# Some Studies in Hemirings by the Falling Fuzzyk-Ideals

Authors: R. Anjum; F. Sarfraz; N. Kausar; Y. U. Gaba; H. Aydi; M. Munir; Salahuddin
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this article, we establish the idea of falling fuzzyk-ideals in hemirings through the falling shadow theory and fuzzy sets. We shall express the relations between fuzzy k-ideals and falling fuzzy k-ideals in hemirings. In particular, we shall establish different characterizations of k-hemiregular hemirings in the perfect positive correlation and independent probability space by means of falling fuzzy k-ideals.


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## Body

## 1. Introduction

Semiring theory had been developed by Vandiver in order to generalize rings [1]. Semirings are used in major applied mathematical fields, like automata and computer sciences [2, 3]. In the structural theory of hemirings and other algebraic structures, like BCI-algebras and KU-algebras, ideals play a significant role in their studies. Several types of ideals like p-ideals, a-ideals, and q-ideals in BCI-algebras and KU-algebras have been discussed by Senapati and others in [4, 5]. The type of ideals, viz., m-ideals, was studied by Munir and others in [6] for semigroups and in [7] for semirings. In [8], Henriksen defined finite category of k-ideals related to the ideals in semirings. In [9], La Torre examined the h-ideals and k-ideals in hemirings.Several authors have studied fuzzyh-ideals and fuzzy k-ideals in semirings [10, 11]. In [12], Shabir and Anjum have examined the fuzzy right k-ideals by using the characterizations of right k-ideals in hemirings. Prime h-ideals and semiprime k-bi-ideals are also defined in hemirings in [13, 14]. The fuzzy sets can be used to study patterns in classification and information processing. The idea of the fuzzy sets has broader applicability, e.g., Goodman worked on the equivalence between a class of random sets and a fuzzy set [15]. In [16–19], Verma used the fuzzy sets in the multiple attribute decision-making and multicriteria decision making environments to study different problems. Some more applications of the fuzzy sets are found in the context of controlled fuzzy metric spaces [20] (see also [21–23]) and pattern recognition [24].The relationship between probability theory and fuzzy sets is one of most contentious matters in an information science and an uncertainty modeling. The literature on this debate is obscure. Dubois and Prade, in their article [25], have addressed some traditional incomprehension between fuzzy sets and probabilities. They considered probabilistic analysis of membership functions that helped in assessments of these functions. In 1981, a statistical experiment was conducted to determine the membership function of the fuzzy concept “young” by three different groups of students under the supervision of Wang in China. From that experiment, they approached the existence of the resulting membership function in the theory of fuzzy sets. After this, Wang developed this approach further and furnished a precise mathematical structure of this theory in his book [26]. In [27], Tan and others established a conceptual perspective to define a fuzzy inference relation based on the theory of falling shadows. In [28, 29], Jun and Yu established some connections between the probability theory and the fuzzy mathematics in algebraic structures, like BCK-algebras and hemirings.In a future work, one can considerk-hemiregular using falling Yuan and Lee, [30], considered a fuzzy subgroup (subring, ideal) as the falling shadow of a cloud of the subgroups (subrings, ideals). By taking inspiration from [30], the notion of falling fuzzy k-ideals of a hemiring is introduced on the basis of the theory of falling shadows and fuzzy sets. This theory is widely applicable in information science and intelligent and fuzzy systems.This research article is arranged such that the first section briefly describes the history and reviews the literature of fuzzy set theory, ideals in hemirings, and the theory of falling shadows. The second section provides an explanation of the fundamental concepts such as hemirings, ideals, fuzzy sets, and probability spaces. The falling fuzzyk-ideals in a probability space and their results are presented in Section III with intriguing examples. Section IV establishes characterizations of the k-hemiregular hemirings by a perfect positive correlation and an independent probability space through falling fuzzy k-ideals. Finally, the research is concluded in the fifth section.


## 2. Preliminaries

Definition 1.
A nonempty setT, together with two binary operations + and ·, is called a semiring if T is a semigroup under both the operations satisfying the following two distributive laws [13]:(1)t1·t2+t3=t1·t2+t1·t3andt1+t2·t3=t1·t3+t2·t3,for all t1,t2,t3∈T.Definition 2.
