GENERALIZATION OF (∈, ∈2q) −FUZZY SUB-NEAR-FIELDS AND IDEALS OF NEAR-FIELDS (GF-NF-IO-NF

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In this paper, We introduce the notion of (∈, ∈ ∨qk) − fuzzy sub-near-field which is a generalization of (∈,∈ ∨q) −fuzzy sub-near-field. We have given examples which are (∈, ∈ ∨qk) −fuzzy ideals but they are not (∈, ∈ ∨q) −fuzzy ideals. We have also

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  International Journal of Mathematical Archive-4(7), 2013, 328-343   Available online through www.ijma.info   ISSN 2229 – 5046 International Journal of Mathematical Archive- 4(7), July – 2013 328    GENERALIZATION OF (   , 2q) −FUZZY SUB-NEAR-FIELDS AND IDEALS OF NEAR-FIELDS (GF-NF-IO-NF) N V Nagendram 1*  Assistant professor, Department of Science & Humanities (Mathematics)  Lakireddy Balireddy College of Engineering Mylavaram 521 230 Andhra Pradesh, India  E-mail:  nvn220463@yahoo.co.in  E S Sudeeshna 2   Student, Department of Electronics and Computer Engineering  P V P Siddhartha Institute of Technology kanuru, Vijayawada-520007, Andhra Pradesh, India (Received on: 19-06-12; Revised & Accepted on: 29-06-13)  ABSTRACT  I  n this paper, We introduce the notion of ( ∈   , ∈   qk) −  fuzzy sub-near-field which is a generalization of ( ∈   , ∈   q) −  fuzzy sub-near-field. We have given examples which are ( ∈   , ∈   qk) −  fuzzy ideals but they are not ( ∈   , ∈   q) −  fuzzy ideals. We have also introduced the notions of ( ∈   , ∈    qk) −  fuzzy quasi-ideals and ( ∈   , ∈   qk) −  fuzzy bi-ideals of near- field. We have characterized ( ∈   , ∈   qk) −  fuzzy quasi-ideals and ( ∈   , ∈    qk) −  fuzzy bi-ideals of near-fields.  Key words:  Near field, Fuzzy sub-near-field, Fuzzy ideal, Fuzzy quasi-ideal, Fuzzy bi-ideal ( ∈   , ∈   q) −fuzzy  sub-near- field, ( ∈   , ∈   q) −fuzzy ideal, ( ∈   , ∈   q) −fuzzy quasi -ideal, ( ∈   , ∈   q) −fuzzy bi -ideal, ( ∈   , ∈   qk) −fuzzy sub -near-field, ( ∈   , ∈   qk) −fuzzy ideal, ( ∈   , ∈   qk) −fuzzy quasi -ideal,( ∈   , ∈   qk) −fuzzy bi -ideal.  Subject classification code 2000: 16Y30, 03E72, 16Y99. SECTION1: INTRODUCTION In 1965 Zadeh [24] introduced the concept of fuzzy subsets and studied their properties on the lines parallel to set theory. In 1971, Rosenfeld [17] defined a fuzzy subgroup and gave some of its properties. Rosenfeld’s definition of a fuzzy group is a turning point for pure mathematicians. Since then, the study of fuzzy algebraic structure has been pursued in many directions such as groups, fields, near-fields, modules, vector spaces and so on. In 1981 Das [6] explained the inter-relationship between a fuzzy subgroups and its t-level subsets. Fuzzy sub-fields, near-fields and ideals were first introduced by Wang-jin Liu [12] in 1982. Subsequently, Mukherjee and Sen [14], Swamy and Swamy [20], Yue [23], Dixit et al  [7] and Rajesh Kumar [10] applied some basic concepts pertaining to ideals from classical field theory and developed a theory of fuzzy