary operations is called semi-group and a set which is semi-group with identity is called monoid. A few analysts additionally work on semi-group hypothesis like [2–7] talks about the real concept of group and commutative group, as well as other major points. Several scholars have expounded about semi-ring in their studies. In their analysis, they had checked at different aspects of semi-ring. Vandiver [8] discusses semi-ring and clarifies several main concepts. The algebraic system was invented by Vandiver. He imagined that an algebraic structure is made up of a nonempty set that can be manipulated using the binary operations and . Semirings play a significant role in geometry, but they also play a role in pure mathematics. Several ideas and results relating rings and semi-rings have been introduced by various researchers like [9–12]. In Applied mathematics and information sciences semi-rings have been set up for tackling the various issues. Semirings with commutative addition and zero element are also very necessary in theoretical computer science. The concept of regular semi-rings was introduced by Von Neumann regularity [13] and Bourne regularity [14]. Von Neumann showed that the ring would also be regular if the semi-group is regular. Bourne showed that if there exists such that then the semi-ring will be regular. Due to their rich structure of Clifford semi-ring, it is obvious to look for classes of regular semi-groups close to Clifford semigroups. Zhu et al. [15] presented the idea of left Clifford semi-groups which sum up Clifford semi-groups. A regular semi-group is called left semi-group of Clifford if . A semi-group will be a left Clifford semi-group, if and only if it will be a semilattice of left groups. Since its presentation, semi-group section shows their advantage to this new class of semi-groups for large and clear interpretation of structure [16–18]. According to more abstraction of left Clifford semigroups, Shum, Guo and Zhu have presented left C-rpp semi-groups [19]. We additionally suggest to visit [20–26] as an application of fuzzy and intutionistic fuzzy structures in decision making.

During the study of semi-rings, the fascinating component is the examination of how the structured characteristics of two reducts are powered by distributive laws to collaborate. Bhuniyaa and Sen carry on their work on the k-regular semi-rings in which the two reducts cooperate more than the normal communication constrained by the distributive laws [27–30]. This research presentes left -semifields and left Clifford semi-rings as abstraction of -semifields and -Clifford semi-rings, respectively. Distinct equivalant characterizations and distributive lattice decompositions have been done likewise. It was in 1937 A. H. Clifford (1908–1992) and G. B. Preston, first describe the relation between representation of group and those of a normal subgroup in order to introduce Clifford semigroups. Bhuniya [28] has initiated left Clifford semi-rings as a simplification of Clifford semi-ring. Being motivated from his work we have introduced Left Clifford semi-rings and also described the basic properties of left -Clifford semi-rings and the equivalent chaacterizations we have committed. We will prove that a semi-group will be a Left Clifford semi-group iff the semi-ring of all subsets of will be Left Clifford semi-ring. We will also characterize the structure of Left Clifford semi-rings. A semi-ring will be left Clifford semi-ring iff it will be distributive lattice of the left h-semifields.

In first section of this article we will present its introduction and litrature review then in section two basic definitions will be depicted. The third and forth sections are the main research and in the last we will conclude our research.

2. Preliminaries

Let then it is said to be semi-ring if it satisfies the following conditions: is a semi-group. is a semi-group. Both (left and right) distributive laws hold i.e. and . Thus; is semi-ring which is denoted by (, , ).

Definition 1. An additive subgroupof a ringthenknown as ideal ofif it satisfies the following properties:
. Ifis left and right ideal ofthenis said to be ideal of ring. A semi-groupwill be known as regular semi-group ifsuch that. Considerbe a semi-ring then for eachsuch thatif and only ifbe a regular semi-ring. Letbe a semi-ring and letis a left ideal ofif the following conditions satisfied:(1) (2) for and .The right ideal ofis defined dually. Ifis left as well as right ideal ofthen I will be called as Ideal of semi-ring.

Definition 2. Letbe a semi-ring andwill be-closure ofdefined as:Suppose that be a semi-ring and be a h-closure of then will be known as -set if .
Let be a semi-ring and is a left ideal of then willl be known as left ideal if:
. The right ideal can be defined dually. Let then is called Band if following axioms satisfied: (i) is semi-group. (ii) each element of is idempotent. .
From [31] we let then will be semilattice if is commutative band. is often a semi-ring the additive reduct is a semilattice and this class of all such semi-ring is denoted by . A semi-ring (,+,.) without zero will be known as (left, right) simple if it has no proper (left, right) -ideal. A semi-ring ( ,+,.)with zero will be known as simple (left, right) if are its only ideals (left, right).

