Vietnam Journal of Mathematics 40:1(2012) 95-105 V i e t n a m j l o u r n a l o f M A T H E M A T I C S
©VAST
Intuitionistic Q-Fuzzy Ideals of Near-Rings
J a n a r d a n D . Y a d a v ' a n d Y a s h a s h r e e S. P a w a r ^ ' 5 . G M. College. Kuiad, Maharashtra, India
^Department of Malhrmotirs, Shwaji University, Kolhapur, Maharashtra, India
Received July 30, 2011 Revised October 23, 2011
Abstract. In this paper we introduce and study the concept of intultionistic Q-fuzzy ideals of near-rings. Also we derive some properties of intultionistic Q-fuzzy ideals of near-rings.
2000 M a t h e m a t i c s S u b j e c t Classification. 03F055. 03E72, 16Y30.
Key words. Fuzzy set. intultionistic fuzzy set, Q-hizzy bet. intultionistic Q-fuzzy set, fuzzy idecJ over Q, intultionistic Q-fuzzy ideal, near-ring.
1. I n t r o d u c t i o n
Near-Ring is a generalized s t r u c t u r e of a ring. The theory of fuzzy sets was intro- duced by Zadeh [16], T h e fuzzy set theory has been developed in many directions by the research scholars. Rosenfeld [15] first introduced the fuzzification of the algebraic structures a n d defined fuzzy subgroups.
The intuitionistic fuzzy sets (IFSs) are substantial extensions of the ordinary fuzzy sets. IFSs are objects having degrees of membership and non-membership such t h a t their sum is less t h a n or equal to 1, T h e most important property of IFSs not shared by t h e fuzzy sets is t h a t model-like operators can be defined over IFSs. T h e IFSs have essentially higher describing possibilities t h a n fuzzy sets
The notion of intuitionistic fuzzy sets was introduced by Atanasov [3] as a generalization of t h e notion of fuzzy sets. Biswas [4] applied the concept of
96 J. D. Yadav, Y. S. Pawar mluili(mi,slic fuzzy sets lo t h e theory of groups a n d studied intuitionistic fuzzy subgroups of a group. T h e riolimi of an intuitionistic fuzzy ideal of a near- ring is given by J u n , Kim a n d Yon [10], Also C h o a n d J u n [5] introduced the notion of intuitionistic fuzzy /^-subgroups in a near-ring a n d related properties are investigated, Kjizaiin, Yainak a n d Yilmaz [11] introduced intuitionistic Q- fuzzy /?-subgroupK of near-rings, Kim [13] introduced sensible fuzzy fi-subgroups of near-rings with res|KTl to fl-iiorm. Kim [12] introduced ituitionistic Q-fuzzy
^('iiii])riirK' ideals of semigroups. Davvaz and Diidek [6] defined intuitionistic fuzzy /7,,-i<l('iils of a ring and derived their properties. Davvaz, Dudek a n d J u n [7]
introduced ml nil ionisl ic fuzzification of //..-submodules in a ff^-module. In this paper wc introduce I lie notion of intuitionistic Q-fuzzy ideals of near-rings and iiivcsligalc some related properties.
^. D e f i n i t i o n s a n d p r e l i m i n a r i e s
\\V recall some definitions for the sake of completeness. T h r o u g h o u t this paper Q denotes any non-empty set and R is a. near-ring.
D e f i n i t i o n 2 . 1 . [9. II] By a near-ring we mean a non-empty set R with two binary operations " -|- " and " • " satisfying the following axioms:
(i) {R,+) IS a group;
(ii) {R, •) is a semi-group;
(iii) X • {y -\- z) = x • y -\- X • z ioT aW X, y. z £ R.
Precisely speaking, it is a left near-ring because it satisfies t h e left distributive law. We will use the word "near-ring" instead of '"left near-ring". We denote xy instead of i • y. Nole t h a t l O = 0 a n d x{-y) = -xy. b u t Oz 7^ 0 for x.y^R.
D e f i n i t i o n 2 . 2 . [1, 2, 8] An ideal / of a near-ring R is a subset of R such t h a t (i) ( / , + ) is a normal subgroup of {R, +);
(ii) RI C / :
(iii) (r + i)s-rs£ I for a l i i G / a n d r,s & R.
Note t h a t if / satisfies (i) a n d (ii) then it is called a left ideal of R.
If / satisfies (i) and (iii) then it is called a right ideal of R.
