New Class of M-Polar Fuzzy Measure Ideals Algebra in BCK2/BCK1/BCI2
Shadia Majeed Noori* | Abd Ghafur Ahmad
© 2022 IIETA. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
OPEN ACCESS
In this work, we introduced the concepts of fuzzy measure algebra of the M-polar electrode ambiguous ideals, and many of them have been investigated properties. Characterizations of the blurry M-polar measure sub-algebra and fuzzy (commutative) ideals of polarity are also looked at. Also, the relationships between M-polar fuzzy measure subalgebras, and M-polar ambiguous and ambiguous pole reciprocal ideals have been discussed. A new Concepts suggested here can be expanded to different types of ideals in BCK2, BCK1 and BCI2-algebras, for instance, a-ideal, implicated, n-fold and n-fold ideals, and commutative ideals. Besides, the properties of BCK2 (resp, BCK1 and BCI2) M-polar fuzzy measure algebra are discussed. Finally, the study also investigates the relationships between the mysterious BCK2 (resp, BCK1 and BCI2) M-polar fuzzy measure ideal. Some examples related to it are also given.
BCK2 ideals, BCK1 ideals, M-Polar fuzzy measure algebra
BCK/BCI-algebras first appeared in the mathematical literature in 1966, as a ramification of general algebra, in work by Iséki and were later formalized in other works [1]. In order to arrive at these concepts, two distinct methodologies were used: propositional calculi and set theory. BCK/BC I-algebras are algebraic patterns of the BCK/BCI-system, which are used in combinatory logic. The name BCK/BC I-algebras is derived from the use of the combinatories B, C, K, and I in combination to form the algebraic structure [1].
Chen et al. [2] expanded the view of bipolar fuzzy groups to get the idea of polar M fuzzy groups and confirmed that polar fuzzy groups and dipolar fuzzy groups are cryptographic mathematical tools. Multipolar information, the theory goes, is consistent with the evolution of value pickers.
BCI/BCK-algebras have been studied by Liu et al. [3] who have demonstrated the extension property of BCI-implicative ideals and described implicative BCI algebras in detail. Borzooei et al. [4] have researched the topic of generalized neutrosophic and suggested a novel concept. Similarly, Jun et al. [5] analysed a neutrosophic quadruple BCK/BCI-number in the context of an established collection.
Al-Masarwah [6] considered the ideal theory of BCK/BCI-algebras, defining and exploring several features. A similar study by Al-Masarwah & Ahmad [7] revealed that these ideals are related to doubt bipolar fuzzy H-ideals. Al-Masarwah [8] supported this, mentioning that bipolar fuzzy H-beliefs with specific homes typically play a crucial position withinside the shape concept of a BCK/BCI algebra. Also, BCK/BCI algebraic notions of homomorphic preimages, and doubt images, were studied by Al-Masarwah & Ahmad [9].
A unified derivation of summation, multiplication, and complex numbers in quantum theory was offered by Skilling & Knuth [10]. Akram [11] addressed the homomorphisms between Lie subalgebras, as well as how they relate to the domains and codomains of M-polar fuzzy Lie subalgebras. According to Ghorai and Pal [12], M-polar fuzzy planar graphs have features that allow for edge crossings that are not allowed in a crisp planar graph as shown in Figure 1. Furthermore, to characterize the relationships between individuals, Ghorai and Pal [13] used M-polar fuzzy set theory as well as to formulate these graphs. Also, an arc of an m-polar fuzzy graph tree is only strong if it is an M-polar fuzzy graph bridge, according to Mandal et al. [14]. In same regard, on topological surfaces, Mandal et al. [15] discussed isomorphism features of the M-polar fuzzy genus graph, as well as an application of this graph. Moreover, Farouk et al. [16] employed the view of the M-polar group to fuzzy graph theory.
Figure 1. 3-polar fuzzy graph [12]
The current study discusses an idea for perfect M-polar fuzzy scaling groups with BCK2 (resp, BCK1 and BCI2)-algebras, and introduces concepts for fuzzy M-polar scaling algebras. Then, it investigates several properties and gives M-polar descriptions of fuzzy algebra and the fuzzy (mutual) ideals of the pole. Their relations are also considered. Finally, the study combines the ideas of M-polar haze clusters and M-polar haze points to introduce a new concept in BCK2, BCK1, and BCI2-algebras termed M-polar (α, β)-ambiguous ideals.
