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Fuzzy relations on rings and groups
Fuzzy Sets and Systems 43 (1991) 117-123 North-Holland
117
Fuzzy relations on rings and groups D.S. Malik and John N. Mordeson Department of Mathematics and Computer Science, Creighton University, Omaha, NE 68178, USA Received February 1989 Revised May 1989
Abstract: Let S be any nonempty set. A fuzzy relation on S is a fuzzy subset of S × S. In this paper we study fuzzy relations on rings and groups. In particular we show that if/z and o are fuzzy left (right)
ideals of a ring R, then/~ x e is a fuzzy left (right) ideal of R x R and conversely if/* x o is a fuzzy left (right) ideal of R × R, then either/~ or o is a fuzzy left (right) ideal of R. An example is given to show that if/* x o is a fuzzy left (right) ideal of R x R, then /~ and a both need not be fuzzy left (right) ideals of R. An example is also given to show that if/~ is a fuzzy left (right) ideal of R x R, then o~,. the weakest fuzzy subset of R on which/~ is a fuzzy relation, need not be a fuzzy left (right) ideal of R. We obtain similar results for groups. We also show that certain results of Bhattacharya and Mukherjee (1985) are not true.
Keywords: Fuzzy relation; strongest fuzzy relation; weakest fuzzy subset; Cartesian product; fuzzy subgroup; fuzzy ideal; group; ring.
1. Introduction T h e notion of a fuzzy subset was i n t r o d u c e d by Z a d e h [14]. R o s e n f e l d [12] i n t r o d u c e d the c o n c e p t o f a fuzzy s u b g r o u p of a g r o u p and established m a n y i m p o r t a n t properties. F u z z y s u b g r o u p s have b e e n studied by several authors; some examples are [ 1 - 5 , 10]. T h e n o t i o n of a fuzzy ideal of a ring was i n t r o d u c e d by Liu [6]. F u z z y ideals have b e e n studied further by Malik [7], Malik and M o r d e s o n [8, 9] and M u k h e r j e e and Sen [11]. T h e c o n c e p t o f a fuzzy relation o n a set was i n t r o d u c e d by Z a d e h [14, 15]. F u z z y relations on a g r o u p have b e e n studied by B h a t t a c h a r y a and M u k h e r j e e [4]. In this p a p e r we study fuzzy relations on a ring. W e p r o v e that if/2 and o are fuzzy left (right) ideals o f a ring R, t h e n / 2 × o is a fuzzy left (right) ideal o f R × R ( T h e o r e m 3.1(i)). C o n v e r s e l y we show that if/2 × o is a fuzzy left (right) ideal o f R x R, then e i t h e r / 2 or o is a fuzzy left (right) ideal o f R ( T h e o r e m 3.2). W e give an e x a m p l e to s h o w that if /2 x o is a fuzzy left (right) ideal o f R x R, t h e n / 2 and o b o t h n e e d n o t be fuzzy left (right) ideals o f R ( E x a m p l e 3.5). R e m a r k 3.6 shows that the c o n v e r s e o f Corollary 3.3 in [4] is not true in general. W e give an e x a m p l e to show that if/2 is a fuzzy left (right) ideal o f R x R and o~, the w e a k e s t fuzzy subset o f R o n w h i c h / 2 is a fuzzy relation, then o r n e e d n o t be a fuzzy left (right) ideal o f R ( E x a m p l e 3.14). W e also give a similar e x a m p l e for g r o u p s ( E x a m p l e 3.15). This E x a m p l e 3.15 shows that T h e o r e m 4.7 in [4] is n o t true in general. R e m a r k 3.13 shows that L e m m a s 4.5 and 4.6 in [4] are not true. 0165-0114/91/$03.50 I~) 1991--Elsevier Science Publishers B.V. All rights reserved
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2. Definitions, notations and preliminaries In this section we review some definitions and results for the sake of completeness. Throughout the paper S will denote a nonempty set, G will denote a group with identity element denoted by e, and R will denote a ring (not necessarily with unity). Definition 2.1. A f u z z y subset of S is a function/~ : S---> [0, 1]. Definition 2.2. A fuzzy subset/~ of G is called a f u z z y subgroup of G if (i) Iz(xy) >>-min(/t(x), #(y)),
(ii) p(x -~) =/z(x), for all x, y • G. Definition 2.3. A fuzzy subset/z of R is called a f u z z y left (right) ideal of R if (i) #(x - y) 1> min(~u(x), ~(y)),
(ii) I~(xy) >I t~(y) (t~(xy) >t It(x)), for all x, y • R. A fuzzy subset # of R is called a f u z z y ideal of R if/~ is a fuzzy left and fuzzy right ideal of R. Definition 2.4. If/~ is a fuzzy subset of S, then for any t • [0, 1] the set /ut = {x • S [/z(x) i> t} is called a level subset of S with respect to/~. [,emma 2.5. Let tz be a f u z z y subset of G. Then (i) if l~ is a f u z z y subgroup o f G, then Iz(e) ~ It(x) for all x • G and Itt is a subgroup of G for all t • [0, p(e)]; (ii) if Izt is a subgroup of G for all t • Ira(p), then I~ is a f u z z y subgroup of G. [,emma 2.6. Let I~ be a f u z z y subset o f R. Then (i) if Iz is a f u z z y left (right) ideal of R, then/t(0)/> ~(x) for all x • R and ~t iS a left (right) ideal o f R for all t • [0,/t(0)]; (ii) if I~t is a left (right) ideal o f R for all t • Im(p), then I~ is a f u z z y left (right) ideal of R. Definition 2.7. A f u z z y relation it on S is a fuzzy subset of S × S. Definition 2.8. Let/~ be a fuzzy relation on S and let tr be a fuzzy subset of S. Then/~ is called a f u z z y relation on o if #(x, y) ~
Vx, y e S.
