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Generalized product of fuzzy subsets of a ring
Fuzzy Sets and Systems 117 (2001) 419–429
www.elsevier.com/locate/fss
Generalized product of fuzzy subsets of a ring K.C. Gupta, Manoj K. Kantroo∗
Department of Mathematics, University of Delhi, Delhi 110007, India Received April 1995; received in revised form August 1998
Abstract We deÿne T -intrinsic product of fuzzy subsets of an algebraic object more general than a ring and study some of its fundamental properties. Deep connection that exists between T -intrinsic product and fuzzy subsets of a ring is brought to the surface. The concepts of T -fuzzy subring, T -fuzzy ideal and T -fuzzy subgroup are redeÿned, to include quite larger classes within their fold. Interconnection between these classes and T -intrinsic product is studied in detail. In the end we c 2001 Published by Elsevier isolate di erent subclasses of the subsemigroup F(R) and reveal their ideal structure. Science B.V. All rights reserved. Keywords: T -intrinsic product; Fuzzy ideal; Fuzzy subring
1. Introduction In this paper, we introduce the concept of T -intrinsic product of fuzzy subsets of an algebraic object S with two binary compositions, addition (+) and multiplication (·) such that (S, +) is a semigroup and study some of its fundamental properties. This is a generalization of the intrinsic product deÿned by us in [3], and subsequently used in [4,5] to study fuzzy algebraic structures. Deep connection that exists between T -intrinsic product and fuzzy subsets of a ring is brought to the surface. In [1,2] Anthony and Sherwood have redeÿned the concept of T -fuzzy subgroup of a group, using the idea of a triangular norm and studied them. These concepts were further extended by Sessa in [9] to deÿne T -fuzzy subrings and T -fuzzy ideals of a ring. We know that if is a fuzzy subgroup or fuzzy subring then it follows quite naturally from the deÿnitions that (0) assumes the maximum grade. Further, the tip (0) plays a very important role in the study of these structures. However, in [1,9] this fact does not follow directly from the deÿnitions, so a very strong condition that (0) = 1 has been forcibly incorporated in these deÿnitions which restricts these classes considerably and reduces their size. Therefore, it would seem appropriate to modify these deÿnitions to ensure that the basic property of (0) assuming the maximum grade percolates down quite naturally from the new deÿnitions as well. With these objectives in mind we redeÿne the concepts of T -fuzzy subring, T -fuzzy ideal and T -fuzzy subgroup. These modiÿed deÿnitions encompass quite larger classes within their fold. The interconnection that exists between these classes and T -intrinsic product is studied in detail. In the end, we isolate di erent subclasses of the subsemigroup F(R). It turns out that most of these classes form ideal structures. ∗
Correspondence address: Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand.
c 2001 Published by Elsevier Science B.V. All rights reserved. 0165-0114/01/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 8 ) 0 0 3 9 3 - 5
420
K.C. Gupta, M.K. Kantroo / Fuzzy Sets and Systems 117 (2001) 419–429
2. Preliminaries We ÿrst recall some of the basic deÿnitions proposed by pioneers in this ÿeld. Deÿnition 2.1. A mapping of a non-empty set A into closed unit interval [0, 1] is a fuzzy subset of A [10]. The set of all fuzzy subsets of A is denoted by F(A). Deÿnition 2.2. A fuzzy subset of a group G is a fuzzy subgroup of G if (1) min{(a); (b)}6(ab), and (2) (a−1 ) = (a) for all a; b in G; [7]. Equivalently, min{(a); (b)}6(ab−1 ). In that case, (x)6(e) for all x in G. Deÿnition 2.3. A fuzzy subset of a ring R is a fuzzy subring of R if (1) min{(x); (y)}6(x − y), and (2) min{(x); (y)}6(xy) for all x,y in R; [6]. Deÿnition 2.4. A fuzzy subset of a ring R is a fuzzy left ideal of R if (1) min{(a); (b)}6(a − b), and (2) (b)6(ab) for all a; b in R. A fuzzy subset of a ring R is a fuzzy right ideal of R if (1) min{(a); (b)}6(a − b), and (2) (a)6(ab) for all a; b in R. If is both a fuzzy left and right ideal of R, then is a fuzzy ideal of R; [6]. Deÿnition 2.5. A binary composition T on [0, 1] = I is called a triangular norm (t-norm) if (1) (aTb)Tc = aT (bTc) ∀a; b; c in I ; i.e. T is associative in I . (2) aTb = bTa ∀a; b in I ; i.e. T is commutative in I . (3) a6c ⇒ aTb6cTb ∀a; b; c in I . (4) aT 1 = a ∀a in I . Let {ai }i∈A , {bj }j∈B be two sets of real numbers in [0, 1]. Then we say T is inÿnitely distributive if ! T
sup ai ; sup bj i∈A
j∈B
= sup sup(T (ai ; bj )) : i
j
For aTb we shall also use T (a; b). Deÿnition 2.6 (Sessa [9]) (The sup-T product of fuzzy subsets). If and are fuzzy subsets of a groupoid (D; ·), the sup-T product ◦T relative to the composition in D is deÿned by ( ◦T )(x) = x=a·b
sup T {(a); (b)} if x is factorizable in D; 0 otherwise; x ∈ D:
From now on R will denote a ring unless otherwise stated and S a non-empty set with two binary compositions addition (+) and multiplication (·) such that (S; +) is a semigroup, F(S) the set of all fuzzy subsets of S.
