# Generalization of (Q,L)-Fuzzy Soft Subhemirings of a Hemiring

Authors: K. Geetha; N. Anitha; S. Noeiaghdam; U. Fernandez-Gamiz; S.S. Santra; K.M. Khedher
Journal: Advances in Fuzzy Systems
 (2022)

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## 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.


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## Body

## 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 (Table1).Table 1
How the person chooses the laptop.
LaptopGood-lookingQualityStorage spaceModern technologyPricel101011l210100l301010l410100l500001


## 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⇔G is a function K in to these to fall subset of the set U.Example 1.
SupposeU is the set of five laptops under consideration. Here, let U=l1,l2,l3,l4,l5 and K=p1good looking,p2quality,p3storage space,p4modern technology,p5price be the set of parameters.(1)K,G=p1,l2,l4,p2,l1,l3,p3,l4,l2,p4,l3,l1,p5,l1,l5.Definition 2 (see [14]).
LetX, Q≠ϕ. A Q-fuzzy ⊆H of X, then H:X×Q⟶0,1.Definition 3 (see [7]).
Letℜ be a hemiring. A L–FS ⊆S1,J1 of ℜ is called a L-fuzzy soft subhemiring (LFSSHR) of ℜ if the following axioms hold:(1)
jS1,J1uS1,J1+vS1,J1≥jS1,J1uS1,J1∧jS1,J1vS1,J1(2)
jS1,J1uS1,J1vS1,J1≥jS1,J1uS1,J1∧jS1,J1vS1,J1, ∀uS1,J1 and vS1,J1∈ℜ.Example 2.
LetR=A=Z6=0,1,2,3,4,5. Then, consider F:R⟶ℜR given by Fx=y∈R,x.y=0. Then, F0=R,F1=0,F2=0,3,F3=0,2,4,F4=0,3, and F5=0. All these sets are subhemirings of R. Therefore, S1,J1 is a soft subhemiring over R.Definition 4 (see [10]).
Letℜ be a hemiring. A Q,L–FS ⊆H of ℜ is called Q,L-fuzzy soft subhemiring (Q-FSSHR) if the following conditions hold:(1)
jS1,J1uS1,J1+vS1,J1,q≥jS1,J1uS1,J1,q∧jS1,J1vS1,J1,q,(2)
jS1,J1uS1,J1vS1,J1,q≥jS1,J1uS1,J1,q∧jS1,J1vS1,J1,q, for all uS1,J1 and vS1,J1 in ℜ and q∈Q.Definition 5 (see [19]).
If (ℜ, +,.) and ℜ1,+,. are any two hemirings and Q is a nonempty set, then ψ:R×Q⟶R1×Q is called a Q,L-known as a Homomorphism if ψu+v,r=ψu,r+ψv,r and ψuv,r=ψu,rψv,r for all u, v∈ℜ and r in Q.Definition 6 (see [17]).
If (ℜ, +,.) and ℜ1,+,. are any two hemirings and Q is a nonempty set, then ψ:R×Q⟶R1×Q is known as an anti-Q,L-homomorphism if ψu+v,r=ψv,r+ψu,r and ψuv,r=ψv,rψu,r, ∀u, v∈ℜ, and r in Q.Definition 7 (see [9]).
LetS1,J1 be a Q,L-fuzzy soft subset. For μ in 0,1, the sets Ujs1,j1,μ=uS1,J1∈X:jS1,J1uS1,J1,q≥μ and is known as (Q, L)-fuzzy soft level μ-cut.Definition 8 (see [16]).
LetS1,J1 be a Q,L-fuzzy soft subhemiring of X. For μ in 0,1, the level subset of S1,J1 is the set jS1,J1μ=uS1,J1∈X:jS1,J1uS1,J1,q≥μ. This is known as a Q,L-fuzzy soft level subset.Definition 9 (see [18]).
