Neutrosophic Q-fuzzy left N-subgroups of a Near-ring

Global Journal of Mathematical Sciences: Theory and Practical.

ISSN 0974-3200 Volume 9, Number 3 (2017), pp. 311-321 Š International Research Publication House http://www.irphouse.com

Neutrosophic Q-fuzzy left N-subgroups of a Near-ring K. Thirusangu1, S. Poornavel2 and R. Vasuki1 1

Associate Professor of Mathematics, SIVET College, Gowrivakkam, Chennai-73, Tamil Nadu, India.

2 Assistant Prof. in Mathematics, Madha Engineering College, Kundrathur, Chennai-69, Tamil Nadu, India.

3 Associate Prof. of Mathematics, SIVET College, Gowrivakkam, Chennai, Tamil Nadu, India.

Abstract In this paper, the notion of neutrosophic Q- fuzzy left N-subgroups is introduced in a near ring and investigated some related properties. Characterization of neutrosophic Q- fuzzy left N-subgroups with respect to T-norm and S-norm are given. Few homomorphic image and its pre-image on neutrosophic Q- fuzzy are obtained.

Section 1: Introduction Nagarajan and Manermaran [2013] gave few algebraic properties on M-fuzzy factor group, and obtained some its power fuzzy group, homomorphic image and preimage, union and intersection. Makamba [1992] discussed direct product and isomorphism of fuzzy subgroups. Ajmal [1994] studied on homomorphism on factor fuzzy group. They gave an idea to introduce M-fuzzy group, & its normality in a ring. Das [1981] found some new results on fuzzy group and level subgroups. It is also motivated to find Nfuzzy group in hemi-ring. Fang [1994] studied on fuzzy homomorphism & fuzzy isomorphism, and it leads to N-fuzzy subgroup in near-ring. Kim [1997], Kim & Kim [1994], Mukherjee and Bhattacharya [1984, 1986], and analyzed on some characterizations of fuzzy subgroups, and it give an idea to verify also few characterizations on N-fuzzy group in a near-ring. Kumar et. al. [1992] obtained some new structures on fuzzy normal subgroup and fuzzy quotient group which initiate to make an attempt on N-fuzzy normal subgroup in a near-ring. Kumbhoikar and Bapal

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[1991] found correspondence theorem for fuzzy ideals in near-ring. Liu et.al. [2001] and Morsi & Yehia [1994] got few properties on quotient fuzzy group and quotient fuzzy ring induced from fuzzy ideals in near-ring, and same argument will argue quotient N-fuzzy subgroup in near-ring. Further Section 2: Preliminaries and Definitions Definition 2.1: A near ring is a non – empty set R with two binary operations + and . satisfying the following axioms: (1). (R, +) is a group; (2). (R, .) is a semigroup; (3). x. (y + z) = x. y + x. z for all x, y, z, \ in R. Then It is called a left near – ring by (3). In this paper, it will use the word near- ring. Here xy denotes x.y; (2). x.0 = 0, and x(-y) = -(xy for x, y in R. Definition 2.2: A two sided đ?‘… – subgroup of a near – ring đ?‘… is a subset đ??ť of đ?‘… such that (1). (H, +) is a subgroup of (R, +); (2). RH ďƒŒ H; (3). HR ďƒŒ H. If H satisfies (1) and (2), then it is a left R-subgroup of R, while if H satisfies (1) and (3), then it is a right R-subgroup of R. Definition 2.3: A fuzzy set ď ­ in a set R is a function ď ­: R ď‚Ž [0, 1]. If Q is set, then a Q-fuzzy set ď Ź is a functionď Ź: R ď‚´ Q ď‚Ž [0, 1].

Definition 2.4: Let G be any group. A mapping Îź: G → [0, 1] is a fuzzy group if (FG1). đ?œ‡(đ?‘Ľđ?‘Ś) ≼ min{đ?œ‡(đ?‘Ľ), đ?œ‡(đ?‘Ś)} and (FG2). đ?œ‡(đ?‘Ľ −1 ) = đ?œ‡(đ?‘Ľ), for all đ?‘Ľ, đ?‘Ś ∈ đ??ş.

Definition 2.5: Let A be any non-empty set, and X be the universe of discourse. A neutrosophic Q-fuzzy set A on X ď&#x201A;´ Q characterized by a truth membership function TA(x, q), an indeterminacy function IA(x, q) and a falsity membership function FA(x, q) is defined as A = { < (x, q), TA(x, q), IA(x, q), FA(x, q) > : x ď&#x192;&#x17D; X, q ď&#x192;&#x17D; Q}, where TA , IA ,FA : X ď&#x201A;´ Q ď&#x201A;Ž [0. 1] and 0 ď&#x201A;Ł TA(x, q) ď&#x201A;Ł 1; 0 ď&#x201A;Ł IA(x, q ) ď&#x201A;Ł 1; 0 ď&#x201A;Ł FA(x, q) ď&#x201A;Ł 1, for all x ď&#x192;&#x17D; X, and q ď&#x192;&#x17D; Q. Definition 2.6: Let X, Y be two non-empty sets and đ?&#x2018;&#x201C;: đ?&#x2018;&#x2039; â&#x;ś đ?&#x2018;&#x152; be a function. (i). If B ={ < (y, q), TB(y, q), IB(y, q), FB(y, q) > : y ď&#x192;&#x17D; Y, q ď&#x192;&#x17D; Q} is a neutrosophic Q-fuzzy set in Y, then the preimage of B under đ?&#x2019;&#x2021;, denoted by đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (đ??ľ), is the neutrosophic fuzzy set in X defined by f-1(B) = {<(x, q), f-1(TB(x, q)), f-1(IB(x, q)), f-1(FB(x, q)) > : x ď&#x192;&#x17D; X, q ď&#x192;&#x17D; Q} where f-1(TB(x, q) = TB(f(x), q); f-1(IB(x, q) = IB(f(x), q); f-1(FB(x, q) = FB(f(x), q). (ii). If A = { < (x, q), TA(x, q), IA(x, q), FA(x, q) > : x ď&#x192;&#x17D; X, q ď&#x192;&#x17D; Q} is a Neutrosophic fuzzy set in X, then the image f(A) of A under f is the neutrosophic fuzzy set in Y defined by f(A) = { A = { < (y, q), Tf(A)(y, q), If(A)(y, q), Ff(A)(y, q) > : y ď&#x192;&#x17D; Y, q ď&#x192;&#x17D; Q} where Tf(A)(y, q) = đ?&#x2018; đ?&#x2018;˘đ?&#x2018;?(đ?&#x2018;Ľ,đ?&#x2018;&#x17E;)ď&#x192;&#x17D; đ?&#x2018;&#x201C;â&#x2C6;&#x2019;1 (đ?&#x2018;Ś,đ?&#x2018;&#x17E;) (đ?&#x2018;&#x2021;đ??´ (đ?&#x2018;Ľ, đ?&#x2018;&#x17E;)) đ?&#x2018;&#x2013;đ?&#x2018;&#x201C; đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (đ?&#x2018;Ś, đ?&#x2018;&#x17E;) ď&#x201A;š {}; 0, otherwise.

