# Fuzzy Product KM-Subalgebras and Some Related Properties

Authors: K. Kalaiarasi; V. Manimozhi; N. Kausar; D. Pamucar; S. Kousar; Y. U. Gaba
Journal: Complexity
 (2021)

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## Abstract

The concept of KM-algebras has been originated in 2019. KM-algebra is a generalization of some of the B-algebras such as BCK, BCI, BCH, BE, and BV and also d-algebras. KM-algebra serves two purposes in mathematics and computer science as follows: a tool for application in both fields and a strategy for creating the foundations. On the fuzziness of KM-algebras, an innovative perspective on fuzzy product KM-algebras as well as some related features is offered. Moreover, the notion of KMM-ideals is described and also initiated the concept of the KM-Cartesian product of fuzzy KM-algebras, and related outcomes are examined. Some of the innovative results in fuzzy KMM-ideals and KM-Cartesian product of fuzzy KM-subalgebras are analyzed, and some are as follows: arbitrary intersection of fuzzy KMM-ideals is again a fuzzy KMM-ideal, order reversing holds true in every KMM-ideal, every fuzzy KM-subalgebra is a fuzzy KMM-ideal, and KM-Cartesian product of two fuzzy KM-subalgebras is again a fuzzy KM-subalgebra.


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## Body

## 1. Introduction

Ansari et al. [1, 2], Hong et al. [3], Qing-ping Hu [4], and Hwang et al. [5] researched a new class of algebra called BCH-algebras and analyzed that the class of BCI-algebras is a proper subclass of BCH-algebras. BCK-algebra was first introduced in 1966, thanks to the mathematicians Imai and Iséki [6] and has since been used in a variety of fields in mathematics, such as Topology, Probability theory, Functional Analysis, and Group theory. BCI-algebras are a generalization of BCK-algebras and were introduced by Iséki [7] in 1966.Kalaiarasi et al. [8] introduced fuzzy sets in KM-algebras. Kim and Kim [9, 10] introduced the idea of BG-algebras which is a speculation of B-algebras. In 2007, Kim and Kim [11] introduced another generalized BCK-algebras, namely, BE-algebras. Till now, so many B-algebras are developed by the researchers; refer [12–14]. In 2018, Mostafa and Cagman [15, 16] developed the concept of fuzzy dot structure in hyper KU-ideals. After BCK/BCI-algebras were introduced, many researchers developed algebraic structures which are the generalization of BCK/BCI-algebras. In 1999, Neggers and Kim [17–19] proposed a new class of algebraic structure, d-algebra, which is a generalization of BCK/BCI/B-algebras. In 2010, Saeid [20] presented fuzzy dot BCK/BCI-algebras. And in 2017, BI-algebras had been developed by Saei et al. [21]. In 2007, BF-algebras were developed by Walendziak. In 2021, Selvam and Priya [22] developed the idea of fuzzy dot Z-ideals of Z-algebras by applying the idea of fuzzification of sets to Z-algebras. In 2014, Senapati et al. [23] presented fuzzy dot BG-algebras. In 2014, Senapati et al. [24] developed fuzzy dot subalgebras and ideals of B-algebras. In 2007, BF-algebras were developed by Walendziak [25] and all these algebras are generalizations of BCK/BCI-algebras.One of the distinct parts of the set theories is fuzzy set and crisp set. In which, fuzzy set allows the elements to be included partially in the set whereas in a crisp set, the elements either belong to the set or not. Zadeh [26] introduced the concept of fuzzy sets in 1965. Nowadays, there has been a vast development in fuzzy set. Many research works in various fields have been fuzzified. Many researchers who are working in the field of algebraic structures in mathematics have applied fuzzy sets, and the researchers are doing a tremendous job on this. The ideal hypothesis plays a crucial role in the ongoing advancement of KM-algebras (KM named after Kalaiarasi and Mani). Fuzzy product subalgebras are fascinating logarithmic constructions and follow many intriguing outcomes. Additionally, fuzzy ideals are arrived at themes in any logarithmic design. Fuzzy product goals are helpful in numerical designs. Supposedly, no works are accessible on fuzzy product subalgebras of KM-algebras. The old style algebras are utilized to plan consistent circuits in PCs. Consequently, we persuaded to foster these hypotheses for KM-algebras.Set theory supports one of the motivations. Among the several operations in set theory, there are three most vital and essential operations. The union, intersection, and set difference of two sets are performed using these fundamental operations. We get the concept of Boolean algebra if we regard these three operations and their qualities as a development of those three operations and their properties. The notion of Distributive Lattice is obtained by considering the union and intersection as a general algebra. Furthermore, we get the notions of an upper semilattice and a lower semilattice when we explore the concepts of union and intersection. Until now, properties of the set union, set intersection, set difference, and the set symmetric difference have been used to introduce numerous algebras. KM-algebras are one of the different algebra, highlighting the properties of the set theory such as the set union, the set intersection, and the set symmetric difference. The class of KM-algebras consists of a combination of four axioms of which two is from BCI-algebras, one from B-algebras and in addition another axiom has been introduced. KM-algebra is a new class of algebra, and it motivates the algebraist to establish more interesting results and uses.To do so, the rest of the article is organized as follows. In section2, definition of KM-algebra, KMM-ideal, and fuzzy KM-subalgebras with some examples has been given, and in section 3, we characterized fuzzy product subalgebras of KM-algebras and explored a great deal of properties. In section 4, definition of fuzzy KMM-ideal with examples and some of its properties has been discussed. In section 5, a very interesting topic KM-Cartesian product of KM-algebras with some theorems has been presented. A reiteration of every single required definition and properties is given in the rest of this article.Throughout this research paper, we consider ‘•’ as the binary operation of KM-algebras.