fields. The notions of fuzzy sub-near-field and ideal were first introduced by Abou-Zaid [1] in 1991. The concept of quasi-coincidence of a fuzzy point with a fuzzy subset was introduced by Pu Pao-Ming and Liu Ying-Ming [13] in 1980. The idea of quasi-coincidence of a fuzzy point with a fuzzy set was introduced by Bhakat and Das [2] in 1992. In particular, the ( ∈ , ∈  q) − fuzzy subgroup is an important and useful generalization of a fuzzy subgroup. In [4], Bhakat and Das have extended the notion of ( ∈ , ∈  q) − fuzzy subgroups to the notion of ( ∈ , ∈  q) − fuzzy sub-near-fields. Narayanan and Manikantan [15] have extended these results to near-rings. I introduce the notion of an ( ∈ , ∈  qk) − fuzzy sub-near-fields which is a generalization of an ( ∈ , ∈  q) − fuzzy sub-near-field. I give examples which are ( ∈ , ∈  qk) − fuzzy ideals but not ( ∈ , ∈  q) − fuzzy ideals. I introduce the notions of ( ∈ , ∈  qk) −fuzzy quasi -ideals and ( ∈ , ∈  qk) −fuzzy bi -ideals of near-fields extension to near-fields which are the generalization of fuzzy bi-ideals, fuzzy quasi-ideals, ( ∈ , ∈  q) −fuzzy quasi -ideals, ( ∈ , ∈  q) −fuzzy bi -ideals of near-field N. I also characterize ( ∈ , ∈  qk) − fuzzy quasi-ideals and ( ∈ , ∈  qk) − fuzzy bi-ideals of near-fields which are extension to near-fields. Corresponding author:   N V Nagendram 1*  Assistant professor, Department of Science & Humanities (Mathematics)  Lakireddy Balireddy College of Engineering Mylavaram 521 230 Andhra Pradesh, India  N V Nagendram 1* & E S Sudeeshna 2  / GENERALIZATION OF (     , 2q) −FUZZY SUB -NEAR-FIELDS AND IDEALS OF NEAR-FIELDS (GF-NF-IO-NF)/ IJMA- 4(7), July-2013. © 2013, IJMA. All Rights Reserved 329 SECTION 2: PRELI MINARIES For the sake of completeness I fir s t  recall some definitions and results p rop os ed by the early pi onee r s .   Definition 2.1:  A near-field N is a system with two binary operations “ +” and “ ·” such t hat:  (i) (N, +) is a group, not necessarily abelian , (ii) (N, ·) is a se migrou p and  (iii) (x + y)z = xz + yz, for all x, y, z ∈   N .  We will use the word “near-field” to mean “ righ t  distributive near-field” and wri te xy instead of x. y. Note that 0. x = 0 and ( −x)y   = − xy but in general x.0 = 0 f  or some x ∈ N .   Definition 2.2:  Let (N, +, ·) be a near-field. A subse t   I of N is said to be an ideal of N if: (i) ( I , +) is a normal subgroup of (N, +), (ii) I  N ∈ I , (iii) n 1 (n 2   + i) −  n 1  n 2   ∈ I  , for all i ∈ I and n 1 , n 2   ∈   N.  If I satisfies (i) and (ii), then it is called a right ideal of N. If I satisfies (i) and (iii), then it is called a left ideal of N .  Let N be a near-field. Given t wo  subsets A and B of N, the product A B =  { a   b | a ∈   A, b ∈   B } and A* B = { a(a 0 + b) −  a a 0 | a, a 0   ∈   A, b ∈   B }   In w h at follows, N will denote r igh t distributive near-field, unless otherwise s p ec ifi e d.  For the basic t e rmin ology and notation we refer to Pilz [16] and Abou-Zaid [ 1].   