Definition 2.9. Let/~ and ¢r be fuzzy subsets of S. The Cartesian product of p and o is defined by
I~ x o(x, y) = min(p(x), o ( y ) ) ,
Vx, y e S.
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Definition 2.10 [13]. L e t o be a fuzzy subset of S. T h e n the strongest f u z z y relation on S that is a fuzzy relation on o is Ito, defined by
Ito(x, y) = o × o(x, y ) = m i n ( o ( x ) , o ( y ) ) ,
Vx, y • S.
Lemma 2.11 [4]. Let Iz and o be f u z z y subsets o f S. Then (i) It x o is a f u z z y relation on S;
(ii) (it × o ) t = Itt × Or, V t • [ 0 , 1]. Definition 2.12 [13]. If # is a fuzzy relation on S, then the weakest f u z z y subset of S on which It is a fuzzy relation is o~,, defined by
o , ( x ) = sup {max(it(x, y), It(y, x))},
Vx • S.
yeS
Definition 2.13. L e t It and o be fuzzy subsets of G. T h e n It o a ( x ) = sup { m i n ( i t ( y ) , a ( z ) ) } ,
Vx • G.
X=yZ
Lemma 2.14 [6]. Let #, o and rI be f u z z y subsets of G. Then # o ( o o T / ) = (it o o ) o ,7.
If It and o are fuzzy subsets of S, then It ~< o m e a n s It(x) <~ o(x) for all x • S.
Lemma 2.15 [6]. Let It be a f u z z y subset o f G. Then It o # <~It if and only if It(xy) >I min(it(x), I t ( y ) ) ,
Vx, y • G.
It is well k n o w n in g r o u p t h e o r y that if H and K are s u b g r o u p s of G, then H K is a s u b g r o u p of G if and only if H K = KH. W e n o w give an a n a l o g o u s result for fuzzy subgroups.
Theorem 2.16. Let It and a be f u z z y subgroups o f G. Then It o o is a f u z z y subgroup o f G if and only if # o o = o o It.
Proof. O b s e r v e that Vx • G, I t o o ( x ) = ooit(x-1). If I t o o = ooit, then ( i t o o ) o (it o o) = (it o It) o ( o o o) ~< It o o. T h u s by L e m m a 2.15, It o o ( x y ) / > min(it o o(x), It o o ( y ) ) for all x, y • G. Also It o a ( x -1) = o o i t ( x ) = It o o(x). H e n c e It o o is a fuzzy s u b g r o u p of G. C o n v e r s e l y , s u p p o s e that It o a is a fuzzy s u b g r o u p of G and let x • G. T h e n It o a ( x ) = It o o ( x - l ) = o o It(x). H e n c e It o o = o o It.
3. Fuzzy relations on rings and groups Theorem 3.1. (i) Let It and o be f u z z y left (right) ideals o f R. Then It × o is a f u z z y left (right) ideal o f R x R. (ii) Let It and o be f u z z y subgroups o f G. Then It x o is a f u z z y subgroup o f G × G [4].
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Proof. (i) Now /~ x tr(0, 0) = min(/u(0), o(0)). Let t • Im(# x o). Then t ~/~(0) and o ( y ) > a(0). Then /~ x a(x, y) = min(/~(x), o ( y ) ) > min(/~(0), o(0)) =/~ × o(0, 0). This is a contradiction since/~ x o is a fuzzy left ideal of R x R. (ii) Suppose that there exists x , y • R such that /~(x)> tr(0) and a ( y ) > a(0). Then ~ x tr(0, 0) = tr(0). Now /u × tr(x, y) = min(/u(x), o ( y ) ) > o(0) = i~ x o(0, 0). This is a contradiction. Hence the result. (iii) Similar to part (ii). (iv) By part (i), either/u(x) ~ min(/~ x o(x, 0), lz x ~r(y, 0)) = min(/~(x),/~(y)). Also l~(xy) = I~ x o(xy, O) = l~ x o((x, O)(y, 0)) >! # x o ( y , O) = I~(y). Hence in this case/z is a fuzzy left ideal of R. Case 2. Let o(x)<<-a(O), Vx • R. Suppose that Case 1 does not hold. Then there exists y • R such that /z(y) > o(0). Then / t ( 0 ) / > / t ( y ) > o(0). Hence /~ x o(0, x) = min(/~(0), o ( x ) ) = o(x), V x • R. Now as in the proof of Case 1, we can show that tr is a fuzzy left ideal of R. Hence either/u or o is a fuzzy left ideal of R. Analogous to T h e o r e m 3.2, we have a similar result for groups, which can be proved in a similar manner. We state the result without proof. Theorem 3.3. Let l~ and o be f u z z y subsets o f G such that I~ x o is a f u z z y subgroup o f G x G. Then (i) either lt(x) ~ It(e), Vx e G or o ( x ) <<-o(e), V x • G;
