NoteonIntuitionisticFuzzyNormalSubgroupsorVagueNormalSubgroups

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1 Global Journal of Scence Fronter Research: F Mathematcs and Decson Scences Volume 15 Issue 2 Verson 1.0 Year 2015 Type : Double Blnd Peer Revewed Internatonal Research Journal Publsher: Global Journals Inc. (USA Onlne ISSN: & Prnt ISSN: Note on Intutonstc Fuzzy (Normal Subgroups or Vague (Normal Subgroups By K. Lakshm & Dr. G. Vasant Adtya Inattute of technology and management, Inda Abstract- The am of ths paper s bascally to study some of the standard propertes of the ntutonstc fuzzy subgroups under a crsp map. Also, we study some propertes of ntutonstc fuzzy normal subgroups. Keywords: ntutonstc fuzzy or vague subset, ntutonstc fuzzy/vague sub (normal group. GJSFR-F Classfcaton : FOR Code : MSC 2010: 03F55, 06C05, 16D25. NoteonIntutonstcFuzzyNormalSubgroupsorVagueNormalSubgroups Strctly as per the complance and regulatons of : K. Lakshm & Dr. G. Vasant. Ths s a research/revew paper, dstrbuted under the terms of the Creatve Commons Attrbuton-Noncommercal 3.0 Unported Lcense permttng all non commercal use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted.

2 Ref 1. Atanassov, K., Intutonstc fuzzy sets, n: V. Sgurev, Ed., VII ITKR's Sesson, Sofa, June 1983 (Central Sc. and Techn. Lbrary, Bulg. Academy of Scences, Note on Intutonstc Fuzzy (Normal Subgroups or Vague (Normal Subgroups K. Lakshm α & Dr. G. Vasant σ Abstract- The am of ths paper s bascally to study some of the standard propertes of the ntutonstc fuzzy subgroups under a crsp map. Also, we study some propertes of ntutonstc fuzzy normal subgroups. Keywords: ntutonstc fuzzy or vague subset, ntutonstc fuzzy/vague sub (normal group. I. Introducton Zadeh, n hs poneerng paper, ntroduced the noton of Fuzzy Subset of a set X as a functon µ from X to the closed nterval [0,1] of real numbers. The functon µ, he called, the membershp functon whch assgns to each memebr x of X ts membershp value, µx n [0, 1]. In 1983, Atanassov[1] generalzed the noton of Zadeh fuzzy subset of a set further by ntroducng an addtonal functon ν whch he called a nonmembershp functon wth some natural condtons on µ and ν, callng these new generalzed fuzzy subsets of a set, ntutonstc fuzzy subsets. Thus accordng to hm an ntutonstc fuzzy subset of a set X, s a par A = (µ A, ν A, where µ A, ν A are functons from the set X to the closed nterval [0, 1] of real numbers such that for each x X, µx + νx 1, where µ A s called the membershp functon of A and ν A s called the nonmembershp functon of A. Later on n 1984, Atanassov and Stoeva[3], further generalzed the noton ntutonstc fuzzy subset to L-ntutonstc fuzzy subset, where L s any complete lattce wth a complete order reversng nvoluton N. Thus an L-ntutonstc fuzzy subset A of a set X, s a par (µ A, ν A where µ A, ν A : X L are such that µ A Nν A. Let us recall that a complete order reversng nvoluton s a map N: L L such that (1 N0 L = 1 L and N1 L = 0 L (2 α β mples Nβ Nα (3 NNα = α (4 N( I α = I Nα and N( I α = I Nα. Interestngly the same noton of ntutonstc fuzzy subset of set was also ntroduced by Gau and Buehrer[6] n 1993 under a dfferent name called Vague subset. Thus whether we called ntutonstc fuzzy subset of a set or f-subset of a set for short, or vague subset of a set, they are one and the same. In order to make the document more readable, hereonwards we use the phrase f-subset for ntutonstc fuzzy or vague subset of a set. Obvously, f/v-subset only means ntutonstc fuzzy/vague subset, f/v-(normalsubgroup only means ntutonstc fuzzy/vague (normal subgroup. Comng to generalzatons of algebrac structures on to the ntutontc fuzzy/vague sets: as early as 1989, Bswas[7] ntroduced the noton of f/v-subgroup of a group and studed some propertes of the same. 37 Author α : Assstant professor of Mathematcs, Department of Basc Scences and Humantes, Adtya Insttute of Technology and Management, Tekkal, A.P. e-mal: klakshm.bsh@adtyatekkal.edu.n Author σ : Professor n Mathematcs, Department of Basc Scences and Humantes, Adtya Insttute of Technology and Management, An Aoutonomous Insttute, S.Kotturu, Tekkal, Srkakulam(dst A. P, Inda. e-mals: gvasanth.bsh@adtyatekkal.edu.n, vasant_u@yahoo.co.n 2015 Global Journals Inc. (US

3 38 Note on Intutonstc Fuzzy (Normal Subgroups or Vague (Normal Subgroups In 2004, Hur-Jang-Kang[15] ntroduced and studed f/v-normal subgroup of a group and Hur etal.[10,11,16] contnued ther studes of the same. In Hur etal.[16], they establshed a one-one correspondence between, f/v-normal subgroups and f/v-congrunces. In 2003, Banergee-Basnet[6] ntroduced and studed the notons of f/v-subrngs and f/v-deals of a rng. The same year Hur-Jang-Kang[10] ntroduced and studed the noton f/v-subrng of a rng. In Hur etal.[17,18] contnued ther studes of f/v-deals. In Hur etal.[18], they ntroduced and studed the notons of f/v-prme deals, f/v-completely prme deals and f/v-weakly completely prme deals. Comng back to the studes of ntutonstc fuzzy/vague subgroups of a group, Feng[8] and Palanappan etal.[22] ntated the study ntutonstc L-fuzzy/Lvague subgroups of a group. In ths paper we studed some properttes of ntutonstc fuzzy subgrups and ntutonstc fuzzy normal subgrups of an ntutonstc fuzzy subset. For any set X, the set of all f/v-subsets of X be denoted by A(X. By defnng, for any par of f/v-subsets A = (µ A, ν A and B = (µ B, ν B of X, A B ff µ A µ B and ν B ν A, A(X becomes a complete nfntely dstrbutve lattce. In ths case for any famly (A I of f/v-subsets of X, ( I A x = I A x and ( I A x = I A x. For any set X, one can naturally assocate, wth X, the f/v-subset (µ X, ν X = (1 X, 0 X, where 1 X s the constant map assumng the value 1 for each x X and 0 X s the constant map assumng the value 0 for each x X, whch turns out to be the largest element n A(X. Observe that then, the f/v-empty subset φ of X gets naturally assocated wth the f/v-subset (µ φ, ν φ = (0 X, 1 X, whch turns out to be the least element n A(X. Let A = (µ A, ν A be an f/v-subset of X. Then the f/v-complement of A, denoted by A c s defned by (ν A, µ A. Observe that A c = X A = X A c. Throughout ths paper the captal letters X, Y Z stand for arbtrary but fxed (crsp sets, the small letters f, g stand for arbtrary but fxed (crsp maps f : X Y and g : Y Z, the captal letters A, B, C, D, E, F together wth ther suffxes stand for f/v-subsets and the captal letters I and J stand for the ndex sets. Ingeneral whenever P s an f-subset of a set X, always µ P and ν P denote the membershp and nonmembershp functon of the f-subset P respectvely. Also we frequently use the standard conventon that φ = 0 and φ = 1. II. Intutonstc Fuzzy/Vague-subgroups In ths secton, frst we gve some defntons and statements. In the Lemma that follows ths, we gve equvalent statements whch are qute frequently used n several prepostons later on wthout an explct menton. Then analogues of some crsp theoretc results are establshed. In the end, Lagranges theorem s generalzed to fuzzy setup. Defntons and Statements 2.1 (a Let A, B be a par of f/v-subsets of G. Let C be defned by, µ C x = x=yz {µ A y µ B z} and ν C x = x=yz {ν A y ν B z}, for each x G. Then the f/v-subset C of G s called the f/v-product of A by B and s denoted by A B. (b For any f/v-subset A of G, the f/v-nverse of A, denoted by A 1, defned by (µ A 1, ν A 1 s n fact an f/v-subset of G, where for each x G µ A 1(x = µ A (x 1 and ν A 1(x = ν A (x 1. (c For any y G and for any par α, β of [0, 1], the f/v-pont of G, denoted by y α,β, s defned by the f/v-subset y α,β = (χ α y, χ β y where χ α y (x = α, χ β y (x = β when x = y and χ α y (x = χ β y (x = 0 when x y. (d An f/v-subset A of G s called an f/v-subgroup of G ff: (1 µ A (xy µ A (x µ A (y and ν A (xy ν A (x ν A (y, for each x, y G. (2 µ A (x 1 µ A (x and ν A (x 1 ν A (x, for each x G. Ref 6. Banerjee B. and Basnet D.,Kr., Intutonstc fuzzy subrngs and deals, Journal of Fuzzy Mathematcs Vol.11(1(2003, Global Journals Inc. (US