D e f i n i t i o n 2 . 3 . [U] A function n : R x Q-^ [0,1] is called a Q-fuzzy set.
D e f i n i t i o n 2 . 4 . [11] Let fi he a. Q-ftizzy set in fi a n d ( € [0,1], t h e n an upper Mevel cut of /j. is defined by
(/(W t) = {xeR\ ii{x, q)>t,qe Q}.
Intuitionistic Q-fuzzy ideals of near-nngs 97 D e f i n i t i o n 2 . 5 . [11] Let /i b e a Q-fuzzy set in H and t G [0,1], then a lower
t-level cut of fi is defined by
L(ir.t) = {x & R \ H{x.q) < t,q € Q}.
Definition 2 . 6 . [11] A n intuitionistic Q-fuzzy set A is an object having the form A = {{{x,q),i>x{.,•.q),\.^{x,q)) \x€R,q&Q}, where the functions//,4 : i ? x Q - > [0,1] and A 1 : RxQ -^ [0,1] denote the degree of membership a n d the degree of non niemliersliip of each element {x,q) G RxQ to the set A, respect n'cly, such t h a t
0<fiA{x.q)-i-XA{x,q)) < 1 for all J: G fi,q G Q.
Definition 2 . 7 . [11] A Q-fuzzv set /i is called a fuzzy fi-subnear-ring of R over Qif
(i) ii{x -y,q)> IIA{J^-q) A / i ^ ( y , q ) , (ii) ti{xy,q) > HA{X. q) A p,A{y, q) for all x,y € R and q £ Q.
D e f i n i t i o n 2 . 8 . [11] A Q-fuzzy set fi is called a fuzzy ii-subgroup of R over Q if
(i) p{x -y,q)> pLA{x,q) A HA{y, q), (;i\)p{rx,q)>iiA{x,q),
{m) p.{xr,q) > ij.A{x,q) for all I , y, r G i? a n d q e Q.
Definition 2 . 9 . [11] A Q-fuzzy set // is called a fuzzy ideal of R over Q if (i) p{x -y,q)> ^iA(x, q) A fiAiy, q),
{u) ii{rx,q) >HA{x,q), (iii) it{{x + i)y - xy) > ;x^(i, q) for all x,y,i e R and q £ Q.
3. I n t u i t i o n i s t i c Q - f u z z y i d e a l s o f n e a r - r i n g s
Now we introduce t h e intuitionistic Q-fuzzy ideals of near-rings as follows.
Definition 3 . 1 . An I Q F S A = (//>i, A^) of a near-ring R is called an intuition- istic Q-fiizzy subnear-ring of R if
'IS ,/ u. Yadav, Y. S. Pawar [i) li.\{x- !l.q) >l'A{r,<l}AlJ,A{y.']) '"id A ^ ( , T - ? / , g ) < A^(3:, 9) V A^(y,.7), (ii) ii.\(ni.q) > ii,\{.i.q)^ii.\i'h'l) an<l A^(j:y,f/) < A ^ ( x , q ) V AA(y,g) for all .r, // G /? and r/ G Q.
D e f i n i t i o n 3 . 2 . An IC^FS .1 = (/'/i,A/\) in a near-ring Ii is called an intuition- islic Q-fuzzy ideal of R if
(i) ihxlr -V,q) >/i.-\{i-.q)/\iiA{v,q) and A ^ ( x - y , 9 ) < AA(2r,g) V A>i(y,g), (ii)// \{g + x-y,q) = iiA{-'.q} »n<[ A^(y-l-X - ly. g) = A ^ ( x . g ) , (iii) /(.,(7-.r,f/) > //,^(,7:.(7) and XA{i.i.q) < A ^ ( x , g ) ,
(iv) /v ,((.r \' i)g - .ig.ii) >ih\{i.q) and XA{(X + i)y - xy,q)) < \A{i^q) for all .r, //. / e /? and q G Q.
If A = (//^,A^) siilishcs (i), (li) and (iii) then A is called an intuitionistic Q- fuzzy loft ideal of fi a n d H A = (//^, A^) satisfies (i). (ii) a n d (iv) then A is called an intuitionistic Q-fuzzy right ideal of fi.
E x a m p l e 3 . 3 . Let R = {a,b,c,d} b e a non-empty set with two binary opera- tions '•-!-" a n d "•" defined as follows:
- { - a b e d a b e d a a b c d a a a a a b b a d c b a a a a c c d b a c a a a a d d r a b d a a b b T h e n (fi,-l-.-) is a near-ring.