A semiringT,+,· is a hemiring if T,+ is a commutative semigroup with a zero, i.e., 0∈T such that t1+0=0+t1=t1 and t1·0=0·t1=0 for all t1∈T [13].Definition 3.
e´∈T is the identity element of T if e´·t1=t1·e´=t1∀t1∈T [12].
A hemiringT,+,· is commutative if T,· is a commutative semigroup [12].Definition 4.
LetX≠ϕ be a subset of hemiring T. It is called an left (right) ideal of T if(i)
i´+j´∈X for all i´,j´∈X(ii)
ri´∈Xi´r∈X for all i´∈X, r∈T
IfX is both a left and a right ideal of T, then it is called two-sided ideal or simply an ideal of T [12].Definition 5.
Referring to [12, 13], the left (right) ideal X in a hemiring T is called a left (right) k-ideal of T if x´1,y´1∈X and a´∈T, a´+x´1=y´1⇒a´∈X.Definition 6.
As in [12, 13], the k-closure X¯ of X in the hemiring T is defined as(2)X¯=a∈T∣a+xi=xjforallxi,xj∈X.Definition 7.
LetF be a nonempty set. A membership function ϑ:F⟶0,1 is called a fuzzy subset ϑ of F. Also, ϑ≠ϕ if ϑx´>0 for all x´1∈F [12].Definition 8 [13].
A fuzzy setα´ in a hemiring T is the fuzzy left (right) k-ideal of T if for e,f´,t´,t´∈T, and we have the following:(i)
α´e´+f´≥α´e´∧α´f´(ii)
α´e´f´≥α´f´,α´e´f´≥α´e´(iii)
e´+t´=t´⟶α´e´≥α´t´∧α´t´
Also,α is said to be a fuzzy k-ideal of T if it is both a left k-ideal and a right k-ideal of T.
Recall thatα´0´≥α´e´ for all e´∈T.Now, we spread some light on theory of the falling shadows. For more details, the references [15, 27] can be followed.LetV be a universal set and Rv be a power set of V. For v∈V, consider(3)v̇=F∣v∈F,F⊆V.LetF∈RV, and then(4)Ḟ=v̇∣v∈F.IfA´ is a δ-field in R and ⊆A`, then the ordered pair RV,A` is called a hypermeasurable structure on V [27].LetΛ,ß,R be a probability space and RV,A` be a hypermeasurable structure, then a random set on V is considered as a function ζ:Λ⟶RV [27], which is ß−A` measurable, i.e.,(5)ζ−1D=qq∈Λ,ζq∈D∈ß,∀D∈A`.Definition 9.
Referring to [27], let ζ be a random set on V, and then K~ is called a falling shadow of ζ if(6)K~v=Rqv∈ζq,forallv∈V,and ζ is cloud of K~, where R is a probability measure.Example 1.
GivenΛ,ß,R=0,1,ß,n, where ß is a Borel field on 0,1 and n is the Lebesgue measure, suppose K~ is a fuzzy set of V and K~s=v∈VK~v≥s is a s-cut of K~, then(7)ζ:0,1⟶RV,s↦K~sis a random set, and ζ is a cloud of K~. We shall call ζ as a cut cloud of K~. It was defined in [29].


## 3. Falling Fuzzyk-Ideals in a Probability Space

In this section, we define the falling fuzzyk-ideals in a hemiring through a probability space. We shall also study the relation between fuzzy k-ideals and falling fuzzy k-ideals in a probability space.Definition 10 [29].
SupposeΛ,ß,R is a probability space and ζ:Λ⟶RT is a random set, let ζq be a left (right) ideal of T for all q∈Λ, and then the random set of the falling shadow(8)K~x=Rqx∈ζqis known as a falling fuzzy left (right) ideal of T. Also, K~x is said to be a falling fuzzy ideal of T if it is both a falling fuzzy left ideal and a falling fuzzy right ideal of T.
LetΛ,ß,R be a probability space. Given G˘T=g∣g:Λ⟶T, where T is a hemiring. Define the two operations + and · on G˘T by(9)g1♭+g2♭q=g1♭q+g2♭q,
g1♭·g2♭q=g1♭q·g2♭q, where q∈Λ and g1♭,g2♭∈G˘T.