4 Note on Intutonstc Fuzzy (Normal Subgroups or Vague (Normal Subgroups (e For any f/v-subgroup A of a group G, A = {x G/µ A (x = µ A (e and ν A (x = ν A (e} and A = {x G/µ A (x > 0 and ν A (x < 1 }. (f For any f/v-subset A of G and for any α, β [0,1], the (α, β-level subset of A, denoted by A α,β, s defned by A α,β = {g G/µ A g α, ν A g β}. The followng Lemma, whch provdes alternatve equvalent statements for some of the above defntons and statements, s qute useful and s frequently used wthout an explct menton of t n several proofs n later chapters. Lemma 2.2 Let A, B, (A I be f/v-subsets of a group G. Let α = µ A G, β = ν A G, y α,β = (χ α y, χ β y. Then the followng are true: 1. (µ A B (x = y G (µ A (y µ B (y 1 x = y G (µ A (y 1 µ B (yx = y G (µ A (xy 1 µ B (y = y G (µ A (xy µ B (y 1 and (ν A B (x = y G (ν A (y ν B (y 1 x = y G (ν A (y 1 ν B (yx = y G (ν A (x y 1 ν B (y = y G (ν A (xy ν B (y 1, for each x G. In partcular, (µ A B (xy = z G (µ A (xz µ B (z 1 y = z G (µ A (xz 1 µ B (zy and (ν A B (xy = z G (ν A (xz ν B (z 1 y = z G (ν A (xz 1 ν B (zy. 2. A (B C = (A B C. 3. y α,β A = (χ α y µ A, χ β y ν A, (χ α y µ A x = µ A (y 1 x and (χ β y ν A x = ν A (y 1 x, for each x, y G. In partcular e α,β A = A. 4. A y α,β = (µ A χ α y, ν A χ β y, (µ A χ α y x = µ A (xy 1 and (ν A χ β y x = ν A (xy 1, for each x, y G. In partcular A e α,β = A. 5. (A 1 1 = A; 6. A A 1 ff A 1 A ff A = A 1 ; 7. A B ff A 1 B 1 ; 8. ( I A 1 = I A 1 ; 9. ( I A 1 = I A 1 ; 10. (A B 1 = B 1 A 1 ; 11. g α,β h γ,δ = (gh α γ,β δ. Proof : (1: Snce G s a group and hence for each x G, {(a, b G G/x = ab} = {(a, a 1 x G G/a G} = {(a 1, ax G G/a G} = {(xb 1, b G G/b G} = {(xb, b 1 G G/b G}, ths asserton follows. (2: µ A (B C (x = y G (µ A (xy 1 µ (B C (y = y G (µ A (xy 1 ( z G (µ B (yz 1 µ C z = y G z G (µ A (xy 1 µ B (yz 1 µ C (z and µ (A B C (x = z G (µ (A B (xz 1 µ C z = z G ( y G (µ A (xy 1 µ B (yz 1 µ C (z = z G y G (µ A (xy 1 µ B (yz 1 µ C (z, snce α ( j J β j = j J (α β j, when [0,1] s a complete nfnte meet dstrbutve lattce. Hence µ A (B C (x = µ (A B C (x. Smlarly, ν A (B C (x = y G (ν A (xy 1 ν (B C (y = y G (ν A (xy 1 ( z G (ν B (yz 1 ν C z = y G z G (ν A (xy 1 ν B (yz 1 ν C (z = ν (A B C (x, snce α ( j J β j = j J (α β j, when [0, 1] s a complete nfnte jon dstrbutve lattce. Therefore A (B C = (A B C. (3: (χ α y µ A x = x=ba (χ α y (b µ A (a = b G (χ α y (b µ A (b 1 x = α µ A (y 1 x = ( µ A G (µ A (y 1 x = µ A (y 1 x. Smlarly (χ β y ν A x = x=ba (χ β y (b ν A (a = b G (χ β y (b ν A (b 1 x = β ν A (y 1 x = ( ν A G (ν A (y 1 x = ν A (y 1 x. Lettng y = e, e α,β A = A Global Journals Inc. (US

5 Note on Intutonstc Fuzzy (Normal Subgroups or Vague (Normal Subgroups 40 (4: (µ A χ α y x = x=ab (µ A (a χ α y (b = b G (µ A (xb 1 χ α y (b = µ A (xy 1 α = µ A (xy 1 ( µ A G = µ A (xy 1. Smlarly (ν A χ β y x = x=ab (ν A (a χ β y (b = b G (ν A (xb 1 χ β y (b = (ν A (xy 1 β = (ν A (xy 1 ( ν A G = ν A (xy 1. Lettng y = e, A e α,β = A. (5: For each x G, µ A 1(x = µ A (x 1 and ν A 1(x = ν A (x 1. µ (A 1 1(x = µ A 1(x 1 = µ A (x 1 1 = µ A x and ν (A 1 1(x = ν A 1(x 1 = ν A (x 1 1 = ν A x. Hence (A 1 1 = A. (6: Let A A 1. Then for each x G, µ A (x µ A 1(x = µ A (x 1 and ν A (x ν A 1(x = ν A (x 1. Hence µ A 1(x 1 = µ A x µ A (x 1 and ν A 1(x 1 = ν A x ν A (x 1 mples µ A 1 µ A and ν A 1 ν A or A 1 A. Thus A A 1 mples A 1 A. Smlarly A 1 A mples for each x G, µ A 1(x 1 µ A (x 1 and ν A 1(x 1 ν A (x 1 whch mples µ A (x = µ A 1(x 1 µ A (x 1 = µ A 1(x and ν A (x = ν A 1(x 1 ν A (x 1 = ν A 1(x. or A A 1. Thus A 1 A mples A A 1. Now A A 1 ff A 1 A ff A = A 1 s clear. (7: ( : Let A B. Then for each x G, µ A (x 1 µ B (x 1 and ν A (x 1 ν B (x 1. Hence µ A 1(x = µ A (x 1 µ B (x 1 = µ B 1(x and ν A 1(x = ν A (x 1 ν B (x 1 = ν B 1(x or A 1 B 1. ( : Let A 1 B 1. Then for each x G, µ A 1(x µ B 1(x and ν A 1(x ν B 1(x. Hence µ A (x 1 = µ A 1(x µ B 1(x = µ B (x 1 and ν A (x 1 = ν A 1(x ν B 1(x = ν B (x 1 or A B. (8: Let A = (µ A, ν A, A 1 = (µ A 1 = ( I µ A (x 1 = I µ A (x 1 = I µ A 1 (x = ( I ν A (x 1 = I ν A Hence ( I A 1 = I A 1 (9: Let A = (µ A, ν A, A 1. = (µ A 1, ν A 1. Then for each x G, ( I µ A 1 (x (x 1 = I ν A 1 = ( I µ A (x 1 = I µ A (x 1 = I µ A 1 = ( I ν A (x 1 = I ν A (x 1 = I ν A 1 (x = ( I µ A 1(x and ( I ν A 1 (x. (x = ( I ν A 1, ν A 1. Then for each x G, ( I µ A 1 (x (x = ( I µ A 1(x and ( I ν A 1 (x (x. (x = ( I ν A 1 Hence ( I A 1 = I A 1. (10: Let (A B 1 = (µ (A B 1, ν (A B 1, B 1 A 1 = (µ B 1 A 1, ν B 1 A 1. Then for each x G, µ (A B 1(x = µ A B (x 1 = y G (µ A (x 1 y µ B (y 1 and ν (A B 1(x = ν A B (x 1 = y G (ν A (x 1 y ν B (y 1. On the other hand, µ (B 1 A 1 (x = y G (µ B 1(y µ A 1(y 1 x = y G (µ B (y 1 µ A (y 1 x 1 = y G (µ B (y 1 µ A (x 1 y = y G (µ A (x 1 y µ B (y 1 = µ (A B (x 1 = µ (A B 1(x and (ν B 1 A 1(x = y G(ν B 1(y ν A 1(y 1 x = y G (ν B (y 1 ν A (y 1 x 1 = y G (ν B (y 1 ν A (x 1 y = y G (ν A (x 1 y ν B (y 1 = ν (A B (x 1 = ν (A B 1(x. Therefore (A B 1 = B 1 A 1. (11: (χ α g χ γ h (x = z G(χ α g (xz 1 χ γ h (z = χα g (xh 1 γ = χ α gh (x γ = α γ = χ α γ gh (x where the thrd equalty follows because g = xh 1 or gh = x and χ β g χ δ h (x = z G(χ β g (xz 1 χ δ h (z = χβ g (xh 1 δ = χ β gh (x δ = β δ = χ β δ gh (x. Hence g α,β h γ,δ = (gh α γ,β δ. Lemma 2.3 For any f/v-subset A of a group G such that µ A (xy µ A (x µ A (y, ν A (xy ν A (x ν A (y: (1 µ A (x n µ A (x (2 ν A (x n ν A (x for each x G and n N. Proof: (1: µ A (x n = µ A (x n 1 x µ A (x n 1 µ A (x µ A (x µ A (x... µ A (x = µ A (x for each x G. (2: ν A (x n = ν A (x n 1 x ν A (x n 1 ν A (x ν A (x ν A (x... ν A (x = ν A (x for each x G Global Journals Inc. (US

