International Journal of Mathematical Archive-8(11), 2017, 207-212
Available online through www.ijma.info ISSN 2229 – 5046 NEUTROSOPHIC (Q, L)-FUZZY SUBGROUP K. THIRUSANGU*1, S. POORNAVEL2, AND R. VASUKI1 1Associate
Professor of Mathematics, SIVET College, Gowrivakkam, Chennai-73, India. 2Assistant Prof. in Mathematics, Madha Engineering College, Kundrathur, Chennai-69, india. 3Associate
Prof. of Mathematics, SIVET College, Gowrivakkam, Chennai, India. (Received On: 20-09-17; Revised & Accepted On: 21-10-17)
ABSTRACT
Goguen [1967] introduced L-fuzzy set. Murali [1991] analyzed lattice of fuzzy algebras and closure systems. Saibaba [2008] initiated fuzzy lattice ordered groups, and studied many properties on fuzzy lattice. Swamy and Raju [1991] gave a detailed findings on algebraic fuzzy system. In this paper, the notion of neutrosophic (Q, L)-fuzzy subgroup is introduced, and discussed some of its basic algebraic properties .Also the results on homomorphic image, pre image of neutrosophic (Q, L)-fuzzy subgroup are derived. Proved findings are that the direct product of any two neutrosophic (Q, L)-subgroups is a neutrosophic (Q, L)-fuzzy subgroup, and it is extended for finite number of neutrosophic groups. Keywords: neutrosophic Q-fuzzy set, neutrosophic Q-fuzzy subgroup (NQLFG), neutrosophic Q-fuzzy normal subgroup.
INTRODUCTION Smarandache [2002] introduced the notion of Neutrosophy as a new branch of philosophy. Neutrosophic is a base of neutrosophic logic which is an extension of fuzzy logic in which indeterminacy is included .In neutrosophic logic, each proposition is estimated to have the percentage of truth in a subset T, percentage of indeterminancy in a subset I, and the percentage of falsity in a subset. Maji et al. [2001] introduced the notion of fuzzy soft set. Afterwards, many researches were conducted on the notion of fuzzy sets. The theory of neutrosophic set have achieved great success in various fields like medical diagnosis, image processing decision making problem and so on. Arockiarani, and Martina Jency [2014] consider the neutrosophic set with value from the subset of [0, 1] and extended the research in fuzzy neutrosophic set. They [2016] initiated the concept of subgroupoid in fuzzy neutrosophic set. SECTION 2: BASIC DEFINITIONS Definition 2.1: Let X be a non-empty set. A fuzzy set A is a map A: X → [0, 1]. Let (L, ≤) be a lattice with the least element 0 and the greatest element 1. Any element a in L satisfies 0 ≤ a ≤ 1. An L-fuzzy set on a non-empty set X is a map f: X → L. Definition 2.2: Let X, Q be two non-empty sets, and L be a lattice. A mapping T: X × Q → L is a (Q, L)-fuzzy set in X. Definition 2.3: A neutrosophic fuzzy set A on the universe of discourse X characterized by a truth membership function TA(x), an indeterminacy function IA(x) and a falsity membership function FA(x) is defined as A = {< x, TA(x), IA(x), FA(x) > : x ∈ X }, where TA , IA ,FA : X → [0. 1] and 0 ≤ TA(x) ≤ 1; 0 ≤ IA(x) ≤ 1; 0 ≤ FA(x) ≤ 1, for all x ∈ X.
Corresponding Author: K. Thirusangu*1 1Associate Professor of Mathematics, SIVET College, Gowrivakkam, Chennai-73, India.
International Journal of Mathematical Archive- 8(11), Nov. – 2017
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K. Thirusangu* , S. Poornavel , and R. Vasuki / Neutrosophic (Q, L)-Fuzzy Subgroup / IJMA- 8(11), Nov.-2017.