Definition 3. A regular semi-groupwill be called Clifford semi-group ifi.e. idempotent ofcommute with all elements of; whereis set of all idempotents of andIs centre of. Let be a semi-group thenwill be called left Clifford semi-group ifis regular semi-group as well as [18].

Definition 4. A semi-ringwill be known asregular ifsuch thatfor [30]. Ifis a semilattice then we can say that a semi-ringis known asregular if and only if for eachsuch thatfor. Letbe a semi-group and be the hypersemiring of; where,“” and “” is defined as: and thenis a semi-ring whose additive reduct is a semi-lattice that is knwon as hyper semi-ring of semi-group. Letbe a semi-ring and letthen .. will be known asinverse ofifandfor some. Ifthen “” isinverse of. Therefore in anregular semi-ring every component has ainverse. The set ofinverses ofinis denoted by.
We will inagurated Green’s relation on semi-rings in [27], as introduced by Sen and Bhuniya in the following way: for any ; if and only if ; if and only if ; if and only if and . These are the equivalence relations of additive congruences on , where represents the multiplicative right and is just multiplicative left and also is just ideal congruence on . For an element of a semi-ring will be known as idempotent if for some . represents the set of all idempotent. For all , but . Semiring in which for some [30]. The class of semi-rings in constrained by extra character is diversity of . Some sub-classes of are decipated in [29]. Due to their structure we can call them almost-idempotent-semirings. An regular semi-ring in will be knowns as Left semifield if for all and there exists such that for some . It follows that a semi-ring is a left semifield iff for all and there exists such that for some . Right semifield can be defined dually. A semi-ring will be known as semifield if it will be both left and right semifield.

Lemma 1. Letbe a semi-ring inand.(1)If there exists and for some such that then,(2)If and for some and then,(3)If there exists such that then,(4)If and for some and then,(5)If and then,(6)If for some then,(7)If and then,

Proof. (1)As Now to prove Hence desired result is obtained.(2)To prove Taking Similarly to prove. Taking(3)To prove Taking By using and as . Taking By using and as . So we have .(4)To prove . Taking Similarly it is easy to prove that(5)Since and . As adding on both sides we get(6)Since Now; for(7)As Also; Adding we have: say.

Definition 5. Aregular semi-ringwill be known as leftClifford semi-ring if for allsuch thatfor some.otherwise; By (Lemma 1); It shows thatis LeftClifford semi-ring iff such thatforsome.

Lemma 2. Forbe a semi-ring inthen;(1)forfor somethen the following conditions are identical: there aresuch that (A₂) there aresuch that (A₃) there issuch that(2)iffor someare such that and for somethen there issuch that(3)iffor someare such that and for somethen there issuch that and

Proof. (1)As and are obvious. Now; for , suppose that For Now for i.e. Thus Hence has been proved.(2)Put For , taking Now, By and we have . Hence Proved.(3)Put ForNow forHence proved.

Theorem 1. The hypersemiringisregular if and only ifis regular semi-group.

Proof. Suppose that is regular. We have to prove that is regular semi-group. Now let then and then also there is s.t. for some i.e. such that for some . Thus is a regular semi-group.
Conversely, suppose that be a regular semi-group and let then such that . We will choose one such “ ” and denote it by then such that which implies that . Thus is regular semi-ring.

3. Characterizations of Left Clifford Semiring

This section is comprised of some characterizations of Left Clifford Semiring.

Proposition 1. Letbe a semi-ring then the semi-ringis the leftsemifieldis a left-group.

Proof. Suppose that be a left-group. We need to show that is left semifield. Since is a left-group then is regular and left-simple-semigroup then is regular semi-ring. Now let and then there exist . Also let . Since is left simple so such that and for some . Let then we have for some . Thus is left semifield.
Conversely, suppose that be a left semifield then regularity of shows that is a regular semi-group. Let then so such that where and so . Hence such that is left-simple-semigroup. Thus is a left-group.

Proposition 2. Letbe a leftsemifield withandifthen.

Proof. Let then such that for some . Also such that for some then for some otherwise would be zero.

Theorem 2. A semi-ringwith “0″ is a leftsemifield iffisregular and leftsimple.