LetS1,J1 be a Q,L-FSSHR of a hemiring (ℜ, +,.). The level subhemiring jS1,J1μ, for μ in 0,1 with the end goal that μ≤jS1,J10,μ≤jS1,J11, is known as a Q,L-fuzzy soft level soft subhemiring.


## 3. Some Properties ofQ,L-Fuzzy Soft Subhemirings of a Hemiring

In this section, we provide main results using properties ofQ,L-FSSHR of a hemiring.Theorem 1.
IfS1,J1is a (Q, L)-FSSHR of a hemiring ℜ, then H=uS1,J1/uS1,J1∈ℜ:jS1,J1uS1,J1,q=0 is either empty or is a subhemiring of ℜ.Proof.
If no elements satisfy these conditions, thenH is empty.
LetuS1,J1,vS1,J1 in H, then(2)jS1,J1uS1,J1+vS1,J1,q≥jS1,J1uS1,J1,q∧jS1,J1vS1,J1,q=0∧0=0.
Thus,jS1,J1uS1,J1+vS1,J1,q=0. For every uS1,J1 and vS1,J1 in R and q in Q. We get uS1,J1+vS1,J1 in H. (3)juS1,J1vS1,J1,q≥jS1,J1uS1,J1,q∧jS1,J1vS1,J1,q=0∧0=0.
So,jS1,J1uS1,J1vS1,J1,q=0, for all uS1,J1, and vS1,J1 in R and q in Q. We get uS1,J1,vS1,J1 in H. Thus, H is a subhemiring of ℜ.Theorem 2.
LetS1,J1 be a (Q, L)-FSSHR of a hemiring ℜ. Then,(1)
IfjS1,J1uS1,J1+vS1,J1,q=1, then either jS1,J1uS1,J1,q=1 or jS1,J1vS1,J1,q=1, for uS1,J1and vS1,J1 in R and q in Q.(2)
IfjS1,J1uS1,J1vS1,J1,q=1, then either jS1,J1uS1,J1,q=1 or jS1,J1vS1,J1,q=1, for every uS1,J1 and vS1,J1 of ℜ and q∈Q.Proof.
LetuS1,J1 and vS1,J1∈ℜ. By the definition,(4)jS1,J1uS1,J1+vS1,J1,q≥jS1,J1uS1,J1,q∧jS1,J1vS1,J1,q.
We have(5)1≥jS1,J1uS1,J1,q∧jS1,J1vS1,J1,q.
Thus, whicheverjS1,J1uS1,J1,q=1 or jS1,J1vS1,J1,q=1, for every uS1,J1 and vS1,J1 in R and q in Q,(6)jS1,J1uS1,J1vS1,J1,q≥jS1,J1uS1,J1,q∧jS1,J1vS1,J1,q.
We have(7)1≥jS1,J1uS1,J1,q∧jS1,J1vS1,J1,q.
jS1,J1uS1,J1,q=1 or jS1,J1vS1,J1,q=1, for every uS1,J1 and vS1,J1 of ℜ and q∈Q.Theorem 3.
LetS1,J1 and (ℜ, D) be two (Q, L)-FSSHR of a hemiring (ℜ, +D). Then, their intersection S1,J1∩R,D is a (Q, L)-FSSHR of ℜ.Proof.
Letu and v be in the right place to ℜ. Then,(8)S1,J1=uS1,J1,q,jS1,J1uS1,J1,q/uS1,J1ofℜ,q∈Q,and(9)R,D=uR,D,q,jR,DuR,D,q/uR,D∈N,q∈Q.