Definition 2.4: Let X, Y be two non-empty sets and đ?‘“: đ?‘‹ â&#x;ś đ?‘Œ be a function. (i) If B = {< đ?‘Ś, đ?‘‡B (y), IB (y), FB (y): y ∈ Y } is a neutrosophic fuzzy set in Y then the preimage of B under đ?‘“, denoted by đ?‘“ −1 (đ??ľ), is the neutrosophic fuzzy set in X defined by đ?‘“ −1 (đ??ľ) = {< đ?‘Ľ, đ?‘“ −1 ďż˝đ?‘‡đ??ľ (đ?‘Ľ)ďż˝, đ?‘“ −1 ďż˝đ??źđ??ľ (đ?‘Ľ)ďż˝, đ?‘“ −1 ďż˝đ??šđ??ľ (đ?‘Ľ)ďż˝ >: đ?‘Ľ ∈ đ?‘‹} where đ?‘“ −1 ďż˝đ?‘‡đ??ľ (đ?‘Ľ)ďż˝ = đ?‘‡đ??ľ (đ?‘“(đ?‘Ľ)). (ii) If A = {< đ?‘Ľ, đ?‘‡A (x), IA (x), FA (x): x ∈ X } 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 đ?‘“(A) = {< đ?‘Ś, đ?‘“ (TA (y)), đ?‘“(IA (y)), đ?‘“(FA (y)) > : đ?‘Ś ∈ đ?‘Œ}, where sup T (x) if f −1 (y) ≠0N −1 đ?‘“(TA (y)) = ďż˝ x∈f (y) A 0 otherwise sup (x) I if f −1 (y) ≠0N −1 đ?‘“(IA (y)) = ďż˝ x∈f (y) A 0 otherwise inf (x) if f −1 (y) ≠0N F −1 , wheređ?‘“~ ďż˝FA (y)ďż˝ = ďż˝1 − đ?‘“(1 − đ??šđ??´ )ďż˝đ?‘Ś đ?‘“~ (FA (y)) = ďż˝ x∈f (y) A 1 otherwise
Definition 2.5: A neutrosophic (Q, L)-fuzzy set is an object having the form A = { < (x, q), TA(x, q), IA(x, q), FA(x, q) > : x ∈ X, q ∈ Q }, where TA: X Ă— Q → L, IA: X Ă— Q → L, FA: X Ă— Q → L denote the degree of truth membership function, degree of indeterminacy membership function and the degree of false membership function for each element (x, q) to the set A respectively, and 0 ≤ TA(x, q) ≤ 1; 0 ≤ IA(x, q) ≤ 1; 0 ≤ FA(x, q) ≤ 1, for all x ∈ X, and q ∈ Q. Definition 2.6: A = < (x, q), TA(x, q), IA(x, q), FA(x, q) > and B = < ( x, q), TB(x, q), IB(x, q), FB(x, q) > are two neutrosophic (Q, L)-fuzzy sets on X. Then, (i) đ??´ ⊆ đ??ľ, if TA(x, q) ≤ TB(x, q), IA(x, q) ≤ IB(x, q), FA(x, q) ≼ FB(x, q), for all x ∈ X, and q ∈ Q. (ii) đ??´ âˆŞ đ??ľ = < (đ?‘Ľ, đ?‘ž), max (TA (x, q), TB (x, q)), max (IA (x, q), IB (x, q)), min( (FA (x, q), FB (x, q)) >. (iii) đ??´ ∊ đ??ľ = < (đ?‘Ľ, đ?‘ž), min (TA (x, q), TB (x, q)), min(IA (x, q), IB (x, q)), max ( (FA (x, q), FB (x, q)) >. Definition 2.7: The complement Ac of a neutrosophic Q-fuzzy subset A on X is defined by Ac = {< (x, q), TAc (x, q), IAc (x, q), FAc (x, q) > : đ?‘Ľ ∈ đ?‘‹, đ?‘ž ∈ đ?‘„ , where TAc (x, q) = FA (x, q), IAc (x, q) = 1 − IA (x, q), FAc (x, q) = TA (x, q) , for all x ∈ X, and q ∈ Q. Throughout the paper, (L, ≤) be a lattice.
SECTION 3: PROPERTIES ON NEUTROSOPHIC (Q, L)-FUZZY SUBGROUP Definition 3.1: Let (G, .) be a group and A be a neutrosophic (Q, L)-fuzzy subset in G. Then A is called a neutrosophic (Q, L)-fuzzy sub group of G (NQLFG) if it satisfies the conditions (i) TA (xy, q) ≼ TA (x, q) ∧ (Tđ??€ (y, q), IA (xy, q) ≼ IA (x, q) ∧ (Iđ??€ (y, q), FA (xy, q) ≤ FA (x, q) ∨ (Fđ??€ (y, q)). (ii) TA (x −1 , q) = TA (x, q), IA (x −1 , q) = IA (x, q), FA (x −1 , q) = FA (x, q) ,for all x , đ?‘Ś ∈ G, q∈ đ?‘„.
Theorem 3.2: A neutrosophic (Q, L)-fuzzy subset A of G is a neutrosophic (Q, L)-fuzzy subgroup of a group G if and only if TA (xy −1 , q) ≼ (TA (x, q) ∧ (Tđ??€ (y, q), IA (xy −1 , q) ≼ (IA (x, q) ∧ (Iđ??€ (y, q) FA (xy −1 , q) ≤ (FA (x, q) ∨ (Fđ??€ (y, q)). Proof: Let A be a neutrosophic (Q, L)-fuzzy subgroup of G. ⇔ TA (xy, q) ≼ TA (x, q) ∧ Tđ??€ (y, q), IA (xy, q) ≼ IA (x, q) ∧ IA (y, q), FA (xy, q) ≤ FA (x, q) ∨ Fđ??€ (y, q)), and TA (x −1 , q) = TA (x, q), IA (x −1 , q) = IA (x, q), FA (x −1 , q) = FA (x, q) for all x, đ?‘Ś ∈ G, and q ∈ đ?‘„. ⇔ TA (xy −1 , q) ≼ TA (x, q) ∧ Tđ??€ (y, q), IA (xy −1 , q) ≼ IA (x, q) ∧ Iđ??€ (y, q), FA (xy −1 , q) ≤ FA (x, q) ∨ Fđ??€ (y, q).