Proof. Suppose that is a left semifield. Let be a left ideal and let be an element of . Let then such.that for some . Since is left ideal so and so . Thus is a left simple.
Conversely, suppose that be an regular and left simple. Let and , since is regular so for some then and hence such that for some . Thus is left semifield.

Theorem 3. Letbe a semi-group thenis a leftClifford semi-ringis a Left Clifford semi-group.

Proof. Let then , this implies that s.t. for some , . Hence there exists such that for some . Thus and so is a left Clifford semi-group.
Conversely, let and let and . Since is left Clifford semi-group so there exist such that for some . For each and we choose one such “ ” and denote it by “ ”. Since both and are finite so . Also and . Hence where . Thus is a left Clifford semi-ring.

Theorem 4. Letbe a semi-ring then the following conditions are equivalent:(1) is left Clifford.(2) is regular and for all .(3) is regular and for all .(4).(5)All left ideals are two sided and for any two ideals of .(6) for any two ideals of .

Proof. . This is trivial.
. Let then such thatAnd hencethen and implies that there exist such that then;Hence .
Let such that then such thatAgain by such thatimplies thatThus, and so .
Let be a left ideal of . Consider and . let be such thatthenThis implies that there is s.t.Then implies that .
Hence there exist s.t. thus, i.e. hence is two-sided ideal of .
Let , be two sided ideal of then , implies that and so . Now let since is a regular semi-ring s.t. then whence and so .
Thus, Trivial.
Let since is regular semi-ring there exist s.t.Which implies thatThenSo .
Hence there are such that . Thus is left Clifford semi-ring.

Theorem 5. Letbe aregular semi-ring then the following conditions are equivalent:(i) is left Clifford semi-ring(ii)for all and such that for some (iii)for all and such that for some (iv)for all and such that for some

Proof. Let and . Since is a left Clifford semi-ring so which implies that there exisst such that . Again regularity of implies that there is such thatWhich implies that Let and then the regularity of together with implies that such thatSince such thatImplies that Let and thenAnd soWhich implis thatWhere then .
Hence such that thus we get Let then such that.
Also, hencewhere then and so there exist such thatThen implies thatWhich again implies that .
Thus is a Clifford semi-ring.

Theorem 6. A semi-ringis leftClifford if and only if for allthere existssuch thatfor some.

Proof. Let be a left Clifford semi-rings and then such that for again, implies that i.e. for
Then we have and where then, and which implies that for , for .
Conversely, suppose that then s.t.HenceAgain we have s.t.Then , s.t. for .
Thus, is a left Clifford semi-ring.

4. Structure of Left Clifford Semiring

In this section we will work on the structure of Clifford semi-ring. The resolved by the least Distributive Lattice Congruence “ ” on are left semifields for a Left Clifford semi-ring . Hence each left Clifford semi-ring will be distributive lattice of left semifields.

Definition 6. Letbe a semi-ring andthen leftcentralizer ofis defined by

Theorem 7. A semi-ringis a leftClifford semi-ring if and only ifisregular andfor all.

Proof. Let is left Clifford semi-ring then s.t. thenThus .
Conversely, suppose that . Consider then similarly we can get s.t.Then , and so . Since is a subsemilattice of the additive reduct . Now implies that there exist such thatThen we haveThus, is a left Clifford semi-ring.

Definition 7. Suppose thatis a class of semi-rings we will call the semi-rings inassemi-rings. A semi-ringwill be known as a distributive lattice ofsemi-rings ifcongruenceonsuch thatis a distributive lattice and eachclass is a semi-ring in.

Theorem 8. Letbe aregular semi-ring thenis a LeftClifford semi-ringis the least distributive lattice congruences on.

Proof. Suppose that is a left Clifford semi-ring then . and so [27]. Again in a regular semi-ring Therefore .
Let be such that and then s.t.Since is a left Clifford semi-ring so s.t. for some , then we getand similarlyWhich implies that .Thus is a congruence on .
Now let then such thatSince is a left Clifford semi-ring so there exists such thatThen we have and hence .
Let then s.t.Since is a left Clifford semi-ring so there exists such that and hence such thatthen we haveSimilarly; such that ; for thus; .
Now, let then s.t.then we haveagain implies that.
Hence, . Thus is a distributive lattice congruence on .
Now let “ ” be a distributive lattice congruence on . Let be s.t. then s.tThen,Shows that .
Thus is the least distributive lattice congruence on .
For the converse, let then implies that such that ; for .
This shows that is left Clifford semi-ring.