Definition 2.1.[8]
A functional μ: T → R^+ is called a σ-additive measure if whenever a set $A \in T$ is a disjoint union of an at most countable sequence {A_k} _(k=1) ^M (where N is either finite or M = ∞) then u(A)=∑_(k=1) ^Mu (A_k). If M = ∞ Then the above sum is understood as a string. If this property applies only to the finite values of M, then μ is a final additive measure.
Definition 2.2.[9]
If X represents a universe of discourse, then A represents a fuzzy set A that is characterized by a membership function that accepts values in the range [0, 1].
Definition 2.3.[4]
Let $t J \neq \varnothing \subseteq F$, where F is BCK/BCI algebra. Then J is a sub algebra of F if $\forall \zeta, \eta \in J$ then $\zeta * \eta \in J$.
Definition 2.4.[4]
Let $J \neq \varnothing \subseteq F$, where F is BCK/BCI algebra. Then J is an ideal of F If it achieves:
1) $0 \in F$
2) $\forall \zeta, \eta \in F, \zeta * \eta \in J, \eta \in J \Rightarrow \zeta \in J$.
Definition 2.5.[11]
Let $F \neq \varnothing$. An M-polar fuzzy set G on F is a map $\psi: \mathrm{F} \rightarrow[0,1]^{\wedge} \mathrm{z} .$ Then,$\forall \zeta \in F$ is characterized by:
$\psi(\zeta)=\left(\mathrm{P}^{\circ} \psi(\zeta), \mathrm{P} \quad 2^{\circ} \psi(\zeta), \ldots, \mathrm{P} z^{\circ} \psi(\zeta)\right)$
where $\mathrm{P}_{-} \mathrm{k}^{\circ} \psi(\zeta):[0,1]^{\wedge} \mathrm{Z} \longrightarrow[0,1]$ is identified as the k-th function of projection.
Three concepts of BCK2, BCK1 and BCI2 are given in fuzzy measure algebra and with a study of its most prominent characteristics.
Definition 3.1.
Let $J \neq \varnothing \subseteq F$, where $\mathrm{F}$ is fuzzy measure algebra. Then $\mathrm{J}$ is a BCK2-sub algebra of $F$ if $\zeta * \eta \in J \forall \zeta, \eta \in J$.
Definition 3.2.
Let $J \neq \varnothing \subseteq F$, where $\mathrm{F}$ is fuzzy measure algebra. Then $\mathrm{J}$ is a $\mathrm{BCK} 1$-sub algebra of $\mathrm{F}$ if $\eta \in J, \zeta * \eta \in J \forall \zeta \in J$.
Definition 3.3.
Let $J \neq \varnothing \subseteq F$, where $F$ is fuzzy measure algebra. Then $\mathrm{J}$ is a BCI2-sub algebra of $\mathrm{F}$ if $\eta \in J,(\zeta * \eta) * \zeta \in J \forall \zeta \in J$.
Definition 3.4.
Let $J \neq \varnothing \subseteq F$, where F is BCK2, BCK1 and BCI2 fuzzy measure algebra. Then, J is an ideal of F If it achieves:
1) $0,1 \in F$
2) $\forall \zeta, \eta \in F, \zeta * \eta \in J, \eta \in J \Longrightarrow \zeta \in J$.
Definition 3.5.
Let $F \neq \varnothing$. An M-polar fuzzy measure set $\psi$ on $\mathrm{F}$ is a mapping $\psi: F \rightarrow[0,1]^{\wedge} \mathrm{z}$. The membership value of $\forall \zeta \in F$ is defined by:
$\psi(\zeta)=\left(\mathrm{P}_{-} 1^{\circ} \psi(\zeta), \mathrm{P}_{-} 2^{\circ} \psi(\zeta), \ldots, \mathrm{P}_{-} \mathrm{Z}^{\circ} \psi(\zeta)\right)$
where, $\mathrm{P}_{-} \mathrm{k}^{\circ} \psi(\zeta):[0,1]^{\wedge} \mathrm{Z} \rightarrow[0,1]$ is identified as the k-th function of projection.