Neutrosophic Q-fuzzy left N-subgroups of a Near-ring

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If(A)(y, q) = đ?&#x2018; đ?&#x2018;˘đ?&#x2018;?(đ?&#x2018;Ľ,đ?&#x2018;&#x17E;)ď&#x192;&#x17D; đ?&#x2018;&#x201C;â&#x2C6;&#x2019;1 (đ?&#x2018;Ś,đ?&#x2018;&#x17E;) (đ??źđ??´ (đ?&#x2018;Ľ, đ?&#x2018;&#x17E;)) đ?&#x2018;&#x2013;đ?&#x2018;&#x201C; đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (đ?&#x2018;Ś, đ?&#x2018;&#x17E;) ď&#x201A;š {}; 0, otherwise.

Ff(A)(y, q) = đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;&#x201C;(đ?&#x2018;Ľ,đ?&#x2018;&#x17E;)ď&#x192;&#x17D; đ?&#x2018;&#x201C;â&#x2C6;&#x2019;1 (đ?&#x2018;Ś,đ?&#x2018;&#x17E;) đ??š(đ?&#x2018;Ľ, đ?&#x2018;&#x17E;)) đ?&#x2018;&#x2013;đ?&#x2018;&#x201C; đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (đ?&#x2018;Ś, đ?&#x2018;&#x17E;) ď&#x201A;š {},where Ff(A)(y, q) = (1-f(1FA))(y, q) ; 1, otherwise.

Definition 2.7: A neutrosophic left R-fuzzy subgroup of a near ring R is a neutrosophic fuzzy set A = (TA, IA, FA) on X in definition (2.3) such that (1). (a) TA(x-y) ď&#x201A;ł min { TA(x), TA(y)}; (b). IA(x-y) ď&#x201A;ł min { IA(x), IA(y)}; (c). FA(x-y) ď&#x201A;Ł max { FA(x), FA(y)}; (2). (a) TA(xy) ď&#x201A;ł min { TA(x), TA(y)}; (b). IA(xy) ď&#x201A;ł min { IA(x), IA(y)}; (c). FA(xy) ď&#x201A;Ł max { FA(x), FA(y)} for all x, y ď&#x192;&#x17D; R.

Definition 2.8: A = < x, TA(x), IA(x), FA(x) > and B = < x, TB(x, IB(x), FB(x) > are two neutrosophic fuzzy sets on X. Then, (ii). A â&#x160;&#x2020; đ??ľ , if TA(x) ď&#x201A;Ł TB(x), IA(x) ď&#x201A;Ł IB(x), FA(x) ď&#x201A;ł FB(x), for all x ď&#x192;&#x17D; X. (ii).đ??´ â&#x2C6;Ş đ??ľ = < đ?&#x2018;Ľ, max(TA (x), TB (x)), max(IA (x), IB (x)), min((FA (x), FB (x)) >. (iii). đ??´ â&#x2C6;Š đ??ľ = < đ?&#x2018;Ľ, min(TA (x), TB (x)), min(IA (x), IB (x)), max ((FA (x), FB (x)) >.

Definition 2.9: The complement Ac of a neutrosophic fuzzy subset A = (TA, IA, FA) on X is defined by Ac = {< x, TAc (x), IAc (x), FAc (x) > : đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; } where TAc (x) = FA (x), IAc (x) = 1 â&#x2C6;&#x2019; IA (x), FAc (x) = TA (x) , for all x ď&#x192;&#x17D; X. Definition 2.10: Let f be a mapping from a near ring R1 to a near ring R2 and A = (TA, IA, FA) be a neutrosophic fuzzy set in R1. Then f(A) the image of ď ­ is neutrosophic fuzzy set (Tf(A), I f(A), F f(A)) in R2 defined by Tf(A)(y) = sup ( TA(x): x ď&#x192;&#x17D; f-1(y) ď&#x201A;š {}; 0, otherwise. for all y ď&#x192;&#x17D; R2; Ff(A)(y) = sup ( FA(x): x ď&#x192;&#x17D; f-1(y) ď&#x201A;š {}; 0, otherwise. for all y ď&#x192;&#x17D; R2; Tf(A)(y) = inf ( IA(x): x ď&#x192;&#x17D; f-1(y) ď&#x201A;š {}; 0, otherwise. for all y ď&#x192;&#x17D; R2; Definition 2.11: Let A = (TA, IA, FA) be a neutrosophic fuzzy set in a near ring R. For each t ď&#x192;&#x17D; [0, 1], upper cut-set of A is defined as the set {x ď&#x192;&#x17D; R: TA(x) ď&#x201A;ł t; FA(x) ď&#x201A;ł t; IA(x) ď&#x201A;Ł t}.