Theorem 9. Letberegular semi-ring thenis a distributive lattice of leftSemifieldsis leftClifford semi-ring.

Proof. Suppose that is distributive lattice of left semifields . Let then such that and hence, since, is a left semifield there exist s.t.Implies thatSince, is semilattice i.e . Thus, is a left Clifford semi-ring.
Conversely, suppose that be a left Clifford semi-ring then is a distributive lattice congruence on . So, each class is a subsemiring of . Let be a class and then and hence which implies that there exist such thatAlso there exist such thatThen there exist such that and Take then we haveWhich shows that i.e. and such that hence is a left semifield. Thus each class is a left semifield. Therefore is a distributive lattice of left semifields.

5. Conclusion

This research represents the structure and characterizations of Left Clifford Semirings. This article also explains some proofs relevant to the specific properties of regular semi-rings. Several properties of left Clifford semi-rings and the equivalent characterizations have been proved in this research. We have shown that a semi-group is left Clifford semi-group the semi-ring of all subsets of is Left Clifford semi-ring. Lastly, we worked on the structure of Left Clifford semi-ring. We prove that a semi-ring is left Clifford semi-ring iff it is distributive lattice of left semifield. In future a similar work can be done for other algebraic structures and one can apply these concepts in decision making problems.

Data Availability

No data is used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to thank the Deanship of Scientific Research of King Abdulaziz University, Jeddah, Saudi Arabia, for technical and financial support.