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(ii) if it(x) <~it(e), Vx • G, then either it(x) <~ o(e), Vx ~ G or o(x) <- o(e), VxeG; (iii) if o(x) <- o(e), Vx • G, then either o(x) <~It(e), Vx • G or it(x) <- it(e), VxeG; (iv) either # or o is a f u z z y subgroup o f G. Corollary 3.4. (i) Let # and o be f u z z y subsets o f R such that # × o is a f u z z y left. (right) ideal o f R x R. I f #(0) = o(0), #(0) >>-#(x), Vx • R and o(0)/> o(x):. Vx ~ R, then # and o both are f u z z y left (right) ideals o f R. (ii) Let # and o be f u z z y subsets o f G such that # x o is a f u z z y subgroup o f G x G. I f #(e) = o(e), #(e) >i #(x), Vx ~ G and o(e) >1 o(x), Vx ~ G, then # and o both are f u z z y subgroups o f G. We now give an example to show that if # x o is a fuzzy left (right) ideal of R × R, then # and a both need not be fuzzy left (right) ideals of R. E x a m p l e 3.5. Let R be any ring with at least two elements and let 0 <~ t <~s < 1. Let # and o be fuzzy subsets of R such that # ( x ) = t, Vx e R and o ( 0 ) = s , o ( x ) = l if O=k x e R. Then # × o(x, y) = m i n ( # ( x ) , a ( y ) ) = t, Vx, y e R. Thus # x o is a fuzzy left (right) ideal of R × R, since # x o is a constant function. Now # is a fuzzy ideal of R but o is not a fuzzy left (right) ideal of R, since o(0) < o(x), V0 =/=x • R. Note that if s = t, then o ( 0 ) = #(0). R e m a r k 3.6. Let G be a group with at least two elements and let # and o be fuzzy subsets of G as defined in E x a m p l e 3.5, that is # ( x ) = t, Vx e G and a(e)=s, a(x)=l if e q : x e G , where 0 ~ < t ~ < s < l . Then I t × o is a fuzzy subgroup of G × G, but o is not a fuzzy subgroup of G. This shows that converse of Corollary 3.3 in [4] is not true in general. Note that if s = t, then o(e) = #(e). Corollary 3.7. (i) Let o be a f u z z y subset o f R. Then Ito = o x o is a f u z z y left (right) ideal of R x R if and only if o is a f u z z y left (right) ideal o f R. (ii) Let o be a f u z z y subset of G. Then Izo is a f u z z y subgroup o f G x G if and only if o is a f u z z y subgroup o f G [4]. L e m m a 3.8. (i) Let o be a f u z z y left (right) ideal of R and let # = #o. Then O~=O.
(ii) Let o be a f u z z y subgroup o f G and let # = #o. Then o r = o. T h e o r e m 3.9. (i) Let t~ be a f u z z y left (right) ideal o f R x R. Then Vx, y, z ~ R, either It(x - y, z) = # ( y , O) or #(x - y, z) = #(x, z) or #(x, z) = It(y, 0). (ii) Let I~ be a f u z z y subgroup of G × G . Then Vx, y, z e G , either # ( x y - ' , z) = # ( y , e) or # ( x y - ' , z) = #(x, z) or #(x, z) =/~(y, e). Proof. Let tt = #(x - y, z), t2 = It(y, 0) and t 3 = ~(X, Z). Using the fact that # is a
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fuzzy left ideal of R × R we deduce that t I/> min(t2, t3),
t 2/> min(t3, tl),
t3/> min(tl, t2).
The result now follows. (ii) Similar to (i).