LetS,T=S1,J1∩R,D and S,T=uS,T,q,jS,TuS,T,q/uS,TinℜqinQ, where jS,TuS,T,q=jS1,J1uS1,J1,q∧jR,DuR,D,q. Now,(10)jS,TuS,T+vS,T,q=jS1,J1uS1,J1+vS1,J1,q∧jR,DuR,D+vR,D,q≥jS1,J1uS1,J1,q∧jS1,J1vS1,J1,q∧jR,DvR,D,q∧jR,DvR,D,q≥jS1,J1uS1,J1,q∧jR,DuR,D,q∧jS1,J1uS1,J1,q∧jR,DvR,D,q≥jS,TvS,T,q∧jS,TvS,T,q,for every uS,T and vS,T in ℜ and q in Q. Again,(11)jS,TuS,T+vS,T,q=jS1,J1uS1,J1+vS1,J1,q∧jR,DuR,D+vR,D,q≥jS1,J1uS1,J1,q∧jS1,J1vS1,J1,q∧,jR,DvR,D,q∧jR,DvR,D,q≥jS1,J1uS1,J1,q∧jR,DuR,D,q∧,jS1,J1uS1,J1,q∧jR,DvR,D,q≥jS,TvS,T,q∧jS,TvS,T,q.
Thus,jS,TuS,TvS,T,q≥jS,TvS,T,q∧jS,TvS,T,q, for each uS,T and vS,T in R and q in Q.Theorem 4.
The∩of aQ,L-FSSHR of a hemiring (ℜ, +, .) is a Q,L-FSSHR of ℜ.Proof.
ConsiderjS1,J1ii∈I as a family of Q,L-FSSHR of a hemiring ℜ and jS1,J1=∏i∈IjS1,J1i. Here, u,v∈R, we have the following two cases:Case 1.
(12)jS1,J1uS1,J1+vS1,J1,q=jS1,J1iuS1,J1+vS1,J1,q≥infi∈IjS1,J1iuS1,J1,q∧jS1,J1ivS1,J1,q≥infi∈IjS1,J1iuS1,J1,q∧infi∈IjS1,J1ivS1,J1,q=jS1,J1iuS1,J1,q∧jS1,J1ivS1,J1,q.
Therefore,(13)jS1,J1uS1,J1+vS1,J1,q≥jS1,J1uS1,J1,q∧jS1,J1vS1,J1,q,for each uS1,J1 and vS1,J1 of ℜ and q∈Q.Case 2.
(14)jS1,J1uS1,J1+vS1,J1,q=jS1,J1iuS1,J1+vS1,J1,q≥infi∈IjS1,J1iuS1,J1,q∧jS1,J1ivS1,J1,q≥infi∈IjS1,J1iuS1,J1,q∧infi∈IjS1,J1ivS1,J1,q=jS1,J1iuS1,J1,q∧jS1,J1vS1,J1,q.
Thus,jS1,J1uS1,J1vS1,J1,q≥jS1,J1uS1,J1,q∧jS1,J1vS1,J1,q, for each uS1,J1 and vS1,J1 of ℜ and q∈Q.Theorem 5.
IfS1,J1 and (R, D) are any two Q,L-FSSHR of a hemiring (ℜ, +, .), then S1,J1∪R,D is a Q,L-FSSHR of ℜ.Proof.
ConsideruS1,J1 and vS1,J1 belonging to ℜ and q∈Q,(15)jS1,J1=uR,D,q,jR,DuR,D,quR,Dinℜ,qinQuS1,J1,q,jS1,J1uS1,J1,quS1,J1inℜandqinQ,jR,D=uR,D,q,jR,DuR,D,quR,DofR,q∈Q.
LetS,T=S1,J1∪R,D and S,T=uS,T,q,jS,TuS,T,q/uS,Tinℜ,qinQ, where jS,TuS,T,q=jS1,J1uS1,J1,q∨jR,DuR,D,q. Now,(16)jS1,J1uS1,J1+vS1,J1,q=jS1,J1iuS1,J1+vS1,J1,q≥jR,DiuR,D,q∧jR,DivR,D,q≥jS1,J1iuS1,J1,q∧jR,DivR,D,q=jS1,J1iuS1,J1,q∧jR,DvR,D,q.