Letφ∈G˘T be defined by φq=0 for each q∈Λ, and then G˘T,+,·,φ is a hemiring.
For any subsetB of T and g∈G˘T, consider(10)Bg=q∈Λgq∈B.
Let(11)ζ:Λ⟶RG˘Tbe given as(12)q↦g∈G˘Tgq∈B.
Then,Bg∈ß.Definition 11.
SupposeΛ,ß,R is a probability space and ζ:Λ⟶RT is a random set, if ζq is a k-ideal of T for all q∈Λ, then the falling shadow of the random set ζ given as(13)K~v=Rqv∈ζqis called the falling fuzzy k-ideal of T.Proposition 1.
We have thatζq=g∈G˘Tgq∈B is a k-ideal of G˘T for any k-ideal B of T.Proof.
Forq∈Λ, let e,f∈G˘T be such that e,f∈ζq, then eq,fq∈B. As B is a k-ideal of T, then eq+fq∈B. Note that(14)e+fq=eq+fq∈B.
This implies thate+f∈ζq.
Lete∈ζq and s∈G˘T, then eq∈B. Since B is a k-ideal of T, one has(15)s·eq=sq·eq∈B.
Therefore,s·e∈ζq, which shows that G˘T·ζq∈ζq. Similarly, we can also show that ζq·G˘T∈ζq. Let ,f∈ζq, and s∈G˘T, then fq∈B. Hence,(16)s+eq=sq+eq=fq,where B is a k-ideal of T; so, sq∈B and also s∈ζq. Hence, ζq is a k-ideal of G˘T.☐By the above result,K~ is a falling fuzzy ideal of G˘T in Λ,ß,R, where(17)K~f=Rqfq∈B.Since(18)ζ−1f=q∈Λf∈ζq=q∈Λfq∈B=Bf∈ß,one asserts that ζ is a random set on G˘T; so, K~ is a falling fuzzy k-ideal on T by Proposition 1.We present the following examples.Example 2.
Consider the setT=0,e♭,f♭,g♭ equipped with the two operations + and · defined by (see Table 1)
Then,T,+,· is a hemiring [29]. Given Λ,ß,R=0,1,ß,n. Let ζ:0,1⟶RT be defined by(19)ζs=0,ifs∈0,0.3,0,e♭,ifs∈0.3,0.5,0,f♭ifs∈0.5,0.7,0,e♭,f♭,ifs∈0.7,0.9,T,ifs∈0.9,1.
Then,ζs is a k-ideal of T for all s∈0,1. Hence, K~=Rs∣w∈ζs is a falling fuzzy k-ideal of T.Example 3.
LetT=0,e♭,x♭,y♭,z♭ be a hemiring with the two operations + and · defined by the following Cayley table (see Table 2):
Then,T,+,· is a hemiring. Given Λ,ß,R=0,1,ß,n. Define ζ:0,1⟶RT by(20)ζs=0,ifs∈0,0.1,0,x♭ifs∈0.1,0.3,0,z♭,ifs∈0.3,0.5,0,x♭,z♭,ifs∈0.5,0.7,0,e♭,x♭,y♭,ifs∈0.7,0.9,T,ifs∈0.9,1.
Then,ζs is a k-ideal of T for all s∈0,1. Hence, K~=Rsw∈ζs is a falling fuzzy k-ideal of T.Theorem 1.
Each fuzzyk-ideal of T is also a falling fuzzy k-ideal of T, but the converse is not true.Proof.