K.C. Gupta, M.K. Kantroo / Fuzzy Sets and Systems 117 (2001) 419–429
421
3. The T-intrinsic product We now deÿne T -intrinsic product of fuzzy subsets of S. Deÿnition 3.1 (The T-intrinsic product). Let , be any two fuzzy subsets of S, and T any t-norm on I . Let x ∈ S. Deÿne ∗T , the T -intrinsic product of and by ( ∗T )(x) =
x=
sup
P
ÿnite
ai bi
T {(a1 ); : : : ; (am ); (b1 ); : : : ; (bm )};
if we can express x = a1 b1 + · · · + am bm for some ai ; bi ∈ S and m ∈ N . Otherwise, deÿne ( ∗T )(x) = 0. Clearly, the product ∗T is commutative if multiplication in S is commutative. Theorem 3.2. The T-intrinsic product; where T is an inÿnitely distributive T-norm; is associative in E(S); where E(S) = { ∈ F(S): (x)T(x) = (x); for all x ∈ S} provided that multiplication in S is associative and is distributive over addition. Proof. Let ; ; ∈ E(S), and let x ∈ S. If x has no reduction in S, then (( ∗T ) ∗T )(x) = ( ∗T ( ∗T ))(x) = 0: Suppose that x does have reductions in S. Let x = t1 c1 + t2 c2 + · · · + tm cm ; (ti ; ci ∈ S) be any reduction of x in S. Firstly, suppose that some tj possesses no reduction in S. Then, we get ( ∗T )(tj ) = 0 and so T {( ∗T )(t1 ); : : : ; ( ∗T )(tm ); (c1 ); : : : ; (cm )} = 06( ∗T ( ∗T ))(x): Secondly, suppose that each ti has a reduction in S. Let ti =
ri X
aij bij ;
16i6m
j=1
be a reduction of ti in S. Then we get T {(a11 ); : : : ; (a1 r1 ); (b11 ); : : : ; (b1 r1 ); (c1 ); (a21 ); : : : ; (a2 r2 ); (b21 ); : : : ; (b2 r1 ); (c2 ); : : : ; (am1 ); : : : ; (am rm ); (bm1 ); : : : ; (bm rm ); (cm )} 6T {(a11 ); : : : ; (a1 r1 ); ( ∗T )(b11 c1 ); : : : ; ( ∗T )(b1r1 c1 ); (a21 ); : : : ; (a2 r2 ); ( ∗T )(b21 c2 ); : : : ; ( ∗T )(b2r2 c2 ); : : : ; (am1 ); : : : ; (am rm ); ( ∗T )(bm1 cm ); : : : ; ( ∗T )(bmrm cm )} 6 ∗T ( ∗T )(a11 b11 c1 + · · · + a1r1 b1r1 c1 + a21 b21 c2 + · · · + a2r2 b2r2 c2 + · · · + am1 bm1 cm + · · · + amrm bmrm cm ) = ∗T ( ∗T )(x):
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K.C. Gupta, M.K. Kantroo / Fuzzy Sets and Systems 117 (2001) 419–429
Taking the supremum over all possible reductions of each ti (16i6m), we obtain T { ∗T (t1 ); (c1 ); ∗T (t2 ); (c2 ); : : : ; ∗T (tm ); (cm )} 6 ( ∗T ( ∗T ))(x): Taking the supremum over all possible reductions of x in S, we ÿnally get (( ∗T ) ∗T )(x) = ( ∗T ( ∗T ))(x): This is sucient to claim that ( ∗T ) ∗T = ∗T ( ∗T ): Lemma 3.3. If ; ; ∈ F(S); such that ⊆ ; then ∗T ⊆ ∗T ;
∗T ⊆ ∗T :
Proof. Let X ∈ S. If x does not have any reduction in S, then the result is trivial. Otherwise, ( ∗T )(x) = 6
x= x=
sup
P
ÿnite
sup
P
ÿnite
ai bi ai bi
T {(a1 ); : : : ; (am ); (b1 ); : : : ; (bm )} T {(a1 ); : : : ; (am ); (b1 ); : : : ; (bm )}
