Ratio Mathematica Volume 44, 2022 Antagonistic Intuitionistic Fuzzy Sub Commutative Ideals of Subtraction G-Algebra B. Lena 1 C. Ragavan 2 Abstract The notions of over antagonistic intuitionistic fuzzy sub commutative ideals of subtraction G-algebras are introduced. The characterization properties of antagonistic intuitionistic fuzzy sub commutative ideals are obtained. We investigate the relations between antagonistic intuitionistic fuzzy sub implicative ideals and antagonistic intuitionistic fuzzy sub commutative ideals of subtraction G-algebra. Keywords: Subtraction G-algebra, AIFSI ideals, AIFSC ideals. 2020 AMS Subject Classifications: 05A15, 11B68, 34A05. 3 1 Department of Mathematics, Trinity College for Women, Namakkal, Tamil Nadu, India;lenasenthil1977@gmail.com 2 Department of Mathematics, SVM College, Uthangarai, Krishnagiri, Tamil Nadu, India. 3 Received on APRIL 12th, 2022.Accepted on Sep 1st, 2022.Published on Nov 30th, 2022.doi: 10.23755/rm.v44i0.913. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors.This paper is published under the CC-BY license agreement. 260 mailto:lenasenthil1977@gmail.com B. Lena, C. Ragavan, S. Jeyanthi 1. Introduction The introduction of fuzzy sets by Zadeh [17], there have been a number of generalizations of this fundamental concept. BCK-algebras and BCI-algebras are two classes of logical algebra. Extensive applications of fuzzy set theory have been found in various fields, for example, artificial intelligence, computer science, control engineering, expert system, management science, operation research and many others which were initiated by K. Iseki [3, 4]. The notion of fuzzy sets, invented by L. A. Zadeh [17], has been applied to many fields. Since then, fuzzy BCI/BCK algebras have been extensively investigated by several researchers. For BCK-algebras, Y. B. Jun et al. [7, 10] introduced the notions of fuzzy positive implicative ideals. Bandaru and Rafi [1] introduced a new notion called G-algebra, since the fuzzy G-algebras have been extensively investigated by several researches. The properties of fuzzy sub commutative ideals and fuzzy sub implicative ideals are obtained. Ragavan and Solairaj [14] some new results on intuitionistic fuzzy H-ideals in BCI algebra. 