LetΛ,ß,R=0,1,ß,n be a probability space, i.e., R is a probability measure, ß is a Bored field in 0,1, and n is the Lebesgue measure. Let μ be a fuzzy k-ideal of T, and then μs is a k-ideal of T for all s∈0,1. Let ζ:0,1⟶RT be a random set given as ζs=μs for all s∈0,1. Hence, μ is also a falling fuzzy k-ideal of T.☐Table 1+0e♭f♭g♭00e♭f♭g♭e♭e♭e♭f♭g♭f♭f♭f♭f♭g♭g♭g♭g♭g♭f♭.0e♭f♭g♭00000e♭0e♭e♭e♭f♭0e♭e♭e♭g♭0e♭e♭e♭Table 2+0e♭x♭y♭z♭00e♭x♭y♭z♭e♭e♭y♭e♭x♭e♭x♭f♭f♭f♭g♭x♭y♭g♭g♭g♭f♭y♭z♭z♭e♭x♭y♭z♭.0e♭x♭y♭z♭000000e♭0e♭e♭z♭z♭x♭0e♭e♭0z♭y♭0y♭x♭e♭z♭z♭0z♭z♭z♭0For the converse, consider the hemiringT=0,x♭,y♭,z♭ with the two following binary operations + and · (see Table 3):Table 3+0x♭y♭z♭00x♭y♭z♭x♭x♭0z♭y♭y♭y♭z♭0x♭z♭z♭y♭x♭0.0x♭y♭z♭00000x♭0000y♭00z♭z♭z♭00z♭z♭GivenΛ,ß,R=0,1,ß,n. Define ζ:0,1⟶RT by(21)ζs=0,ifs∈0,0.3,0,x♭,ifs∈0.3,0.5,0,z♭,ifs∈0.5,0.9,T,ifs∈0.9,1.Then,ζs is a k-ideal of T for all s∈0,1. Hence, K~=Rs∣e∈ζs is the falling fuzzy k-ideal of T, which is represented by(22)K~e=1,ife=0,0.3ife=x♭,0.5ife=z♭,0.1ife=y♭,K~s=0,ifs∈0.5,1,0,z♭,ifs∈0.3,0.5,0,x♭,z♭,ifs∈0.1,0.3,T,ifs∈0,0.1.Ifs∈0.1,0.3, then K~s=0,x♭,z♭ is not a k-ideal of T since x♭+z♭=y♭Ú0,x♭,z♭. Thus, K~ is not a fuzzy k-ideal of T.Consider the probability spaceΛ,ß,R and the falling shadow of a random set ζ:Λ⟶RT. Let Λe,ζ=q∈Λw∈ζq, then Λe,ζ∈R, where e∈T.Theorem 2.
If the falling shadowK~ of the random set ζ:Λ⟶RT is the falling fuzzy left (right) k-ideal in T for all e,f,x,y∈T, then we have the following assertions:(i)
Λe,ζ∩Λf,ζ⊆Λe+f,ζ(ii)
Λf,ζ⊆Λef,ζ, Λe,ζ⊆Λef,ζ(iii)
e+x´=y´ implies that Λx´,ζ∩Λy´,ζ⊆Λe,ζProof.
(i)
Letq∈Λe,ζ∩Λf,ζ where e,f∈ζq. Since ζq is a left k-ideal in T, we have e+f∈ζq, so(23)q∈Λe+f,ζ⇒Λe,ζ∩Λf,ζ⊆Λe+f,ζ.
Similarly, the statement is true whenζq is a right k-ideal.(ii)
Supposeq∈Λf,ζ and f∈ζq, then ζq is a left k-ideal on T, so ef∈ζq and also q∈Λef,ζ. It shows that(24)Λf,ζ⊆Λef,ζ.
We can also prove(25)Λe,ζ⊆Λef,ζis for right k-ideals.(iii)
Lete+x´=y´. Suppose q∈Λx´,ζ∩Λy´,ζ and e∈T, then x´,y´∈ζq. As ζq is a left k-ideal on T, so e∈ζq and q ∈Λe,ζ. We have(26)Λx´,ζ∩Λy´,ζ⊆Λe,ζ.☐Theorem 3.
IfK~ is a falling fuzzy k-ideal of T, then K~ef≥K~e∨K~f.Proof.
SinceΛef,ζ⊇Λe,ζ∩Λf,ζ, one writes(27)K~ef=Rqef∈ζq≥Rqe∈ζq∩qf∈ζq≥Rqe∈ζq+Rqf∈ζq−Rqe∈ζq∩qf∈ζq≥Rqe∈ζq∨Rqf∈ζq=K~e∨K~f.☐


## 4. Characterizations ofk-Hemiregular Hemirings

In this section, we shall characterize thek-hemiregular hemirings on a perfect positive correlation probability space (PPS) and an independent probability space (IPS) through falling fuzzy k-ideals.Definition 12 [13].
A hemiringT is called a k-hemiregular if for all x∈T, there are a1,a2∈T such that(28)x+xa1x=xa2x.Lemma 1 [13].