= ( ∗T )(x): The other case follows similarly. Theorem 3.4. Let S be an additive monoid in which addition is commutative and multiplication is distributive over addition. If ; ; ∈ F(S); such that Im(); Im(); Im() ⊆ DT where DT = {x | xTx = x} and T is an inÿnitely distributive T -norm; then (1) ∗T ( +T ) ⊆ ∗T +T ∗T . (2) If max((x); (x)) 6 (0) = (0); then equality holds in (1). (3) ( +T ) ∗T ⊆ ∗T +T ∗T . (4) If max((x); (x)) 6 (0) = (0); then equality holds in (3). Proof. (1) Let x ∈ R. Suppose that x has a reduction x = a1 b1 +a2 b2 + · · · + am bm in R. Let bj = uj + vj , 1 6 j 6 m. We then have T {(a1 ); : : : ; (am ); T {(u1 ); (v1 )}; T {(u2 ); (v2 )}; : : : ; T {(um ); (vm )}} = T {T {(a1 ); (a2 ); : : : ; (am ); (u1 ); (u2 ); : : : ; (um )}; T {(a1 ); (a2 ); : : : ; (am ); (v1 ); (v2 ); : : : ; (vm )}} 6 T {( ∗T )(a1 u1 + a2 u2 + · · · + am um ); ( ∗T )(a1 v1 + a2 v2 + · · · + am vm )} 6 (( ∗T ) +T ( ∗T ))(x): Now, ( ∗T ( +T ))(x) =
x=
sup
P
ÿnite
ai bi
T {(a1 ); : : : ; (am ); ( +T )(b1 ); ( +T )(b2 ); : : : ; ( +T )(bm )}
(1)
K.C. Gupta, M.K. Kantroo / Fuzzy Sets and Systems 117 (2001) 419–429
=
x=
sup P ÿnite
sup
b1 =u1 +v1
ai bi
sup
b2 =u2 +v2
423
···
sup
bm =um +vm
T {(a1 ); : : : ; (am ); T {(u1 ); (v1 )};
T {(u2 ); (v2 )}; : : : ; T {(um ); (vm )}} · · ·
6 (( ∗T ) +T ( ∗T ))(x)
using Eq: (1):
If x has no reductions in S, then ( ∗T ( +T ))(x) = 0. Consequently, ∗T ( +T ) ⊆ ∗T +T ∗T . (2) For equality let us assume that max((x); (x)) 6 (0) = (0) for all x in S. Let x ∈ S, and let x = a + b. If neither a nor b has any reduction in S, then of course we get T {( ∗T )(a); ( ∗T )(b)} = 0 6 ( ∗T ( +T ))(x): So, suppose that a and b has any reduction in S. Let a = u1 v1 + · · · + um vm , and b = p1 q1 + · · · + pn qn be any reductions in S. Now, we obtain T {T {(u1 ); : : : ; (um ); (v1 ); : : : ; (vm )}; T {(p1 ); : : : ; (pn ); (q1 ); : : : ; (qn )}} = T {(u1 ); : : : ; (um ); (p1 ); : : : ; (pn ); T {(v1 ); (0)}; : : : ; T {(vm ); (0)}; T {(0); (q1 )}; : : : ; T {(0); (qn )}} 6 T {(u1 ); : : : ; (um ); (p1 ); : : : ; (pn ); ( +T )(v1 ); : : : ; ( +T )(vm ); ( +T )(q1 ); : : : ; ( +T )(qn )} 6 ( ∗T ( +T ))(u1 v1 + · · · + um vm + p1 q1 + · · · + pn qn ) = ( ∗T ( +T ))(x): Consequently, we obtain T {( ∗T )(a); ( ∗T )(b)} = sup P x=
ui vi
T {(u1 ); : : : ; (um ); (v1 ); : : : ; (vm )}; T {(p1 ); : : : ; (pn ); (q1 ); : : : ; (qn )} sup P
x=
pi qi
6 ( ∗T ( +T ))(x): Taking the supremum over all the sums of x = a + b of x we get ( ∗T +T ∗T )(x) 6 ( ∗T ( +T ))(x): This completes (2). One can apply a similar technique to prove (3) and (4). Lemma 3.5. If is a fuzzy subset of S; then the following two statements are equivalent: (a) ∗T ⊆ ; (b) T {(a1 ); : : : ; (am ); (b1 ); : : : ; (bm )}6 (a1 b1 + · · · + am bm ) for all ai ; bi ∈S. Proof. (a) ⇒ (b) Let ai ; bi ∈ S; 16i6m, we have T {(a1 ); : : : ; (am ); (b1 ); : : : ; (bm )} 6
x=
sup
P
ÿnite
!