6 Note on Intutonstc Fuzzy (Normal Subgroups or Vague (Normal Subgroups Lemma 2.4 Whenever A s an f/v-subgroup of a group G, for each x G, µ A (x 1 = µ A (x and ν A (x 1 = ν A (x. Proof: Let A be an f/v-subgroup of G. Then for each x G, µ A (x 1 µ A (x, ν A (x 1 ν A (x. µ A (x = µ A ((x 1 1 µ A (x 1 and ν A (x = ν A ((x 1 1 ν A (x 1. Hence µ A (x 1 = µ A (x and ν A (x 1 = ν A (x. Corollary 2.5 For any f/v-subgroup A of a group G, the followng are true for each x G: 1. µ A (e µ A (x and ν A (e ν A (x; 2. µ A A µ A and ν A A ν A. Proof: (1: µ A (e = µ A (xx 1 µ A (x µ A (x 1 = µ A (x µ A (x = µ A (x and ν A (e = ν A (xx 1 ν A (x ν A (x 1 = ν A (x ν A (x = ν A (x. (2: µ A A (x = y G (µ A (xy 1 µ A (y µ A (xe µ A (e µ A (x and ν A A (x = y G (ν A (xy 1 ν A (y ν A (xe ν A (e ν A x for each x G. Lemma 2.6 For any f/v-subset A of a group G, A s an f/v-subgroup ff µ A (xy 1 µ A (x µ A (y and ν A (xy 1 ν A (x ν A (y for each x, y G. Proof: ( : Suppose A s an f/v-subgroup. Then by 2.4, µ A (xy 1 µ A (x µ A (y 1 = µ A (x µ A (y and ν A (xy 1 ν A (x ν A (y 1 = ν A (x ν A (y for each x, y G. ( : Frst, by hypothess and 2.5(1, µ A (x 1 = µ A (ex 1 µ A (e µ A (x = µ A (x and ν A (x 1 = ν A (ex 1 ν A (e ν A (x = ν A x for each x G. Lettng x 1 nplace of x, µ A (x µ A (x 1 and ν A (x ν A (x 1 for each x G or µ A (x = µ A (x 1 and ν A (x = ν A (x 1 for each x G. Next, µ A (xy = µ A (x(y 1 1 µ A (x µ A (y 1 = µ A (x µ A (y. Smlarly ν A (xy ν A (x ν A (y. Therefore A s an f/v-subgroup of G. Lemma 2.7 For any f/v-subgroup A of a group G, 1. A = {x G/µ A (x = µ A (e, ν A (x = ν A (e} s a subgroup of G; 2. A = {x G/µ A (x > 0, ν A (x < 1} s a subgroup of G whenever L s strongly regular. Proof: (1: Let x, y A. Then µ A (xy 1 µ A (x µ A (y = µ A (e, ν A (xy 1 ν A (x ν A (y = ν A (e. By 2.5(1, µ A (xy 1 µ A e, ν A (xy 1 ν A e for each x, y G. So, µ A (xy 1 = µ A e and ν A (xy 1 = ν A e or xy 1 A mplyng A s a subgroup of G. (2: Snce L s strongly regular, by 2.1(f, for each x, y A, µ A (xy 1 µ A (x µ A (y > 0 and ν A (xy 1 ν A (x ν A (y < 1 or xy 1 A mplyng, A s a subgroup of G. Lemma 2.8 For any f/v-subset A of a group G, A s an f/v-subgroup of G ff A satsfes the followng condtons: (1 µ A A = µ A and ν A A = ν A or equvalently A A = A. (2 µ A 1 = µ A and ν A 1 = ν A or equvalently A 1 = A. Proof: ( : Let A be an f/v-subgroup of G. Then for each x, y G, µ A x = µ A (xy 1 y µ A (xy 1 µ A (y, ν A x = ν A (xy 1 y ν A (xy 1 ν A (y, µ A (x 1 = µ A (x and ν A (x 1 = ν A (x. (1: µ A A (x = y G (µ A (xy 1 µ A (y y G µ A (x = µ A (x or µ A A µ A and (ν A A (x = y G (ν A (xy 1 ν A (y y G ν A (x = ν A (x or ν A A ν A. Now by 2.5(2, we get that µ A A = µ A and ν A A = ν A. (2: 2.4 mples for each x G, µ A 1(x = µ A (x 1 = µ A (x or µ A = µ A 1 and ν A 1(x = ν A (x 1 = ν A (x or ν A 1 = ν A Global Journals Inc. (US