Define an intuitionistic Q-fuzzy ,scl. .4 = (/(4.A.1) in fi as follows fiA{(i,q) = 1, PA{b,q) = 1/3, ^lA{c,q) ^ 0, / i ^ ( d , q ) :^ 0, XA{a,q) = 0 . XA{b,q) = 1/3. A^(c.q) = 1. A,,((/.g) = 1 for all 9 € Q.
Then clearly A = (/x^. A..,) is an intuitionistic Q-fuzzy ideal of a near-ring R.
Basic properties of an intuitionistic Q-fuzzy ideal of R are proved in the following theorem.
T h e o r e m 3 . 4 . If A = (///j. A \) is an intuitionistic Q-fuzzy ideal of R then (i) p.A{0,q) >HA{x,q) and Ayi(0,g) <\A{x,q),
{ii) II.\{-x.q) - / ' . i ( . r , g ) andXA{-x,q) = A ^ ( x , g ) , (iii) P.A{X -y,q)> M/5(0,9) =» / / ^ ( x , q) = fiAiy, 9), (iv) A ^ ( i - y , 9 ) < A^(0,g) => XA{x,g) = XA{y,q) for all x,y € R and q £Q.
Intultionistic Q-fuzzy ideals of near-nngs ()9 Proof Let A = {iiA,X.\) be an intuitionistic Q-fuzzy ideal of R.
(i) liA{0,q) =fiA{x-x,q) > / / , i ( , r . 9 ) A ,/.,( r, y).
Therefore / i ^ ( 0 , 9 ) > /i/i(T,9) for all x G R and (/ G Q.
Similarly XA{0,q) < XA{x,q) for all a: G fi and q € Q (ii) |U,i(-a;,9) = / / ^ ( 0 - x . 9 ) > / M ( 0 , ( / ) A/i .1 (.;•,</) = / ( 4 ( a : . 9 ) .
Now/i^(x,9) = liA{-(-x).q) > in{-x,q) > fi \{r.q). Thus wc get/i>i(-3:,9) = pA{x,q) for all X G fi, 9 G Q.
Similarly A ^ ( - j , 9 ) = A.4(.r.q) for all x e R,q eQ.
(ill) \ V c h a v e / J A ( 3 - - y , q ) > t'.A{0.q). But ^.4(0,9) >^A{x~y,q). Thus PA{X- y.q) = liA{0.q).
Now consider
/i.4(x.9) = / M ( j - - y + y,9)
^ / ' . - i ( ( . i " - y ) + y,9)
> ^ A ( a : - y , 9 ) A / i ^ ( y , 9 ) - ^ A ( 0 , 9 ) A / i ^ ( y , 9 ) - / i A ( y , 9 ) -
Shnilarly we can prove t h a t fiA{y,q) > M.A(-1".9)- Hence fiA{x,q) — iJ.A{y,q) for
all I , y G fi. 9 e Q. • A necessary and sufficient condition for x — (x/>Xj) to be an intuitionistic
Q-fuzzy ideal of fi is given m t h e following theorem.
T h e o r e m 3 . 5 . Let R be a near-nng and xi be a charactenstic function of a subset I of fi. Then x = ( x / i X/) " a " intuitionistic Q-fuzzy ideal of R if and only if I IS an ideal of R.
Proof Let 7 be a right ideal of R.
(1) Let 9 G Q and x,y € R. T h e n we have the following three cases.
Case 1. Let x,y € I. T h e n xii^^Q) = l . X / ( y - 9 ) = l , X / ( a : - y , 9 ) = 1 and X'i{x,q) = 0 , x ; ( y , 9 ) = 0,xK^-y'l) = 0 . Therefore ^ / ( ^ ^ - y , 9 ) > X/(3^.9) ^ X/(y,9) and x^iix - y.q) < XJix,q)'^ X'jiy^Q) for all i , y G fi and 9 G Q.
Similarly we can easily obtain the following two cases Case 2. Let x ^ I a n d y ^ I.
Case 3. Let i . y ^ / .
T h u s i n a n y c a s e w e h a v e x / ( 3 : - y , 9 ) >Xi{Xiq)^Xi{y^Q) andx'iix-y.q) <
X%x,9) V X / ( y , 9 } for all x,y € R a n d 9 € Q.