## 2. Preliminaries

Some preliminary definitions for KM-algebra, KMM-ideal, and fuzzy KM-subalgebras have been given in this section.Definition 1 (see [8]).
Let K be a set, together with a binary operation ‘•’ and a constant ‘0’. Then, K = (K, 0) is called a KM-algebra [8] satisfying the following conditions:(i)
(KM1)k•m = 0 if k = m(ii)
(KM2)k•m = k if m = 0(iii)
(KM3)(k•m)•n = n•(k•m)(iv)
(KM4)k•m = 0 and m•k = 0 imply k = m, for all•k,m,n∈KExample 1.
LetM=m00,m01,m02 be a set with the binary operation • and a constant m00 with Table 1.
Here, in this example, the constant ‘0’ in the definition given is ‘m00.’(i)
(KM1)k•m = m00 if k = m. Here, m00 appears in the diagonal from upper left to lower right. The first condition KM1 is true in all the three cases.(ii)
(KM2)k•m = k if m = m00. From the first row of Table 1, KM2 is true.(iii)
(KM3) (k•m)•n = n•(k•m), for all k,m,n∈K. From Table 2, the condition for KM3 is satisfied.(iv)
(KM4)k•m = m00 and m•k = m00 imply k = m, for all k,m,n∈K.
Therefore, the condition KM4 is true. (M,•,m00) is a KM-algebra.Table 1
Binary operation.
•m00m01m02m00m00m01m02m01m01m00m02m02m02m02m00Table 2
(k•m)•n = n•(k•m).
(k, m, n)(k•m)•n(k•n)•mm00,m00,m00m00m00m00,m00,m01m01m01m00,m00,m02m02m02m00,m01,m00m01m01m00,m01,m01m00m00m00,m01,m02m02m02m00,m02,m00m02m02m0??,m02,m01m02m02m00,m02,m02m00m00m01,m00,m00m01m01m01,m00,m01m00m00m01,m00,m02m02m02m01,m01,m00m00m00m01,m01,m01m00m00m01,m01,m02m02m02m01,m02,m00m02m02m01,m02,m01m02m02m01,m02,m02m00m00m02,m00,m00m02m02m02,m00,m01m02m02m02,m00,m02m00m00m02,m01,m00m02m02m02,m01,m01m02m02m02,m01,m02m00m00m02,m02,m00m00m00m02,m02,m01m01m01m02,m02,m02m00m00Definition 2 (see [8]).
Let K be a KM-algebra andId be a subset of K which is nonempty. Then, we call Id as the KMM-ideal [8] of K if the following conditions are satisfied:(1)
0∈Id(2)
k•m∈Idandm∈Idimplyk∈Idor allk,m∈KExample 2.
For Example1, the KMM-ideals of M are m00,m01,m00,m01,m02.Definition 3.
Let B be a fuzzy set in a KM-algebra K. Then, B is said to be a fuzzy KM-subalgebra [8] of K if the following condition holds: λk•m≥∧λk,λm,for allk,m∈B. Here, λ is the function on K such that λk≤1 for all k∈B.Example 3.
In Example1, define a fuzzy set λ:B⟶0,1λm00=0.5,λm01=0.3,λm02=0.1.
From Table3, obviously the condition for fuzzy KM-subalgebras is satisfied. Thus, B satisfies the conditions of the fuzzy KM-subalgebras of M.Table 3
λk•m≥∧λk,λm.
k•mλk•m∧λk,λmm00•m00=m000.50.5m00•m01=m010.30.3m00•m02=m020.10.1m01•m00=m010.30.3m01•m01=m000.50.3m01•m02=m020.10.1m02•m00=m020.10.1m02•m01=m020.10.1m02•m02=m000.50.1