ai bi
T {(a1 ); : : : ; (am ); (b1 ); : : : ; (bm )}
424
K.C. Gupta, M.K. Kantroo / Fuzzy Sets and Systems 117 (2001) 419–429
= ( ∗T )(x);
where x = a1 b1 + a2 b2 + · · · + am bm
6 (x) = (a1 b1 + · · · + am bm ): (b) ⇒ (a) Let x ∈ S. If x has no reductions in S, then ( ∗T )(x) = 0; so ( ∗T )(x) 6 (x): Suppose x has a reduction x = a1 b1 + a2 b2 + · · · + am bm in S. Then T {(a1 ); : : : ; (am ); (b1 ); : : : ; (bm )}6 (a1 b1 + · · · + am bm ) = (x): Taking the supremum over all reductions of x in S, we get ( ∗T )(x) 6 (x);
consequently ( ∗T )(x) ⊆ (x):
4. T-fuzzy subrings, ideals and T-intrinsic product In this section, we deÿne T -fuzzy subrings and ideals and analyze their T -intrinsic product. Deÿnition 4.1. Let ∈ F(R). We say that is a T-fuzzy subring of R, if (1) T ((x); (y)) 6 (x − y), (2) T ((x); (y)) 6 (xy) for all x; y in R, (3) Im ⊆ DT where DT = {x | xTx = x}. If is a T -fuzzy subring of a ring R, then for x ∈ R (x) = T ((x); (x)) = T ((x); (−x)) 6 (x − x) = (0): Deÿnition 4.2. Let ∈ F(R). We say that is a T-fuzzy left ideal of R, if (1) T ((x); (y)) 6 (x − y), (2) (y) 6 (xy) for all x; y in R, (3) Im ⊆ DT where DT = {x | xTx = x}. Similarly, is called a T-fuzzy ideal of R, if (1) T ((x); (y)) 6 (x − y), (2) (x) 6 (xy) for all x; y in R, (3) Im ⊆ DT where DT = {x | xTx = x}. is called a T-fuzzy ideal of R if it is both T -fuzzy left and right ideal of R, i.e. (1) T ((x); (y)) 6 (x − y), (2) max((x); (y)) 6 (xy) for all x; y in R, (3) Im ⊆ DT where DT = {x | xTx = x}. Since T ((x); (y)) 6 min((x); (y)) 6 (x); (y) for all x; y in R. Therefore if is either a T -fuzzy left ideal or a T -fuzzy right ideal of R, then is a T -fuzzy subring of R. Deÿnition 4.3. ∈ F(G), is called T-fuzzy subgroup of a group G if (1) T ((x); (y)) 6 (x − y) for all x; y in G. (2) Im ⊆ DT where DT = {x | xTx = x}. We note that from Deÿnitions 4.1–4.3 (x) 6 (0) follows quite naturally.
K.C. Gupta, M.K. Kantroo / Fuzzy Sets and Systems 117 (2001) 419–429
425
Theorem 4.4. If is a fuzzy subset of a ring R; then the following two statements are equivalent: (a) is a T-fuzzy subring of R. (b) is a T-fuzzy subgroup of the additive group (R; +) and ∗T ⊆ . Proof. (a) ⇒ (b) If is a T -fuzzy subring of R, then clearly is a T -fuzzy subgroup of (R, +). Secondly if ai ; bi ∈ R; 1 6 i 6 m, we have T {(a1 ); : : : ; (am ); (b1 ); : : : ; (bm )} = T {T ((a1 ); (b1 )); : : : ; T ((am ); (bm ))} 6 T {(a1 b1 ); : : : ; (am bm )} 6 (a1 b1 + · · · + am bm ): By Lemma 3.5 this implies ∗T ⊆ . (b) ⇒ (a) It follows by using the deÿnition of a T -fuzzy subring and Lemma 3.5. Theorem 4.5. If ; are T-fuzzy subrings of a ring R such that ∗T = ∗T ; where T is an inÿnitely distributive t-norm; then ∗T is a T-fuzzy subring of R. Proof. Let x; y ∈ R. If either x or y has no reduction in R; then either ( ∗T )(x) = 0
or
( ∗T )(y) = 0:
Therefore, T (( ∗T )(x); ( ∗T )(y)) = 0 6 ( ∗T )(x − y): So we assume that x and y have reductions in R. Let x=
m X
ai bi ;
y=
i=1
n X
ci di :
i=1
P P Hence x − y = ai bi + (− ci )di . We then have T {(a1 ); : : : ; (am ); (− c1 ); : : : ; (− cn ); (b1 ); : : : ; (bm ); (d1 ); : : : ; (dn )} 6 ( ∗T )(x − y): Therefore, T (( ∗T )(x); ( ∗T )(y)) ! =
x=
sup
P
ÿnite ai bi
x=
sup
P
ÿnite ci di
6 ( ∗T )(x − y):