(2) Case 1. Let y e fi and x€ I.
Sincey-|-3:-y G 7 for a l l y G fi and a; G / , T h e r e f o r e x / ( y + a ^ - y ) = 1 = X/(^).
i-e. X/(y + x-y,q) = l = Xi{x,l) a n d xliv + x-y,q) = 0 = x'i{x,q)-
100 J. D. Yadav, Y. S. Pawar fVf.vr 2. Let ,(/ G R and x ^ I. T h e n v/('/ + x - y,q) = 0 = Xiix,q) a n d
\/(,'/ + -'"-.'/.</} -- I ^ \y(,c,(/). H c i i c e i n a n y c a s e w e g e t ) i : / ( y + 3 : - y , 9 ) = X/{3:>q) and \){!/ + x y.q) = \'i{x,q) for all x,y e R and q &Q.
(3) Lei . r , / / € fi, ("H,S( / Let / G / .
SiiiiT / is a right ideal of K. (,i I- ?)y - xy £ I for all x,y £ R a n d i G / . TiieicfoH' \ , ( [ c -h /)// - .•7:y,9) = \,xii'-'l} = 1 a n d X ; ( ( ^ + *)y " ^ ^ - 9 ) = 0, \ / ( ' . g ) = '), T h u s \/((,( I i)y - xg.q) ^ \iii.q) a n d ^ / ( ( a : + i)y - xy,q) <
\'l{i.q) for all x.y.i € R.q € Q.
Co.'^r 2. Let ; ^ / , Then X / ( ( ^ + O/V - ^ y . ? ) = X / ( ^ 9 ) = 0 a n d X;((a; + i ) y -
•Ul.'l) = \;{'-'l)= I
ii<'i"<' \ / ( ( . ' ( ')y-•'•,'/.9) > \ / ( i , q ) a n d x / ( ( i + 0 y - 3 ; y , g ) < \',ii.q) for all .(•.i/,t £ a (I E Q. T h u s x = ( x / t X y ) i" an intuitionistic Q-fuzzy right ideal of fi.
Similarly if / is a lefl ideal of R then we can easily prove t h a t x = ( x / ' X / ) is an intuitionistic Q-fuzzy left ideal of R.
Conversely let x = ( \ ; , \',) be an intuitionistic Q-fuzzy ideal in R. Then we shall prove t h a t I is an ideal of R.
(i) Since \ = ( \ / , \ ; ) is an intuitionistic Q-fuzzy ideal of R. Therefore xii.^ ~ y,Q) > \l{-r.<l) A \ / ( y , 9 ) and \'',{x -y,q) < x'j{x,q)V \'/{y.q} for all J . y 6 fi and q & Q.
Ux,y € I then \ / ( , r , 9 ) A xiiy.q) = 1 implies t h a t \ / ( j - - y , 9 ) = 1. Therefore x,y £ I ^ X - y e I. Hence (/, + ) is a subgroup of (fi.-t-).
(ii) Since X/(y + 3: - y , 9 ) = X/(^>9) a n d \ ' ; ( y + x -y,q) = x'jix.q). therefore clearly if x G / a n d y G fi t h e n X/(y + -i" - y . q ) = Xi{x,q) = 1 and \'j{y + x - y^ Q) = X;(x. q) = 0 implies t h a t y -I- J- - y € 7 for all i G / a n d y € R.
Hence (7, -h) is a normal subgroup of {R, -h).
(in) Since xi{rx,q) > Xii^^l) and \/(''-''-9) ^ xUx.q), therefore if i G / then Xii''x,q) > Xi{x,q) = 1 =* Xiirx,q) = 1 => r x G 7 for all r G R,x G 7.
Similarly if x ^ I then rx ^ I.
(iv) Since \,{{x -\- i)y - xy,q) > \,{i.q) a n d \){{x -\- i)y - xy,q) < x'iihq), therefore if j ' e / then
Xi{{x + i)y-.iy.q) > \i{i.q) - 1
=*• Xi{{x + i)y-xy,q) = 1
=^ {x + i)y- xye I for all x, y G fi a n d i e 7.
Similarly we can prove t h a t if z ^ 7 then {x + i)y- xy ^ I for all x, y e 7.
Heiicf 7 IS an ideal of R. g
For intersection of any number of intuitionistic Q - f u z z y ideals of R, we have
Intuitionistic Q-fuzzy ideals of near-nngs 10, T h e o r e m 3 . 6 . / / { A , [ i e / } is a family of intuitionistic Q-fuzzy tdeaU of R then
r\{A.\i e I) IS an intuitionistic Q-fuzzy ideal of R where I is an index set.