2. Preliminaries Definition 2.1. [8] A Subtraction G-algebra we mean a nonempty set X with a binary operation - and a constant 0 satisfying the following conditions: (F1) x- x = 0, (F2) x – (x - y) = y, for all x, y ϵ X. Definition 2.2. [15] (G-subalgebra) A non-empty subset S of a subtraction G-algebra X is called a subtraction G-subalgebra of X if x-y ∈S [0,1]. Definition 2.3. [16] A fuzzy set f of a universe X is a function from X to the unit closed interval [0, 1], that is f: X → [0, 1]. Definition 2.4. [2] An intuitionistic fuzzy set A in a finite universe of discourse X = {x1, x2, x3, x4, ··· xn} is given by A = {: x ∈X}, Where ΨA: X → [0, 1] and ΩA : X → [0, 1] such that 0 ≤ ΨA(x) + ΩA (x) ≤ 1. The number ΨA(x) and ΩA (x) denote the degree of membership and non-membership of x ∈X to A, respectively. For each IFS A in X, if πA(x) = 1 − ΨA(x)− ΩA(x),for all x ∈X. Definition 2.5. A nonempty subset I of X is called an ideal of X if (I1): 0 ∈ I, and (I2): x - y ∈ I and y ∈ I imply x ∈ I. Definition 2.6. A fuzzy subset ΨA of X is said to be a fuzzy ideal of X if it satisfies (F3) ΨA(0) ≥ ΨA (x) for all x ∈ X, (F4) ΨA (x) ≥ min {ΨA (x - y), ΨA (y)} for all x, y ∈ X. 261 Antagonistic Intuitionistic Fuzzy Sub Commutative Ideals of Subtraction G-Algebra Definition 2.7. An intuitionistic fuzzy set A = {x, ΨA(x), ΩA(x): x ∈ X} in X is called an intuitionistic fuzzy ideal of X if it satisfies (F5) ΨA (0) ≥ ΨA (x), ΩA(0) ≤ ΩA (x), (F6) ΨA (x) ≥ min {ΨA (x - y), ΨA (y)} and (F7) ΩA(x) ≤ max {ΩA (x - y), ΩA(y)} for all x, y ∈X. Definition 2.8. An intuitionistic fuzzy set A = {x, ΨA(x), ΩA(x): x ∈X} in X is called an antagonistic -intuitionistic fuzzy ideal of X if it satisfies (F8) ΨA (0) ≤ ΨA (x), ΩA (0) ≥ ΩA (x), (F9) ΨA(x) ≤ max { ΨA (x - y), ΨA (y)} and (F10) ΩA(x) ≥ min {ΩA (x - y), ΩA(y)} for all x,y ∈X. Definition 2.9. [9, 11] A nonempty subset I of X is called a positive implicative ideal (i.e., weakly positive implicative ideal) of X if it satisfies (I1) 0 ∈ I and (I3): ((x - z) - z) - (y - z) ∈ I and y ∈ I imply x – z ∈ I. Definition 2.10. [9, 14] A nonempty subset I of X is called a sub-implicative ideal of X if it satisfies (I1) 0 ∈ I and (I3): ((x - (x - y)) - (y - x)) - z ∈ I and z ∈ I imply y - (y - x) ∈ I. Definition 2.11. [9, 13] A nonempty subset I of X is called a sub-commutative ideal of X if it satisfies (I1) 0 ∈ I and (I4): (y - (y - (x - (x - y)))) - z ∈ I and z ∈ I imply x - (x - y) ∈ I. Definition 