Thus,jS,TuS,T+vS,T,q≥jS,TuS,T,q∧jS,TvS,T,q, for each uS,T and vS,T in R and q in Q.
Now,(17)jS1,J1uS1,J1+vS1,J1,q=jS1,J1uS1,J1+vS1,J1,q≥jR,DiuR,D,q∧jR,DivR,D,q≥jS1,J1iuS1,J1,q∧jR,DivR,D,q=jS1,J1iuS1,J1,q∧jR,DvR,D,q.
So,jS,TuS,T+vS,T,q≥jS,TuS,T,q∧jS,TvS,T,q, for each uS,T and vS,T in R and q in Q.Theorem 6.
The∪ of a Q,L-FSSHR of a hemiring ℜ is a Q,L-FSSHR of ℜ.Proof.
Given as a chance considerjS1,J1i to be a family of a (Q, L)-FSSHR of a hemiring ℜ and jS1,J1=∪i∈IjS1,J1i. Then, uS,T and vS,T belong to ℜ and q in Q,(18)jS1,J1uS1,J1+vS1,J1,q=supi∈IjS1,J1iuS1,J1+vS1,J1,qsupi∈I≥jS1,J1iuS1,J1,q∧supi∈IjS1,J1ivR,D,q≥supi∈IjS1,J1iuS1,J1,q∧supi∈IjS1,J1ivS1,J1,q=jS1,J1iuS1,J1,q∧jS1,J1vS1,J1,q.
Thus,jS1,J1uS1,J1vS1,J1,q≥jS1,J1uS1,J1,q∧jS1,J1vS1,J1,q, for each uS1,J1 and vS1,J1 in ℜ and q in Q.(19)jS1,J1uS1,J1+vS1,J1,q=supi∈IjS1,J1iuS1,J1+vS1,J1,qsupi∈I≥jS1,J1iuS1,J1,q∧supi∈IjS1,J1ivR,D,q≥supi∈IjS1,J1iuS1,J1,q∧supi∈IjS1,J1ivS1,J1,q=jS1,J1iuS1,J1,q∧jS1,J1vS1,J1,q.
Thus,jS1,J1uS1,J1vS1,J1,q≥jS1,J1uS1,J1,q∧jS1,J1vS1,J1,q for each uS1,J1 and uS1,J1 of ℜ and q∈Q.
Similarly, the union of a family of aQ,L-FSSHR of a hemiring ℜ is a Q,L-FSSHR of ℜ.Theorem 7.
IfS1,J1 and (ℜ, D) be two Q,L-FSSHR of the hemirings ℜ and S, correspondingly, then the product S1,J1×R,D is a Q,L-FSSHR of ℜ×S.Proof.
ConsiderS1,J1 and R,D to be two Q,L-FSSHR of the hemirings ℜ and S correspondingly. Let uS1,J11 and uS1,J12 be in ℜ, and vS1,J11 and vR,D2 be in S. Then, uS1,J1×R,D1,vS1,J1×R,D1 and uS1,J1×R,D2,vS1,J1×R,D2 are in ℜ×S. Now,(20)jS1,J1×R,DuS1,J1×R,D1,vS1,J1×R,D1+uS1,J1×R,D2,vR,D×R,D2,q=jS1,J1×R,DuS1,J11+uS1,J12,vR,D1+vR,D2,q=jS1,J1uS1,J11+uS1,J12,q∧jR,D1vR,D1+vR,D2,q≥jS1,J1uS1,J11,q∧jS1,J1uS1,J12,q∧jR,DvR,D1,q∧jR,DvR,D2,q.