Definition 3.3: A neutrosophic (Q, L)-fuzzy subgroup A of a group G is a neutrosophic (Q, L)-fuzzy normal subgroup of G (NQLFNG) if TA (xy, q) = TA (yx, q), IA (xy, q) = IA (yx, q), FA (xy, q) = FA (yx, q) or TA (xyx −1 , q) = TA (y, q), IA (xyx −1 , q) = IA (y, q), FA (xyx −1 , q) = FA (y, q) for all x , đ?‘Ś ∈ G, q∈ đ?‘„.
Definition 3.4: Let A be a neutrosophic (Q, L)-fuzzy subset of X. Let Îą, β, Îł be in L. Then [Îą, β, Îł]-level subset of (Q, L)-of A is defined by [A](Îą, β, Îł) = {đ?‘Ľ ∈ đ?‘‹, đ?‘ž ∈ đ?‘„: TA (x, q) ≼ Îą , IA (x, q) ≼ β , FA (x, q) ≤ Îł}. Theorem 3.5: If A is a NQLFG of G and Îą, β, Îł ∈ [0,1 ], then [Îą, β, Îł]-level subset [A](Îą, β, Îł) of A is a subgroup of G where TA (e, q) ≼ Îą , IA (e, q) ≼ β , FA (e, q) ≤ Îł , where e is the identity element of G, and đ?‘ž ∈ đ?‘„. Proof: Since, TA (e, q) ≼ Îą , IA (e, q) ≼ β , FA (e, q) ≤ Îł , e ∈ [A](Îą, β, Îł).
Therefore [A](Îą, β, Îł) ≠{ }. Let x, y ∈ [A](Îą, β, Îł) and đ?‘ž ∈ đ?‘„. Then it follows that TA (x, q) ≼ Îą , IA (x, q) ≼ β , FA (x, q) ≤ Îł , TA (y, q) ≼ Îą , IA (y, q) ≼ β , FA (y, q) ≤ Îł. Š 2017, IJMA. All Rights Reserved
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⇔ TA (x, q) ∧ TA (y, q) ≼ Îą, IA (x, q) ∧ IA (y, q) ≼ β , FA (x, q) ∨ FA (y, q) ≤ Îł ⇔ TA (xy −1 , q) ≼ Îą , IA (xy −1 , q) ≼ β , FA (xy −1 , q) ≤ Îł ⇔ xy −1 ∈ [A](Îą, β, Îł). ⇔ [A](Îą, β, Îł) is a subgroup of G.
Theorem 3.6: If A is a neutrosophic Q-fuzzy subset of a group G, then A is a NQLFG of G if and only if [A](Îą, β, Îł) is a subgroup of G for Îą, β, Îł ∈ L. Proof: Let x, y ∈ [A](Îą, β, Îł) and đ?‘ž ∈ đ?‘„.
Let A is a neutrosophic Q-fuzzy subgroup of G. ⇔ TA (xy −1 , q) ≼ TA (x, q) ∧ Tđ??€ (y, q), IA (xy −1 , q) ≼ IA (x, q) ∧ Iđ??€ (y, q), FA (xy −1 , q) ≤ FA (x, q) ∨ Fđ??€ (y, q)) ⇔ TA (xy −1 , q) ≼ Îą, , IA (x, q) ≼ β, FA (xy −1 , q) ≤ Îł ⇔ xy −1 ∈ [A](Îą, β, Îł). ⇔ [A](Îą, β, Îł) is a subgroup of G.
Theorem 3.7: A is a neutrosophic Q-fuzzy subset of a group G. Then A is a neutrosophic Q-fuzzy normal subgroup of G if and only if [A](Îą, β, Îł) is a normal subgroup of G. Proof: Let A be a neutrosophic Q-fuzzy normal subgroup of G. Then, TA (xyx −1 , q) = TA (y, q) ≼ Îą, IA (xyx −1 , q) = IA (y, q) ≼ β, FA (xyx −1 , q) = FA (y, q) ≤ Îł, for all x , đ?‘Ś ∈ G, q∈ đ?‘„. Hence [A](Îą, β, Îł) is a normal subgroup of G.
Theorem 3.8: If A1, A2, ‌ , An be neutrosophic Q-fuzzy subgroups of G. Then A = âˆŞđ?‘›đ?‘–=1 đ??´đ?‘– is a neutrosophic Q-fuzzy subgroup of G. Proof: Let A1, A2,....,An be neutrosophic Q-fuzzy subgroups of G.
Let A = âˆŞđ?‘›đ?‘–=1 đ??´đ?‘– , đ?‘Ľ, đ?‘Ś ∈ G, q ∈ đ?‘„.