Definition 2.12: (T-norm): A triangular norm is a function T: [0, 1] ď&#x201A;´ [0., 1] ď&#x201A;Ž [0, 1] that satisfies the following conditions (T1): T(x, 1) = x; (T2). T(x, y) = T(y, x);

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(T3): T(x, T(y, z)) = T(T(x, y), z); (T4). T(x, y)  T(x, z) if y  z for all x, y, z in [0, 1].

Definition 2.13: Let Q, N be non-empty sets, and . be a map from N  R  R. Let A = (TA, IA, FA) be a neutrosophic fuzzy set in a near ring R. Then A is a neutrosophic Q-fuzzy left N-subgroup of R if (1a) TA(n(x-y),q)  T {TA(x, q), TA(y, q)}; (1b).

TA(nx, q)  TA(x, q) (2a) IA(n(x-y),q)  T {IA(x, q), IA(y, q)}; (2b). IA(nx, q)  IA(x, q) (3a) FA(n(x-y),q)  S {IA(x, q), IA(y, q)}; (3b). IA(nx, q)  TA(x, q) for all x, y in R; q in Q, and n in N. Section 3: Homomorphic properties of neutrosophic Q-fuzzy left N-subgroups The following theorem is first started: Theorem 3.1: If A = (TA, IA, FA) and B = (TB, IB, FB) are neutrosophic Q-fuzzy left Nsubgroups of a near ring R, then their intersection (A  B) s a neutrosophic Q-fuzzy left N-subgroup of R. Proof: Let x, y  R. Let n  N, and q  Q. It gets that (1a) T(AB) (n(x-y), q) = min { TA (n(x-y), q), TA ((n(x – y)), q)  min { T { TA (x, q), TA (y, q)}, T {TB((x, q), TB (y, q) } } = min { T { TA (x, q), TB (y, q)}, T {TA (x, q), TB (y, q) } }  T { min { TA (x, q), TB (y, q)}, min {TA (x, q), TB (y, q) } } = T {T(AB).(x, q), T(AB).(x, q) } (2a) I(AB) (n(x-y), q) = min { IA (n(x-y), q), IA ((n(x – y)), q)  min { T { IA (x, q), IA (y, q)}, T {IB((x, q), IB (y, q) } } = min { T { IA (x, q), IB (y, q)}, T {IA (x, q), IB (y, q) } }  T { min { IA (x, q), IB (y, q)}, min {IA (x, q), IB (y, q) } } = T {I(AB).(x, q), I(AB).(x, q) } (3a) F(AB) (n(x-y), q) = min { FA (n(x-y), q), FA ((n(x – y)), q)  min { S { FA (x, q), FA (y, q)}, S {FB((x, q), FB (y, q) } } = min { S { FA (x, q), FB (y, q)}, S {FA (x, q), FB (y, q) } }  S { min { FA (x, q), FB (y, q)}, min {FA (x, q), FB (y, q) } } = S {F(AB).(x, q), F(AB).(x, q) } (1b).T(A B) (nx, q) = min { TA (nx, q), TB (nx, q) }  min { TA (x, q), TB (x, q) }= T(A B) (x, q).

(2b).I(A B) (nx, q) = min { IA (nx, q), IB (nx, q) }  min { IA (x, q), IB (x, q) }= I(A B) (x, q).

(3b).F(A B) (nx, q) = min { FA (nx, q), FB (nx, q) }  min { FA (x, q), FB (x, q) }= F(A B) (x, q).

Thus (A  B) is a neutrosophic Q-fuzzy left N-subgroup of R.

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Theorem 3.2: Every imaginable neutrosophic Q-fuzzy left N-subgroup ď ­ = = (TA, IA, FA) on a near ring (R, +, .) is a neutrosophic fuzzy left N-subgroup of R. Proof: Assume ď ­ is an imaginable Q-fuzzy left N-subgroup of a near ring (S, +,.).

Let x, y ď&#x192;&#x17D; R, q ď&#x192;&#x17D; Q, and n ď&#x192;&#x17D; N. Then it observes that Since â&#x20AC;&#x2DC;đ?&#x153;&#x2021;â&#x20AC;&#x2122; is imaginable, it finds that min { TA(x, q), TA(y, q)} = T (min { TA(x, q), TA(y, q)}, min { TA(x, q), TA(y, q)} } ď&#x201A;ł T { TA(x, q), TA(y, q)} ď&#x201A;ł min { TA(x, q), TA(y, q)} ( by property of T-norm).

Thus T { TA(x, q), TA(y, q)} = min { TA(x, q), TA(y, q)} It follows that (1a) TA(n(x-y),q) ď&#x201A;ł T {TA(x, q), TA(y, q)} = min {TA(x, q), TA(y, q)}.

In addition, (1b). TA(nx, q) ď&#x201A;ł TA(x, q).

Similarly, it implies that min { IA(x, q), IA(y, q)} = T (min { IA(x, q), IA(y, q)}, min { IA(x, q), IA(y, q)} } ď&#x201A;ł T { IA(x, q), IA(y, q)} ď&#x201A;ł min { IA(x, q), IA(y, q)} ( by property of T-norm).

Thus T { IA(x, q), IA(y, q)} = min { IA(x, q), IA(y, q)} (2a) IA(n(x-y),q) ď&#x201A;ł T {IA(x, q), IA(y, q)} = min {IA(x, q), IA(y, q)}.

Also (2b). IA(nx, q) ď&#x201A;ł IA(x, q).

In addition, min { FA(x, q), FA(y, q)} = S (max { FA(x, q), FA(y, q)}, max { FA(x, q), FA(y, q)} } ď&#x201A;Ł S { FA(x, q), FA(y, q)} ď&#x201A;Ł max { FA(x, q), FA(y, q)} ( by property of T-norm).

Thus S { IA(x, q), IA(y, q)} = max { IA(x, q), IA(y, q)} (3a) IA(n(x-y), q) ď&#x201A;Ł S {IA(x, q), IA(y, q)} = max {TA(x, q), TA(y, q)}.