Thus,(21)jS1,J1×R,DuS1,J1×R,D1,vS1,J1×R,D1+uS1,J1×R,D2,vR,D×R,D2,q=jS1,J1×R,DuS1,J11+uS1,J12,vR,D1+vR,D2,q=jS1,J1uS1,J11+uS1,J12,q∧jR,D1vR,D1+vR,D2,q≥jS1,J1uS1,J11,q∧jS1,J1uS1,J12,q∧jR,DvR,D1,q∧jR,DvR,D2,q,for each uS1,J1×R,D1 and uS1,J1×R,D2 in R and vS1,J1×R,D1 and vS1,J1×R,D2 in S and q in Q. Again,(22)jS1,J1×R,DuS1,J1×R,D1,vS1,J1×R,D1+uS1,J1×R,D2,vR,D×R,D2,q=jS1,J1×R,DuS1,J11+uS1,J12,vR,D1+vR,D2,q=jS1,J1uS1,J11+uS1,J12,q∧jR,D1vR,D1+vR,D2,q≥jS1,J1uS1,J11,q∧jS1,J1uS1,J12,q∧jR,DvR,D1,q∧jR,DvR,D2,q.
Thus,(23)jS1,J1×R,DuS1,J1×R,D1,vS1,J1×R,D1uS1,J1×R,D2,vS1,J1×R,D2,q≥jS1,J1×R,D1uS1,J1×R,D1,vS1,J1×R,D,q∧jS1,J1×R,DuS1,J1×R,D2,vS1,J1×R,D2,q,for every uS1,J1×R,D1 and uS1,J1×R,D2 in ℜ and vS1,J1×R,D1 and vS1,J1×R,D2 in S and q in Q.
Similarly, the productS1,J1×R,D is a Q,L-FSSHR of ℜ×S. □


## 4. Properties ofQ,L-Fuzzy Soft Subhemiring of a Hemiring

Some additional properties ofQ,L-FSSHR of a hemiring are discussed as follows:Theorem 8.
LetS1,J1 be a FS subset of a hemiring ℜ and L1,O1 be the strongest Q-fuzzy soft relation of ℜ. Then, S1,J1 is a Q,L-FSSHR subhemiring of ℜ⇔L1,O1 is a Q,L-FSSHR of ℜ×ℜ.Proof.
Assume thatS1,J1 is a Q,L-FSSHR of ℜ. Here, u=uS1,J11,uS1,J12 and v=vS1,J11,vS1,J12 are in R×R. Now,(24)jL1,O1uL1,O1+vL1,O1,r=jL1,O1uL1,O11,uL1,O12+vL1,O11,vL1,O12,r=jL1,O1uL1,O11,uL1,O12+vL1,O11,vL1,O12,r=jS1,J1uS1,J11+vS1,J11,r∧jS1,J1uS1,J12+vS1,J12,r≥jS1,J1uS1,J11,r∧jS1,J1vS1,J11,r∧jS1,J1uS1,J12,r∧jS1,J1vS1,J12,r≥jS1,J1uS1,J11,r∧jS1,J1vS1,J11,r∧jS1,J1uS1,J12,r∧jS1,J1vS1,J12,r=jL1,O1uL1,O11,uL1,O12,r∧jL1,O1vL1,O11,vL1,O12,r=jL1,O1uL1,O1,r∧jL1,O1vL1,O1,r.
Therefore,jL1,O1uL1,O1+vL1,O1,r≥jL1,O1uL1,O1,r∧jL1,O1vL1,O1,r for each uL1,O1 and vL1,O1 in R×R and r in Q. Again,(25)jL1,O1uL1,O1+vL1,O1,r=jL1,O1uL1,O11,uL1,O12+vL1,O11,vL1,O12,r=jL1,O1uL1,O11,uL1,O12+vL1,O11,vL1,O12,r=jS1,J1uS1,J11+vS1,J11,r∧jS1,J1uS1,J12+vS1,J12,r≥jS1,J1uS1,J11,r∧jS1,J1vS1,J11,r∧jS1,J1uS1,J12,r∧jS1,J1vS1,J12,r≥jS1,J1uS1,J11,r∧jS1,J1vS1,J11,r∧jS1,J1uS1,J12,r∧jS1,J1vS1,J12,r=jL1,O1uL1,O11,uL1,O12,r∧jL1,O1vL1,O11,vL1,O12,r=jL1,O1uL1,O1,r∧jL1,O1vL1,O1,r.