7 Note on Intutonstc Fuzzy (Normal Subgroups or Vague (Normal Subgroups 42 ( : 2.2(1 and the facts that µ A 1 = µ A, ν A 1 = ν A, µ A A µ A, ν A A ν A mply, for each x G µ A (xy 1 µ A A (xy 1 µ A (xy 1 y µ A (y 1 = µ A x µ A y and ν A (xy 1 ν A A (xy 1 ν A (xy 1 y ν A (y 1 = ν A x ν A y. Lemma 2.9 For any par of f/v-subgroups A and B of a group G, A B s an f/v-subgroup of G ff A B = B A. Proof: ( : Snce A, B and A B are f/v-subgroups of G, A 1 = A, B 1 = B, A B = (A B 1 = B 1 A 1 = B A. ( : Let A B = B A. Then (a (A B (A B = A (B A B = A (A B B = (A A (B B = A B and (b (A B 1 = (B A 1 = A 1 B 1 = A B. By 2.8, A B s an f/v-subgroup of G. Lemma 2.10 For any par of groups G and H and for any crsp homomorphsm f:g H the followng are true: 1. A s an f/v-subgroup of G mples f(a s an f/v-subgroup of H, whenever [0, 1] s a complete nfnte dstrbutve lattce; 2. B s an f/v-subgroup of H mples f 1 (B s an f/v-subgroup of G. Proof: (1: Let fa = B. Then µ B y = µ A f 1 y, ν B y = ν A f 1 y. Now we show that µ B (xy 1 µ B (x µ B (y and ν B (xy 1 ν B (x ν B (y. Let us recall that µ B (x = µ A f 1 x = a f 1 xµ A a, µ B y = µ A f 1 y = b f 1 yµ A b and ν B (x = ν A f 1 x = a f 1 xν A a, ν B y = ν A f 1 y = b f 1 yν A b. If one of f 1 x or f 1 y s empty, we are done because φ = 0 L and φ = 1 L. So, let both of them be non-empty. a f 1 x, b f 1 y mply fa = x, fb = y whch mples fab 1 = fafb 1 = xy 1 whch n turn mples c = ab 1 f 1 (xy 1. Snce A s an f/v-subgroup of G, µ B (xy 1 = c f 1 (xy 1 µ A c µ A (ab 1 µ A (a µ A (b and smlarly ν B (xy 1 ν A (a ν A (b for each a f 1 x, b f 1 y. Observe that n any complete nfnte dstrbutve lattce, (1 γ α β for each α M [0, 1], for each β N [0, 1] mples γ ( α M α ( β N β = ( M ( N, (2 γ α β for each α M [0, 1], for each β N [0, 1] mples γ ( α M α ( β N β = ( M ( N. So, we wll get that µ B (xy 1 µ B x µ B y and ν B (xy 1 ν B x ν B y. Hence fa = B s an f/v-subgroup of G. (2: Let f 1 B = A. Then µ A x = µ B fx, ν A x = ν B fx. Now we show that µ A (xy 1 µ A (x µ A (y and ν A (xy 1 ν A (x ν A (y. Snce f s a homomorphsm and B s an f/v-subgroup of H, µ A (xy 1 = µ B f(xy 1 = µ B (fx(fy 1 µ B fx µ B fy = µ A x µ A y and ν A (xy 1 = ν B f(xy 1 = ν B (fx(fy 1 ν B fx ν B fy = ν A x ν A y. Hence f 1 B = A s an f/v-subgroup of G. Lemma 2.11 For any famly of f/v-subgroups (A I of a group G, I A s an f/v-subgroup of G. Proof: Let C = I A. Then µ C = I µ A, ν C = I ν A. Now we show that, µ C (xy 1 µ C (x µ C (y and ν C (xy 1 ν C (x ν C (y. Let us recall that n any complete lattce, (1 I (α β = ( I α ( I β (2 I (α β = ( I α ( I β (3 α β for each I mples I α I β (4 α β for each I mples I α I β Now the above and A s an f/v-subgroup of G mply, µ C (xy 1 = ( I µ A (xy 1 = I µ A (xy 1 I (µ A (x µ A (y = ( I µ A x ( I µ A y = µ C x µ C y and ν C (xy 1 = ( I ν A (xy 1 = I ν A (xy 1 I (ν A (x ν A (y = ( I ν A x ( I ν A y = ν C x ν C y. Hence C = I A s an f/v-subgroup of G Global Journals Inc. (US

8 Note on Intutonstc Fuzzy (Normal Subgroups or Vague (Normal Subgroups It may so happen that the I A may be the empty f/v-subset whch s trvally an f/v-subgroup of G as shown n the followng example: Example 2.12 G. A n = ( 1 n, 1 1 n, n=1a n = (0, 1 = φ, the empty subgroup of The A B of f/v-subgroups A, B of a group G need not be an f/v-subgroup as shown n the followng example: Example 2.13 Let A = (χ 2z, 1 χ 2z, B = (χ 3z, 1 χ 3z be the I-f/v-subgroups of Z, the addtve group of ntegers, where I = [0,1], the closed nterval of real numbers. Then A B = (χ 2z χ 3z, (1 χ 2z (1 χ 3z and µ A B (5 = (χ 2z χ 3z 5 = 0 0 = 0. If A B s an f/v-subgroup of G, then 0 = µ A B (3 + 2 µ A B (3 µ A B (2 = (χ 2z χ 3z 3 (χ 2z χ 3z 2 = (0 1 (1 0 = 1 1 = 1, a contradcton. So A B s not an f/v-subgroup of G. Lemma 2.14 For any famly of f/v-subgroups (A I of G, I A s an f/vsubgroup of G whenever (A I s a sup/nf assumng chan of f/v-subgroups. Proof: Let A = I A. Then µ A = I µ A, ν A = I ν A. Now we show that µ A (xy 1 µ A (x µ A (y and ν A (xy 1 ν A (x ν A (y for each x, y G. If one of µ A x or µ A (y = 0 and one of ν A x or ν A (y = 1 then anyway the nequaltes hold good. Let µ A x, µ A y > 0 and ν A x, ν A y < 1. Then I (µ A x, I (µ A y > 0 and I (ν A x, I (ν A y < 1. Then there exsts 0 I such that µ A0 x = I µ A x, ν A0 x = I ν A x and there exsts j 0 I such that µ Aj0 y = I µ A y, ν Aj0 y = I ν A y because (A I s a sup/nf assumng chan. Now (1 A 0 A j0 or (2 A j0 A 0 because (A I s a chan. (1 Suppose A 0 A j0 or µ A0 µ Aj0 and ν Aj0 ν A0. Then µ A (xy 1 µ Aj0 (xy 1 µ Aj0 x µ Aj0 y µ A0 x µ Aj0 y = ( I µ A x ( I µ A y = µ A x µ A y and ν A (xy 1 ν Aj0 (xy 1 ν Aj0 x ν Aj0 y ν A0 x ν Aj0 y = ( I ν A x ( I ν A y = ν A x ν A y. 2 Suppose A j0 A 0 or µ Aj0 µ A0 and ν A0 ν Aj0. Then µ A (xy 1 µ A0 (xy 1 µ A0 x µ A0 y µ A0 x µ Aj0 y = ( I µ Ax ( I µ A y = µ A x µ A y and ν A (xy 1 ν A0 (xy 1 ν A0 x ν A0 y ν A0 x ν Aj0 y = ( I ν A x ( I ν A y = ν A x ν A y. If/V-Cosets And If/V-Index Of An If/V-Subgroup Defntons 2.15 (1 For any f/v-subgroup A of a group G and for any g G, the f/v-subset ga = (µ ga, ν ga of G, where µ ga, ν ga : G [0, 1], are defned by µ ga x = µ A (g 1 x and ν ga x = ν A (g 1 x, s called the f/v-left coset of A by g n G. The f/v-subset Ag = (µ Ag, ν Ag of G, where µ Ag x = µ A (xg 1 and ν Ag x = ν A (xg 1 s called the f/v-rght coset of A by g n G. (2 The set of all f/v-left cosets of A n G s denoted by (G/A L. The set of all f/v-rght cosets of A n G s denoted by (G/A R. (3 (Later on we show, as n the crsp set up, that The number of f/v-left cosets of A n G s the same as the number of f/v-rght cosets of A n G and ths common number, denoted by (G : A, s called the f/v-ndex of A n G. Theorem 2.16 For any f/v-subgroup A of a group G and for any par of elements g, h of G, the followng are true: 1. ga = g µa e,ν A e A and Ag = A g µa e,ν A e. 2. ga = ha ff ga = ha. 3. Ag = Ah ff A g = A h Global Journals Inc. (US