## 3. Fuzzy Product KM-Subalgebra

We introduced fuzzy product KM-algebra with some examples, and interesting results are investigated.Definition 4.
A fuzzy set B subset of K is called a fuzzy product KM-subalgebra of a KM-algebra K if the following condition is satisfied:λPdtk•m≥λPdtkpdtλPdtm, for all k,m∈B and λPdt, the functions on K such that 0≤λPdtk≤1 for all k∈B where Pdt denotes the multiplication.Example 4.
(1)
In Example1, we define λpdtm00=0.3,λpdtm01=0.5, and λpdtm02=0.4.In Table4, the value for λpdtkPdtλpdtm is given. The condition of the fuzzy product KM-algebra is verified from Table 5.Therefore,λpdtk•m≥λpdtkpdtλpdtm, for all k,m∈B. Thus, M satisfies the condition and hence a fuzzy product KM-subalgebra. If B=0.090.150.120.150.250.200.120.200.16, then B is a symmetric fuzzy matrix, and the elements of the matrix belong to fuzzy unit interval [0,1].(2)
Let us consider the KM-algebraN=n00,n01,n02,n03, having the following Cayley table. Here, the constant ‘0’ is ‘n00.’ From Table 6, clearly N,•,n00 is a KM-algebra.(N,•,n00) is a KM-algebra.DefineλPdtn01=λPdtn02=0.3 and λPdtn03=0.7.In Table7, product of λPdtk and λPdtm has be given.The condition has been checked.In Table8, the condition for fuzzy product KM-subalgebras is verified. Then, N satisfies the conditions of the fuzzy product KM-subalgebras.(3)
DefineλPdtk=0.73andλPdtm=0.023 and(1)λPdtk⋅m=0,ifk≤m,0.6,otherwise.It is clear thatλPdtk⋅m≥λPdtkPdtλPdtm,for allk,m∈K.Therefore, K is a fuzzy product KM-subalgebra.(4)
Also, we shall define(2)λPdtk=0.3andλPdtm=0.32,λPdtk⋅m=0,ifk≤m,0.6,otherwise.
This does not satisfyλPdtk⋅m≥λPdtkPdtλPdtm,for allk,m∈K. Therefore, K is not a fuzzy product KM-subalgebra.Table 4
ProductPdt.
PdtλPdtm00λPdtm01λPdtm02λPdtm000.090.150.12λPdtm010.150.250.20λPdtm020.120.200.16Table 5
λpdtk•m≥λpdtkpdtλpdtm.
k•mλpdtk•mλpdtkpdtλpdtmm00•m00=m000.30.09m00•m01=m010.50.15m00•m02=m020.40.12m01•m00=m010.50.15m01•m01=m000.30.25m01•m02=m020.40.20m02•m00=m020.40.12m02•m01=m020.40.20m02•m02=m000.30.16Table 6
Operator.
•n00n01n02n03n00n00n01n02n03n01n01n00n03n02n02n02n03n00n01n03n03n02n01n00Table 7
ProductPdt.
PdtλPdtn00λPdtn01λPdtn02λPdtn03λPdtn000.810.270.270.63λPdtn010.270.090.090.21λPdtn020.270.090.090.21λPdtn030.630.210.210.49Table 8
λPdtk•m≥λPdtkpdtλPdtm.
k•mλPdtk•mλPdtkpdtλPdtmn00•n00=n000.90.81n00•n01=n010.30.27n00•n02=n020.30.27n00•n03=n030.70.63n01•n00=n010.30.27n01•n01=n000.90.09n01•n02=n030.70.09n01•n03=n020.30.21n02•n00=n020.30.27n02•n01=n030.70.09n02•n02=n000.90.09n02•n03=n010.30.21n03•n00=n030.70.63n03•n01=n020.30.21n03•n02=n010.30.21n03•n03=n000.90.49Remark 1.
Every fuzzy KM-subalgebra is fuzzy product KM-subalgebras of a KM-algebra, but the converse need not be true. Example4λPdtm01•m01=λpdtm00=0.3<λpdtm01=∨λpdtm01,λpdtm01.Theorem 1.
If B is a fuzzy product KM-subalgebra with a membership functionλpdt of a KM-algebra K, then λpdt0≥λpdtkn, for all a∈Kandn≥2 and n is a positive integer.Proof.