Corollary 3.10. (i) Let # be a f u z z y left (right) ideal of R × R. Then Vb ~ R, either #(b, b) = #(b, 0), or #(b, b) = #(0, b) or #(b, O) = #(0, b). (ii) Let # be a f u z z y subgroup of G × G. Then Vb ~ G, either #(b, b) = #(b, e) or #(b, b) = #(e, b) or #(b, e) = #(e, b). Proof. (i) Let x = y = z = b in T h e o r e m 3.9(i). (ii) Let x = y = z = b in T h e o r e m 3.9(ii). We now give an example to show that the three numbers in Corollary 3.10 need not be equal. Example 3.11. Let R = 7/, the ring of integers ( G = 7/, the group of integers under addition). Let o be a fuzzy subset of R ( G ) such that o(x) = 1 if 2 Ix and o(x) = 0 if 24-x. Let # be a fuzzy relation on R ( G ) defined by # ( x , y ) = a(x), Vx, y e R (G). Then it can be easily checked that # is a fuzzy ideal (subgroup) of RxR (G×G). Let b = l . Then #(b, b) = o ( b ) -- 0, #(b, 0 ) - - a ( b ) = 0 and #(0, b) -- o(0) -- 1. H e n c e #(b, b) -- g ( b , 0) < #(0, b). If # is a fuzzy left (right) ideal of R × R, then #(0, O) >1#(x, y), Vx, y ~ R. We now give an example to show that there may exist elements x, y, z in a ring R such that #(x, z) > # ( y , 0). Example 3.12. Let R be any ring with at least four elements such that R has a proper left (right) ideal A. Let o be a fuzzy subset of R such that o(x) = 1 if x e A and o(x) = 0 if x ~ A. Then it can be easily checked that o is a fuzzy left (right) ideal of R. Let # = #o = o x o. T h e n # is a fuzzy left (right) ideal of R × R. Let x, z c A and let y ~ A . Then # ( y , 0 ) = m i n ( o ( y ) , a ( 0 ) ) = 0 , but # ( x , z ) = min(o(x), o(z)) = 1. Thus #(x, z) > # ( y , 0). Also #(0, y) = 0 = / ~ ( y , 0). Thus /t(x, z) > #(0, y). R e m a r k 3.13. Let G be a group with at least four elements such that G has a proper subgroup H. Let o and ~ be defined as in E x a m p l e 3.12, that is o(x) = 1 if x e H and o(x) = 0 if x ~ H, and # = o × o. Then a is a fuzzy subgroup of G and thus # is a fuzzy subgroup of G x G . Now if x, z e H and y ~ H . T h e n #(x, z) = 1 > 0 = # ( y , e) = #(e, y). This show that L e m m a s 4.5 and 4.6 in [4] are not true. We close this section by giving two examples showing that if # is a fuzzy left (right) ideal of R × R, then o, need not be a fuzzy left ideal of R in general and if /~ is a fuzzy subgroup of G × G, then o~ need not be a fuzzy subgroup of G in general.
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Example 3.14. Let R = 7/6, the ring of integers mod 6, and let H = {0, 3} and K = {0, 2, 4}. Then H and K are ideals of R. Let H0 -- {(0, 0)}, //1 = H x K and HE ----R x R. Then Hi is an ideal of R x R, V0 ~< i ~< 2. Let 1/> to > tl > t2 ~ 0. Let /z be a fuzzy subset of R × R such that
Iz(Ho) = to,
tz(H, \Ho) = t,,
/z(H2\H1) = t2.
Now/zt, = Hi, V0 ~ i <~ 2. Thus/~t is an ideal of R × R, Vt ~ Im(/u). Hence g is a fuzzy ideal of R x R. Since ( 5 , x ) , (x, 5) $ Hi, Vx ~ R, or,(5 ) = t2. Also, tit,(3 ) i> max(~(3, 0),/z(0, 3)) = tl, since (3, 0) c/-/1, and o~(2) I> max(/z(2, 0), g(0, 2)) = tl, since (0, 2) ~ HI. Thus min(ot,(2), or,(3))/> tl > t2 = tr~(5) = t%,(2 + 3). Consequently try, is not a fuzzy ideal of R. Example 3.15. Let G = { e , a , b , a b } where a 2 = b z = e and a b = b a . G is the Klein's four group. Let H = {e, a} and K = {e, b}. Then H and K are subgroups of G. Let H0 = {(e, e)}, Hi = H x K, and H 2 = G x G. Let 1/> to > tl > t2/> 0. Let: /z be a fuzzy subset of G x G as defined in Example 3.14. Then /t is a fuzzy subgroup of G x G. As in Example 3.14, it can be easily checked that
tr~(ab)
= t2
< tl <~ min(o~,(a), tr~,(b)).
Hence o~, is not a fuzT.y subgroup of G. This show that Theorem 4.7 in [4] is not true. References [1] J.M. Anthony and H. Sherwood, Fuzzy subgroups redefined, J. Math. Anal. Appl. 69 (1979) 124-130. [2] J.M. Anthony and H. Sherwood, A characterization of fuzzy subgroups, Fuzzy Sets and System~ 7 (1982) 297-305. [3] P. Bhattacharya, Fuzzy subgroups: some characterizations, J. Math. Anal. Appl. 128 (1987) 241-252. [4] P. Bhattacharya and N.P. Mukherjee, Fuzzy relations and fuzzy groups, Inform. Sci. 36 (1985) 267-282. [5] P.S. Das, Fuzzy groups and level subgroups, J. Math. Anal. Appl. 84 (1981) 264-269. [6] W.J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 (1982) 133-139. [7] D.S. Malik, Fuzzy ideals of Artinian rings, Fuzzy Sets and Systems 37 (1990) 111-115. [8] D.S. Malik and J.N. Mordeson, Fuzzy prime ideals of a ring, Fuzzy Sets and Systems 37 (1990) 93-98. [9] D.S. Malik and J.N. Mordeson, Fuzzy maximal, radical, and primary ideals of a ring, Inform. Sci. 53 (1991) 237-250. [10] N.P. Mukherjee and P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Inform. Sci. 34 (1984) 225-239. [11] T.K. Mukherjee and M.K. Sen, On fuzzy ideals of a ring I, Fuzzy Sets and Systems 21 (1987) 99-104. [12] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517. [13] A. Rosenfeld, Fuzzy graphs, in: L.A. Zadeh, K,S. Fu and M. Shimira, Eds, Fuzzy Sets and Their Applications (Academic Press, New York, 1975). [14] L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338-353. [15] L.A. Zadeh, Similarity relations and fuzzy ordering, Inform Sci. 3 (1971) 177-220.