(3b). IA(nx, q) ď&#x201A;ł IA(x, q).

Hence ď ­ is a neutrosophic Q-fuzzy left N-subgroup of R. Theorem 3.3: If A = (TA, IA, FA) is a neutrosophic Q-fuzzy left N-subgroup of a near ring Rď&#x201A;˘ and f is an endomorphism (near ring) from R ď&#x201A;Ž R, then Af = (đ?&#x2018;&#x2021;đ??´đ?&#x2018;&#x201C; , đ??źđ??´đ?&#x2018;&#x201C; , đ??šđ??´đ?&#x2018;&#x201C; ) is a neutrosophic Q-fuzzy left N-subgroup of R. Proof: Let x, y ď&#x192;&#x17D; R. Let n ď&#x192;&#x17D; N, and q ď&#x192;&#x17D; Q. It gets that (1a). đ?&#x2018;&#x2021;đ??´đ?&#x2018;&#x201C; (n(x-y), q) = TA (f (n(x â&#x20AC;&#x201C; y)), q) = TA (f (x, q), f(y, q) ď&#x201A;ł T { TA (f (x, q), TAf(y, q)} ď&#x201A;ł T { đ?&#x2018;&#x2021;đ??´đ?&#x2018;&#x201C; (x, q), đ?&#x2018;&#x2021;đ??´đ?&#x2018;&#x201C; (y, q) }.

(2a). đ??źđ??´đ?&#x2018;&#x201C; (n(x-y), q) = IA (f (n(x â&#x20AC;&#x201C; y)), q) = IA (f (x, q), f(y, q) ď&#x201A;ł T { IA (f (x, q), IAf(y, q)} ď&#x201A;ł T { đ??źđ??´đ?&#x2018;&#x201C; (x, q), đ??źđ??´đ?&#x2018;&#x201C; (y, q) }.

(3a). đ??šđ??´đ?&#x2018;&#x201C; (n(x-y), q) = FA (f (n(x â&#x20AC;&#x201C; y)), q) = FA (f (x, q), f(y, q) ď&#x201A;Ł S { FA (f (x, q), FAf(y, q)} ď&#x201A;ł S { đ??šđ??´đ?&#x2018;&#x201C; (x, q), đ??šđ??´đ?&#x2018;&#x201C; (y, q) }.

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(1b). đ?&#x2018;&#x2021;đ??´đ?&#x2018;&#x201C; (nx), q)

= TA (f (nx, q) = TA (f (x, q) ď&#x201A;ł đ?&#x2018;&#x2021;đ??´đ?&#x2018;&#x201C; (x, q).

(2b). đ??źđ??´đ?&#x2018;&#x201C; (nx, q)

= IA (f (nx, q) = IA (f (x, q) ď&#x201A;ł đ??źđ??´đ?&#x2018;&#x201C; (x, q).

(3b). đ??šđ??´đ?&#x2018;&#x201C; (nx, q)

= FA (f (nx, q) = FA (f (x, q) ď&#x201A;ł đ??šđ??´đ?&#x2018;&#x201C; (x, q).

Thus Af = (đ?&#x2018;&#x2021;đ??´đ?&#x2018;&#x201C; , đ??źđ??´đ?&#x2018;&#x201C; , đ??šđ??´đ?&#x2018;&#x201C; ) is a neutrosophic Q-fuzzy left N-subgroup of R. Theorem 3.4: Let ď Ş: R ď&#x201A;Ž Rď&#x201A;˘ be an onto homomorphism on near rungs. The pre-image of a neutrosophic Q-fuzzy left N-subgroup of near ring Rď&#x201A;˘ under f is a neutrosophic Qfuzzy left N-subgroup of near ring R. Proof: Let ď ­ = (Tď ­, Iď ­, Fď ­) be a neutrosophic Q-fuzzy left N-subgroup of a near ring Rď&#x201A;˘.

Let ď Ź = (Tď Ź, Iď Ź, Fď Ź) be the pre-image of ď ­ under f. Then it finds that (1a).Tď Ź (n(x-y), q) = Tď ­ (ď Ş (n(x â&#x20AC;&#x201C; y)), q) = Tď ­ (ď Ş (x, q), f(y, q) ď&#x201A;ł T { Tď ­ (ď Ş (x, q), Tď ­ ď Ş(y, q)} ď&#x201A;ł T { Tď Ź (x, q), Tď Ź (y, q) }.

(2a). Iď Ź (n(x-y), q) = Iď ­ (ď Ş (n(x â&#x20AC;&#x201C; y)), q) = Iď ­ (ď Ş (x, q), ď Ş(y, q) ď&#x201A;ł T { Iď ­ (ď Ş(x, q), Iď ­ď Ş(y, q)} ď&#x201A;ł T { Iď Ź(x, q), Iď Ź(y, q) }.

(3a). Fď Ź (n(x-y), q) = Fď ­ (ď Ş (n(x â&#x20AC;&#x201C; y)), q) = Fď ­ (ď Ş (x, q), ď Ş(y, q)) ď&#x201A;Ł S { Fď ­ (ď Ş (x, q), Fď ­f(y, q)} ď&#x201A;ł S { Fď Ź (x, q), Fď Ź (y, q) }.

(1b). Tď Ź (nx), q) = Tď ­ (ď Ş (nx, q) = Tď ­ (ď Ş (x, q) ď&#x201A;ł Tď Ź(x, q).

(2b). Iď Ź (nx, q) = Iď ­ (ď Ş (nx, q) = Iď ­ (ď Ş (x, q) ď&#x201A;ł Iď Ź (x, q).

(3b). Fď Ź(nx, q) = Fď ­ (ď Ş (nx, q) = Fď ­ (ď Ş (x, q) ď&#x201A;ł Fď Ź (x, q).