Thus,jS1,J1uL1,O1vL1,O1,r≥jL1,O1uL1,O1,r∧jS1,J1vL1,O1,r for all uL1,O1 and vL1,O1 in R×R and r in Q. Hence, L1,O1 is a Q-fuzzy subhemiring of R×R.
Furthermore, we consider thatL1,O1 is a Q-fuzzy subhemiring of R×R, then u=uL1,O11,uL1,O12 and v=vL1,O11,vL1,O12 are in R×R.(26)jL1,O1uL1,O1+vL1,O1,r=jL1,O1uL1,O11,uL1,O12+vL1,O11,vL1,O12,r=jL1,O1uL1,O11,uL1,O12+vL1,O11,vL1,O12,r=jS1,J1uS1,J11+vS1,J11,r∧jS1,J1uS1,J12+vS1,J12,r≥jS1,J1uS1,J11,r∧jS1,J1vS1,J11,r∧jS1,J1uS1,J12,r∧jS1,J1vS1,J12,r≥jS1,J1uS1,J11,r∧jS1,J1vS1,J11,r∧jS1,J1uS1,J12,r∧jS1,J1vS1,J12,r=jL1,O1uL1,O11,uL1,O12,r∧jL1,O1vL1,O11,vL1,O12,r=jL1,O1uL1,O1,r∧jL1,O1vL1,O1,r.
If(27)jS1,J1uS1,J11+vS1,J11,r≥jS1,J1uS1,J12+vS1,J12,r,jS1,J1uS1,J11,r≥jS1,J1uS1,J12,r,and(28)jS1,J1vS1,J11,r≥jS1,J1vS1,J12,r,we get S1,J1uS1,J11+vS1,J11,r≥jS1,J1uS1,J11,r∧jS1,J1vS1,J11,r for every uS1,J11 and vS1,J11 in ℜ and r in Q. Again,(29)jS1,J1uS1,J11vS1,J11,r∧jS1,J1uS1,J11vS1,J11,r=jL1,O1uL1,O11vL1,O11,uL1,O12vL1,O12,r=jL1,O1uL1,O11,uL1,O12vL1,O11,vL1,O12,r=jL1,O1uL1,O1vL1,O1,r≥jL1,O1uL1,O1,r∧jL1,O1vL1,O1,r=jL1,O1uL1,O11,uL1,O12,r∧jS1,J1vL1,O11,vL1,O12,r=jS1,J11uS1,J11,r∧jS1,J1uS1,J11,r∧jS1,J1vS1,J11,r∧jS1,J1vS1,J11,r.
If(30)jS1,J1uS1,J11vS1,J1,r≥jS1,J12uS1,J1vS1,J1,r,jS1,J12uS1,J1,r≥jS1,J1uS1,J12,r,and(31)jS1,J1vS1,J11,r≥jS1,J1vS1,J12,r,we get jS1,J1uS1,J11vS1,J11,r≥jS1,J1uS1,J11,r∧jS1,J1vS1,J11,r for every uS1,J11 and vS1,J11 in R and r∈Q. We have S1,J1 which is a Q,L-fuzzv soft subhemiring of ℜ.In the following theorem, we use the composition operation of functions.SupposeUS1,J1,VS2,J2,ZS3,J3∈R.We can define a composition mappingF:US1,J1⟶VS2,J2,G:VS2,J2⟶ZS3,J3,F.G=H:US1,J1⟶ZS3,J3Also, using some properties ofQ,L-FSSHR of a hemiring with the and composition operation of functions with the some more theorem are derived.Theorem 9.
LetS1,J1 be a Q,L-FSSHR of a hemiring R′ and g is Q-isomorphism from a hemiring N onto R′. Then, S1,J1°ψ is a Q,L-FSSHR of ℜ.Proof.