By mathematical induction onn, this theorem can be proved. For n = 2,(3)λpdt0=λpdtk•k≥λpdtkpdλpdtk=λpdtk2.
Therefore, forn = 2, the result is true. Assume that the result is true for n=r>2.That is, λpdt0≥λpdtkr, for all r≥2. Now, to prove for n = r + 1, the result is true.(4)λPdt0≥λPdtkr−λPdtkpdtλPdt0≥λPdtar+1λPdtk•0≥λPdtkr+1λPdt0≥λPdtkr+1.
For everyn≥2, the result is true.Theorem 2.
Let B be a fuzzy product KM-subalgebra with the membership functionλPdt of a KM-algebra K. If there exists a sequence uminK such that λPdtumn⟶u,thenu≤λPdt0≤1.Proof.
By Theorem1, λPdt0≥λPdtkn for every k∈Kandn≥2, and n is a positive integer.(5)limm⟶∞λPdt0≥limm⟶∞λPdtumn=u⟹limm⟶∞λPdt0≥u.since1≥λPdt0⟹limm⟶∞λPdt0≥u.sinceλPdt0≤1⟹u≤limm⟶∞λPdt0≤1.
In particular, if the limitu = 1, then λPdt0≤1. and limm⟶∞λPdt0≥limm⟶∞λPdtumn=u=1 imply that λPdt0=1.
The intersection of two fuzzy product KM-subalgebras is also a fuzzy product KM-subalgebra, and the result is proved in the following theorem.Theorem 3.
IfS,σPdt and G,ρPdt are the fuzzy product KM-subalgebra of a KM-algebra K, then S∩G,σPdt∧ρPdt.Proof.
Leta∈S,b∈G. Clearly k,m∈K. Now,(6)σPdt∧ρPdtk•m=∧σPdtk•m,ρPdtk•m≥∧σPdtkpdσPdtm,ρPdtkpdρPdtm≥∧σPdtkρdtk,pdt∧σPdtm,ρPdtm=σPdt∧ρdtkρPdtσPdt∧ρdtm.
Hence, the result is obtained. Also, if B is a fuzzy product KM-subalgebra with the membership functionsλPdt of a KM-algebra K, for i∈N, then ∩Bi is also a fuzzy product KM-subalgebra of K.Theorem 4.
Bss≥1 is fuzzy product KM-subalgebra of K. If B is a fuzzy product KM-subalgebra of K, then Bss≥1 is also a fuzzy product KM-subalgebra of K.Proof.
Letk,m∈K. Bss≥1 is a fuzzy set of K defined by λPdtk=λPdtks,s≥1. Also, given B is a fuzzy product KM-subalgebra of K.
Now, fors≥1,(7)λPdtk•m=λPdtk•ms,≥λPdtkpdtλPdtms,=λPdtkspdtλPdtms,=λPdtkpdtλPdtm.
Therefore,λPdtk•m≥λPdtkpdλPdtm.Theorem 5.
The setMX,1=k∈K/λPdtk=1 is either empty or is a KM-subalgebra of K, where X is a fuzzy product KM-algebra.Proof.
SupposeMX,1 is nonempty.
Letk,m∈MX,1⟹λPdtk=1andλPdtm=1.(8)λPdtk•m≥λPdtkpdtλPdtm=1⟹k•m∈MX,1.
Therefore, M (X, 1) is a KM-subalgebra.Theorem 6.
Letg:U⟶V be an onto homomorphism of a KM-algebra K. If B is a fuzzy product KM-subalgebra of U, then the image g(B) of B under g is a fuzzy product KM-subalgebra of V.Proof.
Letd11,d12∈V. Let C1=g−1d11andC2=g−1d12 and C12=g−1d11•d12.
Consider(9)C1•C2=c•dc∈U,d∈V=g−1d11•d12∈C12C1•C2⊆C12gBc•d=Supc•d∈g−1d11•d12gc•d=Supc•d∈C12gc•d≥Supc•d∈C1•C2gc•d≥Supc∈C1,d∈C2gc•d≥Supc∈C1,d∈C2gc•gd.
Therefore,0,1×0,1⟶0,1 is continuous; then, for any ε>0, there exists a δ>0 such that c˜≥supc∈C1gc−δandd˜≥supd∈C2gd−δ. Then, c˜•d˜≥supc∈C1gcpdtsupd∈C2gd−ε. Choose c1∈C1andc2∈C2 such that(10)gc1≥supc∈C1gc−δandgc2≥supd∈C2gd−δgc1pdgc2≥supc∈C1gcpdsupd∈C2gd−εgBc•d≥Supc∈C1,d∈C2gc•gd≥Supc∈C1gcpdtSupd∈C2gd=gBcpdtgBd.
Hence, g (B) is a fuzzy product KM-subalgebra.