Definition 3.6.
A fuzzy measure effect algebra is a system (F,M,O, u, $\oplus$) consisting of a set F,M is fuzzy measure on bolean algebra, special elements 0_F called the zero and the unit respectively, and a totally defined binary operation $\oplus$ on F, called the ortho sum if for all h, l, ℷ $\in$ F:
If h$\oplus$l and (h$\oplus$l) $\oplus$ℷ are defined, then l$\oplus$ℷ and p $\oplus$ (l$\oplus$ℷ) are defined and h $\oplus$ (l $\oplus$ ℷ) = (l $\oplus$ q) $\oplus$ ℷ.
If h $\oplus$ l, then h $\oplus$ l = l $\oplus$h, also l $\oplus$ h is fuzzy. ∀ h $\in$ F, there is a unique l $\in$ F such that h$\oplus$l is fuzzy and h $\oplus$ l = u.
If h $\oplus$ u is fuzzy defined, then h = 0_F.
Definition 3.7.
Consider the case of Θ ̌ an M-polar fuzzy measure. Set of F is referred to as an M-polar fuzzy measure sub-algebra if and only if the following conditions are met:
∀μ, ν$\in$F (Θ ̌(μ*ν)) ≥inf {Θ ̌(μ), Θ ̌(ν)},
where Θ ̌(μ), Θ ̌(ν) are fuzzy measure point of μ and y. So ∀ μ, ν$\in$F.
p_i∘Θ ̌(μ*ν) ≥inf {p_i∘Θ ̌(μ), p_i∘Θ ̌(ν)} ∀ i=1, 2…, ζ.
Example 3.8.
Let F= {0, ι,κ} be BCK2,BCK1 and BCI2- fuzzy measure algebra.
Define a mapping Θ ̌: F⟶ [0,1] ^3 by:
Θ ̌(μ)={((0.1,0.6,0.7) if μ=0 @(0.3,0.4,0.5) if μ=ι @(0.4,0.5,0.2) if μ=κ )┤
Theorem 3.9.
Assume Θ ̌ is an M-polar fuzzy measure set of F. Also, Θ ̌ is an M-polar fuzzy measure sub-algebra of F if Θ ̌_ [σ ̌] $\neq \varnothing$ is a fuzzy measure sub- algebra of F for all σ ̌〖= {σ_1, σ_2…, σ_m} $\in$ [0,1] 〗^m.
Proof. Let Θ ̌ is an M-polar fuzzy measure sub-algebra of F and
σ ̌〖$\in$ [0,1]〗^m be Θ ̌_[σ ̌ ] $\neq \varnothing$. Let μ, ν$\in$Θ ̌_ [σ ̌].
Then Θ ̌(μ)≥σ ̃. It follows that Θ ̌(μ*ν) ≥inf {Θ ̌(μ), Θ ̌(ν)} ≥σ ̃, so that μ*ν∈Θ ̌_ [σ ̌]. Therefore Θ ̌_ [σ ̌] is a fuzzy measure sub-algebra of F.
Vise versa, assume that Θ ̌_ [σ ̌] is a fuzzy measure sub-algebra of F. Suppose that ∃
ι, κ $\in$ F s.t, Θ ̌(ι*κ) <inf {Θ ̌(ι), Θ ̌(κ)}. Thus ∃ σ ̌〖= {σ_1, σ_2…, σ_m} $\in$ [0,1] 〗^m
Hence ι, κ∈Θ ̌_ [σ ̌], but ι,κ∉Θ ̌_[σ ̌] and its contradiction, so that Θ ̌(μ*ν)≥ inf {Θ ̌(μ),Θ ̌(ν)},∀ μ,ν$\in$F. Thus Θ ̌ is an M-polar fuzzy measure sub- algebra of F.
Lemma 3.10.