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Fuzzy relations on rings and groups D.S. Malik and John N. Mordeson Department of Mathematics and Computer Science, Creighton University, Omaha, NE 68178, USA Received February 1989 Revised May 1989
Abstract: Let S be any nonempty set. A fuzzy relation on S is a fuzzy subset of S × S. In this paper we study fuzzy relations on rings and groups. In particular we show that if/z and o are fuzzy left (right)
ideals of a ring R, then/~ x e is a fuzzy left (right) ideal of R x R and conversely if/* x o is a fuzzy left (right) ideal of R × R, then either/~ or o is a fuzzy left (right) ideal of R. An example is given to show that if/* x o is a fuzzy left (right) ideal of R x R, then /~ and a both need not be fuzzy left (right) ideals of R. An example is also given to show that if/~ is a fuzzy left (right) ideal of R x R, then o~,. the weakest fuzzy subset of R on which/~ is a fuzzy relation, need not be a fuzzy left (right) ideal of R. We obtain similar results for groups. We also show that certain results of Bhattacharya and Mukherjee (1985) are not true.
Keywords: Fuzzy relation; strongest fuzzy relation; weakest fuzzy subset; Cartesian product; fuzzy subgroup; fuzzy ideal; group; ring.
1. Introduction T h e notion of a fuzzy subset was i n t r o d u c e d by Z a d e h [14]. R o s e n f e l d [12] i n t r o d u c e d the c o n c e p t o f a fuzzy s u b g r o u p of a g r o u p and established m a n y i m p o r t a n t properties. F u z z y s u b g r o u p s have b e e n studied by several authors; some examples are [ 1 - 5 , 10]. T h e n o t i o n of a fuzzy ideal of a ring was i n t r o d u c e d by Liu [6]. F u z z y ideals have b e e n studied further by Malik [7], Malik and M o r d e s o n [8, 9] and M u k h e r j e e and Sen [11]. T h e c o n c e p t o f a fuzzy relation o n a set was i n t r o d u c e d by Z a d e h [14, 15]. F u z z y relations on a g r o u p have b e e n studied by B h a t t a c h a r y a and M u k h e r j e e [4]. In this p a p e r we study fuzzy relations on a ring. W e p r o v e that if/2 and o are fuzzy left (right) ideals o f a ring R, t h e n / 2 × o is a fuzzy left (right) ideal o f R × R ( T h e o r e m 3.1(i)). C o n v e r s e l y we show that if/2 × o is a fuzzy left (right) ideal o f R x R, then e i t h e r / 2 or o is a fuzzy left (right) ideal o f R ( T h e o r e m 3.2). W e give an e x a m p l e to s h o w that if /2 x o is a fuzzy left (right) ideal o f R x R, t h e n / 2 and o b o t h n e e d n o t be fuzzy left (right) ideals o f R ( E x a m p l e 3.5). R e m a r k 3.6 shows that the c o n v e r s e o f Corollary 3.3 in [4] is not true in general. W e give an e x a m p l e to show that if/2 is a fuzzy left (right) ideal o f R x R and o~, the w e a k e s t fuzzy subset o f R o n w h i c h / 2 is a fuzzy relation, then o r n e e d n o t be a fuzzy left (right) ideal o f R ( E x a m p l e 3.14). W e also give a similar e x a m p l e for g r o u p s ( E x a m p l e 3.15). This E x a m p l e 3.15 shows that T h e o r e m 4.7 in [4] is n o t true in general. R e m a r k 3.13 shows that L e m m a s 4.5 and 4.6 in [4] are not true. 0165-0114/91/$03.50 I~) 1991--Elsevier Science Publishers B.V. All rights reserved
118
D.S. Malik, J.N. Mordeson
2. Definitions, notations and preliminaries In this section we review some definitions and results for the sake of completeness. Throughout the paper S will denote a nonempty set, G will denote a group with identity element denoted by e, and R will denote a ring (not necessarily with unity). Definition 2.1. A f u z z y subset of S is a function/~ : S---> [0, 1]. Definition 2.2. A fuzzy subset/~ of G is called a f u z z y subgroup of G if (i) Iz(xy) >>-min(/t(x), #(y)),
(ii) p(x -~) =/z(x), for all x, y • G. Definition 2.3. A fuzzy subset/z of R is called a f u z z y left (right) ideal of R if (i) #(x - y) 1> min(~u(x), ~(y)),
(ii) I~(xy) >I t~(y) (t~(xy) >t It(x)), for all x, y • R. A fuzzy subset # of R is called a f u z z y ideal of R if/~ is a fuzzy left and fuzzy right ideal of R. Definition 2.4. If/~ is a fuzzy subset of S, then for any t • [0, 1] the set /ut = {x • S [/z(x) i> t} is called a level subset of S with respect to/~. [,emma 2.5. Let tz be a f u z z y subset of G. Then (i) if l~ is a f u z z y subgroup o f G, then Iz(e) ~ It(x) for all x • G and Itt is a subgroup of G for all t • [0, p(e)]; (ii) if Izt is a subgroup of G for all t • Ira(p), then I~ is a f u z z y subgroup of G. [,emma 2.6. Let I~ be a f u z z y subset o f R. Then (i) if Iz is a f u z z y left (right) ideal of R, then/t(0)/> ~(x) for all x • R and ~t iS a left (right) ideal o f R for all t • [0,/t(0)]; (ii) if I~t is a left (right) ideal o f R for all t • Im(p), then I~ is a f u z z y left (right) ideal of R. Definition 2.7. A f u z z y relation it on S is a fuzzy subset of S × S. Definition 2.8. Let/~ be a fuzzy relation on S and let tr be a fuzzy subset of S. Then/~ is called a f u z z y relation on o if #(x, y) ~
Vx, y e S.