T {(a1 ); : : : ; (ar ); (− c1 ); : : : ; (− ct ); (b1 ); : : : ; (br ); (d1 ); : : : ; (dt )}
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K.C. Gupta, M.K. Kantroo / Fuzzy Sets and Systems 117 (2001) 419–429
We now show that Im ⊆ DT ( ∗T )(x)T ( ∗T )(x) ( =T
x=
x=
sup
P
ÿnite ai bi
sup
P
ÿnite cj dj
= sup P x=
T {(a1 ); : : : ; (an ); (b1 ); : : : ; (bn )}; )
T {(c1 ); : : : ; (cm ); (d1 ); : : : ; (dm )}
sup T {T ((a1 ); : : : ; (an ); (b1 ); : : : ; (bn )); T ((c1 ); : : : ; (cm ); (d1 ); : : : ; (dm ))} P
ai bi x=
cj dj
T {T ((a1 ); : : : ; (an ); (b1 ); : : : ; (bn )); T ((a1 ); : : : ; (an ); (b1 ); : : : ; (bn ))} ¿ sup P x=
ai bi
¿ T {(a)T(a1 )T · · · T(an )T(b1 )T(b2 )T · · · T(bn ) T(a1 )T(a2 )T · · · T(an )T(b1 )T(b2 )T · · · T(bn )} = T {(a1 ); (a2 ); : : : ; (an ); (b1 ); (b2 ); : : : ; (bn )}: P Taking the supremum over all reductions x = ÿnite ai bi , we get ( ∗T )(x)T ( ∗T )(x) ¿ ( ∗T )(x): But we know that ( ∗T )(x)T ( ∗T )(x) 6 ( ∗T )(x): Thus ( ∗T )(x)T ( ∗T )(x) = ( ∗T )(x). If x has no reduction in R, then ( ∗T )(x) = 0 = 0T 0 = ( ∗T )(x)T ( ∗T )(x): Hence ∗T is T -fuzzy subgroup of (R; +). Also, ( ∗T ) ∗T ( ∗T ) = ∗T ( ∗T ) ∗T = ∗T ( ∗T ) ∗T = ( ∗T ) ∗T ( ∗T ) ⊆ ∗T : Therefore by Theorem 4.4 ∗T is a T -fuzzy subring of R. Theorem 4.6. If is a T-fuzzy right ideal of R; and a T-fuzzy left ideal of R then ∗T ⊆ T; where T is an inÿnitely distributive T-norm. Proof. Let x ∈ R. If x has no reduction in R, then ( ∗T )(x) = 0 6 (T)(x). Let x = a1 b1 + a2 b2 + · · · + am bm ; ai ; bi ∈ R. Then T {(ai ); (bi )} 6 (ai ) 6 (ai bi ); 1 6 i 6 m.
K.C. Gupta, M.K. Kantroo / Fuzzy Sets and Systems 117 (2001) 419–429
427
We have T {(a1 ); : : : ; (am ); (b1 ); : : : ; (bm )} 6 T {(a1 b1 ); : : : ; (am bm )} 6 (a1 b1 + a2 b2 + · · · + am bm ) = (x): Taking the supremum over all reductions of x in R we get (i) ( ∗T )(x) 6 (x). Since is a fuzzy left ideal of R, we can similarly prove that (ii) ( ∗T )(x) 6 (x). Since Im , Im ⊆ DT , as proved in Theorem 4.5. Therefore from (i) and (ii) we get ( ∗T )(x) = ( ∗T )(x)T ( ∗T )(x) 6 (x)T(x) ∀x ∈ R: Hence ∗T ⊆ T. Theorem 4.7. If is a T-fuzzy left ideal of a ring R and is a T-fuzzy right ideal of R; then ∗T is a T-fuzzy ideal of R; where T is an inÿnitely distributive T-norm. Proof. From the ÿrst part of Theorem 4.5 it follows that ∗T is a fuzzy subgroup of (R; +). Secondly let x; y ∈ R. We show that ( ∗T )(x) 6 ( ∗T )(xy)
and
( ∗T )(y) 6 ( ∗T )(xy):
The above inequalities are obvious if x or y has no reduction in R. We may assume that x and y have reduction in R. Let x = a1 b1 + · · · + am bm ; ai ; bi ∈ R. Then xy = a1 (b1 y) + · · · + am (bm y). We have T {(a1 ); : : : ; (am ); (b1 ); : : : ; (bm )} 6 T {(a1 ); : : : ; (am ); (b1 y); : : : ; (bm y)} 6 ( ∗T )(xy): Therefore, ( ∗T )(x) 6 ( ∗T )(xy). Similarly, ( ∗T )(y) 6 ( ∗T )(xy). Consequently, max{( ∗T )(x); ( ∗T )(y)}6 ( ∗T )(xy): 5. The classiÿcation In this section, we deÿne E(R) a subset of F(R) and isolate its certain subclasses. It is shown that some of these subclasses are ideals of semigroup (E(R); ∗T ). Deÿnition 5.1. We deÿne E(R) as E(R) = { ∈ R: Im ⊆ DT }; where T is an inÿnitely distributive T -norm and DT = {x | xTx = x}. Theorem 5.2. (E(R); ∗T ) is a subsemigroup of the semigroup (F(R); ∗T ).