Let M-polar fuzzy measure sub-algebra Θ ̌ of F satisfies the following inequality:
∀ μ∈F, Θ ̌ (0) ≥Θ ̌(μ)
Proof: Since μ*μ=0 ∀ μ $\in$ F. So,
Θ ̌(0)=Θ ̌(μ*μ)≥inf {Θ ̌(μ),Θ ̌(μ)}=Θ ̌(μ) ∀ μ$\in$F.
Proposition 3.11.
If M-polar fuzzy measure sub-algebra Θ ̌ of F satisfies:
∀ μ, ν∈F, Θ ̌(μ*ν) ≥Θ ̌(ν), then Θ ̌(x)=Θ ̌ (0).
Proof. Let μ∈F, so Θ ̌(μ)≥Θ ̌(μ*0) ≥Θ ̌ (0). Thus Θ ̌(μ)=Θ ̌ (0).
Definition 3.12.
An M-polar fuzzy measure set Θ ̌ of F is named an M-polar fuzzy measure ideal if satisfies:
∀ μ, ν∈F, (p_i ∘Θ ̌ (0) ≥p_i ∘Θ ̌(x)≥inf {p_i ∘Θ ̌(μ*ν), p_i ∘Θ ̌(ν)})
∀ i= 1, 2..., ζ.
Example 3.13.
Let F= {0, ι,8,9} be BCK2, BCK1 and BCI2-fuzzy measure algebra with Cayley table defines a mapping Θ ̌: F⟶ [0,1] ^3 by:
Θ ̌(μ)={((0.6,0.7,0.4) if μ=0 @(0.3,0.5,0.6) if μ=ι,8 @(0.1,0.4,0.5) if μ=9)┤
then Θ ̌ is a 3-polar fuzzy measure ideal of F. because for any M-polar fuzzy measure set
Θ ̌ on F and σ ̌〖= {σ_1, σ_2…, σ_m} ∈ [0,1] 〗^m, the following satisfies:
Table 1. BCK2, BCK1 and BCI2-*-operation
|
* |
0 |
a |
1 |
2 |
|
0 |
0 |
0 |
2 |
2 |
|
a |
a |
0 |
2 |
1 |
|
1 |
1 |
1 |
0 |
2 |
|
2 |
2 |
2 |
0 |
1 |
If Θ ̌ is an M-polar fuzzy measure ideal of F:
∀ μ, ν∈F,μ≤ν⇒Θ ̌(μ)≥Θ ̌(ν).
Proof. Let μ, ν ∈ F be s.t, μ ≤ ν. Then μ * ν=0 and so
Θ ̌(μ)≥inf {Θ ̌(μ * ν),Θ ̌(ν)}=inf {Θ ̌(0),Θ ̌(ν)}=Θ ̌(ν). Thus Θ ̌(μ)≥Θ ̌(ν).
Theorem 3.15.
Let ω ∈F. If Θ ̌ is an M-polar fuzzy measure ideal of F, then F_ω is a fuzzy measure ideal of F.
Proof. Let ω ∈F_ω. Let μ, ν ∈F be s.t, μ * ν∈F_ω and ν∈F_ω.
Then Θ ̌ (μ * ν) ≥Θ ̌(ω) and Θ ̌(ν)≥Θ ̌(ω). Since Θ ̌ is an M-polar fuzzy measure ideal of F, so Θ ̌(μ)≥inf {Θ ̌ (μ * ν), Θ ̌(ν)} ≥Θ ̌(ω), ω ∈F_ω. Hence, F_ω is a fuzzy measure ideal of F.
Proposition 3.16.
Assume Θ ̌ is an M-polar fuzzy measure ideal of F. If F satisfies then:
∀ μ, ν,ξ∈F,μ * ν≤ξ,
then Θ ̌(μ)≥inf {Θ ̌(ν), Θ ̌(ξ)} ∀ μ,ν,ξ∈F.
Proof.
μ * ν≤ξ is satisfied in F ∀ μ,ν,ξ∈F, . So
Θ ̌(μ * ν)≥inf {Θ ̌((μ * ν) *ξ),Θ ̌(ξ)}=inf {Θ ̌(0),Θ ̌(ξ)}=Θ ̌(ξ) ∀ μ,ν,ξ∈F.