Then , A(xy −1 , q) =âˆŞđ?‘›đ?‘–=1 đ??´đ?‘– (đ?‘Ľđ?‘Ś −1 , đ?‘ž)
=< (đ?‘Ľ, đ?‘ž), ďż˝TâˆŞđ?‘›đ?‘–=1Ai (xy −1 , q)ďż˝ , ďż˝IâˆŞđ?‘›đ?‘–=1Ai (xy −1 , q)ďż˝ , (F∊đ?‘›đ?‘–=1Ai (xy −1 , q) >
TâˆŞđ?‘›đ?‘–=1Ai (xy −1 , q) = ∨ TAi (xy −1 , q) ≼ ∨ (TAi (x, q) ∧ TAi (y, q)) = (∨ TAi (x, q)) ∧ (∨ TAi (y, q)) = TâˆŞđ?‘›đ?‘–=1Ai (x, q) ∧ TâˆŞđ?‘›đ?‘–=1Ai (y, q)
IâˆŞđ?‘›đ?‘–=1Ai (xy −1 , q) = ∨ IAi (xy −1 , q) ≼ ∨ (IAi (x, q) ∧ IAi (y, q)) = (∨ IAi (x, q)) ∧ (∨ IAi (y, q)) = IâˆŞđ?‘›đ?‘–=1Ai (x, q) ∧ IâˆŞđ?‘›đ?‘–=1Ai (y, q)
FâˆŞđ?‘›đ?‘–=1Ai (xy −1 , q) = ∧ FAi (xy −1 , q) ≤∧ (FAi (x, q) ∨ FAi (y, q)) = (∧ (FAi (x, q)) ∨ (∧ (FAi (y, q)) = FâˆŞđ?‘›đ?‘–=1Ai (x, q) ∨ FâˆŞđ?‘›đ?‘–=1Ai (y, q) đ?‘› Hence, A = âˆŞđ?‘–=1 đ??´đ?‘– is a neutrosophic Q-fuzzy subgroup of G.
Theorem 3.9: If A1, A2,....,An be neutrosophic Q-fuzzy subgroups of G. Then A = ∊đ?‘›đ?‘–=1 đ??´đ?‘– is a neutrosophic Q-fuzzy subgroup of G. Proof: Let A1, A2,...., An be neutrosophic Q-fuzzy subgroups of G. Let A = ∊đ?‘›đ?‘–=1 đ??´đ?‘– , đ?‘Ľ, đ?‘Ś ∈ G, q∈ đ?‘„.
Then A(xy −1 , q) = ∊đ?‘›đ?‘–=1 đ??´đ?‘– (đ?‘Ľđ?‘Ś −1 , đ?‘ž)
= < (đ?‘Ľ, đ?‘ž), ďż˝T∊đ?‘›đ?‘–=1Ai (xy −1 , q)ďż˝ , ďż˝I∊đ?‘›đ?‘–=1Ai (xy −1 , q)ďż˝ , (FâˆŞđ?‘›đ?‘–=1Ai (xy −1 , q) > .
T∊đ?‘›đ?‘–=1Ai (xy −1 , q) = ∧ TAi (xy −1 , q) ≼ ∧ (TAi (x, q) ∧ TAi (y, q)) = (∧ TAi (x, q)) ∧ (∧ TAi (y, q)) = T∊đ?‘›đ?‘–=1Ai (x, q) ∧ T∊đ?‘›đ?‘–=1Ai (y, q). Š 2017, IJMA. All Rights Reserved
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I∊đ?‘›đ?‘–=1Ai (xy −1 , q) = ∧ IAi (xy −1 , q) ≼ ∧ (IAi (x, q) ∧ IAi (y, q)) = (∧ IAi (x, q)) ∧ (∧ IAi (y, q)) = I∊đ?‘›đ?‘–=1Ai (x, q) ∧ I∊đ?‘›đ?‘–=1Ai (y, q).
F∊đ?‘›đ?‘–=1Ai (xy −1 , q) = ∨ FAi (xy −1 , q) ≤ ∨ (FAi (x, q) ∨ FAi (y, q)) = (∨ (FAi (x, q)) ∨ (∨ (FAi (y, q)) = F∊đ?‘›đ?‘–=1Ai (x, q) ∨ F∊đ?‘›đ?‘–=1Ai (y, q). đ?‘› Hence, A = ∊đ?‘–=1 đ??´đ?‘– is a neutrosophic Q-fuzzy subgroup of G.
SECTION 4: HOMOMORPHISM OF NEUTROSOPHIC Q-FUZZY SUBGROUPS Definition 4.1: Let G, G’ be any two groups .The function f: G Ă— Q → G’ Ă— Q is a group Q-homomorphism if (i). f: G → G’ is a group homomorphism, and (ii). f(xy, q) = f(x. q) f(y, q) for all x, y ∈ G and q ∈ Q. Theorem 4.2: Let G, G’ be any two groups and f be a homomorphism of G onto G’. If A is a NQLFG into of G’, then f-1(A) is a NQLFG of G. Proof: Let A be a neutrosophic fuzzy subgroup of đ??ş ′ .