Proof. Let x.y.a.r e R a n d q eQ. Then (1) {nitA,)(x-»,?) = A { M / I , ( I - y , g ) \ i e l }
> (.A{llA,(x,q) \iSl))A{A{,n,(y,g) | i e / ) )
= {l^ifA.lx, q)\ieI])A {r\{nA,(y, 9) I i E / } ) , {U\Ai){x-y,q) = V{\A,{x-y,q) | i 6 / )
< (v{A.4,(x,g) I i 6 / ) ) v{v{A4,(!/,<;) I > e / } )
= {u{>.A:{.r.g) I i e /}) V iu{\A,{y,q) I . e /)), (2) (nMA.lCy + i - ! / , i ; ) = A{(ij,{!/ + . r - ! / , ? ) | t e / }
= A{/iA,(i,g) I ! € /}
= n{>ij,(i,9) l i e / } , (UA^i)to + 1 - !/, g) = v{Aj,(!, + 1 - !/, g) I i e /}
= V{Aj,(i,g) l i e / }
= U{A^,(i,5) l i e / } , (3) (n»<j,)(ri,5) = A{iiA,(rx,q) 1 . 6 / }
>A{^x,(i,?) l i e / )
= n{,i^,(i,5) | i € / ) , (UAj,)(rx,?) = V{A^,(ri,9) | i e / }
<v{\A,{x,g) l i e / }
= U{Aj,{i,g) l i e / } ,
(•*) (l''liAi)((x-l-a)y-xy,q) = /\{^IA,^^x-^a)y-xy,q) | i 6 / }
> A { M J , ( I , « ) I J E / }
= n{fi^,(i,g) I i € / } ,
(UAAi){(i + a)y - xy, g) = V{AA,((X + a)y -xy,q)\ie /}
< V { A A , ( I , « ) | i € / }
= UlXA,(x,q) l i e / )
for all x, y, a, 7- e /? and g €Q.
Hence n{v4, | i 6 / } is an intuitionistic Q-fuzzy ideal of R. • An intuitionistic Q-fuzzy ideal A oi R induces another intuitionistic Q-fuzzy
ideal of R is shown in the following theorem.
Theorem 3.7. If A = (jtA, ^A) IS on intuitionistic Q-fuzzy ideal of R then
(i) oA — {tiA,f'\) is also an intuitionistic Q-fuzzy ideal of R;
(ii) *A = (A^, A.4) is also an intuitionistic Q-fuzzy ideal m R.
Proof. Let x , y , a , r e / i and g € Q.
102 J. D. Yadav, Y. S. Pawar (i) (1) II'A{J -»,<;) = i - / ' / i ( j ' --fi,g)< 1 -UiA{x,g)'\iiA{y,q))-
= { l - / M ( j , ' i ) ) V ( l - / M ( ! / , ? ) ) = M ' A ( x , ' 7 ) V r t ( ! / . 9 ) . [2] li',i[!l + x- II.q] ^ I - liA(y + x-y,q) = 1 - llA{x,q) = r t ( x , 9 ) - (:l) i,\(rx.q) = 1 - ,.,,(r,r,./) < 1 - tiA(r.q) = rt(x,g).
(4) ^ ( ( . r I a)ii .rg.g) ^ I -/IAUX •{-a)y-xy,q) < 1 - , 1 4 ( 0 , j ) = Ai'^Cu.g).
(ii) (I) A',,{r- ,;/,5) = l - A ^ ( , r », ?) > 1 - ( A ^ ( x , g ) V A.4(»,g)).
= (1 A,,(.i.(;)l A (1 - XA('l.g)) = A:,(x,g) A A^(!;,g).
(^1 X'A(V+ •'' - »-l) = ' - >'A(y + X - y,g) = 1 - A j ( x , g ) = A^(T.g).
(••1) A:,,(r,r,,) - 1 - A A ( r i , , ) > 1 - A ^ ( i , g ) = A5,(x,g).