Thus ď Ź = (Tď Ź, Iď Ź, Fď Ź) is a neutrosophic Q-fuzzy left N-subgroup of R. Theorem 3.5: Let f: R ď&#x201A;Ž Rď&#x201A;˘ be an onto homomorphism on near rungs. The pre-image of a neutrosophic Q-fuzzy left N-subgroup of near ring Rď&#x201A;˘ under f with supreme property is a neutrosophic Q-fuzzy left N-subgroup of near ring R. Proof: Let ď ­ = (Tď ­, Iď ­, Fď ­) be a neutrosophic Q-fuzzy N-subgroup of a near ring Rď&#x201A;˘ with supreme property. Let ď Ź = (Tď Ź, Iď Ź, Fď Ź) be a neutrosophic Q-fuzzy N-subset in Rď&#x201A;˘.

Let x1, y1 ď&#x192;&#x17D; Rď&#x201A;˘ and x0 ď&#x192;&#x17D; f-1(x1), y0 ď&#x192;&#x17D; f-1(y1) be such that đ?&#x2018; đ?&#x2018;˘đ?&#x2018;?

Tď ­(đ?&#x2018;Ľ0 , đ?&#x2018;&#x17E;) = đ?&#x2018;&#x2021; (â&#x201E;&#x17D;, đ?&#x2018;&#x17E;) â&#x201E;&#x17D; â&#x2C6;&#x2C6; đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (đ?&#x2018;Ľ 1 ) ď ­ đ?&#x2018; đ?&#x2018;˘đ?&#x2018;?

Tď ­(đ?&#x2018;Ś0 , đ?&#x2018;&#x17E;) = đ?&#x2018;&#x2021; (â&#x201E;&#x17D;, đ?&#x2018;&#x17E;) respectively â&#x201E;&#x17D; â&#x2C6;&#x2C6; đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (đ?&#x2018;Ś1 ) ď ­ Iď ­(đ?&#x2018;Ľ0 , đ?&#x2018;&#x17E;) = Iď ­(đ?&#x2018;Ś0 , đ?&#x2018;&#x17E;) =

đ?&#x2018; đ?&#x2018;˘đ?&#x2018;? đ??ź (â&#x201E;&#x17D;, đ?&#x2018;&#x17E;) â&#x201E;&#x17D; â&#x2C6;&#x2C6; đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (đ?&#x2018;Ľ1 ) ď ­

đ?&#x2018; đ?&#x2018;˘đ?&#x2018;? đ??ź (â&#x201E;&#x17D;, đ?&#x2018;&#x17E;) respectively â&#x201E;&#x17D; â&#x2C6;&#x2C6; đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (đ?&#x2018;Ś1 ) ď ­ đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;&#x201C; Fď ­(đ?&#x2018;Ľ0 , đ?&#x2018;&#x17E;) = đ??š (â&#x201E;&#x17D;, đ?&#x2018;&#x17E;) â&#x201E;&#x17D; â&#x2C6;&#x2C6; đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (đ?&#x2018;Ľ 1 ) ď ­

Neutrosophic Q-fuzzy left N-subgroups of a Near-ring 𝑖𝑛𝑓 𝐹 (ℎ, 𝑞) respectively. ℎ ∈ 𝑓 −1 (𝑦1 )  Then it can deduce that 𝑠𝑢𝑝 (1a). 𝑇 𝑓 (𝑛(𝑥 1 − 𝑦1 ), 𝑞) = 𝑇 (𝑧, 𝑞) 𝑧 ∈ 𝑓 −1 (𝑛(𝑥 1 − 𝑦1 ), 𝑞)  ≥ 𝑇{𝑇 (𝑥0 , 𝑞), 𝑇 (𝑦0 , 𝑞)} 𝑆𝑢𝑝 𝑆𝑢𝑝 ≥ 𝑇 { 𝑇(𝑥0 , 𝑞), 𝑇 (𝑦 , 𝑞)} −1 (ℎ, 𝑞) ∈ 𝑓 −1 (𝑦)  0 ℎ ∈ 𝑓 (𝑥0 ) = 𝑇 {𝑇 𝑓 (𝑥, 𝑞) , 𝑇 𝑓 (𝑦, 𝑞)}. 𝑠𝑢𝑝 (2a). 𝐼 𝑓 (𝑛(𝑥 1 − 𝑦1 ), 𝑞) = 𝐼 (𝑧, 𝑞) 𝑧 ∈ 𝑓 −1 (𝑛(𝑥 1 − 𝑦1 ), 𝑞)  ≥ 𝑇{𝐼 (𝑥0 , 𝑞), 𝐼 (𝑦0 , 𝑞)} 𝑆𝑢𝑝 𝑆𝑢𝑝 ≥ 𝑇 { 𝐼 (𝑥0 , 𝑞), 𝐼 (𝑦 , 𝑞)} −1 (ℎ, ℎ ∈ 𝑓 (𝑥0 ) 𝑞) ∈ 𝑓 −1 (𝑦)  0 = 𝑇 {𝐼 𝑓 (𝑥, 𝑞) , 𝐼 𝑓 (𝑦, 𝑞)}. 𝑖𝑛𝑓 (3a). 𝐹 𝑓 (𝑛(𝑥 1 − 𝑦1 ), 𝑞) = 𝐹 (𝑧, 𝑞) −1 𝑧 ∈ 𝑓 (𝑛(𝑥 1 − 𝑦1 ), 𝑞)   𝑆{𝐹 (𝑥0 , 𝑞), 𝐹 (𝑦0 , 𝑞)} 𝑖𝑛𝑓 𝑖𝑛𝑓 ≥ 𝑆{ 𝐹(𝑥0 , 𝑞), 𝐹 (𝑦 , 𝑞)} −1 (ℎ, 𝑞) ∈ 𝑓 −1 (𝑦)  0 ℎ ∈ 𝑓 (𝑥0 ) = 𝑇 {𝐹 𝑓 (𝑥, 𝑞) , 𝐹 𝑓 (𝑦, 𝑞)}. 𝑆𝑢𝑝 (1b). 𝑇 𝑓 (𝑛𝑥, 𝑞) = 𝑇 ( 𝑧, 𝑞 ) (𝑧, 𝑞) ∈ 𝑓 −1 (𝑛𝑥, 𝑞)  ≥ 𝑇 (𝑦0 , 𝑞) 𝑆𝑢𝑝 = 𝑇 ( 𝑥0 , 𝑞 ) (ℎ, 𝑞) ∈ 𝑓 −1 (𝑦, 𝑞)  = 𝜇𝑡 (𝑥, 𝑞 ). 𝑆𝑢𝑝 (2b). 𝐼 𝑓 (𝑛𝑥, 𝑞) = 𝐼 ( 𝑧, 𝑞 ) (𝑧, 𝑞) ∈ 𝑓 −1 (𝑛𝑥, 𝑞)  ≥ 𝐼 (𝑦0 , 𝑞) 𝑆𝑢𝑝 = 𝐼 ( 𝑥0 , 𝑞 ) (ℎ, 𝑞) ∈ 𝑓 −1 (𝑦, 𝑞)  = 𝐼 𝑓 (𝑥, 𝑞 ). 𝑖𝑛𝑓 (3b). 𝐹 𝑓 (𝑛𝑥, 𝑞) = 𝐹 ( 𝑧, 𝑞 ) (𝑧, 𝑞) ∈ 𝑓 −1 (𝑛𝑥, 𝑞)  ≥ 𝐹 (𝑦0 , 𝑞) 𝑖𝑛𝑓 = 𝐹 ( 𝑥0 , 𝑞 ) (ℎ, 𝑞) ∈ 𝑓 −1 (𝑦, 𝑞)  = 𝐹 𝑓 (𝑥, 𝑞 ).