By definition, đ?‘“ −1 (đ??´) = (đ?‘“ −1 (đ?‘‡đ??´ ), đ?‘“ −1 (đ??źđ??´ ), đ?‘“ −1 (đ??šđ??´ ))
Now for x, y ∈ G and q ∈ q, it gets that đ?‘“ −1 (đ?‘‡đ??´ )(xy −1 , q) = đ?‘‡đ??´ (đ?‘“(xy −1 , q)) = đ?‘‡đ??´ (đ?‘“(đ?‘Ľ)đ?‘“(y −1 ), đ?‘ž) (since f is a homomorphism) ≼ đ?‘‡đ??´ (đ?‘“(đ?‘Ľ), đ?‘ž) ∧ đ?‘‡đ??´ (đ?‘“(y −1 ), đ?‘ž) = đ?‘“ −1 (đ?‘‡đ??´ )(x, q) ∧ đ?‘“ −1 (đ?‘‡đ??´ )(y −1 , q).
đ?‘“ −1 (đ??źđ??´ )(xy −1 , q) = đ??źđ??´ (đ?‘“(xy −1 , q)) = đ??źđ??´ (đ?‘“(đ?‘Ľ)đ?‘“(y −1 ), đ?‘ž) (since, f is a homomorphism) ≼ đ??źđ??´ (đ?‘“(đ?‘Ľ), đ?‘ž) ∧ đ??źđ??´ (đ?‘“(y −1 ), đ?‘ž) = đ?‘“ −1 (đ??źđ??´ )(x, q) ∧ đ?‘“ −1 (đ??źđ??´ )(y −1 , q) = đ?‘“ −1 (đ??źđ??´ )(x, q) ∧ đ?‘“ −1 (đ??źđ??´ )(y, q).
đ?‘“ −1 (đ??šđ??´ )(xy −1 , q) = đ??šđ??´ (đ?‘“(xy −1 , q)) = đ??šđ??´ (đ?‘“(đ?‘Ľ)đ?‘“(y −1 ), đ?‘ž) (since, f is a homomorphism) ≤ đ??šđ??´ (đ?‘“(đ?‘Ľ), đ?‘ž) ∨ đ??šđ??´ (đ?‘“(y −1 ), đ?‘ž) = đ?‘“ −1 (đ??šđ??´ )(x, q) ∧ đ?‘“ −1 (đ??šđ??´ )(y −1 , q) = đ?‘“ −1 (đ??šđ??´ )(x, q) ∧ đ?‘“ −1 (đ??šđ??´ )(y, q).
Hence đ?‘“ −1 (đ??´)is a NQLFG of G.
Theorem 4.3: Let X, Y be two groups and f be a homomorphism of X onto Y. If A is a neutrosophic (Q, L)-fuzzy subgroup of X, then f(A) is a neutrosophic (Q, L)-fuzzy subgroup of Y. Proof: Let A be a neutrosophic Q-fuzzy subgroup of đ?‘‹. By definition, đ?‘“(đ??´) = (đ?‘“(TA ), đ?‘“(IA ), đ?‘“(FA ))
Now for x1 ,x2 ∈ đ?‘‹, y1 , y2 ∈ Y and q ∈ Q, đ?‘“(đ?‘‡đ??´ )(đ?‘Ś1 đ?‘Ś2 , đ?‘ž) = supđ?‘Ľ1đ?‘Ľ2 ∈đ?‘“−1(đ?‘Œ) đ?‘‡đ??´ (đ?‘Ľ1 đ?‘Ľ2 , đ?‘ž) ≼ supđ?‘Ľ1 ,đ?‘Ľ2 ∈đ?‘“−1(đ?‘Œ) (đ?‘‡đ??´ (đ?‘Ľ1 , đ?‘ž) ∧ đ?‘‡đ??´ (đ?‘Ľ2 , đ?‘ž)) (since A is a NQFSG ) = sup đ?‘‡đ??´ (đ?‘Ľ1 , đ?‘ž) ∧ sup đ?‘‡đ??´ (đ?‘Ľ2 , đ?‘ž) đ?‘Ľ1 ∈đ?‘“ −1 (đ?‘Œ)
đ?‘Ľ2 ∈đ?‘“ −1 (đ?‘Œ)
= đ?‘“(đ?‘‡đ??´ )(đ?‘Ś1 , đ?‘ž) ∧ đ?‘“(đ?‘‡đ??´ )(đ?‘Ś2 , đ?‘ž).
đ?‘“(đ?‘‡đ??´ )(đ?‘Ś −1 , đ?‘ž) =
sup
đ?‘Ľ −1 ∈đ?‘“ −1 (đ?‘Œ)
đ?‘‡đ??´ (đ?‘Ľ −1 , đ?‘ž) =
sup đ?‘‡đ??´ (đ?‘Ľ, đ?‘ž) = đ?‘“(đ?‘‡đ??´ )(đ?‘Ś, đ?‘ž).