## 4. Fuzzy KMM-Ideals

A new notion of fuzzy KMM-ideal has been introduced with an example, and some related results have been investigated.Definition 5.
LetF˜ be a fuzzy set in K. Then, we call F˜ as a fuzzy KMM-ideal of K if the following inequality hold:
(KMI1)F˜0=F˜k,(KMI2) F˜k≥∧F˜k•m,F˜m,for allk,m∈K.Example 5.
DefineF˜ in M of Example 1 by F˜k=0.1k>0inK0.7ifk=0. Thus, F˜ is a fuzzy KMM-ideal of K.Theorem 7.
Order reversing holds true in every fuzzy KMM-ideal of K.Proof.
Letk,m∈K be such that k=m⟹k•m=0. Now,(11)F˜k≥∧F˜k•m,F˜m=∧F˜0,F˜m=F˜m.
Thus,F˜ is order reversing.Theorem 8.
Every fuzzy KM-subalgebraF˜ is a fuzzy KMM-ideal of K.Proof.
GivenF˜is a fuzzy KM-subalgebra of K.(12)F˜0=F˜k•k≥∧F˜k,F˜k=F˜k.
Therefore,(13)F˜0≥F˜k.
AndF˜k≥F˜m by Theorem 1(14)=∧F˜0,F˜mby1=∧F˜k•m,F˜m.
Thus,F˜k≥∧F˜k•m,F˜m.Theorem 9.
∩F˜i,i∈N of fuzzy KMM-ideals in K is again a fuzzy KMM-ideal.Proof.
Letk,m∈K.(15)∩F˜i0=∧i∈NF˜i0≥∧i∈NF˜ik=∩F˜ik∩F˜ik=∧i∈NF˜ik≥∧i∈N∧F˜ik•m,F˜im=∧∧i∈NF˜ik•m,infF˜im=∧∩F˜ik•m,∩F˜ik.∩F˜i is a fuzzy KMM-ideal.


## 5. KM-Cartesian Product of Fuzzy KM-Algebras

Here, an interesting notion KM-Cartesian product in KM-algebras has been introduced, and some of its properties have been analyzed.Definition 6.
LetF˜andE˜ be the fuzzy KM-algebras. The fuzzy KM-Cartesian product F˜⊗E˜:K×K⟶0,1 is defined by F˜⊗E˜k,m=∧F˜k,E˜m, for all k,m∈K. Here, the symbol ⊗ represents the operator of fuzzy KM-Cartesian product.Theorem 10.
F˜⊗E˜ is a fuzzy KMM-ideal of K×K where F˜ and E˜ are fuzzy KMM-ideals of a KM-algebra K.Proof.
Letk,m∈K×KF˜⊗E˜0,0=∧F˜0,E˜0 by Definition 5.
≥∧F˜k,E˜m since F˜andE˜ are fuzzy KMM-ideals of a KM-algebra K.(16)=F˜⊗E˜k,m.
Letk1,m1andk2,m2∈K×K.(17)F˜⊗E˜k1,k2=∧F˜k1,E˜k2≥∧∧F˜k1•m1,F˜m1,∧E˜k2•m2,E˜m2.
