View of Intuitionistic Fuzzy Ideals of M

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Intuitionistic Fuzzy Ideals of M

𝛤groups in Near Rings as Maximal Product of Graphs

S K Mala1, M M Shanmugapriya2, S Santhosh Kumar3

1_{Assistant Professor of Mathematics, KG College of Arts and}

Science,Tamil Nadu – 6410351, India. & Research Scholar in Mathematics, Karpagam Academy of Higher Education , Coimbatore. mala.sk@kgcas.com

2_{Professor in Mathematics, Karpagam Academy of Higher Education ,}

Coimbatore -641021. priya.mirdu@gmail.com

3_{Assistant Professor of Mathematics, Sri Ramakrishna Mission Vidyalaya}

College of Arts and Science, Coimbatore – 641020. fuzzysansrmvcas@gmail.com

Article History: Received: 11 January 2021; Revised: 12 February 2021; Accepted: 27 March 2021; Published

online: 16 April 2021

Abstract: This paper explains about the degree of the vertices in the Maximal product of graphs which represents the IF Ideals of M𝛤groups in Near rings. It also describes various theorems about the characteristics of the vertices in calculating the degrees with an example

Keywords: IF Ideals of M𝛤groups in Near rings, Maximal product of graphs, Degrees of vertices in Maximal product of graphs

1. Introduction

In 1983. G. Pilz [11] introduced and explained the concept of Near rings. After the introduction of Fuzzy set by Zadeh L.A[15], many extended the algebraic concept of Near rings to Fuzzy Near rings. In 1996, S. D. KIM &H. S. KIM [4] extended Fuzzy Ideals of Near rings and also explained their various characteristics by theorems.K. T. Atanassov [1], in 1986 extended Fuzzy sets to IF sets by introducing IF sets .Later A.Jianming et al[2] discussed about IF ideals in near rings in 2005,M.G.Karunambigai et al[6] in 2012, explained various properties of IF graphs with its properties.S.K.Mala et al [7]described IF ideals of M𝛤groups in Near rings in 2018 and later represented them as graph byS.K.Mala et al[8] by explaining its properties in 2019.In 2019, M. Sitara et al [10] introduced Fuzzy graph structures with applications in detail .The maximal product of graphs of IF Ideals of M𝛤groups in Near rings has been discussed by S.K.Mala et al[ 9] in 2020 .

2. Preliminaries Definition: 2.1

Near ring is a non-empty set with two binary operation satisfying i. Group with respective to first operation

ii. Semi group with respect to second operation

iii. Second operation is distributive over the first operation.

Definition: 2.2

Fuzzy set is a crisp set with its elements having membership function. If they have non-membership value along with it satisfying the condition that their sum lies between 0 and 1,is called as anIntuitionistic Fuzzy set.

Definition: 2.3

A Fuzzy set in a near-ring R is called a fuzzy ideal of R if it satisfies: (i) μ (x − y) ≥ min{μ(x), μ(y)}

(ii) μ (y + x − y) ≥ μ(x) iii) μ(xy) ≥ μ(y)

iv. μ ((x + z) y − xy) ≥ μ(z) for all x, y, z ∈ R.

Definition: 2.4

An IF set A of a Near ringis said to beIntuitionistic fuzzy ringif it obeys (i) μA (x – y) > Min {μA(x), μA (y)}

(ii) μA (xy) > Min {μA (x), μA (y)}

(iii) γA (x – y) < Max {γA (x), γA (y)}

(iv) γA (xy) < Max {γA (x), γA (y)}, for all x, y in near ring. Definition: 2.5

Let GI1(VI1, EI1, µI1, γI1) and GI2(VI2, EI2, µI2, γI2) be 2 graphs of intuitionistic fuzzy ideals of M𝛤group in

near rings (IFIM𝛤GNR) I1 and I2 then GI1* GI2 = (VI, EI, µI, γI) is called maximal product graph of intuitionistic

fuzzy ideal of M𝛤group in near rings with structure vertices –

VI = VI1 * VI2 and, edges -EI = {((u1,v1) (u2,v2)) / u1=u2 and v1,v2 ϵ EI2 (or)

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Here µI (u,v) = µI1(u) ˅ µI2 (v) for all (u,v) ϵ VI

and γI (u,v) = γI1 (u) ˄ γI2 (v) for all (u,v) ϵ VI.