đ?‘Ľâˆˆđ?‘“ −1 (đ?‘Œ)
đ?‘“(đ??źđ??´ )(đ?‘Ś1 đ?‘Ś2 , đ?‘ž) = supđ?‘Ľ1đ?‘Ľ2 ∈đ?‘“−1 (đ?‘Œ) đ??źđ??´ (đ?‘Ľ1 đ?‘Ľ2 , đ?‘ž) ≼ supđ?‘Ľ1 ,đ?‘Ľ2 ∈đ?‘“−1(đ?‘Œ) (đ??ź(đ?‘Ľ1 , đ?‘ž) ∧ đ??źđ??´ (đ?‘Ľ2 , đ?‘ž)) (since A is a NQFSG ) = sup đ??źđ??´ (đ?‘Ľ1 , đ?‘ž) ∧ sup đ??źđ??´ (đ?‘Ľ2 , đ?‘ž) đ?‘Ľ1 ∈đ?‘“ −1 (đ?‘Œ)
đ?‘Ľ2 ∈đ?‘“ −1 (đ?‘Œ)
= đ?‘“(đ??źđ??´ )(đ?‘Ś1 , đ?‘ž) ∧ đ?‘“(đ??źđ??´ )(đ?‘Ś2 , đ?‘ž).
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𝑓(𝐼𝐴 )(𝑦 −1 , 𝑞) =
sup
𝑥 −1 ∈𝑓 −1 (𝑌)
𝐼𝐴 (𝑥 −1 , 𝑞) =
sup 𝐼𝐴 (𝑥, 𝑞) = 𝑓(𝐼𝐴 )(𝑦, 𝑞).
𝑥∈𝑓 −1 (𝑌)
𝑓(𝐹𝐴 )(𝑦1 𝑦2 , 𝑞) = inf𝑥1𝑥2 ∈𝑓−1(𝑌) 𝐹𝐴 (𝑥1 𝑥2 , 𝑞) ≤ inf𝑥1 ,𝑥2∈𝑓−1(𝑌) (𝐹𝐴 (𝑥1 , 𝑞) ∨ 𝐹𝐴 (𝑥2 , 𝑞)) (since A is a NQFSG ) = inf 𝐹𝐴 (𝑥1 , 𝑞) ∨ inf 𝐹𝐴 (𝑥2 , 𝑞) −1 −1 𝑥1 ∈𝑓
(𝑌)
𝑥2 ∈𝑓
(𝑌)
= 𝑓(𝐹𝐴 )(𝑦1 , 𝑞) ∧ 𝑓(𝐹𝐴 )(𝑦2 , 𝑞).
𝑓(𝐹𝐴 )(𝑦 −1 , 𝑞) =
inf
𝑥 −1 ∈𝑓 −1 (𝑌)
𝐹𝐴 (𝑥 −1 , 𝑞) =
inf
𝑥∈𝑓 −1 (𝑌)
𝐹𝐴 (𝑥, 𝑞) = 𝑓(𝐹𝐴 )(𝑦, 𝑞).
Therefore f(A) is a neutrosophic (Q, L)-fuzzy subgroup of Y.
Lemma 4.4: For all 𝑎, 𝑏 ∈ 𝐼 and 𝑖 is any positive integer, if 𝑎 ≤ 𝑏, 𝑡ℎ𝑒𝑛 (𝑖)(𝑎)𝑖 ≤ (𝑏)𝑖 (𝑖𝑖) (𝑎 ∧ 𝑏)𝑖 = (𝑎)𝑖 ∧ (𝑏)𝑖 (𝑖𝑖𝑖) (𝑎 ∨ 𝑏)𝑖 = (𝑎)𝑖 ∨ (𝑏)𝑖
Theorem 4.5: Let A be a NQLFG of a group G. Then 𝐴𝑖 = {< (𝑥, 𝑞), (𝑇𝐴 (𝑥, 𝑞))𝑖 , (𝐼𝐴 (𝑥, 𝑞))𝑖 , (𝐹𝐴 (𝑥, 𝑞))𝑖 >: 𝑥 ∈ 𝐺, 𝑞 ∈ 𝑄} is a NQLFG of G, where i is a positive integer. Proof: Let A be a NQLFG of a group G.
Now for x, y ∈ G and q ∈ Q, 𝑇𝐴𝑖 (𝑥𝑦 −1 , 𝑞) = (𝑇𝐴 (𝑥𝑦 −1 , 𝑞))𝑖 ≥ (𝑇𝐴 (𝑥, 𝑞) ∧ 𝑇𝐴 (𝑦, 𝑞))𝑖 = (𝑇𝐴 (𝑥, 𝑞))𝑖 ∧ (𝑇𝐴 (𝑦, 𝑞))𝑖 = 𝑇𝐴𝑖 (𝑥, 𝑞) ∧ 𝑇𝐴𝑖 (𝑦, 𝑞).