Definition 2.3: Let S be any set. A mapping µ: S →   [0, 1] is called a fuzzy su bset of S .  A fuzzy subset µ: S →   [0, 1] is n onem p t y  if µ is not the constant map with   value 0. For any t wo  fuzzy subsets λ   and µ of S, λ    ≤   µ means that λ(a)   ≤   µ(a) for all a ∈   S. The c haracte r is t ic function of N is denoted by N and the c har ac t e ri s ti c function of a subset A is denoted by f A. The image of a fuzzy s u bse t  µ is denoted b y I  m(µ) = { µ  ( n )  | n ∈   N } and | I  m(µ)| denotes the cardinali t y  of I  m(µ). H e reaft e r , we consider only nonempty fuzzy s u bse t s of N .   Definition 2.4:  [12] Let µ be a fuzzy subset of a field N Then µ is called a fuzzy sub-(resp. ideal) of N if for all x, y ∈ N :  (1) µ(x −  y) ≥   min { µ  ( x ) ,   µ  ( y ) } ,  (2) µ(xy) ≥   min { µ  ( x ) ,   µ  ( y ) } (resp, µ(xy) ≥   max { µ  ( x ) ,   µ  ( y ) } ). Definition 2.5: Let µ be a fuzzy subset of N. Then µ is called a fuzzy left (ri ght ) N − subgroup of N if for all x, y ∈   N: (1) µ(x −   y) ≥   min { µ  ( x ) ,   µ  ( y ) } ,  (2) µ(xy) ≥   µ(y)(µ(xy) ≥   µ  ( x)). If µ is both le f  t  and r igh t  fuzzy N –subgroup of N, then it is called a fuzzy N − subgroup of N .   Definition 2.6:  Let µ be any fuzzy s u bse t  of N. For t ∈   [0, 1], the se t µ  t   ={ x   ∈   N | µ(x) ≥   t}   is called a level subset of µ  .   Definition 2.7:  Let f and g be any t wo  fuzzy subsets of N. Then f ∩   g, f ∪   g , f    + g, f g and f ∗   g are fuzzy subsets of N defined b y: (f ∩   g)(x) = min { f   (x), g ( x ) } and   (f ∪   g)(x) = max { f   (x), g ( x ) }   (f + g ) (x) = sup (min {f (z), g(z)}) if x expressed as x = y + z = 0 otherwise (f g ) (x) = sup (min{f (z), g(z)}) if x expressed as x = y = 0 otherwise ( f * g )(x) = abcba x  −+= )( sup { min { f   (a), g ( c ) }} ,  if x = a(b + c) – ab = {0 otherwise    N V Nagendram 1* & E S Sudeeshna 2  / GENERALIZATION OF (     , 2q) −FUZZY SUB -NEAR-FIELDS AND IDEALS OF NEAR-FIELDS (GF-NF-IO-NF)/ IJMA- 4(7), July-2013. © 2013, IJMA. All Rights Reserved 330 Definition 2.8: For any x ϵ   N and t ϵ   (0, 1], define a fuzzy point xt as Xt(y) = t if y = x = 0 if y ≠ x . If xt is a fuzzy p oi n t  and µ is any fuzzy subset of N and xt ≤   µ, then we write x t ϵ   µ. Note that xt ϵ   µ if and only if x ϵ   µ  t  where µ  t  is a level s u bse t  of µ    Definition 2.9: A fuzzy subset µ of N is called a fuzzy sub-near-field of N if for al l x, y ∈   N; (1) µ(x −  y) ≥   min { µ  ( x ) ,   µ  ( y ) } , (2) µ(xy) ≥   min { µ  ( x ) ,   µ  ( y ) } .   Definition 2.10:   Let µ be a n onem p t y  fuzzy subset of N. µ is a fuzzy ideal of N if for all x, y, i in N; (1) µ(x −  y) ≥   min { µ  ( x ) ,   µ  ( y ) } , (2) µ(x) = µ(y + x −   y ),  (3) µ(xy) ≥   µ  ( x),  (4) µ(x(y + i) −  xy) ≥   µ(i) Definition 2.11: A fuzzy p oi n t   xt is said to belong to (resp.be quasi-c oinci den t with) a fuzzy s u bse t  µ, written as xt ∈   µ (resp. xt qµ ) if µ(x) ≥  t (resp. µ(x) + t > 1). If xt ∈   µ or xt qµ, then we write