Definition 2.9. Let/~ and ¢r be fuzzy subsets of S. The Cartesian product of p and o is defined by
I~ x o(x, y) = min(p(x), o ( y ) ) ,
Vx, y e S.
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Definition 2.10 [13]. L e t o be a fuzzy subset of S. T h e n the strongest f u z z y relation on S that is a fuzzy relation on o is Ito, defined by
Ito(x, y) = o × o(x, y ) = m i n ( o ( x ) , o ( y ) ) ,
Vx, y • S.
Lemma 2.11 [4]. Let Iz and o be f u z z y subsets o f S. Then (i) It x o is a f u z z y relation on S;
(ii) (it × o ) t = Itt × Or, V t • [ 0 , 1]. Definition 2.12 [13]. If # is a fuzzy relation on S, then the weakest f u z z y subset of S on which It is a fuzzy relation is o~,, defined by
o , ( x ) = sup {max(it(x, y), It(y, x))},
Vx • S.
yeS
Definition 2.13. L e t It and o be fuzzy subsets of G. T h e n It o a ( x ) = sup { m i n ( i t ( y ) , a ( z ) ) } ,
Vx • G.
X=yZ
Lemma 2.14 [6]. Let #, o and rI be f u z z y subsets of G. Then # o ( o o T / ) = (it o o ) o ,7.
If It and o are fuzzy subsets of S, then It ~< o m e a n s It(x) <~ o(x) for all x • S.
Lemma 2.15 [6]. Let It be a f u z z y subset o f G. Then It o # <~It if and only if It(xy) >I min(it(x), I t ( y ) ) ,
Vx, y • G.
It is well k n o w n in g r o u p t h e o r y that if H and K are s u b g r o u p s of G, then H K is a s u b g r o u p of G if and only if H K = KH. W e n o w give an a n a l o g o u s result for fuzzy subgroups.
Theorem 2.16. Let It and a be f u z z y subgroups o f G. Then It o o is a f u z z y subgroup o f G if and only if # o o = o o It.
Proof. O b s e r v e that Vx • G, I t o o ( x ) = ooit(x-1). If I t o o = ooit, then ( i t o o ) o (it o o) = (it o It) o ( o o o) ~< It o o. T h u s by L e m m a 2.15, It o o ( x y ) / > min(it o o(x), It o o ( y ) ) for all x, y • G. Also It o a ( x -1) = o o i t ( x ) = It o o(x). H e n c e It o o is a fuzzy s u b g r o u p of G. C o n v e r s e l y , s u p p o s e that It o a is a fuzzy s u b g r o u p of G and let x • G. T h e n It o a ( x ) = It o o ( x - l ) = o o It(x). H e n c e It o o = o o It.
3. Fuzzy relations on rings and groups Theorem 3.1. (i) Let It and o be f u z z y left (right) ideals o f R. Then It × o is a f u z z y left (right) ideal o f R x R. (ii) Let It and o be f u z z y subgroups o f G. Then It x o is a f u z z y subgroup o f G × G [4].
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Proof. (i) Now /~ x tr(0, 0) = min(/u(0), o(0)). Let t • Im(# x o). Then t ~/~(0) and o ( y ) > a(0). Then /~ x a(x, y) = min(/~(x), o ( y ) ) > min(/~(0), o(0)) =/~ × o(0, 0). This is a contradiction since/~ x o is a fuzzy left ideal of R x R. (ii) Suppose that there exists x , y • R such that /~(x)> tr(0) and a ( y ) > a(0). Then ~ x tr(0, 0) = tr(0). Now /u × tr(x, y) = min(/u(x), o ( y ) ) > o(0) = i~ x o(0, 0). This is a contradiction. Hence the result. (iii) Similar to part (ii). (iv) By part (i), either/u(x) ~ min(/~ x o(x, 0), lz x ~r(y, 0)) = min(/~(x),/~(y)). Also l~(xy) = I~ x o(xy, O) = l~ x o((x, O)(y, 0)) >! # x o ( y , O) = I~(y). Hence in this case/z is a fuzzy left ideal of R. Case 2. Let o(x)<<-a(O), Vx • R. Suppose that Case 1 does not hold. Then there exists y • R such that /z(y) > o(0). Then / t ( 0 ) / > / t ( y ) > o(0). Hence /~ x o(0, x) = min(/~(0), o ( x ) ) = o(x), V x • R. Now as in the proof of Case 1, we can show that tr is a fuzzy left ideal of R. Hence either/u or o is a fuzzy left ideal of R. Analogous to T h e o r e m 3.2, we have a similar result for groups, which can be proved in a similar manner. We state the result without proof. Theorem 3.3. Let l~ and o be f u z z y subsets o f G such that I~ x o is a f u z z y subgroup o f G x G. Then (i) either lt(x) ~ It(e), Vx e G or o ( x ) <<-o(e), V x • G;