Also, µI ((u1,v1) (u2,v2)) = {µI1(u1) ˅ µI2(v1 v2) where u1 = u2& v1 v2 ϵ EI2

µI2 (v2) ˅ µI1(u1 u2) where v1=v2& u1 u2 ϵ EI1

and γI ((u1,v1) (u2,v2)) = {γI1 (u1) ˄ γI2 (v1 v2) where u1 = u2& v1 v2 ϵ EI2

γI2 (v2) ˄ γI1 (u1 u2) where v1=v2& u1 u2 ϵ EI1.

Here EI(edges set) has edges only if either the first coordinates are same, or the second coordinates are same

with an edge existing already in GI1 or GI2.

3. Intuitionistic Fuzzy Ideals of M𝛤groups in Near Rings as Maximal Products of Graph

Let GI1(V_I1,E_I1,µ_I1,γ_I1) and GI2(V_I2, E_I2, µ_I2, γ_I2) be two graphs of IFIM𝛤GNR I1 and I2 in near ring N*then,

GI1*GI2 = (VI, EI, µI, γI) is called maximal product structure of IFIM𝛤GNR.

The following theorems explains the degree and total degree of vertices VI of GI1 * GI2. Theorem: 3.1

If GI1 (V_I1,E_I1,µ_I1,γ_I1) and GI2 (V_I2, E_I2, µ_I2, γ_I2) are the graphs of IFIM𝛤GNR such that µ_I1 (ui) ≤ µ_I2 (uivj) γ_I1

(ui) ≥ γ_I2 (vivj) therefore, the vertex degree of maximal product

GI1* GI2 (VI, EI, µI, γI) is given by

D_{GI1* GI2}µI(ui, vj) = D_GI1*µ_I1 (ui) µ_I2 (vj) + D_GI2 µ_I2 (vj)

D_{GI1* GI2}γI(ui, vj) = D_GI1*γ_I1 (ui) γ_I2 (vj) + D_GI2γ_I2 (vj) Proof

Let G1 (V_I1,E_I1,µ_I1,γ_I1) and G2 (V_I2, E_I2, µ_I2, γ_I2) are the graphs of IFIM𝛤GNR such that

µ_I1(ui) ≤ µ_I2 (vivj) then µ_I1 (uivj) ≤ µ_I2 (vj) and γ_I1 (ui) ≥ γ_I2 (vivj) then γ_I1 (uiuj) ≥ γ_I2 (vj) for ui∈V_I1, uiuj∈E_I1,

vi∈V_I2, vivj∈E_I2.

Therefore, the vertex degree of GI1*GI2 maximal product are:

D_{GI1* GI2}µI (ui, vj) = ∑ µ_I1 (uiuj) ˅ µ_I2 (vj) + ∑ µ_I2 (vivj) ˅ µ_I1 (ui) and

D_{GI1* GI2}γI (ui, vj) = ∑ γ_I1 (uiuj) ˄ γ_I2 (vj) + ∑ γ_I2 (vivj) ˄ γ_I1 (ui)

⟹ D_{GI1* GI2}µI (ui, vj) = ∑ µ_I2 (vj) + ∑ µ_I2 (vivj) and

D_{GI1* GI2}γI (ui, vj) = ∑ γ_I2(vj) + ∑ γ_I2 (vivj)

⟹ D_{GI1* GI2}µI (ui, vj) = D_GI1*µ_I1 (ui)µ_I2 (vj) + D_GI2µ_I2 (vj) and

D_{GI1* GI2}γI (ui, vj) = D_GI1*γ_I1 (ui)γ_I2 (vj) + D_GI2γ_I2 (vj) Example: 3.2

Consider GI1 (V_I1,E_I1,µ_I1,γ_I1) for I1 = {0} of Z3 and GI2 (V_I2, E_I2, µ_I2, γ_I2) for I2 = {0} of Z4 therefore, GI = GI1

* GI2 is a maximal product of GI1 and GI2. This is explained in the following example.