(4) A',,((.i ) " ) , ' / - .'•;';.«)= 1 - A : i ( { x - K i ) ! ; - i ! ; , , ) > 1 - A^Ca.g) = A ^ ( a , g ) . llenre <i.\ - (/i/i,/i^) a n d • 1 = ( A ^ , A ^ ) are intuitionistic Q-fuzzy ideals of
R. m
Kioiii t h e above theorem we olitain the following
T h e o r e m 3 . 8 . .1 = {I^A.^A) IS on intuitionistic Q-fuzzy ideal of R if and only if oA mid • 1 ajT intultionistic Q-fuzzy ideals of R.
For a given A = {HA, ^A), we define
RtiA = {xeR\ HA{x,q) = (</l(0,rj)) a n d /lA^ = {x € /J I X.iix.q) = A A ( 0 , ? ) ) . We prove an i m p o r t a n t p r o p e r t y of R^IA and R\A in t h e following theorem.
T h e o r e m 3 . 9 . / / an IQFS .1 = {pA, ^A) IS an intuitionistic Q-fuzzy ideal of R then the sets RjiA and RXA are ideals of R for all q E Q.
Proof. Let .4 = ((ZA, XA) b e an intuitionistic Q-fuzzy ideal of R. Let x.y e R/IA a n d g e Q . T h e n in(x.q) = ltA(0,g),ltA{y,q) = / i . i ( 0 , g ) .
(i) Now 0 e R therefore 0 G RfiA. Hence R/IA is a n o n - e m p t y subset of R.
(ii) Since A = {fiA,XA) is an intuitionistic Q-fuzzy ideal of R, C/i(x - .V. g) > / u ( j - , q) A iiAiy, g) = ( i j ( 0 , g).
But/1.4(0,9) > / i . 4 ( x - y , 9). Therefore M A ( i - y , g ) = / I A ( 0 , 9 ) . T h u s x - y € R/IA for all x,y € R. Hence (RtiA,-^) is a s u b g r o u p of {R,-h).
(iii) Again since A = [IIA,XA) IS an intuitionistic Q-fuzzy ideal of R, fiAiy-hx- y.q) =^tA{x,q) = ( 1 ^ ( 0 , ? ) .
Therefore J - l - x - y e RIIA for all x,y e R. Hence (/J/i^, + ) is a n o r m a l subgroup of ( / ? , + ) .
(iv) Since f<A(rx,9) > /iA(x,g) for all x , r 6 /J a n d 9 e Q, therefore iiAlrx,q) >
C A ( X , 9 ) . B u t iiA{0,q) > IZA{rx,q). Hence iiA(rx,q) = f i x ( 0 , 9 ) implies t h a t rx e /i/i>i for all r e / ? , x e /?/i.4, i.e., i?/i/x.4 C Rfz^.
Intuitionistic Q-fuzzy ideals of near-nngs 103 Hence RfiA is a left ideal of R.
( v ) N e x t ^ > i ( ( u - | - i ) u - u u , 9 ) > fiA{i,q) = ^ 4 ( 0 . 9 } for all u,w e fi,i G fi/i^. But PA{Q,Q)>l^A{{u + i)v-vr.q) T h e r e f o r e / ; . i ( ( i ( - I - ? ) ' ' - ('",9) = / t ^ ( 0 , 9 ) for all u,v G R,i G R/iA- Hence (it + i)v - iic G R[IA for all u,v € R and v S fi/i^.
Thus fi//>i is a right ideal of R. B Theorem 3 . 1 0 . If an IQFS A = (IIA.XA) is an niliuliom.slir Q-fuzzy ideal
of R then the sets t / ( ^ ^ , ( ) and L(A^.f) are ideals of R for all q ^ Q,t e
7^(w)n7„(A^).
Proof Lei 9 G Q and t € I,n{fiA) C\ 7,„(A^), Wc luw'e U{ti.t) =^ {x € R \ iiA{x.q) > t,q € Q}, L{fi,t)^{xeR I XA{x,q) <t.q€Q}.
(i) Let x,y € U{ii,t). T h e n HA(-i'-q) > t and HA{y,q) > t. Since P,A{X -y,q) >
PA{x.q) A / i ^ ( y , 9 ) > t. therefore x - y G U{fi,t) for all x . y G U{p,t). Hence lV[p,,t).-\-) is a subgroup of (fi, -I-).
(ii) Let I G U{fi,t) and y ^ R. Again since /i/i(y + x - y,q) = HA{x,q) > t, therefore y - 1 - i — y G U{f.i. t) for all x G (7(^,() and y ^ R Hence (t/(/(, t ) , + ) is a normal subgroup of (fi. + ) .
