Abstract
This paper investigates the properties and results of (Q,L)-fuzzy soft subhemirings ((Q,L)-FSSHR) of a hemiring R. The motivation behind this study is to utilize the concept of L-fuzzy soft set of a hemiring and to derive a few specific outcomes on (Q, L)-FSSHR. The concepts of strongest Q-fuzzy soft set relation, Q-isomorphism, pseudo-Q-fuzzy soft coset, and some of their related properties are implemented while analyzing the results. Finally, the properties are verified with a numerical example from the 2000 AMS subject classification: 05C38, 05A15, and 15A18.
1. Introduction
The pioneering work on fuzzy sets was presented by Zadeh in [1]. Following this, intuitionistic fuzzy sets (IFS) were introduced by Attansov [2] and Agarwal et al. [3] who studied the generalized IFS with applications in decision making. Goguen [4] generalized fuzzy sets to L-fuzzy sets. Molodtsov introduced soft sets and Maji et al. [5] did a combined study on fuzzy sets and soft sets and introduced fuzzy soft sets. After that, Maji et al. [6] started a combined work on IFS and soft sets called intuitionistic fuzzy soft sets (IFSS). Hooda et al. [7] introduced the intuitionistic fuzzy soft set theory and its applications in medical diagnosis.
Anjan et al. [8] introduced fuzzy soft multiset and presented some results on this. The concept of an approximation space associated with each parameter in a soft set is discussed in [9]. The intuitionistic neutrosophic soft set was studied in [10] and reformulated by Feng [11] and introduced into soft semiring by means of level soft sets and an adjustable approach to the fuzzy soft set. The notion of semiring was introduced by Vandiver and Anjum et al. [12] to characterize the hemirings by falling fuzzy k-ideals. In [13], vague sets were introduced and analyzed the various operations. Under a vulnerability climate, the neutrosophic soft sets [16] have been effectively applied, and numerical models have been effectively applied in dynamic issues. By using these definitions, the applications of the soft set hypothesis have been concentrated progressively.
In this paper, by introducing the concepts of hemiring and subhemiring for fuzzy soft sets along with L-fuzzy soft sets, the logarithmic structure of (Q, L)-fuzzy soft subhemiring concept is developed and investigated for their major properties. Fuzzy soft sets were also applied as an operational tool of (Q, L)-fuzzy soft relation homomorphic prepicture and synthesis activity and not many of its connected properties are broken down [16]. Molodtsov [17] presented the soft set hypothesis which draws in many creators since it has a wide scope of utilizations in fields of dynamic, gauging, and information examination. Presently, many researchers attempt to hybridize the soft set with various numerical models such as in [18], Shabir et al. [19] studied the characterization of hemirings by the properties of their k-ideals. Muhammed [21] discussed fuzzy ideals in nearing with respect to t-norm and investigated quotients near rings. Dudek [22] introduced the notion of intuitionistic fuzzy left k-ideals of semirings and min-max-plus semiring connections with left k-ideals of the corresponding semirings. Shabir et al. [23] studied on k-bi-ideals in hemirings. The ambiguous soft set [24] and the reluctant fuzzy soft set [25] are presented and then further augmentations of soft sets such as the span esteemed fuzzy soft set [26], the multifuzzy soft set [27], and the trapezoidal fuzzy soft set [28] were carried out. Xu et al. [29] presented the vague delicate sets and their properties.
This universe is stacked with second thoughts, imprecision, and vagueness. In actuality, most of the thoughts we bargain contain muddled data in contrast to accurate. Overseeing the second thought, vulnerability is a vital issue in various regions, for instance, financial aspects, designing, regular science, medicinal science, and social sciences. Countless researchers have ended up with the same conclusion, exhibiting a lack of definition (Table 1).
2. Preliminaries
In this section, we provide some basic definitions those are related to this study.
Definition 1 (see [8]). A pair is identified as a soft set is a function in to these to fall subset of the set .
Example 1. Suppose is the set of five laptops under consideration. Here, let and be the set of parameters.
Definition 2 (see [14]). Let , . A -fuzzy of , then .
Definition 3 (see [7]). Let be a hemiring. A –FS of is called a -fuzzy soft subhemiring (LFSSHR) of if the following axioms hold:(1)(2), and .
Example 2. Let . Then, consider given by . Then, , and . All these sets are subhemirings of . Therefore, is a soft subhemiring over .
Definition 4 (see [10]). Let be a hemiring. A –FS of is called -fuzzy soft subhemiring (Q-FSSHR) if the following conditions hold:(1)(2) for all and in and qQ.
Definition 5 (see [19]). If (, +,.) and are any two hemirings and is a nonempty set, then is called a -known as a Homomorphism if and for all , and in .
Definition 6 (see [17]). If (, +,.) and are any two hemirings and is a nonempty set, then is known as an anti--homomorphism if and , , , and in .
Definition 7 (see [9]). Let be a -fuzzy soft subset. For in , the sets and is known as (Q, L)-fuzzy soft level -cut.
Definition 8 (see [16]). Let be a -fuzzy soft subhemiring of . For in , the level subset of is the set . This is known as a -fuzzy soft level subset.