xt ∈  V q µ  . Definition 2.12:   A fuzzy subset µ of a field R is said to be an −   fu zzy sub-field of R if for all x, y ∈   R and t, r ∈   (0, 1]; (1) xt , y r   ∈   µ implies (x + y ) mi n { t,   r } ∈ V q µ  , (2) xt ∈   µ implies ( − x ) t   ∈  V q µ  ,  (3) xt , y r   ∈   µ implies ( xy ) m in { t ,   r } ∈ V q µ  . Definition 2.13:   [4] A fuzzy subset µ of a field R is said to be an ( ϵ , ϵ  V q ) − fu zzy ideal of R if: (1) µ is an ( ∈ , ∈  qk)   − fu zz y  sub-near-field of R ,  (2) xt µ and y ϵ   R implies ( xy ) t  , ( y x ) t   ∈  V q µ  ,  for all x, y R and t ∈   (0, 1].   Example 2.14: [4] Consider the field R = J  /   (4).  L e t   λ:  R →  [0, 1] be defined b y λ(0) = 0.6, λ(1) = λ(3) = 0.4, λ(2) = 0.7. Then λ   is an ( ∈ , ∈  qk)   − fu z zy ideal of R but not a fuzzy ideal. Note 2.15:  Refer to Definition 2.4. Definition 2.16: [15] A fuzzy s u bse t  µ of N is said to be an ( ∈ , ∈  qk)   − fu zzy   ideal  of N if for all x, y, z ∈   N and for all r, t ∈  (0, 1]: (1) xr , y t   ∈   µ implies (x −   y ) mi n( r ,   t) V q µ  ,  (2) xt ∈   µ and y ∈   N implies (y + x −   y ) t   ∈  V q µ  ,  (3) xt ∈   µ and y ∈   N implies ( x y ) t   ∈  V q µ  ,  (4) z t   ∈   µ and x, y ∈  N implies (x(y + z) −   x y ) t   ∈   V q µ N implies (x(y + z) −   x y ) t   ∈  V q µ  .  If µ satisfies (1), (2) and (3), then it is called an ( ∈ , ∈  qk)   − fu zzy  right ideal of N. If µ satisfies (1), (2) and (4), then it is called an ( ∈ , ∈  qk) − fu zzy  left ideal of N . Example 2.17:   ([15]) Let N = { 0 ,  a, b, c }  be Klein’s four group. Define m ult iplication in N as follows: Then (N, +, ⋅ } is a near-field ( see [ [16], P-408 ] scheme 19). Let µ: N →  [0, 1] be a fuzzy s u bse t  of N such that µ(0) = 0.7, µ(a) = µ(c) = 0.4, µ(b) = 0.8. Then µ is an ( ∈ , ∈  q) − fu z zy sub-near-field of N. But, since 0.7 = µ(0) = µ(b −  b) 6 ≥   min { µ  ( b ) ,   µ  ( b ) } = 0.8, µ is not a fuzzy sub-near-field of N. Further, µ is an ( ∈ , ∈  qk)   − fu z zy ideal of N. Since 0.7 = µ(0) = µ(b0) 6 ≥   µ(b) = 0.8, µ is not a f  uzzy   ideal of N .    N V Nagendram 1* & E S Sudeeshna 2  / GENERALIZATION OF (     , 2q) −FUZZY SUB -NEAR-FIELDS AND IDEALS OF NEAR-FIELDS (GF-NF-IO-NF)/ IJMA- 4(7), July-2013. © 2013, IJMA. All Rights Reserved 331 2   2   2   Lemma 2.18:   [1] Let µ be a fuzzy subset of N. µ is a fuzzy left (right) N − sub gr oup of N if and only if the level subset µ  t  , t ∈   I  m µ, is a left (right) N − subgroup of N .   Lemma 2.19:   [1] Let I be a subset of N. I is an (left or right) ideal of N iff f  I   is a fuzzy (left or right) ideal of N .   Lemma 2.20:   [1] Let µ be a fuzzy subset of N. µ is a fuzzy (left or right) ideal of N if and only if the level subset µ  t  , t ∈   I  m µ, is an ideal of N .   SECTION 3: (   ,   qk) − fuzzy sub-near-fields and (   , qk) − fuzzy Ideals in N. In this section, we introduce the notion of V q k    − fuzzy sets which are a ge n e r al ization of fuzzy sets .   