LetuS1,J1 and vS1,J1 in ℜ and S1,J1 be a Q-FSSHR of R′. Then,(32)bjS1,J1pu+v,r=pbjS1,J1uS1,J1+vS1,J1,r≥pbjS1,J1uS1,J1,r,jS1,J1vS1,J1,r=pbjS1,J1uS1,J1,r∧pbjS1,J1vS1,J1,r=bjS1,J1puS1,J1,r∧bjS1,J1pvS1,J1,r,for each uS1,J1 and vS1,J1 in ℜ and(33)bjS1,J1pu+v,r=pbjS1,J1uS1,J1+vS1,J1,r≥pbjS1,J1uS1,J1,r,jS1,J1vS1,J1,r=pbjS1,J1uS1,J1,r∧pbjS1,J1vS1,J1,r=bjS1,J1puS1,J1,r∧bjS1,J1pvS1,J1,r,for each uS1,J1 and vS1,J1≠0 in ℜ.Theorem 10.
IfS1,J1 is a (Q, L)-FSSHR of ℜ, then the pseudo-Q-fuzzy soft coset bjS1,J1p is a (Q, L)-FSSHR of ℜ, for each a∈ℜ.Proof.
LetS1,J1 be a Q,L-FSSHR of ℜ, for each u,v∈N. Then,(34)bjS1,J1pu+v,r=pbjS1,J1uS1,J1+vS1,J1,r≥pbjS1,J1uS1,J1,r,jS1,J1vS1,J1,r=pbjS1,J1uS1,J1,r∧pbjS1,J1vS1,J1,r=bjS1,J1puS1,J1,r∧bjS1,J1pvS1,J1,r.
Thus,bjS1,J1pu+v,r≥bjS1,J1puS1,J1,r∧bjS1,J1pvS1,J1,r for every uS1,J1 and vS1,J1 in ℜ and r in Q and for every uS1,J1 and vS1,J1≠0 in ℜ.(35)bjS1,J1puS1,J1vS1,J1,r=pbjS1,J1uS1,J1vS1,J1,r≥pbjS1,J1uS1,J1,r∧jS1,J1vS1,J1,r=pbjS1,J1uS1,J1,r∧pbjS1,J1vS1,J1,r=bjS1,J1puS1,J1,r∧bjS1,J1pvS1,J1,r.
Thus,jS1,J1uS1,J1+vS1,J1,r≥jS1,J1uS1,J1,r∧jS1,J1vS1,J1,r for all uS1,J1 and vS1,J1≠0 in ℜ. In this manner, bjS1,J1p is a Q,L-FSSHR subhemiring of a hemiring ℜ.Theorem 11.
IfS1,J1 is a (Q, L)-FSSHR of a hemiring ℜ, then H=uS1,J1,r,juS1,J1,r:0<juS1,J1,r≤1 is either empty or a (Q, L)-fuzzy soft subhemiring of ℜ.Proof.
In the event that no segment holds the axioms, by thenH is unfilled. If uS1,J1 and vS1,J1 satisfy this condition, then(36)jS1,J1uS1,J1vS1,J1,r,≥jS1,J1uS1,J1,r∧jvS1,J1,r,≥0∧0.
Thus,jS1,J1uS1,J1+vS1,J1,r≥jS1,J1uS1,J1,r∧jS1,J1vS1,J1,r for each uS1,J1 and vS1,J1 in R and in Q.(37)jS1,J1uS1,J1vS1,J1,r,≥jS1,J1uS1,J1,r∧jvS1,J1,r≥0∧0.
Thus,jS1,J1uS1,J1vS1,J1,r≥jS1,J1uS1,J1,r,∧jS1,J1vS1,J1,r for each uS1,J1 and vS1,J1≠e in ℜ. In this way, H is either empty or a Q,L-fuzzy soft subhemiring of ℜ.


## 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.



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*Source: 6102211-2022-09-30.xml*