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Figure 2: Graph GI2

Figure 3: Graph GI1 * GI2

The µI value for GI1* GI2 both by theorem and direct calculation are obtained as follows.

By theorem,

D_{GI1* GI2}µI(0, 0) = D_GI1* (0)µ_I2 (0)+ D_GI2µ_I2 (0)

= 2(0.3) + ( 0.3 + 0.25 + 0.4) = 1.55

= 2(0.4) + (0.3) = 1.1

D_{GI1* GI2}µI(0, 2) = D_GI1* (0)µ_I2 (2) +D_GI2µ_I2 (2)

= 2(0.4) + (0.25) = 1.05

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= 1(0.3) + (0.3 + 0.25 + 0.4) = 1.25

= 1(0.4) + (0.3)= 0.7

= 1(0.4) + (0.25) = 1.65

= 1(0.5) + (0.4) = 0.9

= 1(0.3) + (0.3 + 0.25 + 0.4) = 1.25

D_{GI1* GI2}µI(2, 1) = D_GI1* (2)µ_I2 (1) +D_GI2µ_I2 (1)

= 1(0.4) + (0.3) = 0.7

= 1(0.4) + (0.25) = 1.65

= 1(0.5) + (0.4) = 0.9 By direct calculation, D_{GI1* GI2}µI(0, 0) = 0.25 + 0.3 + 0.4 + 0.3 +0.3 = 1.55 D_{GI1* GI2}µI(0, 1) = 0.3 + 0.4 + 0.4 = 0.1 D_{GI1* GI2}µI(0, 2) = 0.4 + 0.25 +0.4 = 1.05 D_{GI1* GI2}µI(0, 3) = 0.5 + 0.5 + 0.4 = 1.4 D_{GI1* GI2}µI(1, 0) = 0.25 + 0.4 + 0.3 + 0.3 = 1.25 D_{GI1* GI2}µI(1, 1) = 0.3 + 0.4 = 0.7 D_{GI1* GI2}µI(1, 2) = 0.25 + 0.4 = 0.65 D_{GI1* GI2}µI(1, 3) = 0.4 + 0.5 = 0.9 D_{GI1* GI2}µI(2, 0) = 0.4 + 0.25 + 0.3 + 0.3 = 1.25 D_{GI1* GI2}µI(2, 1) = 0.4 + 0.3 = 0.7 D_{GI1* GI2}µI(2, 2) = 0.4 + 0.25 = 0.65 D_{GI1* GI2}µI(2, 3) = 0.5 + 0.4 = 0.9

Similarly, the γI value forGI1* GI2 both by theorem and direct calculation are obtained as follows:

By theorem,

D_{GI1* GI2}γI(0, 0) = D_GI1* (0)γ_I2 (0)+ D_GI2γ_I2 (0)

= 2(0.2) + (0.15 + 0.2 + 0.3) = 1.05

= 2(0.2) + (0.15) = 0.55

= 2(0.3) + (0.2) = 0.8

= 2(0.4) + (0.3) = 1.1

= 1(0.2) + (0.15 + 0.2 + 0.3) = 0.85

= 1(0.2) + (0.15) = 0.35

= 1(0.3) + (0.2) = 1.5

= 1(0.4) + (0.3)= 0.7

= 1(0.2) + (0.15 + 0.2 + 0.3) = 0.85

D_{GI1* GI2}γI(2, 1) = d_GI1* (2)γ_I2 (1)+ d_GI2γ_I2 (1)