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K.C. Gupta, M.K. Kantroo / Fuzzy Sets and Systems 117 (2001) 419–429
Proof. Let ; ∈ E(R), then (x)T(x) = (x) and (x)T(x) = (x) for all x ∈ R. Let x ∈ R. As in Theorem 4.5, we have T {( ∗T )(x); ( ∗T )(x)} = ( ∗T )(x): Hence ∗T ∈ E(R). We now isolate the following subclasses of E(R). Deÿnition 5.3. (1) A−1 (R) = { ∈ E(R): +T ⊇ }: (2) A−s (R) = { ∈ E(R): +T ⊇ ; (− x) = (x) ∀x ∈ R}. (3) As (R) = { ∈ E(R): (− x) = (x) ∀x ∈ R}. (4) A0 (R) = { ∈ E(R): (x) 6 (0) ∀x ∈ R}. (5) A1 (R) = { ∈ E(R): +T ⊆ }. (6) A2 (R) = { ∈ E(R): +T = }. (7) A3 (R) = { ∈ E(R): +T ⊆ ; (x) 6 (0) ∀x ∈ R}. (8) A4 (R) = { ∈ E(R): +T ⊆ ; (− x) = (x) ∀x ∈ R}. Theorem 5.4. (1) A−1 (R) is an ideal of (E(R); ∗T ). (2) A−s (R) is an ideal of (E(R); ∗T ). (3) As (R) is an ideal of (E(R); ∗T ). (4) A0 (R) is an ideal of (E(R); ∗T ). (5) A3 (R) is an ideal of (E(R); ∗T ). (6) A4 (R) is an ideal of (E(R); ∗T ). Proof. (1) Let ∈ E(R); ∈ A−1 (R), then by Theorem 3.4 ∗T ⊆ ∗T ( +T ) ⊆ ∗T + ∗T : Hence ∗T ∈ A−1 (R). Similarly ∗T ∈ A−1 (R). (2) Let ∈ E(R); ∈ A−s (R). Let x ∈ R: x has a reduction in R i − x has a reduction in R. If x has no reduction in R, then ( ∗T )(x) = 0 = ( ∗T )(−x): Let x have a reduction in R. Then for x ∈ R T {(a1 ); : : : ; (an ); (b1 ); : : : ; (bn )} ( ∗T )(x) = sup P x=
ai b i
T {(a1 ); : : : ; (an ); (− b1 ); : : : ; (− bn )} = sup P x=
ai b i
= ( ∗T )(− x): Also from (1) ∗T ∈ A−1 (R). Hence A−s (R) is an ideal of E(R). (3) Obvious. (4) Let ∈ E(R); ∈ A0 (R). Let x ∈ R. If x has no reduction in R, then 0 = ( ∗T )(x) 6 ( ∗T )(0):
K.C. Gupta, M.K. Kantroo / Fuzzy Sets and Systems 117 (2001) 419–429
So let x have a reduction in R. Then ( ∗T )(x) =
sup T {(a1 ); : : : ; (an ); (b1 ); : : : ; (bn )} P
x=
ai bi
T {(a1 ); : : : ; (an ); (0); : : : ; (0)} 6 sup P x=
ai bi
6 ( ∗T )(0): Consequently, ∗T ∈ A0 (R). (5) Let ∈ E(R); ∈ A3 (R), then (x) 6 (0) for all x ∈ R. Therefore by Theorem 3.4 ∗T +T ∗T = ∗T ( +T ) ⊆ ∗T : Also ( ∗T )(x) 6 ( ∗T )(0) from (4). Hence ∗T ∈ A3 (R). (6) Let ∈ E(R); ∈ A4 (R). From (3) ( ∗T )(x) = ( ∗T )(− x); for all x ∈ R. Also, (x) = T ((x); (− x)); 6 ( +T )(x − x) = ( +T )(0) 6 (0)
for all x ∈ R:
Therefore by Theorem 3.4 it follows that ∗T + ∗T = ∗T ( + ) ⊆ ∗T : Consequently, ∗T ∈ A4 (R). Similarly, ∗T ∈ A4 (R). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
J.M. Anthony, H. Sherwood, Fuzzy groups redeÿned, J. Math. Anal. Appl. 69 (1979) 124 –130. J.M. Anthony, H. Sherwood, A characterization of fuzzy subgroups, Fuzzy Sets and Systems 7 (1982) 297–305. K.C. Gupta, M.K. Kantroo, The intrinsic product of fuzzy subsets of a ring, Fuzzy Sets and Systems 57 (1993) 103–110. K.C. Gupta, M.K. Kantroo, The nil radical of a fuzzy ideal, Fuzzy Sets and Systems 59 (1993) 87–93. K.C. Gupta, M.K. Kantroo, Classiÿcation of fuzzy subrings of a division ring, Fuzzy Sets and Systems 72 (1995) 103–117. W.-J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 (1982) 133–139. A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512–517. B. Schweizer, A. Sklar, Statistical metric spaces, Paciÿc J. Math. 10 (1966) 313–334. S. Sessa, On fuzzy subgroups and fuzzy ideals under triangular norms, Fuzzy Sets and Systems 13 (1984) 95–100. L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338–353.