F(𝑦0 , 𝑞) =

Hence  is a neutrosophic Q-fuzzy left N-subset of R.

317

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Section 4: Other properties of neutrosophic Q-fuzzy left N-subgroups Theorem 4.1: Let f: R ď&#x201A;Ž Rď&#x201A;˘ be an onto homomorphism on a near rings. If ď Ź = = (Tď Ź, Iď Ź, Fď Ź) be a neutrosophic Q-fuzzy N-subset in R. Then its image ď Źf = (Tď Źf, Iď Źf, Fď Źf) is a neutrosophic Q-fuzzy left N-subgroup of Rď&#x201A;˘.

Proof: Here f(R) = Rď&#x201A;˘. Let y1, y2 ď&#x192;&#x17D; Rď&#x201A;˘, and q ď&#x192;&#x17D; Q. Assume that A1 = f-1(y1, q); A2 = f-1(y2, q); and A12 = f-1(n(y1- y2), q)).

Consider A1 â&#x20AC;&#x201C; A2 = {x ď&#x192;&#x17D; R: (x, q) = (a1, q) â&#x20AC;&#x201C; (a2, q) for some (a1, q) ď&#x192;&#x17D; A1, and (a2, q) ď&#x192;&#x17D; A2.

If (x, q) ď&#x192;&#x17D; A1 â&#x20AC;&#x201C; A2, then (x, q) = (a1, q) â&#x20AC;&#x201C; (a2, q) for some (a1, q) ď&#x192;&#x17D; A1, and (a2, q) ď&#x192;&#x17D; A2. implies that f(x, q) = f(x1, q) â&#x20AC;&#x201C; f(x2, q) = y1 â&#x20AC;&#x201C;y2.

Thus (x, q) ď&#x192;&#x17D; f -1( (y1, q) â&#x20AC;&#x201C; (y2, q)) = f -1( n(y1 - y2), q)) = A12.

This gives that A1 â&#x20AC;&#x201C; A2 ď&#x192;? A12.

It follows that Tď Źf(n(y1-y2), q) = sup { Tď Ź(x, q) : (x, q) ď&#x192;&#x17D; f -1( n (y1-y2), q) = sup { Tď Ź(x, q) : (x, q) ď&#x192;&#x17D; A12} ď&#x201A;ł sup { Tď Ź(x, q) : (x, q) ď&#x192;&#x17D; A1 â&#x20AC;&#x201C; A2} ď&#x201A;ł sup { Tď Ź(x1, q) - Tď Ź(x2, q) : (x1, q) ď&#x192;&#x17D; A1, (x2, q) ď&#x192;&#x17D; A2} Since the norm T is continuous, and every â&#x2C6;&#x2C6; > 0, đ?&#x2018;&#x2020;đ?&#x2018;˘đ?&#x2018;? {đ?&#x2018;&#x2021;ď Ź (đ?&#x2018;Ľ1 , đ?&#x2018;&#x17E;): (đ?&#x2018;Ľ1 , đ?&#x2018;&#x17E;) â&#x2C6;&#x2C6; đ??´1 } â&#x2C6;&#x2019; (đ?&#x2018;Ľ1 â&#x2C6;&#x2014;, đ?&#x2018;&#x17E;) â&#x2C6;&#x2C6; đ??´2 } â&#x2030;¤ đ?&#x203A;ż and đ?&#x2018;&#x2020;đ?&#x2018;˘đ?&#x2018;?{ đ?&#x2018;&#x2021;ď Ź (đ?&#x2018;Ľ2 , đ?&#x2018;&#x17E;): (đ?&#x2018;Ľ2 , đ?&#x2018;&#x17E;) â&#x2C6;&#x2C6; đ??´2 } â&#x2C6;&#x2019; (đ?&#x2018;Ľ2 â&#x2C6;&#x2014;, đ?&#x2018;&#x17E;) â&#x2C6;&#x2C6; đ??´2 } â&#x2030;¤ đ?&#x203A;ż implies that đ?&#x2018;&#x2021; {đ?&#x2018;&#x2020;đ?&#x2018;˘đ?&#x2018;?{ đ?&#x2018;&#x2021;ď Ź (đ?&#x2018;Ľ1 , đ?&#x2018;&#x17E;) â&#x2C6;ś (đ?&#x2018;Ľ1 , đ?&#x2018;&#x17E;) â&#x2C6;&#x2C6; đ??´1 }, đ?&#x2018;&#x2020;đ?&#x2018;˘đ?&#x2018;?{đ?&#x2018;&#x2021;ď Ź (đ?&#x2018;Ľ2 , . đ?&#x2018;&#x17E;): (đ?&#x2018;Ľ2 , đ?&#x2018;&#x17E;) â&#x2C6;&#x2C6; đ??´2 } â&#x2C6;&#x2019; đ?&#x2018;&#x2021; {(đ?&#x2018;Ľ1 â&#x2C6;&#x2014; , đ?&#x2018;&#x17E;), (đ?&#x2018;Ľ2 â&#x2C6;&#x2014;, đ?&#x2018;&#x17E;)} â&#x2030;¤ â&#x2C6;&#x2C6;.