References

A. R. Wallis, “Semigroup and Category-Theoretic Approaches to Partial Symmetry,” 2017, https://arxiv.org/pdf/1707.02066.
View at: Google Scholar
A. H. Clifford and G. B. Preston, Algebraic Theory of Semigroups, American Mathematical Soc, Providence, RI, USA, 1967.
J. M. Howie, Fundamentals of Semigroup Theory (No. 12), Oxford University Press, Oxford,UK, 1995.
J. S. Milne, Algebraic groups: the theory of group schemes of finite type over a field, Cambridge University Press, Cambridge UK, vol. 170, 2017.
N. Kausar, M. Munir, M. Gulzar, and M. Alesemi, “Ordered LA-groups and ideals in ordered LA-semigroups,” Italian Journal of Pure and Applied Mathematics, vol. 44, pp. 723–730, 2020.
View at: Google Scholar
N. Kausar, M. Munir, B. Islam, M. Alesemi, and M. G. Salahuddin, “Ideals in LA-rings,” Ital. J. Pure Appl. Math, vol. 44, pp. 731–744, 2020.
View at: Google Scholar
N. Kausar, M. Munir, M. Gulzar, G. M. Addis, and R. Anjum, “Anti fuzzy bi-ideals on ordered AG-groupoids,” Journal of the Indonesian Mathematical Society, vol. 26, no. 3, pp. 299–318, 2020.
View at: Publisher Site | Google Scholar
H. S. Vandiver, “Note on a simple type of algebra in which the cancellation law of addition does not hold,” Bulletin of the American Mathematical Society, vol. 40, no. 12, pp. 914–920, 1934.
View at: Publisher Site | Google Scholar
T. Vasanthi and N. Sulochana, “On the additive and multiplicative structure of semirings,” Annals of Pure and Applied Mathematics, vol. 3, no. 1, pp. 78–84, 2013.
View at: Google Scholar
A. Kaya and M. Satyanarayana, “Semirings satisfying properties of distributive type,” Proceedings of the American Mathematical Society, vol. 82, no. 3, pp. 341–346, 1981.
View at: Publisher Site | Google Scholar
N. Kausar, M. Munir, M. Gulzar, G. Mulat Addis, and M. Gulistan, “Study on left almost-rings by anti fuzzy bi-ideals,” International Journal of Nonlinear Analysis and Applications, vol. 11, no. 2, pp. 483–498, 2020.
View at: Publisher Site | Google Scholar
D. Alghazzawi, W. H Hanoon, M. Gulzar, G. Abbas, and N. Kausar, “Certain properties of ω-Q-fuzzy subrings,” Indonesian Journal of Electrical Engineering and Computer Science, vol. 21, no. 2, pp. 822–828, 2021.
View at: Publisher Site | Google Scholar
J. Von Neumann, “On regular rings,” Proceedings of the National Academy of Sciences, vol. 22, no. 12, pp. 707–713, 1936.
View at: Publisher Site | Google Scholar
S. Bourne, “The Jacobson radical of a semiring,” Proceedings of the National Academy of Sciences, vol. 37, no. 3, pp. 163–170, 1951.
View at: Publisher Site | Google Scholar
P. Y. Zhu, Y. Q. Guo, and K. P. Shum, “Structure and characteristics of left Clifford semigroups,” Science in China - Series A: Mathematics, Physics, Astronomy & Technological Sciences, vol. 35, no. 7, pp. 791–805, 1992.
View at: Google Scholar
G. U. O. Yuqi, “Structure of the weakly left C-semigroups,” Chinese Science Bulletin, 1996.
View at: Google Scholar
M. K. Sen, S. Ghosh, and S. Pal, On a Class of Subdirect Products of Left and Right Clifford Semigroups, Taylor & Francis, UK, 2004.
K. P. Shum, M. K. Sen, and Y. Q. Guo, “Subdirect product structure of left Clifford semigroups,” in Proceedings of the Words, Languages & Combinatorics III, pp. 428–433, Kyõto, Japan, March 2000.
View at: Google Scholar
G. Yuqi, K. P. Shum, and Z. Pinyu, “The structure of left C-rpp semigroups,” Semigroup Forum, vol. 50, no. 1, pp. 9–23, 1995.
View at: Google Scholar
L. Du and K. P. Shum, “On left C-wrpp semigroups,” Semigroup Forum, vol. 67, no. 3, pp. 373–387, 2003.
View at: Google Scholar
D. Bozanic, D. Tešić, D. Marinković, and A. Milić, “Modeling of neuro-fuzzy system as a support in decision-making processes,” Reports in Mechanical Engineering, vol. 2, no. 1, pp. 222–234, 2021.
View at: Publisher Site | Google Scholar
R. E. Precup, S. Preitl, E. Petriu et al., “Model-based fuzzy control results for networked control systems,” Reports in Mechanical Engineering, vol. 1, no. 1, pp. 10–25, 2020.
View at: Publisher Site | Google Scholar
K. Mohanta, A. Dey, and A. Pal, “A study on picture dombi fuzzy graph,” Decision Making: Applications in Management and Engineering, vol. 3, no. 2, pp. 119–130, 2020.
View at: Publisher Site | Google Scholar
I. Badi and D. Pamucar, “Supplier selection for steelmaking company by using combined Grey-MARCOS methods,” Decision Making: Applications in Management and Engineering, vol. 3, no. 2, pp. 37–48, 2020.
View at: Publisher Site | Google Scholar
E. K. Zavadskas, Z. Turskis, Ž. Stević, and A. Mardani, “Modelling procedure for the selection of steel pipes supplier by applying fuzzy AHP method,” Operational Research in Engineering Sciences: Theory and Applications, vol. 3, no. 2, pp. 39–53, 2020.
View at: Publisher Site | Google Scholar
A. Blagojević, S. Vesković, S. Kasalica, A. Gojić, and A. Allamani, “The application of the fuzzy AHP and DEA for measuring the efficiency of freight transport railway undertakings,” Operational Research in Engineering Sciences: Theory and Applications, vol. 3, no. 2, pp. 1–23, 2020.
View at: Publisher Site | Google Scholar
M. K. Sen, S. K. Maity, and K. P. Shum, “On completely regular semirings,” Bulletin of the Calcutta Mathematical Society, vol. 98, no. 4, pp. 319–328, 2006.
View at: Google Scholar
M. K. Sen and A. K. Bhuniya, “On additive idempotent k-clifford semirings,” Southeast Asian Bulletin of Mathematics, vol. 32, no. 6, 2008.
View at: Google Scholar
M. K. Sen and A. K. Bhuniya, “The structure of almost idempotent semirings,” Algebra Colloquium, vol. 17, no. 1, pp. 851–864, 2010.
View at: Publisher Site | Google Scholar
M. K. Sen and A. K. Bhuniya, “On semirings whose additive reduct is a semilattice,” Semigroup Forum, vol. 82, no. 1, pp. 131–140, 2011.
View at: Google Scholar
A. K. Bhuniya and K. Jana, “Semigroup of k-bi-Ideals of a semiring with semilattice additive reduct,” Demonstratio Mathematica, vol. 49, no. 2, pp. 119–128, 2016.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2022 Rukhshanda Anjum 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.

PDF Download Citation

Download other formats

Order printed copies

Views

162

Downloads

286

Citations