𝐼𝐴𝑖 (𝑥𝑦 −1 , 𝑞) = (𝐼𝐴 (𝑥𝑦 −1 , 𝑞))𝑖 ≥ (𝐼𝐴 (𝑥, 𝑞) ∧ 𝐼𝐴 (𝑦, 𝑞))𝑖 = (𝐼𝐴 (𝑥, 𝑞))𝑖 ∧ (𝐼𝐴 (𝑦, 𝑞))𝑖 = 𝐼𝐴𝑖 (𝑥, 𝑞) ∧ 𝐼𝐴𝑖 (𝑦, 𝑞).
𝐹𝐴𝑖 (𝑥𝑦 −1 , 𝑞) = (𝐹𝐴 (𝑥𝑦 −1 , 𝑞))𝑖 ≤ (𝐹𝐴 (𝑥, 𝑞) ∨ 𝐹𝐴 (𝑦, 𝑞))𝑖 = (𝐹𝐴 (𝑥, 𝑞))𝑖 ∨ (𝐹𝐴 (𝑦, 𝑞))𝑖 = 𝐹𝐴𝑖 (𝑥, 𝑞) ∨ 𝐹𝐴𝑖 (𝑦, 𝑞). Therefore 𝐴𝑖 is a NQFSG of G.
SECTION 5: DIRECT PRODUCT OF NEUTROSOPHIC Q-FUZZY SUBGROUPS
Definition 5.1: Let A, B be neutrosophic (Q, L)–fuzzy subsets of X and Y respectively. Then the cartesian product 𝐴 × 𝐵 of A and B is defined by 𝐴 × 𝐵 = {< �(𝑥, 𝑦), 𝑞�, 𝑇𝐴×𝐵 �(𝑥, 𝑦), 𝑞�, 𝐼𝐴×𝐵 �(𝑥, 𝑦), 𝑞�, 𝐹𝐴×𝐵 �(𝑥, 𝑦), 𝑞� >: 𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌, 𝑞 ∈ 𝑄} Where 𝑇𝐴×𝐵 �(𝑥, 𝑦), 𝑞� = 𝑇𝐴 (𝑥, 𝑞) ∧ 𝑇𝐵 (𝑦, 𝑞)), 𝐼𝐴×𝐵 �(𝑥, 𝑦), 𝑞� = 𝐼𝐴 (𝑥, 𝑞) ∧ 𝐼𝐵 (𝑦, 𝑞)), and 𝐹𝐴×𝐵 �(𝑥, 𝑦), 𝑞� = 𝐹𝐴 (𝑥, 𝑞) ∨ 𝐹𝐵 (𝑦, 𝑞). Theorem 5.2: If A, B are neutrosophic (Q, L)-fuzzy subgroups of groups X and Y respectively, then A × B is a neutrosophic Q-fuzzy subgroup of X × Y. Proof: Let A and B be neutrosophic Q-fuzzy subgroups of the group X and Y respectively. Now for �𝑥1, 𝑦1 �, �𝑥2 , 𝑦2 � ∈ 𝐴 × 𝐵 , 𝑞 ∈ 𝑄
𝑇𝐴×𝐵 ��𝑥1, 𝑦1 ��𝑥2 , 𝑦2 �, 𝑞� = 𝑇𝐴×𝐵 ��𝑥1 𝑥2 , 𝑦1 𝑦2 �, 𝑞�
= 𝑇𝐴 ��𝑥1 𝑥2 , 𝑞�� ∧ 𝑇𝐵 ��𝑦1 𝑦2 , 𝑞�� ≥ [𝑇𝐴 (𝑥1 , 𝑞) ∧ 𝑇𝐴 (𝑥2 , 𝑞)] ∧ [𝑇𝐵 (𝑦1 , 𝑞) ∧ 𝑇𝐵 (𝑦2 , 𝑞)] (Since A and B are NQFSG) = 𝑇𝐴 (𝑥1 , 𝑞) ∧ 𝑇𝐵 (𝑦1 , 𝑞) ∧ 𝑇𝐴 (𝑥2 , 𝑞) ∧ 𝑇𝐵 (𝑦2 , 𝑞) = 𝑇𝐴×𝐵 (�𝑥1, 𝑦1 �, 𝑞) ∧ 𝑇𝐴×𝐵 (�𝑥2, 𝑦2 �, 𝑞).
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K. Thirusangu* , S. Poornavel , and R. Vasuki / Neutrosophic (Q, L)-Fuzzy Subgroup / IJMA- 8(11), Nov.-2017.
đ?‘‡đ??´Ă—đ??ľ ďż˝(đ?‘Ľ, đ?‘Ś)−1 , đ?‘ž)ďż˝ = đ?‘‡đ??´Ă—đ??ľ ((đ?‘Ľ −1 , đ?‘Ś −1 ), đ?‘ž) = đ?‘‡đ??´ (đ?‘Ľ −1 , đ?‘ž) ∧ đ?‘‡đ??ľ (đ?‘Ś −1 , đ?‘ž) = đ?‘‡đ??´ (đ?‘Ľ, đ?‘ž) ∧ đ?‘‡đ??ľ (đ?‘Ś, đ?‘ž) = đ?‘‡đ??´Ă—đ??ľ ((đ?‘Ľ, đ?‘Ś), đ?‘ž).