(iii) Let X G U{p.,t) a n d r G fi. Then ^ ^ ( r x . 9 ) > A*/t(a:,9) > t implies rx G (/(//,() for aU I G f/(/i,t) and r G 71, i.e., RU{ti,t) C t l ( / x , t ) . Hence U{ii,t) is a left ideal of R.
(iv) Let i e E7(/z,f) and x , y G R. Then /(^((j: + i)y - 3:y,9) > M/i(*i9) > *•
Therefore ( i - i - i ) y - x y G t / ( / i , 0 for all i G C/j^.f) and x . y e fi. Hence C/(fi,i)
is a right ideal of R. • As in Theorem 3.10 we have
Theorem 3 . 1 1 . If A = {IIA,XA) IS an intuitionistic Q-fuzzy subset of R such that all non-empty level sets U{iiA,t) and L{XA,t) are ideals of R then A ^ {I^A,XA) IS an intuitionistic Q-fuzzy ideal of R.
Proof Suppose A = {(IAAA) is an intuitionistic Q-fuzzy set of R such t h a t all non-empty level sets U{p.A,t) a n d L{XA,t) are ideals of R.
{i) Let to = HA{x,q) A HA{yyq) a n d i i = A A ( X , 9 ) V A,l(y,9) for x . y G R,q&Q.
Then x,y G f7(^A,*o) also x , y G L{^AM)- Therefore x - y G t7(/t^,io) and s - y € L ( / i ^ , f i ) . T h u s / i ^ ( x - y , 9) > to = M ^ ( x , 9 ) A ^ ^ ( y , 9 ) a n d / i A ( x - y , 9 ) <
f i = / i v i ( x , 9 ) V / / ^ ( y , 9 ) .
(ii) Let to = iiA{x,q) and ti = A ^ ( x , 9 ) . T h e n x G U{IXAM) also x € I(A>i,ti) and y G 7?.
Since U{pA,t) a n d L{XA,t) are ideals of fi, they are normal. Therefore y+x-y G
^[p-A.ta) and y - l - x - y G L ( A ^ , f i ) for all x G t / ( / i ^ , t o ) - Now X G L ( A ^ , t i ) a n d y G fi
104 J. D. Yadav, Y. S. Pawar
=S'/M(l7 + . ' ' - y , 9 ) > / M ( X . 9 ) and A/i(y-|-x - y . 9 ) < A ^ ( x , 9 ) . (iii) I,cl I2 - / / 4 ( ' ' . ' / ) a i K U 3 = : A ^ ( x . 9 ) f o r . r 6 fi and 9 € Q. T h e n x € (7(/zyi,t2) and ,(•€ L{XA.I\)
Since U{nA,t2) find L{XA,U\) are loft ideals of R,TX G U{{iA,t2) a n d rx G L{XA.I.'.) for r € fi. Hence / ' , i ( ' ' f , 9 ) > '2 = PA{x,q) a n d A.4(rx,g) < (3 = A i ( ' - , 9 )
(iv) Let (2 =tiA{i,Q) and ty = A A ( * , 9 ) Tor X G fi a n d 9 G Q , T h e n i G t/(/i^,^2}
and i G L{XA, I:\).
Since V(11 A.I) is a right ideal of fi,
and
(,r -I /)y - xy G t / ( / M , ( 2 ) for all .r, y € R a n d i G t/(/iA,t2)
(X -f i)y - xy e L(A^.(3) for all x,y € R a n d i G L ( A ^ , i 3 ) . T h e r e f o r e / j A ( ( . r - I - i ) y - x y , 9) > (2 = / i v i ( i . 9 ) and
A^((x -I- i)y — xy,q) < t^ = XA{i,q) for all x.y,i 6 R a n d g G Q.
Hence A = {fiA, XA) is an intuitionistic Q-fuzzy ideal of R. •
4. C o n c l u s i o n
Near-ring theory has m a n y appUcations in t h e s t u d y of p e r m u t a t i o n groups, block schemes a n d projective geometry Near-rings provide a non-linear ana- logue to the development of Linear Algebra, combinatorial problems a n d useful for agricultural experiments. In this p a p e r we have presented t h e notion of intu- itionistic Q-fuzzy ideals of near-rings and derived t h e properties of these ideals.
A c k n o w l e d g e m e n t . T h e a u t h o r s are greateful to t h e referees for their con- structive comments for improving the p a p e r .
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