2.12. An anti-Fuzzy Subset ΨAof X is called an antagonistic -fuzzy sub- implicative ideal of X if it satisfies (F11) ΨA(0) ≤ ΨA (x) for all x ∈ X, and (F12) ΨA(y - (y - x)) ≤ max { ΨA(((x - (x - y)) - (y - x)) - z), ΨA(z)} for all x, y, z ∈ X. Definition 2.13. An anti-fuzzy subset ΨA of X is called an antagonistic -fuzzy sub- commutative ideal (briefly, AFSC-ideals) of X if it satisfies (F1) and (F13) ΨA(x - (x - y)) ≤ max { ΨA((y - (y - (x - (x - y)))) - z), ΨA(z)} for all x, y, z ∈ X. Definition 2.14. An anti-Fuzzy Subset (ΨA, ΩA) of X is called an antagonistic - intuitionistic fuzzy sub-implicative ideal of X if it satisfies (F16) ΨA(0) ≤ ΨA (x), ΩA (0) ≥ ΩA (x) for all x ∈ X, and (F17) ΨA(y - (y - x)) ≤ max { ΨA (((x - (x - y)) - (y - x)) - z), ΨA(z)} for all x, y, z ∈ X. (F18) ΩA (y - (y - x)) ≥ min {ΩA (((x - (x - y)) - (y - x)) - z), ΩA(z)} for all x, y, z ∈X. Definition 2.15. An anti-fuzzy subset (ΨA, ΩA) of X is called an antagonistic intuitionistic fuzzy sub-commutative ideal (briefly, AFSC-ideals) of X if it satisfies (F19) ΨA (0) ≤ ΨA (x), ΩA (0) ≥ ΩA (x) for all x ∈ X, and (F20) ΨA (x - (x - y)) ≤ max { ΨA((y - (y - (x - (x - y)))) - z), ΨA(z)} for all x, y, z ∈ X. 262 B. Lena, C. Ragavan, S. Jeyanthi (F21) ΩA (x - (x - y)) ≥ min {ΩA ((y - (y - (x - (x - y)))) - z), ΩA(z))} for all x, y, z ∈ X. 3. Over Antagonistic-Intuitionistic Fuzzy Sub Commutative Ideal of subtraction G-algebras Theorem 3.1. If Gα, β(A) and Gα, β(B) are antagonistic -intuitionistic fuzzy sub- commutative ideal of subtraction G-algebra then Gα, β(A+B) is also antagonistic - intuitionistic fuzzy sub-commutative ideal of subtraction G-algebra. Proof: Given Gα, β(A) and Gα, β(B) are antagonistic -intuitionistic fuzzy sub- commutative ideal of X. (1) ΨA (0) ≤ ΨA(x), ΩA (0) ≥ ΩA (x) for all x ∈ X, and (2) ΨA (x - (x - y)) ≤ max {ΨA ((y - (y - (x - (x - y)))) - z), ΨA (z)} for all x, y, z ∈ X. (3) ΩA (x - (x - y)) ≥ min { ΩA ((y - (y - (x - (x - y)))) - z), ΩA (z))} for all x, y, z ∈ X. (4) ΨB (0) ≤ ΨB (x), ΩB (0) ≥ ΩB (x) for all x ∈ X, and (5) ΨB (x - (x - y)) ≤ max {ΨB ((y - (y - (x - (x - y)))) - z), ΨB (z)} for all x, y, z ∈ X. (6) ΩB (x - (x - y)) ≥ min {ΩB ((y - (y - (x - (x - y)))) - z), ΩB (z))} for all x, y, z ∈ X. Case1: ΨA (0) + ΨB (0) - ΨA (0). ΨB (0) ≤ ΨA (x) + ΨB (x) - ΨA (x). ΨB (x) Gα, β {ΨA (0) + ΨB (0) - ΨA (0). ΨB (0)} ≤ Gα, β {ΨA (x) + ΨB (x) - ΨA (x). ΨB (x)} {αΨA (0) + αΨB (0) - αΨA (0). α ΨB (0)} ≤ {α ΨA(x) + α ΨB (x) - α ΨA (x). α ΨB (x)} {αΨA (0) + αΨB (0) – α 2 ΨA (0). ΨB (0)} ≤ {αΨA (x)+ αΨB (x)- α 2 ΨA (x). ΨB (x)} Gα, β (A + B) (0) ≤ Gα, β (A + B) (x) ΩA (0). ΩB (0) ≥ ΩA (x). ΩB (x) Gα, β {ΩA (0). ΩB (0)} ≥ Gα, β {ΩA (x). ΩB (x)} {β ΩA (0). β ΩB (0)} ≥ {β ΩA(x). β ΩB(x)} {β 2 ΩA (0). ΩB (0)} ≥ {β 2 ΩA (x). ΩB (x)} Gα, β (A + B) (0) ≥ Gα, β (A + B) (x) Case 2: { ΨA (x - (x - y)) + ΨB (x - (x - y)) - ΨA (x - (x - y)). ΨB (x - (x - y))} ≤ max {[ ΨA ((y -(y - (x - (x - y))) - z), ΨA(z)]} + max [ΨB ((y - (y - (x - (x - y))) - z), ΨB (z)]- max {ΨA ((y - (y - (x - (x - y))) - z), ΨA (z)}. max {ΨB ((y - (y - (x - (x - y))) - z), ΨB (z)}} Gα, β {ΨA (x - (x - y)) + ΨB (x - (x - y)) - ΨA (x - (x - y)). ΨB (x - (x - y))} ≤ Gα, β max{{μA ((y - (y - (x - (x - y))) - z) + μB ((y - (y - (x - (x - y))), μA(z) + μB (z)}- max{ μA ((y - (y - (x - (x - y))) - z).μB ((y - (y - (x - (x - y))), μA (z).μB (z)}} {αΨA(x-(x-y)) + αΨB(x-(x-y)) - αΨA(x-(x-y)). αΨB(x-(x-y))} ≤ max {{α ΨA ((y-(y-(x- (x-y)))-z) + α ΨB ((y-(y-(x-(x-y)))-z), α ΨA (z) + α ΨB(z)}- max {α ΨA ((y-(y-(x-(x-y)))- z). α ΨB((y-(y-(x-(x-y)))-z)-, α ΨA (z). α ΨB (z)}} {αΨA(x-(x-y)) + αΨB(x-(x-y)) – α 2 ΨA(x-(x-y)). ΨB(x-(x-y))} ≤ max {{ α ΨA ((y-(y-(x- (x-y)))-z) + αΨB ((y-(y-(x-(x-y)))-z) – α 2 ΨA ((y-(y-(x-(x-y)))-z). ΨB ((y-(y-(x-(x-y)))- z)}, {α ΨA (z) + αΨB (z)- α 2 ΨA (z). ΨB (z)}} Gα, β(A+B) (x-(x-y)) ≤ max {Gα, β (A+B) ((y-(y-(x-(x-y)))-z), Gα, β (A+B) (z)} ΩA (x - (x - y)). ΩB (x - (x - y)) ≥ min {ΩA ((y - (y - (x - (x - y)))) - z), ΩA (z)}. min {ΩB ((y - (y - (x - (x - y)))) - z), ΩB (z)} 263 Antagonistic Intuitionistic Fuzzy Sub Commutative Ideals of Subtraction G-Algebra ΩA (x - (x - y)). ΩB (x - (x - y)) ≥ min {ΩA ((y - (y - (x - (x - y)))) - z). ΩB ((y - (y - (x - (x - y))))-z}, min {ΩA (z). ΩB(z)} Gα,β{ ΩA(x - (x - y)). ΩB (x - (x - y))}≥ Gα,β{ min { ΩA ((y - (y - (x - (x - y)))) - z). ΩB ((y - (y - (x - (x - y))))-z}, min{ ΩA (z). ΩB(z)}} {β ΩA(x - (x - y)). β ΩB (x - (x - y))} ≥min {β ΩA ((y - (y - (x - (x - y)))) - z). β ΩB ((y - (y - (x - (x - y))))-z}, min {α ΩA(z)).α ΩB (z)} { β 2 ΩA (x - (x - y)). ΩB (x - (x - y))} ≥min {{β 2 ΩA((y - (y - (x - (x - y)))) - z). ΩB ((y - (y - (x - (x - y))))-z},{ β 2 ΩA(z)).