IfX and Y are both a right k-ideal and a left k-ideal in a hemiring T, then(29)XY¯⊆X∩Y.Lemma 2. [13].
The hemiringT is a hemiregular hemiring if for each right and left k-ideal X and Y, we have XY¯=X∩Y.Definition 13.
The probability space is called a perfect positive correlation [29] if(30)Λe,ζ⊆Λf,ζorΛf,ζ⊆Λe,ζ∀e,f∈T.Definition 14.
SupposeΛ is a PPS, let K~ϕ and K~ψ be the falling fuzzy k-ideals of T, and then the I product of K~ϕ and K~ψ is defined by(31)K~ϕeIK~ψe=e+∑r=1usrtr=∑m=1vs′mt′mK~ϕsr∧K~ϕs´m∧K~ψtr∧K~ψt´mand K~ϕeIK~ψe=0ifecannotexpressedase+∑r=1usrtr=∑m=1vs´mt´m, where r=1,⋯,u and ,m=1,⋯,v.Theorem 4.
SupposeΛ is a PPS and K~ is the falling fuzzy left (right) k-ideal on T for all e,f,x,y∈T, then(1)
K~e+f≥K~e∧K~f(2)
K~ef≥K~f,K~ef≥K~e(3)
e+x=y⇒K~e≥K~x∧K~yProof.
(1)
Letζq be a k-ideal on T for any q ∈Λ. Since(32)Λe,ζ∩Λf,ζ⊆Λe+f,ζ,one writes(33)K~e+f=Rqe+f∈ζq≥Rqe∈ζq∩qf∈ζq.
Ifqe∈ζq⊇qf∈ζq, then K~e+f≥K~e. While if qe∈ζq⊆qf∈ζq, then K~e+f≥K~f. Hence,(34)K~e+f≥K~e∧K~f.(2)
Asζq is a left k-ideal on T, for any q ∈Λ, so Λf,ζ⊆Λef,ζ. Hence, we have(35)K~ef=Rqef∈ζq≥Rqf∈ζq=K~f.
Similarly, for a rightk-ideal, the statement is true.(3)
Ifζq is a left k-ideal on T for any q ∈Λ, then(36)e+x=yimpliesthatΛx,ζ∩Λy,ζ⊆Λe,ζ.
So, we have(37)K~e=Rqe∈ζq≥Rqx∈ζq∩qy∈ζq.
Ifqx∈ζq⊇qy∈ζq, then K~e≥K~x. While if qx∈ζq⊆qy∈ζq, then K~e≥K~y. Hence,(38)K~e≥K~x∧K~y.☐Proposition 2 [27].
IfΛ is a perfect positive correlation probability space, then(39)K~ϕ∩K~ψe=K~ϕe∧K~ψe.Proposition 3.
LetΛ be a (PPS) perfect positive correlation probability space. Suppose that K~ϕ and K~ψ are falling fuzzy left (right) k-ideals on T, then K~ϕ∩K~ψ is the falling fuzzy left (right) k-ideal on T.Proof.
(1)
We have(40)K~ϕ∩K~ψe+f=K~ϕe+f∧K~ψe+f≥K~ϕe∧K~ϕf∧K~ψe∧K~ψf=K~ϕe∧K~ψe∧K~ϕf∧K~ψf=K~ϕ∩K~ψe∧K~ϕ∩K~ψf.(2)
Now,(41)K~ϕ∩K~ψef=K~ϕef∧K~ψef≥K~ϕf∧K~ψf=K~ϕ∩K~ψf.(3)
Lete+x=y, then(42)K~ϕ∩K~ψe≥K~ϕx∧K~ϕy∧K~ψx∧K~ψy=K~ϕx∧K~ψx∧K~ϕy∧K~ψy=K~ϕ∩K~ψx∧K~ϕ∩K~ψy.
For rightk-ideals, the proof is similar.☐Theorem 5.
If the falling fuzzy setK~ is the falling fuzzy left (right) k-ideal on T, then(43)χT⨀IK~⊆K~K~⨀IχT⊆K~.Proof.