It follows that Θ ̌(μ)≥inf {Θ ̌ (μ * ν), Θ ̌(ν)} ≥inf {Θ ̌(ν), Θ ̌(ξ)} ∀ μ, ν,ξ∈F. Therefore,
Θ ̌(μ)≥inf {Θ ̌(ν),Θ ̌(ξ)}
Theorem 3.17.
For any BCK2, BCK1 and BCI2 fuzzy measure algebra F, then each measure ideal is M-polar fuzzy measure sub-algebra.
Proof.
Θ ̌ is an M-polar fuzzy measure ideal of BCK2,BCK1 and BCI2- fuzzy measure algebra F and let μ ,ν ∈ F. Then,
Θ ̌(μ * ν)≥inf {Θ ̌((μ * ν)*μ),Θ ̌(μ)}=inf {Θ ̌((μ * μ)*ν),Θ ̌(μ)}
=inf {Θ ̌(0*ν), Θ ̌(μ)} =inf {Θ ̌ (0), Θ ̌(μ)} ≥inf {Θ ̌(μ), Θ ̌(ν)}. Thus, Θ ̌ is an M-polar fuzzy measure sub-algebra of F.
Example 3.18.
Consider BCK2, BCK1 and BCI2 -fuzzy measure algebra F= {0, ι,κ} which is characterize a 2-polar fuzzy measure set Θ ̌:F→[0,1]^2 by:
Θ ̌(μ)={((0.2,0.9) if μ=0@(0.5,0.3) if μ=ι)┤
Then Θ ̌ is a 3-polar fuzzy measure sub-algebra of F. But it is not a 2-polar fuzzy measure ideal of F, because Θ ̌(μ)= (0.5,0.3) < (0.2,0.9) = inf {Θ ̌(ι*κ), Θ ̌(κ)}.
Remark 3.19. If F is a BCI2-fuzzy measure algebra, then M-polar fuzzy measure ideal. Then, Θ ̌: F→ [0,1] ^m by:
Θ ̌(μ)={ ((0.3,0.3,0.3) ,μ∈Γ@ (0.1,0.1,0.1), μ∉Γ)┤
Then Θ ̌ is M-polar fuzzy measure ideal of F. And μ= (0,0) and ν= (0,1/5), then ξ=μ * ν= (0,0) *(0,1/5) = (0, -1/5), thus Θ ̌ (μ * ν) =Θ ̌(ξ)= (0.1,0. 1…,0.1) < (0.3,0. 3…,0.3) =inf {Θ ̌(μ), Θ ̌(ν)}. Hence, Θ ̌ is not measure sub-algebra of F.
Definition 3.20.
Assume that F is a fuzzy measure algebra composed of BCK2, BCK1, and BCI2. The closed state of an M-polar fuzzy measure ideal Θ ̌ of F is achieved when the ideal of F is an M-polar fuzzy measure sub-algebra of F.
Example 3.21.
Let BCK2, BCK1 and BCI2 fuzzy measure algebra, F= {0, ι,8,9} which is define a mapping Θ ̌: F→ [0,1] ^3 by:
Θ ̌(μ)={((0.5,0.6,.0.8) if μ=0 @(0.3,0.4,0.6) if μ=ι,9 @(0.2,0.3,.0.5) if μ=1,8)┤
And, Θ ̌ is an ideal of F with a closed 3-polar fuzzy measure.
Theorem 3.22.
Assume F is a BCK2, BCK1, and BCI2 -fuzzy measure algebra, and that is the M-polar fuzzy measure set of F’s:
Θ ̌(μ)={(t ̌=(t_1,t_2,…,t_m), μ∈F @s ̌=(s_1,s_2,…,s_m ), otherwise)┤
where t ̌, s ̌∈ [0,1] ^m with t ̌>s ̌ and F= {μ∈ F:0 ≤μ}. Then Θ ̌ is the ideal of F's closed M-polar fuzzy measure.
Proof. Let 0∈F, then Θ ̌ (0) = t ̌ = (t_1, t_2…, t_m) ≥Θ ̌(μ) ∀ μ∈F.