α ΩB (z}} Gα,β(A+B)(x- (x - y))≥min{ Gα,β(A+B)((y - (y - (x - (x - y)))), Gα,β(A+B) (A+B)(z)} Therefore, the two numbers of A+B is an antagonistic- intuitionistic fuzzy sub- commutative ideal in subtraction G-algebra of X. Theorem 3.2. If Gα,β(A) and Gα,β(B) are antagonistic -intuitionistic fuzzy sub- commutative ideal of subtraction G-algebra then Gα,β(A•B) is also antagonistic - intuitionistic fuzzy sub-commutative ideal of subtraction G-algebra. Proof: Given Gα,β(A) and Gα,β(B) are antagonistic -intuitionistic fuzzy sub- commutative ideal of X. (1) ΨA (0)≤ ΨA (x), ΩA(0)≥ΩA(x) for all x ∈ X, and (2) ΨA(x - (x - y)) ≤ max { ΨA ((y - (y - (x - (x - y)))) - z), ΨA(z)} for all x, y, z ∈ X. (3) ΩA (x - (x - y)) ≥min { ΩA((y - (y - (x - (x - y)))) - z), ΩA (z))} for all x, y, z ∈ X. (4) ΨB (0)≤ ΨB (x), ΩB(0)≥ ΩB(x) for all x ∈ X, and (5) ΨB (x - (x - y)) ≤ max { ΨB ((y - (y - (x - (x - y)))) - z), ΨB (z)} for all x, y, z ∈ X. (6) ΩB (x - (x - y))≥ min { ΩB ((y - (y - (x - (x - y)))) - z), ΩB (z))} for all x, y, z ∈ X. Case1: {ΨA(0). ΨB(0)} ≤ {ΨA(x). ΨB (x)} Gα,β{ΨA(0). ΨB(0)}≤ Gα,β{ΨA (x). ΨB (x)} {αΨA(0). αΨB(0)}≤ { αΨA (x). αΨB (x)} {α 2 ΨA(0). ΨB(0)}≤ { α 2 ΨA (x). ΨB (x)} Gα,β(A•B)(0)≤ Gα,β(A•B)(x) {ΩA(0)+ ΩB(0) - ΩA(0). ΩB(0)} ≥ {ΩA(x)+ ΩB (x)- ΩA (x). ΩB (x)} Gα,β{ ΩA(0)+ ΩB(0) - ΩA(0). ΩB(0)} ≥ Gα,β{ ΩA(x)+ ΩB (x)- ΩA (x). ΩB (x)} {β ΩA(0)+ β ΩB(0) - β ΩA(0). β ΩB(0)} ≥{ β ΩA(x)+ β ΩB (x)- β ΩA (x). β ΩB (x)} {β ΩA(0)+ β ΩB(0) – β 2 ΩA(0). ΩB(0)} ≥ { β ΩA(x)+ β ΩB (x)- β 2 ΩA (x). ΩB (x)} Gα,β(A•B)(0) ≥ Gα,β(A•B)(x) Case2: {ΨA(x- (x- y)). ΨB (x- (x- y))} ≤ max {ΨA ((y-(y-(x-(x-y)))-z), ΨA (z)}. max {ΨB ((y-(y- (x-(x-y)))-z), ΨB (z)}} Gα,β{ ΨA(x-(x-y)). ΨB(x-(x-y))} ≤ Gα,β max{ ΨA ((y-(y-(x-(x-y)))-z). ΨB((y-(y-(x-(x- y))), ΨA(z). ΨB(z)}} {αΨA(x-(x-y)). αΨB(x-(x-y))} ≤ max {α ΨA ((y-(y-(x-(x-y)))-z). α ΨB((y-(y-(x-(x-y)))- z)-, α ΨA (z). α ΨB (z)}} {α 2 ΨA(x-(x-y)). ΨB(x-(x-y))} ≤ max {{α 2 ΨA ((y-(y-(x-(x-y)))-z). ΨB ((y-(y-(x-(x-y)))- z)}, {α ΨA (z) + αΨB (z)- α 2 ΨA (z). ΨB (z)}} 264 B. Lena, C. Ragavan, S. Jeyanthi Gα,β(A•B)( x-(x-y))≤ max{ Gα,β (A•B) ((y-(y-(x-(x-y)))-z), Gα,β (A•B) (z)} { ΩA(x- (x- y)) + ΩB (x- (x- y)) - ΩA(x- (x- y)). ΩB (x- (x- y))} ≥min {[ ΩA((y-(y-(x-(x- y)))-z), ΩA(z)]} + max [ΩB((y-(y-(x-(x-y)))-z), ΩB (z)]- max {ΩA ((y-(y-(x-(x-y)))-z), ΩA (z)}. max { ΩB ((y-(y-(x-(x-y)))-z), ΩB (z)}} { ΩA(x- (x- y)) + ΩB (x- (x- y)) - ΩA(x- (x- y)). ΩB (x- (x- y))} ≥{min { ΩA((y-(y-(x-(x- y)))-z) +ΩB((y-(y-(x-(x-y)))-z) - ΩA ((y-(y-(x-(x-y)))-z). ΩB ((y-(y-(x-(x-y)))-z)}, min{ ΩA (z)+ ΩB (z)- ΩA (z). ΩB (z)}} ΩA(x- (x- y)) + ΩB (x- (x- y)) - ΩA(x- (x- y)). ΩB (x- (x- y))} ≥{min { ΩA((y-(y-(x-(x- y)))-z) +ΩB((y-(y-(x-(x-y)))-z) - ΩA ((y-(y-(x-(x-y)))-z). ΩB ((y-(y-(x-(x-y)))-z)}, {ΩA (z)+ ΩB (z)- ΩA (z). ΩB (z)}} Gα,β{ΩA(x-(x-y))+ ΩB(x-(x-y)) - ΩA(x-(x-y)). ΩB(x-(x-y))} ≥min Gα,β {{ΩA ((y-(y-(x- (x-y)))-z) + ΩB ((y-(y-(x-(x-y))), ΩA (z) + ΩB (z)}- max{ΩA((y-(y-(x-(x-y)))-z). ΩB((y- (y-(x-(x-y))), ΩA(z). ΩB(z)}} {αΩA(x-(x-y))+ αΩB(x-(x-y)) - αΩA(x-(x-y)). αΩB(x-(x-y))} ≥min{{αΩA ((y-(y-(x-(x- y)))-z) + αΩB ((y-(y-(x-(x-y)))-z), αΩA (z) + αΩB(z)}- max {αΩA ((y-(y-(x-(x-y)))-z). αΩB((y-(y-(x-(x-y)))-z), αΩA (z). αΩB (z)}} {α ΩA (x - (x - y)) + α ΩB (x - (x - y)) – α 2 ΩA (x - (x - y)). ΩB (x - (x - y))} ≥ min {{ α ΩA ((y - (y - (x - (x - y))) - z) + α ΩB ((y - (y - (x - (x - y))) - z) – α 2 ΩA ((y - (y - (x - (x - y))) - z). ΩB ((y - (y - (x - (x - y))) - z)}, {αΩA (z) + αΩB (z)- α 2 ΩA (z). ΩB (z)}} Gα, β(A • B) (x - (x - y)) ≥ min {Gα, β (A • B) ((y - (y - (x - (x - y))) - z), Gα,β (A • B) (z)} Therefore, the two numbers of A B is an antagonistic- intuitionistic fuzzy sub- commutative ideal in subtraction G-algebra of X. Example 3.1. Let X = {0, a, b, c} be a subtraction G-algebra with the following Cayley table. - 0 a b c 0 0 a b c a 0 0 b c b 0 a 0 c c 0 a b 0 We define an anti- fuzzy set A then A = < X, ΨA, ΩA> by routine calculation A is an antagonistic -intuitionistic fuzzy sub-implicative ideal of X. X 0 a b c ΨA .40 .42 .62 .75 ΩA .50 .45 .29 .21 We define an anti- fuzzy set B then B = < X, ΨB, ΩB > by routine calculation B is an antagonistic -intuitionistic fuzzy sub-implicative ideal of X. X 0 a b C ΨB .51 .53 .61 .72 ΩB .47 42 .31 .26 265 Antagonistic Intuitionistic Fuzzy Sub Commutative Ideals of Subtraction G-Algebra If A and B are antagonistic -intuitionistic fuzzy sub-implicative ideal of X. obviously the union A B, Since x = b, y = c, z = a, α =.41, β=.46. {α 2 ΨA (0). ΨB (0)} ≤ {α 2 ΨA(x). ΨB(x)}. 0343 ≤ .0636 {βΩA (0) +βΩB (0)-β 2 ΩA (0). ΩB (0)} ≥ {βΩA(x)+β ΩB(x)- β 2 ΩA(x). ΩB(x)}. 4605 ≥ .2569 {α 2 ΨA (x - (x - y)). ΨB (x - (x - y))} ≤ max {{α 2 ΨA ((y - (y - (x - (x - y)))) - z). 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