Definition 3.1 :  A fuzzy p oi n t   xt is said to belong to (resp., be k  − quasi-c oinci den t with) a fuzzy s u bse t  µ, written as xt ∈   µ (resp., xt qk µ) if µ(x) ≥  t (resp., µ(x) +   t >  1 −  k, where k ∈   [0, 1) ).  For any t ∈   (0, 1], xt ∈   µ or xt ∈  V   qk µ will be denoted by xt ∈  qk µ. xt ∈   µ, xt ∈  V q k   will resp ec t iv el y  mean xt ∈   µ and xt ∈   V q k   µ do not h old . If k = 0, then xt ∈ V q k   µ if and only if xt ϵ  V q µ  .  Thus the t wo  defin iti ons Definition 2.11 and Definition 3.1 will coincide when k = 0. Throughout thi s paper, k  ∈   [0, 1) is arbitrary, but fi xed.   Definition 3.2:  A fuzzy s u bse t  µ is said to be an ( ∈ , ∈  qk) −  fuzzy sub-near -field   of N if for all x, y ∈  N and t, r ∈   (0, 1] : (1) xt , y r   ∈   µ implies (x + y ) m in { t,   r } ∈ V q k   µ, (2) xt ∈   µ implies ( − x ) t   ∈  V q k    µ  ,  (3) xt , y r   ∈   µ implies ( x y ) mi n { t,   r } ∈ V q k    µ  .   Note 3.3:  Let µ be a fuzzy subset of N and t, r ∈  (0, 1]. T hen:  (1) (a) xt , y r   ∈   µ implies (x + y ) mi n { t,   r } ∈ V q k   µ, an d (b) µ(x + y) ≥   min(µ(x), µ(y), 21  k  − ) for all x, y ∈   N ar e e quiv a l ent .  (2) (c) xt ∈   µ implies ( − x ) t   ∈   V q k   µ, a nd (d) µ( −x)   ≥   min(µ(x), 21 k  − )for all x ∈   N are equiva l ent.  (3) (e) xt , y r   ∈   µ implies ( xy ) m in { t ,   r } ∈ V q k   µ, a nd   ( f   ) µ(xy) ≥   min(µ(x), µ(y), 21  k  − ) for all x, y ∈   N are equiva l ent. Theorem 3.4: A fuzzy subset µ of N is an ( ∈ , ∈  V q k  ) − fuzzy sub-near-field of N if and only if µ(x −  y), µ(xy) ≥   min { µ  ( x ) ,  µ(y) 21 k  − } ,  for all x, y ∈   N. Pr o of:   It   follo ws from  Note 3.3. Corollary 3.5: ( [15], Lemma 3.2.) A fuzzy subset µ of N is an ( ∈ , ∈  qk) − fu zzy sub-near-field of N if and only if µ(x −  y), µ(xy) ≥   min { µ  ( x ) ,  µ(y), 0 . 5 } ,  for all x, y ∈   N .nm   Pr o of:   The   re su lt   follo ws eas i ly fr om Lem ma  3.3 if    we   tak  e  k =  0. Corollary 3.6:  [[4], Theorem 3.3.] µ is an ( ∈ , ∈  qk)   − fu zzy  subfield if and only if µ(x −  y), µ(xy) ≥   min { µ  ( x ) ,  µ(y), 0 . 5 } ,  for all x, y ∈   R .   Remark 3.7:  Every fuzzy sub-near-field and ( ∈ , ∈  qk)   − fu z zy sub-near-field of N is an ( ∈ , ∈  qk) − fuzzy sub-near-field of N, but, as the following example shows, the converse is not necessarily tr ue.  N V Nagendram 1* & E S Sudeeshna 2  / GENERALIZATION OF (     , 2q) −FUZZY SUB -NEAR-FIELDS AND IDEALS OF NEAR-FIELDS (GF-NF-IO-NF)/ IJMA- 4(7), July-2013. © 2013, IJMA. All Rights Reserved 332 2   2   2   2   Example 3.8 : Consider the near-field (N, +, •} as defined in Example 2.17. Defin e a fuzzy subset µ: N →  [0, 1] by µ(0) = 0.42, µ(a) = µ(c) = 0.4, µ(b) = 0 . 44. Then µ is an ( ϵ , ϵ q 0 . 2  ) −fuzzy   sub-near-field of N. But, since µ(0) = µ(b −  b) 6 ≥   min { µ  ( b ) ,   µ  ( b ) }  and µ(0) = µ(b −  b) 6 ≥   min { µ  ( b ) ,  µ(b), 0 . 5 } ,  µ is neither a  fuzzy sub-near-field of N nor an ( ∈ , ∈   V q ) − fu z zy sub-near-field of N .  