Let μ, ν∈F. If μ∈F, then Θ ̌(μ)= t ̌ = (t_1, t_2…, t_m) ≥inf {Θ ̌(μ*ν), Θ ̌(ν)}.
Suppose that, μ∉F. If μ * ν∈F, then ν∉F; if ν∈F, then μ, ν∉F. In other hand,
we get Θ ̌(μ)= s ̌ = (s_1, s_2…, s_m) ≥inf {Θ ̌ (μ * ν), Θ ̌(ν)}. For any μ, ν∈F, if
μ or ν∉ F, then Θ ̌ (μ * ν) ≥ s ̌ = (s_1, s_2…, s_m) = inf {Θ ̌(μ), Θ ̌(ν)}. If
μ, ν∈F, then μ * ν∈F, and so Θ ̌(μ)= t ̌ = (t_1, t_2…, t_m) ≥inf {Θ ̌(μ), Θ ̌(ν)}.
Therefore, Θ ̌ is a closed M- polar fuzzy measure ideal of F.
Proposition 3.23. Specifically, each closed M-polar fuzzy measure ideal(Θ) ̌ of a BCK2-fuzzy measure algebra F meets the following conditions:
∀ μ∈F Θ ̌(0*μ) ≤Θ ̌(μ).
Proof. For any μ ∈ F, we have Θ ̌(0*μ) ≤inf {Θ ̌ (0), Θ ̌(μ)} ≤inf {Θ ̌(x), Θ ̌(μ)} =Θ ̌(μ). Therefore, Θ ̌(0*μ) ≤Θ ̌(μ).
Proposition 3.24. Let F be BCK1 and BCI2-fuzzy measure algebra.
Proof. (μ * ν) *μ≤0*ν ∀ μ, ν∈F. Thus,
Θ ̌ (μ * ν) ≥inf {Θ ̌ (μ), Θ ̌(0*ν)} ≥inf {Θ ̌(μ), Θ ̌(ν)}.
So, Θ ̌ is M-polar fuzzy measure sub-algebra of F and therefore Θ ̌ is a closed M-polar fuzzy measure ideal of F.
Herein, it suggests and discussion this concept M-polar (α, β)- BCK2, BCK1 and BCI2 fuzzy measure ideals, where:
α,β ∈ {∈,δ,∈ ∨δ,∈ ∧δ},α ≠ ∈ ∧q.
Proposition 4-1. Let ℘ be an M-pfm of F, the set 〖℘ 〗_1≠∅ ∀ ι ̂∈〖 (0.25,1] 〗^m is an ideal of F,
then,
(1) inf {℘ (0), (0.25) ̌} ≤ ℘(x),
(2) inf{℘(x), (0.25) ̌} ≤ inf {℘ (x * y), ℘(y)}.
Proof.
〖℘ 〗_1≠∅ be an ideal of F. Let υ ∈ F such that
sup {℘ (0), (0.25) ̌} < ℘(ν). Then, ℘(ν) ∈〖 (0.25,1] 〗^m,so υ ∈℘_(℘(ν) ),hence
℘(0)>℘(ν), thus 0 ∉℘_(℘(ν)) and it’s a contradiction. So that (1) holds.
Now, Assume sup{℘(x), (0.25) ̌}> inf {℘ (x * y), ℘(y)} =ι ̂ for some x, y ∈ F.
So,
ι ̂∈〖 (0.25,1] 〗^m m and y,x * y ∈ 〖℘ 〗_1.
Let, x∉〖℘ 〗_1 since ℘(x) >ι ̂, a contradiction. Hence, (2) holds.
Assume (1) and (2) hold. And, ι ̂∈〖 (0.25,1] 〗^m be such that 〖℘ 〗_1≠∅
For any x ∈ ℘, then (0.25) ̌> ι ̂≤ ℘(x) ≥ sup{℘(x), (0.25) ̌}. Also, ℘ (0) =
sup{℘(x), (0.25) ̌} ≤ι ̂. Thus, 0 ∈ 〖℘ 〗_ι ̂. Let x, y ∈F be such that x * y,y ∈〖℘ 〗_ι ̂.