= 1(0.2) + (0.15)= 0.35

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D_{GI1* GI2}γI(2, 3) = d_GI1* (2)γ_I2 (3)+ d_GI2γ_I2 (3)

= 1(0.4) + (0.3) = 0.7 By direct calculation, D_{GI1* GI2}γI(0, 0) = 0.2 + 0.3 + 0.2 + 0.2 +0.15 = 1.05 D_{GI1* GI2}γI(0, 1) = 0.2 + 0.2 + 0.15 = 0.55 D_{GI1* GI2}γI(0, 2) = 0.3 + 0.3 +0.2 = 0.8 D_{GI1* GI2}γI(0, 3) = 0.3 + 0.4 + 0.4 = 1.1 D_{GI1* GI2}γI(1, 0) = 0.2 + 0.2 + 0.3 + 0.15 = 1.85 D_{GI1* GI2}γI(1, 1) = 0.15 + 0.2 = 0.35 D_{GI1* GI2}γI(1, 2) = 0.2 + 0.3 = 0.5 D_{GI1* GI2}γI(1, 3) = 0.4 + 0.3 = 0.7 D_{GI1* GI2}γI(2, 0) = 0.15 + 0.2 + 0.2 + 0.3 = 0.85 D_{GI1* GI2}γI(2, 1) = 0.2 + 0.15 = 0.35 D_{GI1* GI2}γI(2, 2) = 0.3 + 0.2 = 0.5 D_{GI1* GI2}γI(2, 3) = 0.3 + 0.4 = 0.7 Theorem:3.3

If GI1 (V_I1,E_I1,µ_I1,γ_I1) and GI2 (V_I2, E_I2, µ_I2, γ_I2) are the graphs of IFIM𝛤GNR such that µ_I1(ui) ≤ µ_I2(vivj)

γ_I1(ui) ≥ γ_I2(vivj) and function µ_I2(vj) is a constant “C1” and function γ_I2 is a constant “C2”. Therefore, the vertex

degree in GI1 * GI2 maximal product graph of IFIM𝛤GNR is given by D_{GI1* GI2}µI (ui, vj) = D*_GI1µ_I1 (ui) . C1 +

D_GI2µ_I2 (vj) and D_{GI1* GI2}γI (ui, vj) = D*_GI1γ_I1 (ui). C2 + D_GI2γ_I2 (vj) Proof

If GI1 (V_I1,E_I1,µ_I1,γ_I1) and GI2 (V_I2, E_I2, µ_I2, γ_I2) are the graphs of IFIM𝛤GNR in such a way thatµ_I1 (ui) ≤ µ_I2

(vivj) and γ_I1 (ui) ≥ γ_I2 (vivj) for i, j =1 to n. Here µ_I2 (vj) = C1 and γ_I2 (vj) = C2. Also, µ_I1(ui) ≤ µ_I2 (vivj) ⟹µ_I1

(uiuj) ≤ µ_I2 (vj) and γ_I1 (ui) ≥ γ_I2 (vivj) ⟹γ_I1 (uiuj) ≥ γ_I2 (vj) for i, j =1 to n.

Therefore, the vertex degree in GI1 * GI2 maximal product is

D_{GI1* GI2}µI(ui, vj) = ∑ µ_I1 (uiuj) ˅ µ_I2 (vj) + ∑ µ_I2 (vi, vj) ˅ µ_I1 (ui)

= ∑ µ_I2 (vj) + ∑ µ_I2 ( vivj)

= D*_GI1µ_I2 (ui).C1 + D_GI2µ_I2 (vj)

Also, D_{GI1* GI2}γI(ui, uj) = ∑ γ_I1 (uiuj) ˄ γ_I2 (vj) + ∑ γ_I2 (vi, vj) ˄ γ_I1 (ui)