9 Note on Intutonstc Fuzzy (Normal Subgroups or Vague (Normal Subgroups 44 Proof: (1: From 2.2(3 and 2.15(1, (χ µ Ae g (χ ν Ae g ν A (x = ν A (g 1 x = ν ga x or µ ga = χ µ Ae g Hence ga = g µa e,ν A e A. From 2.2(4 and 2.15(1, (µ A χ µ Ae g µ A (x = µ A (g 1 x = µ ga x and µ A and ν ga = χ ν Ae g ν A. (x = µ A (xg 1 = µ Ag x and (ν A χ ν Ae g (x = ν A (xg 1 = ν Ag x or µ Ag = µ A χ µ Ae g and ν Ag = ν A χ ν Ae g. Hence Ag = A g µa e,ν A e. (2: ( : Suppose ga = ha. Then µ ga = µ ha and ν ga = ν ha or for each x G, µ ga (x = µ ha (x and ν ga x = ν ha x whch mples µ A (g 1 x = µ A (h 1 x and ν A (g 1 x = ν A (h 1 x. Choosng x = h, µ A (g 1 h = µ A (h 1 h = µ A (e and ν A (g 1 h = ν A (h 1 h = ν A (e mplyng g 1 h A, where A = {x G/µ A (x = µ A (e, ν A (x = ν A (e}. Hence ga = ha. ( : From 2.7, A s an f/v-subgroup of G mples A s a subgroup of G. Suppose ga = ha. Then g 1 h A or µ A (g 1 h = µ A (e, ν A (g 1 h = ν A (e. Hence for each z G, µ A (g 1 z = µ A (g 1 hh 1 z µ A (g 1 h µ A (h 1 z = µ A (e µ A (h 1 z = µ A (h 1 z and ν A (g 1 z = ν A (g 1 hh 1 z ν A (g 1 h ν A (h 1 z = ν A (e ν A (h 1 z = ν A (h 1 z because, µ A e s the largest of µ A G and ν A e s the least of ν A G. Smlarly, for each z G, µ A (h 1 z µ A (g 1 z and ν A (h 1 z ν A (g 1 z. Hence for each z G, µ A (g 1 z = µ A (h 1 z, ν A (g 1 z = ν A (h 1 z or µ ga (z = µ ha (z, ν ga (z = ν ha (z for each z or µ ga = µ ha, ν ga = ν ha or ga = ha. (3 ( : Suppose Ag = Ah. Then µ Ag = µ Ah, ν Ag = ν Ah or for each x G, µ Ag (x = µ Ah (x and ν Ag x = ν Ah x whch mples µ A (xg 1 = µ A (xh 1 and ν A (xg 1 = ν A (xh 1. Choosng x = h, µ A (hg 1 = µ A (hh 1 = µ A (e and ν A (hg 1 = ν A (hh 1 = ν A (e mplyng hg 1 A or A g = A h. ( : Suppose A g = A h. Then hg 1 A or µ A (hg 1 = µ A (e and ν A (hg 1 = ν A (e. Hence for each z G, µ A (zg 1 = µ A (zh 1 hg 1 µ A (zh 1 µ A (hg 1 = µ A (zh 1 µ A (e = µ A (zh 1 and ν A (zg 1 = ν A (zh 1 hg 1 ν A (zh 1 ν A (hg 1 = ν A (zh 1. Smlarly, for each z G, µ A (zh 1 µ A (zg 1 and ν A (zh 1 ν A (zg 1. Hence for each z G, µ A (zg 1 = µ A (zh 1, ν A (zh 1 = ν A (zg 1 or µ Ag (z = µ Ah (z, ν Ag (z = ν Ah (z for each z or Ag = Ah. Corollary 2.17 For any f/v-subgroup A of a group G, the followng are true: (1 The number of f/v-left(rght cosets of A n G s the same as the number of left(rght cosets of A n G. (2 (G : A = (G : A. Proof : (1: Let I be the set of all f/v-left cosets of A n G and ℵ be the set of all f/v-left cosets of B n G. Defne φ : I ℵ by φ(ga = ga. Then by 2.16(2, φ s both well defned and one-one. But clearly, φ s onto. Thus φ s a bjecton mplyng our asserton. (2: For any subgroup H of a group G, the number of left coset of H n G s the same as the number of rght coset of H n G. Now the asserton follows from (1. In the crsp set up, when G s a fnte group, for any subgroup H of G, H. If one were to defne the order for an f/v-subgroup of a fnte group, = G (G:H the preceedng equaton suggests that A = G (G:A. But (G : A = (G : A and consequently A = A. Thus the defnton of f/v-order of an f/v-subgroup s as follows: Defnton 2.18 For any f/v-subgroup A of a group G, the order of A, denoted by A, s defned to be the order of A or A. In other words A = A. An f/v-subgroup A of a group G s fnte or nfnte accordng as ts order A s fnte or nfnte Global Journals Inc. (US

10 Note on Intutonstc Fuzzy (Normal Subgroups or Vague (Normal Subgroups Lagranges Theorem Theorem 2.19 For any fnte group G and for any f/v-subgroup A, order of A, A dvdes the order of G, G. III. Intutonstc Fuzzy/Vague-Normal Subgroups In ths secton, we begn wth equvalent condtons for f/v-normalty for a subgroup and several of these condtons wll be used n some subsequent results, sometmes, wthout an explct menton. Later on we proceed to generalze varous crsp theoretc results mentoned n the begnnng of ths chapter. The followng s a theorem whch gves equvalent statements for an f/v-normal subgroup, some what smlarly as n crsp set up. Theorem 3.1 Let A be an f/v-subgroup of G. Then the followng are equvalent: 1. µ A (xy = µ A (yx and ν A (xy = ν A (yx for each x, y G, 2. µ A (xyx 1 = µ A (y and ν A (xyx 1 = ν A (y for each x, y G, 3. µ A [xy] µ A x and ν A [xy] ν A x for each x, y G, where [x, y] = x 1 y 1 xy s the commutator of x, y, 4. µ A (xyx 1 µ A (y and ν A (xyx 1 ν A (y for each x, y G, 5. µ A (xyx 1 µ A (y and ν A (xyx 1 ν A (y for each x, y G, 6. A B = B A for each f/v-subset B of G, 7. Ag Ah = Agh, ga ha = gha, Agh = gha and Ag Ah = Ah Ag for each g, h G, 8. ga = Ag for each g G, 9. A = g µa e,ν A e A g 1 µ A e,ν A e for each g G. Proof : Let x, y G. (1 (2: µ A (xyx 1 = µ A (x 1 xy = µ A (y and ν A (xyx 1 = ν A (x 1 xy = ν A (y. (2 (3: µ A (x 1 y 1 xy = µ A (x 1 (y 1 xy µ A (x 1 µ A (y 1 xy = µ A (x 1 µ A (x = µ A (x and ν A (x 1 y 1 xy = ν A (x 1 (y 1 xy ν A (x 1 ν A (y 1 xy = ν A (x 1 ν A (x = ν A (x, by 2.4 and 2.6. (3 (4: µ A (y 1 xy = µ A (xx 1 y 1 xy µ A (x µ A (x 1 y 1 xy µ A (x and ν A (y 1 xy = ν A (xx 1 y 1 xy ν A (x ν A (x 1 y 1 xy ν A (x. (4 (5: µ A (xyx 1 µ A (x 1 xyx 1 (x 1 1 = µ A (y and ν A (xyx 1 ν A (x 1 xyx 1 (x 1 1 = ν A (y. (5 (1: µ A (xy = µ A (xyxx 1 = µ A (x yx x 1 µ A (yx and µ A (yx = µ A (y xy y 1 µ A (xy, mplyng µ A (xy = µ A (yx. ν A (xy = ν A (xyxx 1 = ν A (x yx x 1 ν A (yx and ν A (yx = ν A (y xy y 1 ν A (xy, mplyng ν A (xy = ν A (yx. (1 (6: µ A B (x = y G (µ A (xy 1 µ B (y = y G (µ A (y 1 x µ B (y = y G (µ B (y µ A (y 1 x = µ B A (x and ν A B (x = y G (ν A (xy 1 ν B (y = y G (ν A (y 1 x ν B (y = y G (ν B (y ν A (y 1 x = ν B A (x, mplyng A B = B A Global Journals Inc. (US