429
www.elsevier.com/locate/fss
Generalized product of fuzzy subsets of a ring K.C. Gupta, Manoj K. Kantroo∗
Department of Mathematics, University of Delhi, Delhi 110007, India Received April 1995; received in revised form August 1998
Abstract We deÿne T -intrinsic product of fuzzy subsets of an algebraic object more general than a ring and study some of its fundamental properties. Deep connection that exists between T -intrinsic product and fuzzy subsets of a ring is brought to the surface. The concepts of T -fuzzy subring, T -fuzzy ideal and T -fuzzy subgroup are redeÿned, to include quite larger classes within their fold. Interconnection between these classes and T -intrinsic product is studied in detail. In the end we c 2001 Published by Elsevier isolate di erent subclasses of the subsemigroup F(R) and reveal their ideal structure. Science B.V. All rights reserved. Keywords: T -intrinsic product; Fuzzy ideal; Fuzzy subring
1. Introduction In this paper, we introduce the concept of T -intrinsic product of fuzzy subsets of an algebraic object S with two binary compositions, addition (+) and multiplication (·) such that (S, +) is a semigroup and study some of its fundamental properties. This is a generalization of the intrinsic product deÿned by us in [3], and subsequently used in [4,5] to study fuzzy algebraic structures. Deep connection that exists between T -intrinsic product and fuzzy subsets of a ring is brought to the surface. In [1,2] Anthony and Sherwood have redeÿned the concept of T -fuzzy subgroup of a group, using the idea of a triangular norm and studied them. These concepts were further extended by Sessa in [9] to deÿne T -fuzzy subrings and T -fuzzy ideals of a ring. We know that if is a fuzzy subgroup or fuzzy subring then it follows quite naturally from the deÿnitions that (0) assumes the maximum grade. Further, the tip (0) plays a very important role in the study of these structures. However, in [1,9] this fact does not follow directly from the deÿnitions, so a very strong condition that (0) = 1 has been forcibly incorporated in these deÿnitions which restricts these classes considerably and reduces their size. Therefore, it would seem appropriate to modify these deÿnitions to ensure that the basic property of (0) assuming the maximum grade percolates down quite naturally from the new deÿnitions as well. With these objectives in mind we redeÿne the concepts of T -fuzzy subring, T -fuzzy ideal and T -fuzzy subgroup. These modiÿed deÿnitions encompass quite larger classes within their fold. The interconnection that exists between these classes and T -intrinsic product is studied in detail. In the end, we isolate di erent subclasses of the subsemigroup F(R). It turns out that most of these classes form ideal structures. ∗
Correspondence address: Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand.
c 2001 Published by Elsevier Science B.V. All rights reserved. 0165-0114/01/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 8 ) 0 0 3 9 3 - 5
420
K.C. Gupta, M.K. Kantroo / Fuzzy Sets and Systems 117 (2001) 419–429
2. Preliminaries We ÿrst recall some of the basic deÿnitions proposed by pioneers in this ÿeld. Deÿnition 2.1. A mapping of a non-empty set A into closed unit interval [0, 1] is a fuzzy subset of A [10]. The set of all fuzzy subsets of A is denoted by F(A). Deÿnition 2.2. A fuzzy subset of a group G is a fuzzy subgroup of G if (1) min{(a); (b)}6(ab), and (2) (a−1 ) = (a) for all a; b in G; [7]. Equivalently, min{(a); (b)}6(ab−1 ). In that case, (x)6(e) for all x in G. Deÿnition 2.3. A fuzzy subset of a ring R is a fuzzy subring of R if (1) min{(x); (y)}6(x − y), and (2) min{(x); (y)}6(xy) for all x,y in R; [6]. Deÿnition 2.4. A fuzzy subset of a ring R is a fuzzy left ideal of R if (1) min{(a); (b)}6(a − b), and (2) (b)6(ab) for all a; b in R. A fuzzy subset of a ring R is a fuzzy right ideal of R if (1) min{(a); (b)}6(a − b), and (2) (a)6(ab) for all a; b in R. If is both a fuzzy left and right ideal of R, then is a fuzzy ideal of R; [6]. Deÿnition 2.5. A binary composition T on [0, 1] = I is called a triangular norm (t-norm) if (1) (aTb)Tc = aT (bTc) ∀a; b; c in I ; i.e. T is associative in I . (2) aTb = bTa ∀a; b in I ; i.e. T is commutative in I . (3) a6c ⇒ aTb6cTb ∀a; b; c in I . (4) aT 1 = a ∀a in I . Let {ai }i∈A , {bj }j∈B be two sets of real numbers in [0, 1]. Then we say T is inÿnitely distributive if ! T
sup ai ; sup bj i∈A
j∈B
= sup sup(T (ai ; bj )) : i
j
For aTb we shall also use T (a; b). Deÿnition 2.6 (Sessa [9]) (The sup-T product of fuzzy subsets). If and are fuzzy subsets of a groupoid (D; ·), the sup-T product ◦T relative to the composition in D is deÿned by ( ◦T )(x) = x=a·b
sup T {(a); (b)} if x is factorizable in D; 0 otherwise; x ∈ D:
From now on R will denote a ring unless otherwise stated and S a non-empty set with two binary compositions addition (+) and multiplication (·) such that (S; +) is a semigroup, F(S) the set of all fuzzy subsets of S.