Definition 9 (see [18]). Let be a -FSSHR of a hemiring (, +,.). The level subhemiring , for in with the end goal that , is known as a -fuzzy soft level soft subhemiring.
3. Some Properties of -Fuzzy Soft Subhemirings of a Hemiring
In this section, we provide main results using properties of -FSSHR of a hemiring.
Theorem 1. If is a (Q, L)-FSSHR of a hemiring , then is either empty or is a subhemiring of .
Proof. If no elements satisfy these conditions, then is empty.
Let in , thenThus, . For every and in R and q in Q. We get in H. So, , for all , and in R and q in Q. We get in . Thus, is a subhemiring of .
Theorem 2. Let be a (Q, L)-FSSHR of a hemiring . Then,(1)If , then either or , for and in R and q in Q.(2)If , then either or , for every and of and qQ.
Proof. Let and . By the definition,We haveThus, whichever or , for every and in R and q in Q,We have or , for every and of and Q.
Theorem 3. Let and (, D) be two (Q, L)-FSSHR of a hemiring (, +D). Then, their intersection is a (Q, L)-FSSHR of .
Proof. Let and be in the right place to . Then,andLet and , where . Now,for every and in and in . Again,Thus, , for each and in and in .
Theorem 4. Theof a-FSSHR of a hemiring (, +, .) is a -FSSHR of .
Proof. Consider as a family of -FSSHR of a hemiring and . Here, , we have the following two cases:
Case 1. Therefore,for each and of and Q.
Case 2. Thus, , for each and of and .
Theorem 5. If and (R, D) are any two -FSSHR of a hemiring (, +, .), then is a -FSSHR of .
Proof. Consider and belonging to and ,Let and , where . Now,Thus, , for each and in and in .
Now,So, , for each and in and in .
Theorem 6. The of a -FSSHR of a hemiring is a -FSSHR of .
Proof. Given as a chance consider to be a family of a (Q, L)-FSSHR of a hemiring and . Then, and belong to and q in Q,Thus, , for each and in and q in Q.Thus, for each and of and Q.
Similarly, the union of a family of a -FSSHR of a hemiring is a -FSSHR of .
Theorem 7. If and (, D) be two -FSSHR of the hemirings and , correspondingly, then the product is a -FSSHR of .
Proof. Consider and to be two -FSSHR of the hemirings and correspondingly. Let and be in , and and be in S. Then, and are in . Now,Thus,for each and in and and in and in . Again,Thus,for every and in and and in and in .
Similarly, the product is a -FSSHR of . □
4. Properties of -Fuzzy Soft Subhemiring of a Hemiring
Some additional properties of -FSSHR of a hemiring are discussed as follows:
Theorem 8. Let be a FS subset of a hemiring and be the strongest -fuzzy soft relation of . Then, is a -FSSHR subhemiring of is a -FSSHR of .
Proof. Assume that is a -FSSHR of . Here, and are in . Now,Therefore, for each and in and in . Again,Thus, for all and in and in . Hence, is a -fuzzy subhemiring of .
Furthermore, we consider that is a -fuzzy subhemiring of , then and are in .Ifandwe get for every and in and in . Again,Ifandwe get for every and in and . We have which is a -fuzzv soft subhemiring of .
In the following theorem, we use the composition operation of functions.
Suppose .
We can define a composition mapping
Also, using some properties of -FSSHR of a hemiring with the and composition operation of functions with the some more theorem are derived.
Theorem 9. Let be a -FSSHR of a hemiring and is -isomorphism from a hemiring onto . Then, is a -FSSHR of .
Proof. Let and in and be a Q-FSSHR of . Then,for each and in andfor each and in .
Theorem 10. If is a (Q, L)-FSSHR of , then the pseudo-Q-fuzzy soft coset is a (Q, L)-FSSHR of , for each .
Proof. Let be a -FSSHR of , for each . Then,Thus, for every and in and in and for every and in .Thus, for all and in . In this manner, is a -FSSHR subhemiring of a hemiring .
Theorem 11. If is a (Q, L)-FSSHR of a hemiring , then is either empty or a (Q, L)-fuzzy soft subhemiring of .
Proof. In the event that no segment holds the axioms, by then is unfilled. If and satisfy this condition, thenThus, for each and in R and in Q.Thus, for each and in . In this way, is either empty or a -fuzzy soft subhemiring of .
5. Conclusion
The main idea of this research work is to briefly explain and establish the results, the properties, and some theorems on the morphism of (Q, L)-FSSHR of a hemiring. This work can be extended in future to ideals of (Q, L)-fuzzy soft subhermiring and inter-valued (Q, L)-FSSHR of a hemiring. We trust that this work will have a profound effect on the forthcoming exploration in this field and other soft algebraic investigations to open up new horizons of premium and advancement.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors are grateful to the assessors for their precious implication supportive in convalescing this manuscript. This research work was sustained by Periyar University. The work of U.F.-G. was supported by the government of the Basque Country for the ELKARTEK21/10 KK-2021/00014 and ELKARTEK22/85 research programs, respectively.