Therefore, sup{℘(x), (0.25) ̌} ≤ inf {℘ (x * y), ℘(y)} ≤ι ̂< (0.25) ̌
hence, ℘(x)= sup{℘(x), (0.25) ̌} ≤ ι ̂, that is, x ∈ 〖℘ 〗_ι ̂. Thus 〖℘ 〗_ι ̂ is an ideal of F.
Definition 4.2.
Let ℘ be an M-pfm-ideal of F. Then ℘ is named an (α, β)- BCK2-fuzzy measure ideal (M-polar) of F if for all x, y ∈ F and ι ̂, κ ̂ ∈ 〖 (0.5,1] 〗^m
If x_ι ̂ α℘ then 〖0.5〗_ι ̂ β℘,
If 〖 (x * y) 〗_ι ̂ α℘ and 〖y 〗_κ ̂ α℘ then x_sup{ι ̂,κ ̂ }β℘.
Definition 4.3.
Let ℘ be an M-pfm-ideal of F. Then ℘ is named an (α, β)- BCK1- fuzzy measure ideal (M-polar) of F if for all x, y ∈F and ι ̂, κ ̂ ∈ 〖 (0.5,1] 〗^m
If x_ι ̂ α℘ then 〖0.075〗_ι ̂ β℘,
If 〖 (x * y) 〗_ι ̂ α℘ and 〖y 〗_κ ̂ α℘ then x_inf {ι ̂, κ ̂} β℘.
Definition 4.4.
Let ℘ be an M-pfm-ideal of F. Then ℘ is named an (α, β)- BCK2-fuzzy measure ideal (M-polar) of F if for all x, y ∈ F and ι ̂, κ ̂ ∈ 〖 (0.5,1] 〗^m
If x_ι ̂ α℘ then 〖0.25〗_ι ̂ β℘,
If 〖 (x * y) 〗_ι ̂ α℘ and 〖y 〗_κ ̂ α℘ then x_inf {ι ̂, κ ̂} β℘.
Theorem 4.5.
Let ℘ be an M-pfm-ideal, and:
(1) ℘(x) = (0.5) ̌, for all x ∉ ξ,
(2) ℘(x) ≥ 0.5, for all x ∈ J.
Then, ℘(x) is an M-polar (α, ∈ ∨q)- BCK2- fuzzy measure ideal of F.
Proof. (1) (For α = q) Let x ∈F and ι ̂ ∈ 〖 (0.5,1] 〗^m such that x_ι ̂ q℘.
Then, ℘(x) +ι ̂> 1 ̌. Since 0.5 ∈ J, so ℘ (0.5) ≥ (0.75) ̌. If ι ̂ ≤ (0.75) ̌, then
℘ (0.5) ≤ι ̂ and so 0.5 ∈℘. ι ̂ ≥ (0.75) ̌, then ℘ (0.5) +ι ̂<1 ̌.
Hence, 〖0.5〗_ι ̂ ∈ ∨q℘.
Let x, y ∈ F and ι ̂, κ ̂ ∈ 〖 (0.5,1] 〗^m be such that 〖 (x * y) 〗_ι ̂ q℘ and y_κ ̂ q℘ thus,
℘ (x * y) +ι ̂ <1 ̌ and ℘(y)+ κ ̂ < 1 ̌.
Therefore x * y, y ∈ J, and x ∈ J,
℘(x) ≤ (0.75) ̌. If inf {ι ̂, κ ̂} ≥ (0.75) ̌, then ℘(x) ≤ (0.75) ̌ ≤ inf {ι ̂, κ ̂}
and so, x_inf {ι ̂, κ ̂} ∈q℘. If inf {ι ̂, κ ̂} < (0.75) ̌, then ℘(x) +inf {ι ̂, κ ̂} < (1) ̌
and we have x_inf {ι ̂, κ ̂} ∈∨q℘. Therefore, ℘(x) is an M-polar (α, ∈ ∨q)- BCK2-fuzzy measure ideal of F.
Theorem 4.6.