đ??źđ??´Ă—đ??ľ ��đ?‘Ľ1, đ?‘Ś1 ��đ?‘Ľ2 , đ?‘Ś2 ďż˝, đ?‘žďż˝ = đ??źđ??´Ă—đ??ľ ��đ?‘Ľ1 đ?‘Ľ2 , đ?‘Ś1 đ?‘Ś2 ďż˝, đ?‘žďż˝
= đ??źđ??´ ��đ?‘Ľ1 đ?‘Ľ2 , đ?‘žďż˝ďż˝ ∧ đ??źđ??ľ ��đ?‘Ś1 đ?‘Ś2 , đ?‘žďż˝ďż˝ ≼ [đ??źđ??´ (đ?‘Ľ1 , đ?‘ž) ∧ đ??źđ??´ (đ?‘Ľ2 , đ?‘ž)] ∧ [đ??źđ??ľ (đ?‘Ś1 , đ?‘ž) ∧ đ??źđ??ľ (đ?‘Ś2 , đ?‘ž)] (Since A and B are NQFSG) = đ??źđ??´ (đ?‘Ľ1 , đ?‘ž) ∧ đ??źđ??ľ (đ?‘Ś1 , đ?‘ž) ∧ đ??źđ??´ (đ?‘Ľ2 , đ?‘ž) ∧ đ??źđ??ľ (đ?‘Ś2 , đ?‘ž) = đ??źđ??´Ă—đ??ľ (ďż˝đ?‘Ľ1, đ?‘Ś1 ďż˝, đ?‘ž) ∧ đ??źđ??´Ă—đ??ľ (ďż˝đ?‘Ľ2, đ?‘Ś2 ďż˝, đ?‘ž).
đ??źđ??´Ă—đ??ľ ďż˝(đ?‘Ľ, đ?‘Ś)−1 , đ?‘ž)ďż˝ = đ??źđ??´Ă—đ??ľ ((đ?‘Ľ −1 , đ?‘Ś −1 ), đ?‘ž) = đ??źđ??´ (đ?‘Ľ −1 , đ?‘ž) ∧ đ??źđ??ľ (đ?‘Ś −1 , đ?‘ž) = đ??źđ??´ (đ?‘Ľ, đ?‘ž) ∧ đ??źđ??ľ (đ?‘Ś, đ?‘ž) = đ??źđ??´Ă—đ??ľ ((đ?‘Ľ, đ?‘Ś), đ?‘ž).
đ??šđ??´Ă—đ??ľ ��đ?‘Ľ1, đ?‘Ś1 ��đ?‘Ľ2 , đ?‘Ś2 ďż˝, đ?‘žďż˝ = đ??šđ??´Ă—đ??ľ ��đ?‘Ľ1 đ?‘Ľ2 , đ?‘Ś1 đ?‘Ś2 ďż˝, đ?‘žďż˝
= đ??šđ??´ ��đ?‘Ľ1 đ?‘Ľ2 , đ?‘žďż˝ďż˝ ∨ đ??šđ??ľ ��đ?‘Ś1 đ?‘Ś2 , đ?‘žďż˝ďż˝ ≤ [đ??šđ??´ (đ?‘Ľ1 , đ?‘ž) ∨ đ??šđ??´ (đ?‘Ľ2 , đ?‘ž)] ∨ [đ??šđ??ľ (đ?‘Ś1 , đ?‘ž) ∨ đ??šđ??ľ (đ?‘Ś2 , đ?‘ž)] (Since A and B are NQFSG) = đ??šđ??´ (đ?‘Ľ1 , đ?‘ž) ∨ đ??šđ??ľ (đ?‘Ś1 , đ?‘ž) ∨ đ??šđ??´ (đ?‘Ľ2 , đ?‘ž) ∨ đ??šđ??ľ (đ?‘Ś2 , đ?‘ž) = đ??šđ??´Ă—đ??ľ (ďż˝đ?‘Ľ1, đ?‘Ś1 ďż˝, đ?‘ž) ∨ đ??šđ??´Ă—đ??ľ (ďż˝đ?‘Ľ2, đ?‘Ś2 ďż˝, đ?‘ž).
đ??šđ??´Ă—đ??ľ ďż˝(đ?‘Ľ, đ?‘Ś)−1 , đ?‘ž)ďż˝ = đ??šđ??´Ă—đ??ľ ((đ?‘Ľ −1 , đ?‘Ś −1 ), đ?‘ž) = đ??šđ??´ (đ?‘Ľ −1 , đ?‘ž) ∨ đ??šđ??ľ (đ?‘Ś −1 , đ?‘ž) = đ??šđ??´ (đ?‘Ľ, đ?‘ž) ∨ đ??šđ??ľ (đ?‘Ś, đ?‘ž) = đ??šđ??´Ă—đ??ľ ((đ?‘Ľ, đ?‘Ś), đ?‘ž). Hence đ??´ Ă— đ??ľ is a neutrosophic (Q,, L)-fuzzy subgroup of X Ă— Y.
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