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(ii) if it(x) <~it(e), Vx • G, then either it(x) <~ o(e), Vx ~ G or o(x) <- o(e), VxeG; (iii) if o(x) <- o(e), Vx • G, then either o(x) <~It(e), Vx • G or it(x) <- it(e), VxeG; (iv) either # or o is a f u z z y subgroup o f G. Corollary 3.4. (i) Let # and o be f u z z y subsets o f R such that # × o is a f u z z y left. (right) ideal o f R x R. I f #(0) = o(0), #(0) >>-#(x), Vx • R and o(0)/> o(x):. Vx ~ R, then # and o both are f u z z y left (right) ideals o f R. (ii) Let # and o be f u z z y subsets o f G such that # x o is a f u z z y subgroup o f G x G. I f #(e) = o(e), #(e) >i #(x), Vx ~ G and o(e) >1 o(x), Vx ~ G, then # and o both are f u z z y subgroups o f G. We now give an example to show that if # x o is a fuzzy left (right) ideal of R × R, then # and a both need not be fuzzy left (right) ideals of R. E x a m p l e 3.5. Let R be any ring with at least two elements and let 0 <~ t <~s < 1. Let # and o be fuzzy subsets of R such that # ( x ) = t, Vx e R and o ( 0 ) = s , o ( x ) = l if O=k x e R. Then # × o(x, y) = m i n ( # ( x ) , a ( y ) ) = t, Vx, y e R. Thus # x o is a fuzzy left (right) ideal of R × R, since # x o is a constant function. Now # is a fuzzy ideal of R but o is not a fuzzy left (right) ideal of R, since o(0) < o(x), V0 =/=x • R. Note that if s = t, then o ( 0 ) = #(0). R e m a r k 3.6. Let G be a group with at least two elements and let # and o be fuzzy subsets of G as defined in E x a m p l e 3.5, that is # ( x ) = t, Vx e G and a(e)=s, a(x)=l if e q : x e G , where 0 ~ < t ~ < s < l . Then I t × o is a fuzzy subgroup of G × G, but o is not a fuzzy subgroup of G. This shows that converse of Corollary 3.3 in [4] is not true in general. Note that if s = t, then o(e) = #(e). Corollary 3.7. (i) Let o be a f u z z y subset o f R. Then Ito = o x o is a f u z z y left (right) ideal of R x R if and only if o is a f u z z y left (right) ideal o f R. (ii) Let o be a f u z z y subset of G. Then Izo is a f u z z y subgroup o f G x G if and only if o is a f u z z y subgroup o f G [4]. L e m m a 3.8. (i) Let o be a f u z z y left (right) ideal of R and let # = #o. Then O~=O.
(ii) Let o be a f u z z y subgroup o f G and let # = #o. Then o r = o. T h e o r e m 3.9. (i) Let t~ be a f u z z y left (right) ideal o f R x R. Then Vx, y, z ~ R, either It(x - y, z) = # ( y , O) or #(x - y, z) = #(x, z) or #(x, z) = It(y, 0). (ii) Let I~ be a f u z z y subgroup of G × G . Then Vx, y, z e G , either # ( x y - ' , z) = # ( y , e) or # ( x y - ' , z) = #(x, z) or #(x, z) =/~(y, e). Proof. Let tt = #(x - y, z), t2 = It(y, 0) and t 3 = ~(X, Z). Using the fact that # is a
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fuzzy left ideal of R × R we deduce that t I/> min(t2, t3),
t 2/> min(t3, tl),
t3/> min(tl, t2).
The result now follows. (ii) Similar to (i).