SinceF˜andE˜ are fuzzy KMM-ideals of a KM-algebra K,(18)=∧∧F˜k1•m1,E˜k2•m2,∧F˜m1,E˜m2=∧F˜⊗E˜k1•m1,k2•m2,F˜⊗E˜m1,m2by definition5.1.=∧∧F˜k1•m1,E˜k2•m2,∧F˜m1,E˜m2≥∧∧∧F˜k1,F˜m1,∧E˜k2,E˜m2,∧F˜m1,E˜m2=∧∧∧F˜k1,E˜k2,∧F˜m1,E˜m2,∧F˜m1,E˜m2=∧∧F˜k1,E˜k2,∧F˜m1,E˜m2=∧F˜⊗E˜k1,k2,F˜⊗E˜m1,m2by definition5.1.Theorem 11.
F˜⊗E˜ is a fuzzy KM-subalgebra of K×K where F˜ and E˜ are fuzzy KM-subalgebras of a KM-algebra K.Proof.
(19)F˜⊗E˜k1,m1⊗k2,m2=F˜⊗E˜k1⊗k2,m1⊗m2=∧F˜k1⊗k2,E˜m1⊗m2≥∧∧F˜k1,F˜k2,∧E˜m1,E˜m2=∧∧F˜k1,E˜k2,∧F˜m1,E˜m2=∧F˜⊗E˜k1,k2,F˜⊗E˜m1,m2.Theorem 12.
IfF˜andE˜ are fuzzy sets in a fuzzy KM-algebra K, where F˜⊗E˜ is a fuzzy KMM-ideal of K×K, then either F˜0=F˜k or E˜0=E˜k, for all k∈K.Proof.
Assume thatF˜0<F˜k and E˜0<E˜k for some k∈K.
Then, for some(20)F˜⊗E˜0×0=∧F˜0,E˜0<∧F˜k,E˜k=F˜⊗E˜k,k.which is a contradiction to F˜⊗E˜, a fuzzy KMM-ideals of K×K. Hence proved.Theorem 13.
IfF˜⊗E˜ is a fuzzy KMM-ideal of K×K where F˜ and E˜ are fuzzy sets of a KM-algebra K and F˜0=F˜k for all k∈K, then either E˜0=F˜k or E˜0=E˜k, for all k∈K.Proof.
Assume thatE˜0<F˜k and E˜0<E˜k for some k∈K.(21)F˜⊗E˜0,0=∧F˜0,E˜0=E˜0F˜⊗E˜k,k=∧F˜k,E˜k>E˜0=F˜⊗E˜0,0F˜⊗E˜k,k>F˜⊗E˜0,0.which is a contradiction to F˜⊗E˜, a fuzzy KMM-ideals of K×K. Hence proved.
Similarly, we can proveE˜0=E˜k for all k∈K, then either F˜0=F˜k or F˜0=E˜k, for all k∈K.Theorem 14.
LetF˜andE˜ be the fuzzy sets in a fuzzy KM-algebra K. If F˜⊗E˜ is a fuzzy KMM-ideal of K×K, then the fuzzy KMM-ideal of K is F˜ or E˜.Proof.
By Theorem12, we assume without losing generality that E˜0≥E˜k for all k∈K. By Theorem 13, either F˜0≥F˜k or F˜0≥E˜k, for all k∈K.
IfF˜0≥E˜k, for all k∈K, then(22)F˜⊗E˜0,k=∧F˜0,E˜k=E˜k.
GivenF˜⊗E˜ is a fuzzy KMM-ideal of K×K.(23)F˜⊗E˜k1,k2≥∧F˜⊗E˜k1,k2•F˜⊗E˜m1,m2,F˜⊗E˜m1,m2=∧F˜⊗E˜k1•m1,k2•m2,F˜⊗E˜m1,m2.
Takingk1=m1=0, we have(24)F˜⊗E˜0,k2≥∧F˜⊗E˜0•0,k2•m2,F˜⊗E˜0,m2E˜k2≥∧F˜⊗E˜0,k2•m2,F˜⊗E˜0,m2.
By (22), E˜k2≥∧E˜k2•m2,E˜m2. Hence proved.