11 Note on Intutonstc Fuzzy (Normal Subgroups or Vague (Normal Subgroups 46 (6 (7: 2.16, 2.8, 2.2(11 mply Ag Ah = A g µa e,ν A e A h µa e,ν A e = A A g µa e,ν A e h µa e,ν A e = A g µa e,ν A e h µa e,ν A e = A (gh µa e,ν A e = Agh. Smlarly ga ha = gha. Now lettng B = (gh µa e,ν A e, by the hypothess, the above mples Agh = gha. Agan by hypothess, Ag Ah = A g µa e,ν A e h µa e,ν A e = A h µa e,ν A e g µa e,ν A e = Ahg = Ah Ag. (7 (8: h = e mples Ag = ga. (8 (9: By 2.16, 2.2(4 and 2.2(11, g µa e,ν A e A gµ 1 A e,ν A e = ga gµ 1 A e,ν A e = Ag gµ 1 A e,ν A e = A g µa e,ν A e gµ 1 A e,ν A e = A (gg 1 µa e,ν A e = A (e µa e,ν A e = Ae =A. (9 (1: By 2.15, µ A (xy = µ A (y 1 yxy = µ yay 1(yx = µ A (yx and ν A (xy = ν A (y 1 yxy = ν yay 1(yx = ν A (yx. Defnton and Statements 3.2 (1 For any f/v-subgroup A of a group G, A s an L-f/v-normal subgroup of G ff t satsfes any one of the prevous nne equvalent condtons. In partcular, A s an f/v-normal subgroup of G ff for each g G, Ag = ga. (2 The set of all f/v-cosets of G, denoted by G/A or G A, whenever A s an f/v-normal subgroup of G, s called the f/v-quotent set of G by A. (3 Whenever G s a fnte group and A s an f/v-normal subgroup of G, from the generalzed Lagranges Theorem 2.19, (G/A = G A. Proposton 3.3 The followng are true for any group G: (a If G s abelan then every f/v-subgroup of G s f/v-normal subgroup of G, but not conversely. (b For an f/v-subgroup A of G and for any z G, the f/v-subset zaz 1 = (µ zaz 1, ν zaz 1 where µ zaz 1x = µ A (z 1 xz and ν zaz 1x = ν A (z 1 xz for each x G, s an f/v-subgroup of G. (c For any f/v-subgroup A of G, for each z G, zaz 1 = z µa e,ν A e A zµ 1 A e,ν A e. Proof: (a: It follows from 3.1(1 and 3.2(1. (b: Snce µ zaz 1x = µ A (z 1 xz Nν A (z 1 xz = Nν zaz 1x, t follows that zaz 1 s an f/v-subset of G. µ zaz 1(xy = µ A (z 1 xyz = µ A (z 1 xzz 1 yz µ A (z 1 xz µ A (z 1 yz = µ zaz 1(x µ zaz 1(y and ν zaz 1(xy = ν A (z 1 xyz = ν A (z 1 xzz 1 yz ν A (z 1 xz ν A (z 1 yz = ν zaz 1(x ν zaz 1(y. µ zaz 1(x = µ A (z 1 xz = µ A (z 1 x 1 z = µ zaz 1(x 1 and ν zaz 1(x = ν A (z 1 xz = ν A (z 1 x 1 z = ν zaz 1(x 1 for each z G.Hence zaz 1 s an f/vsubgroup of G. (c: It follows from 2.16(1. Defnton 3.4 For any par of f/v subgroups A and B of a group G, A s sad to be an -f/v-conjugate of B ff there exsts y G such that A = yby 1 or smply A = B y. It s easy to see that beng conjugate to an arbtrary but fxed f/v-subgroup A, s an equvqlence relaton on the set of all f/v-subgroups of G. Theorem 3.5 For any f/v-normal subgroup A of G, the followng are true: (1 A = {x/µ A (x = µ A (e, ν A (x = ν A (e} s a normal subgroup of G. (2 A = {x G/µ A (x > 0, ν A (x < 1} s a normal subgroup of G, whenever L s a strongly regular complete lattce. Proof: By 2.7, A s subgroup of G and A s a subgroup of G when L s a strongly regular complete lattce. Snce A s an f/v-normal subgroup of G, by 3.1(2, 2015 Global Journals Inc. (US

12 Note on Intutonstc Fuzzy (Normal Subgroups or Vague (Normal Subgroups (1: For each y A, µ A (xyx 1 = µ A y = µ A (e and ν A (xyx 1 = ν A y = ν A (e or xyx 1 A or A s a normal subgroup of G. (2: For each y A, µ A (xyx 1 = µ A y > 0 and ν A (xyx 1 = ν A (y < 1 or xyx 1 A or A s a normal subgroup of G. If/V-Normalzer Theorem 3.6 For any f/v-subgroup A of a group G, N G (A = {x G/µ A (xy = µ A (yx, ν A (xy = ν A (yx, for each y G} s a subgroup of G and the restrcton of A to N G (A, denoted by A N G (A, defned by (µ A N G (A, ν A N G (A, s an f/v-normal subgroup of N G (A. Proof: Snce µ A (ey = µ A (y = µ A (ye and ν A (ey = ν A (ye for each y G, e N G (A. Let x, y N G (A and z G. Then x N G (A mples µ A (x y 1 z = µ A (y 1 z x, ν A (x y 1 z = ν A (y 1 z x and y N G (A mples µ A (x 1 z 1 y = µ A (y x 1 z 1, ν A (x 1 z 1 y = ν A (y x 1 z 1. From the above, µ A (xy 1 z = µ A (x y 1 z = µ A (y 1 z x = µ A ((y 1 zx 1 = µ A (x 1 z 1 y = µ A (y x 1 z 1 = µ A ((z xy 1 1 = µ A (z xy 1 and ν A (xy 1 z = ν A (x y 1 z = ν A (y 1 z x = ν A ((y 1 zx 1 = ν A (x 1 z 1 y = ν A (y x 1 z 1 = ν A ((z xy 1 1 = ν A (z xy 1. Thus xy 1 N G (A and N G (A s a subgroup of G. Now we show that A N G (A s an f/v-normal subgroup of N G (A. But frst A N G (A s an f/v-subgroup of N G (A because for each x, y N G (A, (µ A N G (A(xy 1 = µ A (xy 1 µ A x µ A y = (µ A N G (Ax (µ A N G (Ay and (ν A N G (A(xy 1 = ν A (xy 1 ν A x ν A y = (ν A N G (Ax (ν A N G (Ay. Next for each x, y N G (A, (µ A N G (A(xy = µ A (xy = µ A (yx = (µ A N G (A(yx and (ν A N(A(xy = ν A (xy = ν A (yx = (ν A N G (A(yx mplyng A N G (A s an f/v-normal subgroup of N G (A. Defnton 3.7 For any f/v-subgroup A of a group G, the subgroup N G (A of G defned as above s called the normalzer of A n G and A N G (A s called the f/v-normalzer of A. lemma 3.8 For any f/v-subgroup A of a group G, A s an f/v-normal subgroup of G ff N G (A = G. Proof: ( : Always N G (A G. On the other hand, x G mples for each y G, by 3.1(1, µ A (xy = µ A (yx and ν A (xy = ν A (yx. So, x N G (A. ( : Agan by 3.1(1, we get that A s an f/v-normal subgroup of G. Theorem 3.9 For any f/v-subgroup B of a group G, the number of f/vconjugates of B n G s equal to the ndex (G : N G (B of the normalzer N G (B n G. Proof: Let u, v G. Then v 1 Gu = G. Now ubu 1 = vbv 1 ff for each x G, µ B (u 1 xu = µ B (v 1 xv and ν B (u 1 xu = ν B (v 1 xv ff (put x = vxu 1 µ B (u 1 v x = µ B (x u 1 v and ν B (u 1 v x = ν B (x u 1 v ff u 1 v N G (B ff u 1 N G (B = v 1 N G (B. Hence B u u 1 N G (B s a bjecton from {ubu 1 /u G} onto {un G (B/u G}. Theorem 3.10 For any f/v-subgroup B of a group G, u G ubu 1 s an f/vnormal subgroup of G and s the largest f/v-normal subgroup of G that s contaned n B. Proof: Frst observe that ubu 1 s an f/v-subgroup of G for each u G by 6.1.3(b. So u G ubu 1 s an f/v-subgroup of G, by Snce {ubu 1 /u G} = {(xub(xu 1 /u G} for each x G, u G µ ubu 1(x 1 yx = u G µ B (u 1 x 1 yx u = u G µ B ((xu 1 y(xu = u G µ (xub(xu 1(y = u G µ ubu 1(y and Global Journals Inc. (US