Corollary 3.10. (i) Let # be a f u z z y left (right) ideal of R × R. Then Vb ~ R, either #(b, b) = #(b, 0), or #(b, b) = #(0, b) or #(b, O) = #(0, b). (ii) Let # be a f u z z y subgroup of G × G. Then Vb ~ G, either #(b, b) = #(b, e) or #(b, b) = #(e, b) or #(b, e) = #(e, b). Proof. (i) Let x = y = z = b in T h e o r e m 3.9(i). (ii) Let x = y = z = b in T h e o r e m 3.9(ii). We now give an example to show that the three numbers in Corollary 3.10 need not be equal. Example 3.11. Let R = 7/, the ring of integers ( G = 7/, the group of integers under addition). Let o be a fuzzy subset of R ( G ) such that o(x) = 1 if 2 Ix and o(x) = 0 if 24-x. Let # be a fuzzy relation on R ( G ) defined by # ( x , y ) = a(x), Vx, y e R (G). Then it can be easily checked that # is a fuzzy ideal (subgroup) of RxR (G×G). Let b = l . Then #(b, b) = o ( b ) -- 0, #(b, 0 ) - - a ( b ) = 0 and #(0, b) -- o(0) -- 1. H e n c e #(b, b) -- g ( b , 0) < #(0, b). If # is a fuzzy left (right) ideal of R × R, then #(0, O) >1#(x, y), Vx, y ~ R. We now give an example to show that there may exist elements x, y, z in a ring R such that #(x, z) > # ( y , 0). Example 3.12. Let R be any ring with at least four elements such that R has a proper left (right) ideal A. Let o be a fuzzy subset of R such that o(x) = 1 if x e A and o(x) = 0 if x ~ A. Then it can be easily checked that o is a fuzzy left (right) ideal of R. Let # = #o = o x o. T h e n # is a fuzzy left (right) ideal of R × R. Let x, z c A and let y ~ A . Then # ( y , 0 ) = m i n ( o ( y ) , a ( 0 ) ) = 0 , but # ( x , z ) = min(o(x), o(z)) = 1. Thus #(x, z) > # ( y , 0). Also #(0, y) = 0 = / ~ ( y , 0). Thus /t(x, z) > #(0, y). R e m a r k 3.13. Let G be a group with at least four elements such that G has a proper subgroup H. Let o and ~ be defined as in E x a m p l e 3.12, that is o(x) = 1 if x e H and o(x) = 0 if x ~ H, and # = o × o. Then a is a fuzzy subgroup of G and thus # is a fuzzy subgroup of G x G . Now if x, z e H and y ~ H . T h e n #(x, z) = 1 > 0 = # ( y , e) = #(e, y). This show that L e m m a s 4.5 and 4.6 in [4] are not true. We close this section by giving two examples showing that if # is a fuzzy left (right) ideal of R × R, then o, need not be a fuzzy left ideal of R in general and if /~ is a fuzzy subgroup of G × G, then o~ need not be a fuzzy subgroup of G in general.
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Example 3.14. Let R = 7/6, the ring of integers mod 6, and let H = {0, 3} and K = {0, 2, 4}. Then H and K are ideals of R. Let H0 -- {(0, 0)}, //1 = H x K and HE ----R x R. Then Hi is an ideal of R x R, V0 ~< i ~< 2. Let 1/> to > tl > t2 ~ 0. Let /z be a fuzzy subset of R × R such that
Iz(Ho) = to,
tz(H, \Ho) = t,,
/z(H2\H1) = t2.
Now/zt, = Hi, V0 ~ i <~ 2. Thus/~t is an ideal of R × R, Vt ~ Im(/u). Hence g is a fuzzy ideal of R x R. Since ( 5 , x ) , (x, 5) $ Hi, Vx ~ R, or,(5 ) = t2. Also, tit,(3 ) i> max(~(3, 0),/z(0, 3)) = tl, since (3, 0) c/-/1, and o~(2) I> max(/z(2, 0), g(0, 2)) = tl, since (0, 2) ~ HI. Thus min(ot,(2), or,(3))/> tl > t2 = tr~(5) = t%,(2 + 3). Consequently try, is not a fuzzy ideal of R. Example 3.15. Let G = { e , a , b , a b } where a 2 = b z = e and a b = b a . G is the Klein's four group. Let H = {e, a} and K = {e, b}. Then H and K are subgroups of G. Let H0 = {(e, e)}, Hi = H x K, and H 2 = G x G. Let 1/> to > tl > t2/> 0. Let: /z be a fuzzy subset of G x G as defined in Example 3.14. Then /t is a fuzzy subgroup of G x G. As in Example 3.14, it can be easily checked that
tr~(ab)
= t2
< tl <~ min(o~,(a), tr~,(b)).
Hence o~, is not a fuzT.y subgroup of G. This show that Theorem 4.7 in [4] is not true. References [1] J.M. Anthony and H. Sherwood, Fuzzy subgroups redefined, J. Math. Anal. Appl. 69 (1979) 124-130. [2] J.M. Anthony and H. Sherwood, A characterization of fuzzy subgroups, Fuzzy Sets and System~ 7 (1982) 297-305. [3] P. Bhattacharya, Fuzzy subgroups: some characterizations, J. Math. Anal. Appl. 128 (1987) 241-252. [4] P. Bhattacharya and N.P. Mukherjee, Fuzzy relations and fuzzy groups, Inform. Sci. 36 (1985) 267-282. [5] P.S. Das, Fuzzy groups and level subgroups, J. Math. Anal. Appl. 84 (1981) 264-269. [6] W.J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 (1982) 133-139. [7] D.S. Malik, Fuzzy ideals of Artinian rings, Fuzzy Sets and Systems 37 (1990) 111-115. [8] D.S. Malik and J.N. Mordeson, Fuzzy prime ideals of a ring, Fuzzy Sets and Systems 37 (1990) 93-98. [9] D.S. Malik and J.N. Mordeson, Fuzzy maximal, radical, and primary ideals of a ring, Inform. Sci. 53 (1991) 237-250. [10] N.P. Mukherjee and P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Inform. Sci. 34 (1984) 225-239. [11] T.K. Mukherjee and M.K. Sen, On fuzzy ideals of a ring I, Fuzzy Sets and Systems 21 (1987) 99-104. [12] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517. [13] A. Rosenfeld, Fuzzy graphs, in: L.A. Zadeh, K,S. Fu and M. Shimira, Eds, Fuzzy Sets and Their Applications (Academic Press, New York, 1975). [14] L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338-353. [15] L.A. Zadeh, Similarity relations and fuzzy ordering, Inform Sci. 3 (1971) 177-220.