## 6. Conclusion

We have introduced the new notions of fuzzy product KM-subalgebra, fuzzy KMM-ideal, and fuzzy KM-Cartesian product. Also, we have proved some results like arbitrary intersection of fuzzy product KM-subalgebra is again a fuzzy KM-subalgebra and arbitrary intersection of fuzzy KMM-ideal is again a fuzzy KMM-ideal, and also some interesting results have been investigated, and it laid the foundation of our future research work. The definitions and key conclusions of fuzzy product KM-subalgebras, in our opinion, can be applied to a variety of different algebraic systems, particularly B-algebras such as BV-algebra, BF-algebra, BM-algebra, BN-algebra, BO-algebra, and BT-algebra.(1)
Our findings could potentially be used in engineering.(2)
Fuzzy product KM-subalgebras may be extended in the future to include fuzzy product KM-homomorphism and fuzzy product KM-ideals.(3)
Fuzzy product KM-subalgebras can indeed be extended in our future work(1)
To find the interval-valued fuzzy product KM-subalgebras.(2)
To find the intuitionistic fuzzy product KM-subalgebras.(3)
To find negative-valued fuzzy product KM-subalgebras.(4)
To establish more results in fuzzy product KM-subalgebras and application.



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*Source: 3991871-2021-10-06.xml*