Choose (đ?&#x2018;&#x17D;1 , đ?&#x2018;&#x17E;) â&#x2C6;&#x2C6; đ??´1 and (đ?&#x2018;&#x17D;2 , đ?&#x2018;&#x17E;) â&#x2C6;&#x2C6; đ??´2 such that đ?&#x2018;&#x2020;đ?&#x2018;˘đ?&#x2018;? {đ?&#x2018;&#x2021;ď Ź (đ?&#x2018;Ľ1 , , đ?&#x2018;&#x17E;): (đ?&#x2018;Ľ1 , đ?&#x2018;&#x17E;) â&#x2C6;&#x2C6; đ??´1 } â&#x2C6;&#x2019; đ?&#x2018;&#x2021;ď Ź (đ?&#x2018;&#x17D;1 , đ?&#x2018;&#x17E;) â&#x2030;¤ đ?&#x203A;ż implies that đ?&#x2018;&#x2021;{đ?&#x2018;&#x2020;đ?&#x2018;˘đ?&#x2018;? { đ?&#x2018;&#x2021;ď Ź (đ?&#x2018;Ľ1 , đ?&#x2018;&#x17E;): (đ?&#x2018;Ľ1 , đ?&#x2018;&#x17E;) â&#x2C6;&#x2C6; đ??´1 }, đ?&#x2018;&#x2020;đ?&#x2018;˘đ?&#x2018;? {đ?&#x2018;&#x2021;ď Ź (đ?&#x2018;Ľ2 , . đ?&#x2018;&#x17E;): (đ?&#x2018;Ľ2 , đ?&#x2018;&#x17E;) â&#x2C6;&#x2C6; đ??´2 } â&#x2C6;&#x2019; đ?&#x2018;&#x2021;{đ?&#x2018;&#x2021;ď Ź (đ?&#x2018;&#x17D;1 , đ?&#x2018;&#x17E;) â&#x2C6;&#x2019; đ?&#x2018;&#x2021;ď Ź (đ?&#x2018;&#x17D;2 , đ?&#x2018;&#x17E;)} â&#x2030;¤ â&#x2C6;&#x2C6;.

Consequently it gives that đ?&#x2018;&#x2021;ď Ź đ?&#x2018;&#x201C; (đ?&#x2018;&#x203A;(đ?&#x2018;Ś1 â&#x2C6;&#x2019; đ?&#x2018;Ś2 , đ?&#x2018;&#x17E;) â&#x2030;Ľ sup{đ?&#x2018;&#x2021;(đ?&#x2018;&#x2021;ď Ź (đ?&#x2018;Ľ1 , đ?&#x2018;&#x17E;) â&#x2C6;&#x2C6; đ??´1 đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018; đ?&#x2018;&#x2021;ď Ź (đ?&#x2018;Ľ2 , đ?&#x2018;&#x17E;) â&#x2C6;&#x2C6; đ??´2 ): (đ?&#x2018;Ľ1 , đ?&#x2018;&#x17E;) â&#x2C6;&#x2C6; đ??´1 đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018; (đ?&#x2018;Ľ2 , đ?&#x2018;&#x17E;) â&#x2C6;&#x2C6; đ??´2 } â&#x2030;Ľ đ?&#x2018;&#x2021;{đ?&#x2018; đ?&#x2018;˘đ?&#x2018;?{đ?&#x2018;&#x2021;ď Ź (đ?&#x2018;Ľ1 , đ?&#x2018;&#x17E;): (đ?&#x2018;Ľ1 , đ?&#x2018;&#x17E;) â&#x2C6;&#x2C6; đ??´1 }. sup{đ?&#x2018;&#x2021;ď Ź (đ?&#x2018;Ľ2 , đ?&#x2018;&#x17E;): (đ?&#x2018;Ľ2 , đ?&#x2018;&#x17E;) â&#x2C6;&#x2C6; đ??´2 )}} â&#x2030;Ľ đ?&#x2018;&#x2021;{đ?&#x2018;&#x2021;ď Ź đ?&#x2018;&#x201C; (đ?&#x2018;Ś1 , đ?&#x2018;&#x17E;), đ?&#x2018;&#x2021;ď Ź đ?&#x2018;&#x201C; (đ?&#x2018;Ś2 , đ?&#x2018;&#x17E;)}.

Similarly, it an show đ?&#x2018;&#x2021;ď Ź đ?&#x2018;&#x201C; (đ?&#x2018;&#x203A;đ?&#x2018;Ľ, đ?&#x2018;&#x17E;) â&#x2030;Ľ đ?&#x2018;&#x2021;ď Ź đ?&#x2018;&#x201C; [đ?&#x2018;Ľ, đ?&#x2018;&#x17E;).

All these results are hold for đ??źď Ź đ?&#x2018;&#x201C; and đ??šď Ź đ?&#x2018;&#x201C; by using T-norm , and S-norm respectively.

It concludes that ď Źf = (Tď Źf, Iď Źf, Fď Źf) is a neutrosophic Q-fuzzy left N-subgroup of Rď&#x201A;˘.