13 Note on Intutonstc Fuzzy (Normal Subgroups or Vague (Normal Subgroups 48 u G ν ubu 1(x 1 yx = u G ν B (u 1 x 1 yx u = u G ν B ((xu 1 y(xu = u G ν (xub(xu 1(y = u G ν ubu 1(y for each x, y G. Hence u G ubu 1 s an f/v-normal subgroup of G. Next, let A be an f/v-normal subgroup of G, wth A B. Snce A s an f/v-normal subgroup of G, A = uau 1 for each u G. Snce A B, A = uau 1 ubu 1 for each u G or A u G ubu 1 or u G ubu 1 s the largest f/v-normal subgroup of G that s contaned n B. lemma 3.11 For any f/v-normal subgroup A of a group G and for any x, y G such that xa = ya, µ A (x = µ A (y and ν A (x = ν A (y. Proof: By 2.16(2, xa = ya mples xa = ya whch mples x 1 y A and y 1 x A or µ A (x 1 y = µ A e = µ A (y 1 x and ν A (x 1 y = ν A e = ν A (y 1 x. Snce A s an f/v-normal subgroup of G, µ A (x = µ A (y 1 xy µ A (y 1 x µ A (y = µ A (e µ A (y = µ A (y and ν A (x = ν A (y 1 xy ν A (y 1 x ν A (y = ν A (e ν A (y = ν A (y. Smlarly, µ A (y = µ A (x 1 yx µ A (x 1 y µ A (x = µ A (e µ A (x = µ A (x and ν A (y = ν A (x 1 yx ν A (x 1 y ν A (x = ν A (e ν A (x = ν A (x. Hence µ A (x = µ A (y and ν A (x = ν A (y. Theorem 3.12 For any f/v-normal subgroup A of a group G. The followng are true n G/A: 1. (xa (ya = (xya for each x, y G; 2. (G/A, s a group; 3. G/A = G/A ; 4. Let A ( be an f/v-subset of G/A be defned by µ A ( (xa = µ A (x and ν A ( (xa = ν A (x for each x G. Then A ( s an f/v-normal subgroup of G/A. Proof: (1: Snce A s an f/v-normal subgroup, by 3.1(7, ths follows. (2: By (1, G/A s closed under the operaton. For each x, y, z G, xa (ya za = xa (yza = (xyza = (xya za = (xa ya za. So G/A s assocatve under the operaton. By 2.2(3, ea = A. Further by (1, for each x G, A xa = ea xa = exa = xa and xa A = xa ea = xea = xa or A s the dentty element for G/A. (x 1 A (xa = (x 1 xa = ea = A = (xa (x 1 A or x 1 A s the nverse of xa n G/A. Hence (G/A, s a group. (3: Let η : G/A G/A, defned by η(xa = xa. Then η s well defned and 1-1 because xa = ya ff xa = ya. Now we show that η s a homomorphsm or xya = xa ya. But by 3.5(1, A s a normal subgroup of G and so t follows that η s a homomorphsm. Now we show that η s onto. β G/A mples β = ga, g G. Then ga G A such that η(ga = ga = β or η s onto. (4: Frst we show that A ( s an f/v-subgroup of G/A. Snce A be an f/v-subgroup of G, (a: µ A ( (ga ha = µ A ( (gha = µ A (gh µ A (g µ A (h = µ A ( (ga µ A ( (ha and ν A ( (ga ha = ν A ( (gha = ν A (gh ν A (g ν A (h = ν A ( (ga ν A ( (ha. (b: µ A ( ((ga 1 = µ A ( (g 1 A = µ A (g 1 = µ A (g = µ A ( (ga and ν A ( ((ga 1 = ν A ( (g 1 A = ν A (g 1 = ν A (g = ν A ( (ga. Therefore A ( s an f/v-subgroup of G/A. Now we show that A ( s an f/v-normal subgroup of G/A. Snce A s an f/v-normal subgroup of G, for each g, h G, µ A ( ((ga 1 (ha (ga = µ A ( (g 1 A ha ga = µ A ( (g 1 hga = µ A (g 1 hg µ A (h = µ A ( (ha and ν A ( ((ga 1 (ha (ga = ν A ( (g 1 A ha ga = ν A ( (g 1 hga = ν A (g 1 hg ν A (h = ν A ( (ha. Hence A ( s an f/v-normal subgroup of G/A Global Journals Inc. (US

14 Note on Intutonstc Fuzzy (Normal Subgroups or Vague (Normal Subgroups Theorem 3.13 For any f/v-subgroup B of a group G and for any normal subgroup N of G, the f/v-subset C: G/N L where for each x G, µ C (xn = µ B (xn and ν C (xn = ν B (xn, s an f/v-subgroup of G/N when L s a complete nfnte dstrbutve lattce. Proof: Snce B s an f/v-subgroup of G and N s a normal subgroup of G and hence for each x G, (xn 1 = x 1 N, µ C ((xn 1 = µ C (x 1 N = µ B (x 1 N = z x 1 N µ B z = w 1 x 1 N(=(xN 1 µ B w 1 = w xn µ B w = µ B (xn = µ C (xn and ν C ((xn 1 = ν C (x 1 N = ν B (x 1 N = z x 1 N ν B z = w 1 x 1 N(=(xN 1 ν B w 1 = w xn ν B w = ν B (xn = ν C (xn where the 5 th equalty n both cases s due to the fact that w xn ff w 1 (xn 1. Hence C(xN 1 = C(xN. Snce [0,1] s a complete nfnte dstrbutve lattce and N s a normal subgroup of G, for each x, y G µ C ((xn(yn = µ B (xyn = z xyn µ B z = u xn,v yn µ B (uv u xn,v yn (µ B (u µ B (v = ( u xn µ B (u ( v yn µ B (v = ( µ B (xn ( µ B (yn = (µ C (xn (µ C (yn and ν C ((xn(yn = ν B (xyn = z xyn ν B z = u xn,v yn ν B (uv u xn,v yn (ν B (u ν B (v = ( u xn ν B (u ( v yn ν B (v = ( ν B (xn ( ν B (yn = (ν C (xn (ν C (yn. Hence C s an f/v-subgroup of G/N. Defnton 3.14 For any f/v-subgroup B of a group G and for any normal subgroup N of G, the f/v-subgroup C:G/N L, where L s a complete nfnte dstrbutve lattce, defned by µ C (xn = µ B (xn and ν C (xn = ν B (xn for each x G, s called the f/v-quotent subgroup of G/N relatve to B and s denoted by B/N or B N. In other words when N s a normal subgroup of G and B s any f/v-subgroup of G, and [0,1] s a complete nfnte dstrbutve lattce, B N : G N [0, 1] s defned by µ B (gn = µ B (gn and ν B (gn = ν B (gn for each g G. N N Lemma 3.15 For any par of groups G and H and for any crsp homomorphsm f : G H, the followng are true: 1. A s an f/v-normal subgroup of G mples f(a s an f/v-normal subgroup of H when f s onto. 2. B s an f/v-normal subgroup of H mples f 1 (B s an f/v-normal subgroup of G. Proof: (1: A s an f/v-normal subgroup of G mples µ A (g 1 hg µ A (h and ν A (g 1 hg ν A (h for each h, g G. Let fa = B. Then µ B y = µ A f 1 y and ν B y = ν A f 1 y. Snce the f/v-mage of an f/v-subgroup s an f/v-subgroup, we only show that µ B (g 1 hg µ B (h and ν B (g 1 hg ν B (h for each g, h G. Snce f s onto, for each y H, f 1 y φ. Let a f 1 g, b f 1 h. Then fa = g, fb = h and fa 1 = g 1. Snce f s a homomorphsm, g 1 hg = f(a 1 ba and a 1 ba f 1 (g 1 hg. So, for each b f 1 h, µ B (g 1 hg = µ A f 1 (g 1 hg = c f 1 (g 1 hgµ A c µ A (a 1 ba µ A (b and ν B (g 1 hg = ν A f 1 (g 1 hg = c f 1 (g 1 hgν A c ν A (a 1 ba ν A (b mplyng µ B (g 1 hg b f 1 h µ A (b = µ B (h and ν B (g 1 hg b f 1 h ν A (b = ν B (h or B = f(a s an f/v-normal subgroup of H when f s onto. (2: Let f 1 B = A. Then for each g G, µ A g = µ B fg and ν A g = ν B fg. Snce the f/v-nverse mage of an f/v-subgroup s an f/v-subgroup we only show that µ A (g 1 hg µ A (h and ν A (g 1 hg ν A (h Global Journals Inc. (US