SupposeK~ is a falling fuzzy k-ideal on T, let e∈T. If χT⨀IK~e=0, then χT⨀IK~eK~e. Otherwise, there are sr,tr,s´m,t´m∈T such that e+∑r=1usrtr=∑m=1vs´mt´m, so(44)χT⨀IK~e=⋁e+∑r=1usrtr=∑m=1vs´mt´mK~tr∧K~t´m≤⋁e+∑r=1usrtr=∑m=1vs´mt´mK~srtr∧K~s´mt´m≤⋁e+∑r=1usrtr=∑m=1vs´mt´mK~∑r=1usrtr∧K~∑m=1vs´mt´m≤⋁e+∑r=1usrtr=∑m=1vs´mt´mK~e=K~e.
It implies thatχT⨀IK~⊆K~.
The proof is similar for falling fuzzy rightk-ideals.☐Theorem 6.
LetT be a hemiring and X,Y⊆T, then χX⨀IχY=χXY¯.Proof.
Ife∈XY¯, then χXY¯e=1 and e+∑r=1usrtr=∑m=1vs´mt´m for all sr,s´m∈X,tr,t´m∈Y. Let e∈T. We have(45)χX⨀IχYe=⋁e+∑r=1usrtr=∑m=1vs´mt´mχXsr∧χXs´m∧χYtr∧χYt´m≥χXsr∧χXs´m∧χYtr∧χYt´m=1.
That is,χX⨀IχYe=1=χXY¯e. Hence, χX⨀IχY=χXY¯. If e∉XY¯, then χXY¯e=0. Let χX⨀IχY≠0, and then(46)χX⨀IχYe=⋁e+∑r=1usrtr=∑m=1vs´mt´mχXsr∧χXs´m∧χYtr∧χYt´m≠0.
Hence, there aresr,s´m,tr,t´m∈T such that e+∑r=1usrtr=∑m=1vs´mt´m and(47)χXsr∧χXs´m∧χYtr∧χYt´m≠0imply that(48)χXsr=χXs´m=χYtr=χYt´m=1for all sr,s´m∈X,tr,t´m∈Y and e∈XY¯, which is a contradiction to χXY¯e=0. So, our supposition is wrong, and χX⨀IχYe=0=χXY¯e.
In both cases,(49)χX⨀IχY=χXY¯which completes the proof.☐Theorem 7.
A hemiringT is a k-hemiregular if for any falling fuzzy right k-ideal K~ϕ and any falling fuzzy left k-ideal K~ψ of T, we have(50)K~ϕ⨀IK~ψ=K~ϕ∩K~ψ.Proof.
SupposeT is a k-hemiregular hemiring, let K~ϕ (resp., K~ψ) be a falling fuzzy right (resp., left) k-ideal of T. Then,(51)K~ϕ⨀IK~ψ⊆K~ϕ⨀IχT⊆K~ϕ,K~ϕ⨀IK~ψ⊆χT⨀IK~ψ⊆K~ψThenK~ϕ⨀IK~ψ⊆K~ϕ∩K~ψ.
Conversely, supposee∈T, as T is a k-hemiregular, so there are x,x´∈T such that(52)e+exe=ex´e,K~ϕ⨀IK~ψe=⋁e+∑r=1usrtr=∑m=1vs´mt´mK~ϕsr∧K~ϕs´m∧K~ψtr∧K~ψt´m≥K~ϕex∧K~ϕe¯x´∧K~ψe≥K~ϕe∧K~ψe=K~ϕ∩K~ψe.
This implies that(53)K~ϕ⨀IK~ψ⊇K~ϕ∩K~ψ.
Therefore,(54)K~ϕ⨀IK~ψ=K~ϕ∩K~ψ.
Conversely, supposeζϕq (resp., ζψq is a left k-ideal (resp., right) k-ideal of T, then K~ϕν (resp., K~ψv) is a falling fuzzy left (resp., right) k-ideal of T. So, χζϕq (resp., χζψq) is the characteristic function of the fuzzy left (resp., right) k-ideal of T. We have(55)χζϕqζψq¯=χζϕq⨀Iχζψq=χζϕq∩χζψq=χζϕq∩ζψq,ζϕqζψq¯=ζϕq∩ζψq.
Thus, we have(56)K~ϕK~ψ¯v=Rq∣v∈ζϕqζψq¯=Rq∣v∈ζϕq∩ζψq=K~ϕ∩K~ψv.