Neutrosophic Q-fuzzy left N-subgroups of a Near-ring

319

Theorem 4.2: Let A = (TA, IA, FA) be a neutrosophic Q-fuzzy left N-subgroup of a near ring R. Then < A > is a neutrosophic Q-fuzzy left N-subgroup of R generated by A. In addition, < A > is the smallest neutrosophic Q-fuzzy left N-subgroup containing A. Proof: Let x, y ď&#x192;&#x17D; R, n ď&#x192;&#x17D; N, and q ď&#x192;&#x17D; Q. Assume that <A>(x,q) = t1; <A> (y, q) = t2, and TA(n(x-y), q) = t.

Now t = < A > (n(x-y), q) ď&#x201A;Ł T { < A > (nx, q), , < A >(ny, q)} = T { t1, t2} = t1 (say).

Then t1 = <A>(x, q) = sup {k: (x, q) ď&#x192;&#x17D; <Ak> } ď&#x201A;ł t.

Therefore there exists k1 with (x, q) ď&#x192;&#x17D; <Ak>.

Also t2 = < A> (y, q) = sup { k: (y, q) ď&#x192;&#x17D; < Ak >} ď&#x201A;ł t.

Therefore there exist k2 > t with (y, q) ď&#x192;&#x17D; < Ak >.

Without loss of generality, assume that k1, k2 so that < đ??´đ?&#x2018;&#x2DC;1 > ď&#x192;? < đ??´đ?&#x2018;&#x2DC;1 >. Then x, y ď&#x192;&#x17D; < đ??´đ?&#x2018;&#x2DC; > which is a contradiction since k2 > t. Thus t ď&#x201A;ł t1.

Consequently, A(n(x-y), q) ď&#x201A;ł T { < A > (x, q), < A > (y, q) } â&#x20AC;Ś. (1).

Now let, if possible t3 = { < A> (nx, q)} ď&#x201A;Ł {< A > (x, q) } = t1.

Then t1 = < A > (x, q) = sup {k: (x, q) ď&#x192;&#x17D; < Ak > } > t3.

Therefore there exists k with (x, q) ď&#x192;&#x17D; < Ak > and t1 > k > t3 implies that (nx, q) ď&#x192;&#x17D; < Ak > ď&#x192;? <Ak > which is a contradiction. Thus t3 = {<A>(nx, q) ď&#x201A;ł {<A>(x, q) } = t2 â&#x20AC;Ś(2).

Consequently the equations (1) and (2) yield that <A> is a neutrosophic Q-fuzzy left N-subgroup of R. Finally to show that <A> is the smallest neutrosophic Q-fuzzy left N-subgroup containing A. Let us assume that B is a neutrosophic Q-fuzzy left N-subgroup of R such that A ď&#x192;&#x152; B. Claim that < A > ď&#x192;&#x152; B. Let it possible, t = < A > (x, q) ď&#x201A;ł B (x, q) for some x ď&#x192;&#x17D; R, and q ď&#x192;&#x17D; Q. Let ď Ľ > 0 be given. Then t = At = sup {k: (x, q) ď&#x192;&#x17D; < Ak > and t ď Ľ ď&#x201A;Ł k ď&#x201A;Ł t.

So that (x, q) ď&#x192;&#x17D; <Ak> ď&#x192;&#x152; < Akt - ď Ľ > for all ď Ľ > 0.

Now a = a1x1 + a2x2 +â&#x20AC;Ś+ anxn where ai ď&#x192;&#x17D; R, and xi ď&#x192;&#x17D; At-ď Ľ implies that A(xi, q) ď&#x201A;ł t - ď Ľ.

Then B(x, q) ď&#x201A;ł (t - ď Ľ) for all ď Ľ > 0. Thus B(x, q) ď&#x201A;ł T { B(x1, q), B(x2, q), â&#x20AC;Ś, B(xn, q)} ď&#x201A;ł (t-ď Ľ) for all ď Ľ > 0. Therefore B(x, q) = t, which is a contradiction to our supposition Theorem 4.3: Let A = (TA, IA, FA) be a neutrosophic Q-fuzzy left N-subgroup of a near ring R. Define A+ = (TA+, IA+, FA+) by TA+(x,q) = TA(x,q) + 1 â&#x20AC;&#x201C; TA(0,q), IA+(x,q) = IA(x,q) + 1-TA(0,q), FA+(x, q) = FA(x, q) + 1 â&#x20AC;&#x201C; FA(0, q), for all x in R, and q in Q,.

Then A+ is a neutrosophic normal Q-fuzzy left N-subgroup of R containing A.

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Proof: Let x, y  R, n  N, and q  Q. It implies that TA+ (n(x-y) ,q) = TA(n(x-y), q) + 1 – TA(0, q)  T {TA(x, q), TA(y, q)} + 1 – TA(0, q)  T {TA(x, q) + 1 – TA(0, q), TA(y, q)} + 1 – TA(0, q) }  T { TA+ (x,q), TA+ (y ,q) } TA+ (nx, q) = TA(nx, q) + 1 – TA(0, q)  TA(x, q) + 1 – TA(0, q) = TA+(x, q).

Similarly IA+ (n(x-y) ,q) = IA(n(x-y), q) + 1 – IA(0, q)  T {IA(x, q), IA(y, q)} + 1 – IA(0, q)  T {IA(x, q) + 1 – IA(0, q), IA(y, q)} + 1 – IA(0, q) }  T { IA+ (x,q), IA+ (y ,q) } IA+ (nx, q) = IA(nx, q) + 1 – IA(0, q)  IA(x, q) + 1 – IA(0, q) = IA+(x, q).

Also FA+ (n(x-y) ,q) = FA(n(x-y), q) + 1 – FA(0, q)  S {FA(x, q), FA(y, q)} + 1 – TA(0, q)  S {FA(x, q) + 1 – FA(0, q), FA(y, q)} + 1 – FA(0, q) }  S { FA+ (x, q), FA+ (y ,q) } FA+ (nx, q)

= FA(nx, q) + 1 – FA(0, q)  FA(x, q) + 1 – FA(0, q) = FA+(x, q).

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