15 Note on Intutonstc Fuzzy (Normal Subgroups or Vague (Normal Subgroups 50 Snce f s a homomorphsm and B s an f/v-normal subgroup of H, for each g, h G, µ A (g 1 hg = µ B f(g 1 hg = µ B ((fg 1 (fh(fg µ B fh = µ A h and ν A (g 1 hg = ν B f(g 1 hg = ν B ((fg 1 (fh(fg ν B fh = ν A h or A = f 1 B s an f/v-normal subgroup of G. Defnton 3.16 For any par of f/v-subgroups A and B of a group G such that A B, A s called an f/v-normal subgroup of B ff for each x, y G, µ A (xyx 1 µ A (y µ B (x and ν A (xyx 1 ν A (y ν B (x. Theorem 3.17 For any par of f/v-subgroups A and B of a group G such that A B, the followng are equvalent: 1. A s an f/v-normal subgroup of B. 2. µ A (yx µ A (xy µ B (x and ν A (yx ν A (xy ν B (x for each x, y G. 3. (χ µ Ae x each x G. µ A (µ A χ µ Ae x µ B and (χ ν Ae x ν A (ν A χ ν Ae x ν B for Proof: (1 (2: Snce A s an f/v-normal subgroup of B, for each x, y G, µ A (yx = µ A (x 1 xyx = µ A (x 1 (xyx µ A (xy µ B (x and ν A (yx = ν A (x 1 xyx = ν A (x 1 (xyx ν A (xy ν B (x. (2 (3: By 2.2(3 and 2.2(4, we have (χ µ Ae x µ A y = µ A (x 1 y µ A (yx 1 µ B (y = (µ A χ µ Ae x y µ B y = ((µ A χ µ Ae x µ B y and (χ ν Ae x ν A y = ν A (x 1 y ν A (yx 1 ν B (y = (ν A χ ν Ae x y ν B y = ((ν A χ ν Ae x ν B y or for each x G, (χ µ Ae x µ A (µ A χ µ Ae x µ B and (χ ν Ae x ν A (ν A χ ν Ae x ν B. (3 (1: Lettng z 1 = x 1 y and by 2.2(3 and 2.2(4, we have µ A (x 1 yx = µ A (z 1 x = (χ µ Ae z µ A x (µ A χ µ Ae z x µ B x = µ A (xz 1 µ B (x =µ A (xx 1 y µ B (x = µ A (y µ B (x and ν A (x 1 yx = ν A (z 1 x=(χ ν Ae z ν A x (ν A χ ν Ae z x ν B x = ν A (xz 1 ν B (x = ν A (xx 1 y ν B (x = ν A (y ν B (x or for each x, y G, µ A (x 1 yx µ A (y µ B (x and ν A (x 1 yx ν A (y ν B (x or A s an f/v-normal subgroup of B. Theorem 3.18 For any par of f/v-subgroups A and B of a group G such that A s an f/v-normal subgroup of B: 1. A s a normal subgroup of B. 2. A s a normal subgroup of B whenever [0, 1] s strongly regular. Proof: (1: Snce µ A e s the largest of µ A G, ν A e s the smallest of ν A G and A s an f/v-subgroup of G, we get for each x, y A, µ A (xy 1 µ A x µ A y = µ A e and ν A (xy 1 ν A x ν A y = ν A e, so we have µ A xy 1 = µ A e and ν A xy 1 = ν A e or xy 1 A. Hence A s a subgroup of B. Agan snce µ A e s the largest of µ A G, ν A e s the smallest of ν A G, A s an f/v-normal subgroup of B; we get for each b B and a A, µ A (bab 1 µ A a µ B b = µ A e µ B e = µ A e and ν A (bab 1 ν A a ν B b = ν A e ν B e = ν A e, so we have µ A (bab 1 = µ A e and ν A (bab 1 = ν A e or bab 1 A. Therefore A s a normal subgroup of B. (2: Snce [0, 1] s strongly regular, for each x, y A, µ A (xy 1 µ A x µ A y > 0 and ν A (xy 1 ν A x ν A y < 1 or xy 1 A. Hence A s a subgroup of B. Agan, snce [0,1] s strongly regular, for each b B and a A, we get µ A (bab 1 µ A a µ B b > 0 and ν A (bab 1 ν A a ν B b < 1 or bab 1 A. Hence A s a normal subgroup of B when [0,1] s strongly regular. Lemma 3.19 For any par of f/v-subgroups A and B of a group G such that B A s an f/v normal subgroup of B, the f/v-subset C: A [0, 1] defned by, for each b B µ C ba = µ B ba and ν C ba = ν B ba, s an f/v-subgroup of B A, whenever [0,1] s a strongly regular complete nfnte dstrbutve lattce Global Journals Inc. (US

16 Note on Intutonstc Fuzzy (Normal Subgroups or Vague (Normal Subgroups Proof: Snce [0,1] s strongly regular, by 3.18, A s a normal subgroup of B. Now n 3.13 set G = B, N = A, B = B. Then snce [0,1] s a complete nfnte dstrbutve lattce, C s an f/v-subgroup of B A. Defnton 3.20 For any par of f/v-subgroups A and B of a group G such that A s an f/v normal subgroup of B and L s a strongly regular complete nfnte dstrbutve lattce, the f/v-quotent subgroup of B B relatve to A, denoted by B/A or B A, s defned by B/A: B /A L wth µ B/A (ba = µ B (ba and ν B/A (ba = ν B (ba for each b B and s called L-f/v-quotent subgroup of B relatve to A. In what follows we prove a natural relaton between ( B A and B A whch s used n the Thrd Isomorphsm Theorem. Lemma 3.21 For any par of f/v-subgroups A and B of a group G such that A s an f/v-normal subgroup of B, (B/A = B /A. Proof: Let us recall that (B/A = {ba (B /A /b B, µ B/A (ba > 0 and ν B/A (ba < 1}. So always, (B/A B /A. α B /A mples α = ba for some b B. Now as e A and b B, µ B/A (ba = µ B (ba µ B b > 0 and ν B/A (ba = ν B (ba ν B b < 1 mplyng α (B/A. Hence (B/A = B /A. Theorem 3.22 For any f/v-normal subgroup A of G and an f/v-subgroup B of G, A B s an f/v-normal subgroup of B. Proof: By 3.11, f A, B are f/v-subgroups of G then A B s an f/v-subgroup of G and A B B. Now we show that C = A B s an f/v-normal subgroup of B or for each x, y G, µ C (xyx 1 µ C (y µ B (x and ν C (xyx 1 ν C (y ν B (x. Snce A s an f/v-normal subgroup of G, for each x, y G, µ C (xyx 1 = (µ A µ B (xyx 1 = µ A (xyx 1 µ B (xyx 1 µ A (y µ B (xyx 1 µ A (y µ B (x µ B (y µ B (x 1 = (µ A (y µ B (y µ B (x = µ A B (y µ B (x = µ C (y µ B (x and ν C (xyx 1 = (ν A ν B (xyx 1 = ν A (xyx 1 ν B (xyx 1 ν A (y ν B (xyx 1 ν A (y ν B (x ν B (y ν B (x 1 = (ν A (y ν B (y ν B (x = ν A B (y ν B (x = ν C (y ν B (x. Therefore µ C (xyx 1 µ C (y µ B (x and ν C (xyx 1 ν C (y ν B (x or C = A B s an f/v-normal subgroup of B. IV. Acknowledgments The 1st author would lke to express her heart full thanks to the 2nd author, Prncpal and the management of Adtya Insttute of Technology and Management, Tekkal for ther contnuous encouragement. References 1. Atanassov, K., Intutonstc fuzzy sets, n: V. Sgurev, Ed., VII ITKR's Sesson, Sofa, June 1983 (Central Sc. and Techn. Lbrary, Bulg. Academy of Scences, Atanassov, K. and Stoeva, S., Intutonstc fuzzy sets, n: Polsh Symp. On Interval and Fuzzy Mathematcs, Poznan (Aug Atanassov, K. and Stoeva, S., Intutonstc L-fuzzy sets, n: R. Trappl, Ed., Cybernetcs and Systems Research 2 (Elsever Sc. Publ., Amsterdam, Atanassov, K., Intutonstc fuzzy relatons, n: L. Antonov, Ed., III Internatonal School "Automaton and Scent_c Instrumentaton", Varna (Oct, Atanassov, K., New Operatons de_ned over the Intutonstc fuzzy sets, n: Fuzzy Sets and Systems 61 ( , North - Holland Global Journals Inc. (US

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