Hence,T is a k-hemiregular hemiring.☐Definition 15 [29].
LetΛ be a probability space, and then Λ is called an independent probability space if(57)Λe,ζ∩Λf,ζ=Λe,ζΛf,ζ,∀e,f∈T.Definition 16.
LetΛ be an IPS, and suppose that K~ϕ and K~ψ are falling fuzzy k-ideals on T, then the L-product of K~ϕ and K~ψ is defined by(58)K~ϕ⨀LK~ψe=K~ϕsrK~ϕs´mK~ψtrK~ψt´m,ife+∑r=1usrtr=∑m=1vsm′tm′0,otherwise.Theorem 8.
LetΛ be an IPS and K~ be a falling fuzzy left (right) k-ideal of T for all e,f,x,y∈T. Then,(1)
K~e+f≥K~eK~f(2)
K~ef≥K~fK~ef≥K~e(3)
e+x=y⇒K~e≥K~xK~yProof.
(1)
Letζq be a left k-ideal on T for any q∈Λ. As we know that Λe,ζ∩Λf,ζ⊆Λe+f,ζ, then(59)K~e+f=Rq∣e+f∈ζq≥Rq∣e∈ζq∩q∣f∈ζq=Rq∣e∈ζqRq∣f∈ζq=K~eK~f(2)
Letζq be a left k-ideal of T for any q∈Λ. As Λf,ζ⊆Λef,ζ, so we have(60)K~ef=Rqef∈ζq≥Rqf∈ζq=K~f(3)
Letζq be a left k-ideal of T for any q ∈Λ. So,(61)e+x=y⇒Λx,ζ∩Λy,ζ⊆Λe,ζ
Hence,(62)K~e=Rqe∈ζq≥Rqx∈ζq∩qy∈ζq=Rq∣x∈ζqRq∣y∈ζq,K~e=K~xK~y.☐Proposition 4 [27].
IfΛ is an independent probability space, then(63)K~ϕ∩K~ψe=K~ϕeK~ψe.Proposition 5.
LetΛ be an independent probability space and K~ be a falling fuzzy k-ideal on T, and then(64)K~ef≥K~e+K~f−K~eK~f.Proof.
SinceΛef,ζ⊇Λe,ζ∩Λf,ζ, one gets(65)K~ef=Rqef∈ζq≥Rqe∈ζq∩qf∈ζq≥Rqe∈ζq+Rqf∈ζq−Rqe∈ζq∩qf∈ζq=Rqe∈ζq+Rqf∈ζq−Rqe∈ζqRqf∈ζq≥K~e+K~f−K~eK~f.☐Proposition 6.
SupposeΛ is an IPS, let K~ϕ and K~ψ be falling fuzzy left (right) k-ideals of T, and then K~ϕ∩K~ψ is also a falling fuzzy left (right) k-ideal of T.Proof.
LetK~ϕ and K~ψ be falling fuzzy left k-ideals of T, and then(66)K~ϕ∩K~ψe+f=K~ϕe+fK~ψe+f≥K~ϕeK~ϕfK~ψeK~ψf=K~ϕ∩K~ψeK~ϕ∩K~ψf,K~ϕ∩K~ψef=K~ϕefK~ψef≥K~ϕfK~ψf=K~ϕ∩K~ψf.(1)
Let e+x=y, and then(67)K~ϕ∩K~ψe≥K~ϕxK~ϕyK~ψxK~ψy=K~ϕ∩K~ψxK~ϕ∩K~ψy.
For the case of rightk-ideals, the proof is found on similar lines.☐Theorem 9.
If the falling fuzzy setK~ of T is a falling fuzzy left (right) k-ideal of T, then(68)χT⨀LK~⊆K~K~⨀LχT⊆K~.Proof.
SupposeK~ is a falling fuzzy k-ideal of T and e∈T, if χT⨀LK~e=0, then χT⨀LK~eK~e. Otherwise, there are sr,tr,s´m,t´m∈T such that e+∑r=1usrtr=∑m=1vs´mt´m; so,(69)χT⨀LK~e=K~trK~t´mK~srtrK~s´mt´mK~∑r=1usrtrK~∑m=1vs´mt´mK~e.