Let ℘ M-pfm-ideal, and:
(1) ℘(x) = (0.5) ̌, for all x ∉ ξ,
(2) ℘(x) ≥ 0.5, for all x ∈ J.
And, ℘(x) is an M-polar (α, ∈ ∨q)-BCK1-fuzzy measure ideal of F.
Proof. (1) (For α = q) Let x ∈F and ι ̂ ∈ 〖 (0.25,0.50] 〗^m such that x_ι ̂ q℘.
Then, ℘(x) +ι ̂< (0.50) ̌. Since 0.25 ∈ J, so ℘ (0.25) ≤ (0.35) ̌. If ι ̂ ≥ (0.35) ̌, then
℘ (0.5) ≤ι ̂ and so 0.5 ∈℘. ι ̂ ≥ (0.75) ̌, then ℘ (0.25) +ι ̂> (0.50) ̌.
Hence, 〖0.25〗_ι ̂ ∈ ∨q℘.
Let x, y ∈ F and ι ̂, κ ̂ ∈ 〖 (0.25,0.50] 〗^m be such that 〖 (x * y) 〗_ι ̂ q℘ and y_κ ̂ q℘ thus,
℘ (x * y) +ι ̂ >1 ̌ and ℘(y)+ κ ̂ > 1 ̌.
Therefore x * y, y ∈ J, and x ∈ J,
℘(x) ≥ (0.35) ̌. If inf {ι ̂, κ ̂} ≥ (0.35) ̌, then ℘(x) ≤ (0.35) ̌ ≥ inf {ι ̂, κ ̂}
and so, x_inf {ι ̂, κ ̂} ∈q℘. If inf {ι ̂, κ ̂} < (0.75) ̌, then ℘(x) +inf {ι ̂, κ ̂} < (1) ̌
and we have x_inf {ι ̂, κ ̂} ∈∨q℘. Therefore, ℘(x) is an M-polar (α, ∈ ∨q)- BCK1-fuzzy measure ideal of F.
Theorem 4.7
Let ℘ M-pfm-ideal subset of F and ξ be an ideal of F such that
(1) ℘(x) = (0.5) ̌, for all x ∉ ξ,
(2) ℘(x) ≥ 0.5, for all x ∈ J.
And, ℘(x) is an M-polar (α, ∈ ∨q)- BCI2- fuzzy measure ideal of F.
Proof. As same as Theorem 4.5.
Example 4.8.
Let F= {0,1,2, c,d} be BCK2, BCK1 and BCI2-fuzzy measure algebra with Cayley in Table 2.
Table 2. BCK2, BCK1 and BCI2-*-operation under FUZZY MEASURE IDEALS
|
* |
0 |
1 |
2 |
c |
d |
|
0 |
0 |
0 |
0 |
d |
d |
|
1 |
1 |
0 |
1 |
c |
d |
|
2 |
2 |
2 |
0 |
d |
d |
|
c |
c |
d |
c |
0 |
1 |
|
d |
d |
d |
d |
1 |
2 |
Define a mapping Θ ̌: F⟶ [0,1] ^3 by:
℘(x)= {((0.6,0.7,0.8) if μ=0,1 @ ((0.6,0.6,0.6) if μ=c @ (0.9,0.8,0.8) if μ=2 @ (0.6,0.8,0.7) if μ=d @)) ┤
Then, J = {0, d,1} is an ideal of F. Thus, ℘(x) is a 3-polar (α, ∈ ∨q)-BCK2, BCK1 and BCI2-fuzzy measure ideal of F.
In the recent study, novel concepts BCK2, BCK1, and BCI2 based entirely on M-polar fuzzy modules were studied and added. As well as some properties and ideas of the fuzzy algebra M-polar. The descriptions of the fuzzy M-polar sub-algebra and the ambiguous (mutual) beliefs of polarity were studied. In addition, their relationships were discussed. For example, a completely new idea known as m-polar (α, β)- BCK2, BCK1 and BCI2-fuzzy measure algebras was derived and some results related to these concepts were obtained. Finally, some results for the concepts BCK2, BCK1 and BCI2 were obtained.
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