= ∑ γ_I2 (vj) + ∑ γ_I2 (vivj) =D*_GI1γ_I2 (ui).C2 + D_GI2γ_I2 (vj) Theorem :3.4

If GI1 (V_I1, EI₁, µ_I1, γ_I1) and GI2 (V_I2, E_I2, µ_I2, γ_I2) are the graphs of IFIM𝛤GNR, I1 and I2 such that µ_I2 (vj) ≤

µ_I1(uivj) and γ_I2 (vj) ≥ γ_I1 (vivj) Therefore, the vertex degree in GI1 * GI2 maximal product graph is given by

D_{GI1* GI2}µ(ui, vj) = D*_GI2µ_I2 (vj) µ_I1(ui) + D_GI1µ_I1 (ui)

D_{GI1* GI2}γ(ui, vj) = D*_GI2γ_I2 (vj) γ_I1(ui) + D_GI1γ_I1 (ui) Proof

If GI1 (V_I1,E_I1,µ_I1,γ_I1) and GI2 (V_I2, E_I2, µ_I2, γ_I2) are the graphs of IFIM𝛤GNR, I1 and I2 so that µ_I2 (uj) ≤ µ_I1

(uiuk) ⟹µ_I2 (vivj) ≤ µ_I1 (ui) for i, j =1 to n and

γ_I2 (uj) ≥ γ_I1 (uiuk) ⟹γ_I2 (vivj) ≥ γ_I1 (ui).

Therefore, the vertex degree of maximal product GI1*GI2 is

D_{GI1* GI2}µI (uivj) = ∑ µ_I1 (uiuk) ˅ µ_I2 (vj) + ∑ µ_I2 (vivj) ˅ µ_I1 (ui)

⟹ D_{GI1* GI2}µI (ui, vj) = ∑ µ_I1 (uiuk) + ∑ µ_I1 (ui)

= D_GI1µ_I1 (ui) + D*_GI2µ_I2 (vj) µ_I1 (ui)

Similarly, D_{GI1* GI2} γI (uivj) = ∑ γ_I1 (uiuk) ˄ γ_I2 (vj) + ∑ γ_I2 (vivj) ˄ γ_I1 (ui)

⟹ D_{GI1* GI2}γI (ui, vj) = ∑ γ_I1 (uiuk) + ∑ γ_I1 (ui)

= D_GI1γ_I1 (ui) + D*_GI2γ_I2 (vj) γ_I1 (ui) 4. Conclusion

The degree of maximal product of two graphs of IF Ideals of M𝛤groups in Near rings are explained by theorems and verified through an example .This can be further extended in Intuitionistic fuzzy graph which has wider applications in the modern scientific world.

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5. References

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4. Y. B. Jun, M. Sapanci and M. A. Ozturk Fuzzy ideals in gamma near-rings, Turk.J.Math., 22(1998), 449–459.

5. S. D. Kim and H. S. Kim Fuzzy ideals of near-rings, Bull. Korean Mathematical Society,33(4) (1996), 593–601.

6. M. G. Karunambigai, R. Parvathi and R. Buvaneswari Arcs in Intuitionistic fuzzy graphs, Notes on Intuitionistic fuzzy sets, 18(4) (2012), 48–58.

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8. S. K. Mala, and M. M. Shanmugapriya IF ideals of M𝛤groups, International Journal of Mathematical Archieve, 9(1) (2018), 114–122.

9. S. K. Mala, and M. M. Shanmugapriy A Graph of an IF ideals inM𝛤groups of Near rings.IJITEE 8 (2019) 3154-3157

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13. Rosenfeld Fuzzy graphs, Fuzzy sets and their application to cognitive and decisionprocess. Academic Press, New York, (1975), 77–95.

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17. L. A. Zadeh Fuzzy sets. In Fuzzy Sets, Fuzzy Logic and Fuzzy Systems, (1996), 338–353. 18. J. Zhan And Xueling IF ideals of nearrings, Scientiae MathematicaeJaponicae,