Thus,χT⨀LK~⊆K~.
The proof is similar for rightk-ideals.☐Theorem 10.
IfT is a hemiring and X,Y⊆T, then χXeLχY=χXY¯.Proof.
Lete∈T. If e∈XY¯, then χXY¯e=1 and e+∑r=1usrtr=∑m=1vs´mt´m, for all sr,s´m∈X,tr,t´m∈Y. So, we have(70)χX⨀LχYe=χXsrχXs´mχYtrχYt´m≥χXsrχXs´mχYtrχYt´m=1.
Thus,χX⨀LχYe=1=χXY¯e. If e∉XY¯, then χXY¯e=0. Let us suppose that χX⨀LχY≠0, and then(71)χX⨀LχYe=χXsrχXs´mχYtrχYt´m≠0.
Therefore, there aresr,s´m,tr,t´m∈T such that e+∑r=1usrtr=∑m=1vs´mt´m and(72)χXsrχXs´mχYtrχYt´m≠0,χXsr=χXs´m=χYtr=χYt´m=1.
Hence, for allsr,s´m∈X,tr,t´m∈Y and e∈XY¯, which is a contradiction to χXY¯e=0, our supposition is wrong, and(73)χX⨀LχYe=0=χXY¯e.
This implies that(74)χX⨀LχYe=χXY¯e.☐Theorem 11.
A hemiringT is a k-hemiregular if for any falling fuzzy right k-ideal K~ϕ of T and any falling fuzzy left k-ideal K~ψ on T, we have K~ϕeLK~ψ=K~ϕ∩K~ψ.Proof.
SupposeT is a k-hemiregular hemiring, let K~ϕ (resp., K~ψ) be the falling fuzzy right (resp., left) k-ideal of T. Then,(75)K~ϕ⨀LK~ψ⊆K~ϕeLχT⊆K~ϕ,K~ϕ⨀LK~ψ⊆χTeLK~ϕ⊆K~ψ.
It implies that(76)K~ϕ⨀LK~ψ⊆K~ϕ∩K~ψ.
Supposee∈T, since T is a k-hemiregular, there are x,x´∈T such that e+exe=ex´e where e+∑r=1usrtr=∑m=1vs´mt´m, for all sr,s´m,tr,t´m∈T. We have(77)K~ϕ⊙LK~ψe=K~ϕsrK~ϕs´mK~ψtrK~ψt´m≥K~ϕexeK~ϕex´e≥K~ϕxeK~ψex´e≥K~ϕeK~ψe=K~ϕ∩K~ψe.
It yields thatK~ϕ⨀LK~ψ⊇K~ϕ∩K~ψ.
Conversely, suppose thatζϕq and ζψq are right k-ideal and left k-ideal of T, respectively, then K~ϕυ¯ and K~ψυ¯ are both a falling fuzzy right k-ideal and a falling fuzzy left k-ideal of T. So, χζϕq and χζψq. Hence, the characteristic function of ζϕq and ζψq is a fuzzy right k-ideal and a fuzzy left k-ideal of T, respectively. We have(78)υ¯ζϕqζψq¯=υ¯ζϕq⨀Lυ¯ζψq=υ¯ζϕq∩υ¯ζψq=υ¯ζϕq∩ζψq,ζϕqζψq¯=ζϕq∩ζψq.
Thus,(79)K~ϕK~ψ¯υ¯=Rq∣υ¯∈ζϕqζψq¯=Rq∣υ¯∈ζϕq∩ζψq=K~ϕ∩K~ψυ¯.
Hence,T is a k-hemiregular hemiring.☐


## 5. Conclusion

In this paper, we introduced the notion of falling fuzzyk-ideals of a hemiring and illustrated it with the help of examples. Then, we investigated some characteristics of k-hemiregulars by means of falling fuzzy k-ideals based on a perfect positive correlation and an independent probability space. In future works, one can consider k-hemiregulars using falling fuzzy k-bi-ideals and falling fuzzy k-quasi-ideals. One also can apply fuzzy inference relations to hemirings. Further, one can investigate this theory to information science, intelligent and fuzzy systems, etc. Moreover, the idea of falling fuzzy k-ideals can be extended to BCI-algebras [4] and KU-Algebras [5].



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