Now we generalize the notions of fuzzy ideals of N defined by Zaid [1] and   ( ∈ , ∈ V q ) − fu z zy ideals of N defined by Narayana and Manikantan [15]. Definition 3.9:  A fuzzy s u bse t  µ of N is said to be an ( ∈ , ∈ V q k   ) −fuzzy  sub gr oup of N if xr , y t   ∈  µ implies (x −   y ) mi n( r ,   t) ∈  V q k   µ, for all x, y ∈   N and f  or  all r, t ∈   (0, 1] .   Definition 3.10:   A fuzzy s u bse t  µ of N is said to be an ( ∈ , ∈  Vqk)   − fuzzy ideal of N if for all x, y, z ∈   N and for all r, t ∈   (0, 1] : (1) xr , y t   ∈   µ implies (x −   y ) mi n( r ,   t) ∈  V q k    µ  ,  (2) xt ∈   µ and y ∈   N implies (y + x −   y ) t   ϵ V q k    µ  ,  (3) xt ∈   µ and y ∈   N implies ( xy ) t   ∈  V q k    µ  ,  (4) z t   ∈   µ and x, y ∈   N implies (x(y + z) −   xy ) t   ∈  V q k    µ  . A fuzzy subset µ with conditions (1), (2) and (3) is called an ( ϵ , ϵ V q k  ) − fuzzy right ideal of N. If µ satisfies (1), (2) and (4), then it is called an ( ϵ , ϵ V q k  ) − fuzzy left ideal of N.   Note 3.11:   Let µ be a fuzzy subset of N. T hen:  (1) (a) xr , y t   ϵ   µ implies (x −   y ) mi n( r ,   t) ∈  V q k   µ and  (b) µ(x −  y) ≥   min(µ(x), µ(y), 21  k  −  ) for all x, y ∈ N are e quiv a l ent .  (2) (c) xt ∈   µ and y ∈   N implies (y + x −   y ) t   ϵ V q k   µ and  (d) µ(y + x −  y) ≥   min { µ  ( x ) ,   21 k  −   } for all x, y ∈   N are equival ent .  (3) (e) xt ϵ   µ and y ϵ   N implies ( x y ) t   ϵ V q k   µ and   ( f   ) µ(xy) ≥   min { µ  ( x ) ,   21  k  −   } for all x, y ϵ   N are equiva l ent.  (4) (g) z t   ∈   µ and x, y ∈ N implies (x(y + z) −   xy ) t   ∈  V q k   µ and  (h) µ(x(y + z) −  xy) ≥   min { µ  ( z ) ,   21 k  − } for all x, y, z ∈   N are equiva l ent.   Cor. 3.12:   [[15], Theorem 3.8.] A fuzzy subset µ of N is an ( ∈ , ∈  V q ) − fu zzy ideal of N if and only if for all x, y, z ∈   N :  (1) µ(x −  y) ≥   min { µ  ( x ) ,  µ(y), 0 . 5 } ,  (2) µ(y + x −  y) ≥   min { µ  ( x ) ,   0 . 5 } , (3) µ(xy) ≥   min { µ  ( x ) ,   0 . 5 } ,  (4) µ(x(y + z) −  xy) ≥   min { µ  ( z ) ,   0 . 5 } .   Pr o of  :   The   re su lt   follo ws f  rom  L e mm a  3.11, i f    we   tak  e  k =  0. Cor. 3.13: [[4] BHDF, Theorem 3.5.] µ is an ( ∈ , ∈   V q ) − fu zzy  ideal of field  R if and only if: (1) µ(x −  y) ≥   min { µ  ( x ) ,  µ(y), 0 . 5 } ,  (2) µ(x y), µ(y x) ≥   min { µ  ( x ) ,   0 . 5 } ,  for all x, y ∈   R .   Remark 3.14: A fuzzy ideal and an ( ∈ , ∈ V q ) −  fuzzy ideal of N are ( ∈ , ∈ V q k  ) −fuzzy  ideals of N. However, as the following example shows, the converse is not  necessarily tr ue. Example 3.15:   Consider the near-field (N, +, • )  as defined in Example 2.17. De - fine a fuzzy s u bse t  µ: N →  [0, 1] by µ(0) = 0.42, µ(a) = µ(c) = 0.4, µ(b) = 0 . 44. Then µ is an ( ∈ , ∈  V q 0 . 2 ) − fuzzy ideal of N. But, since µ(b0) = µ(0) 6 ≥   µ  ( b )   and µ(b0) = µ(0) 6 ≥   min { µ  ( b ) ,   0 . 5 } ,  µ is neither a fuzzy ideal nor an ( ∈ , ∈  V q ) − fu z zy ideal of N .  
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