K.C. Gupta, M.K. Kantroo / Fuzzy Sets and Systems 117 (2001) 419–429
421
3. The T-intrinsic product We now deÿne T -intrinsic product of fuzzy subsets of S. Deÿnition 3.1 (The T-intrinsic product). Let , be any two fuzzy subsets of S, and T any t-norm on I . Let x ∈ S. Deÿne ∗T , the T -intrinsic product of and by ( ∗T )(x) =
x=
sup
P
ÿnite
ai bi
T {(a1 ); : : : ; (am ); (b1 ); : : : ; (bm )};
if we can express x = a1 b1 + · · · + am bm for some ai ; bi ∈ S and m ∈ N . Otherwise, deÿne ( ∗T )(x) = 0. Clearly, the product ∗T is commutative if multiplication in S is commutative. Theorem 3.2. The T-intrinsic product; where T is an inÿnitely distributive T-norm; is associative in E(S); where E(S) = { ∈ F(S): (x)T(x) = (x); for all x ∈ S} provided that multiplication in S is associative and is distributive over addition. Proof. Let ; ; ∈ E(S), and let x ∈ S. If x has no reduction in S, then (( ∗T ) ∗T )(x) = ( ∗T ( ∗T ))(x) = 0: Suppose that x does have reductions in S. Let x = t1 c1 + t2 c2 + · · · + tm cm ; (ti ; ci ∈ S) be any reduction of x in S. Firstly, suppose that some tj possesses no reduction in S. Then, we get ( ∗T )(tj ) = 0 and so T {( ∗T )(t1 ); : : : ; ( ∗T )(tm ); (c1 ); : : : ; (cm )} = 06( ∗T ( ∗T ))(x): Secondly, suppose that each ti has a reduction in S. Let ti =
ri X
aij bij ;
16i6m
j=1
be a reduction of ti in S. Then we get T {(a11 ); : : : ; (a1 r1 ); (b11 ); : : : ; (b1 r1 ); (c1 ); (a21 ); : : : ; (a2 r2 ); (b21 ); : : : ; (b2 r1 ); (c2 ); : : : ; (am1 ); : : : ; (am rm ); (bm1 ); : : : ; (bm rm ); (cm )} 6T {(a11 ); : : : ; (a1 r1 ); ( ∗T )(b11 c1 ); : : : ; ( ∗T )(b1r1 c1 ); (a21 ); : : : ; (a2 r2 ); ( ∗T )(b21 c2 ); : : : ; ( ∗T )(b2r2 c2 ); : : : ; (am1 ); : : : ; (am rm ); ( ∗T )(bm1 cm ); : : : ; ( ∗T )(bmrm cm )} 6 ∗T ( ∗T )(a11 b11 c1 + · · · + a1r1 b1r1 c1 + a21 b21 c2 + · · · + a2r2 b2r2 c2 + · · · + am1 bm1 cm + · · · + amrm bmrm cm ) = ∗T ( ∗T )(x):
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Taking the supremum over all possible reductions of each ti (16i6m), we obtain T { ∗T (t1 ); (c1 ); ∗T (t2 ); (c2 ); : : : ; ∗T (tm ); (cm )} 6 ( ∗T ( ∗T ))(x): Taking the supremum over all possible reductions of x in S, we ÿnally get (( ∗T ) ∗T )(x) = ( ∗T ( ∗T ))(x): This is sucient to claim that ( ∗T ) ∗T = ∗T ( ∗T ): Lemma 3.3. If ; ; ∈ F(S); such that ⊆ ; then ∗T ⊆ ∗T ;
∗T ⊆ ∗T :
Proof. Let X ∈ S. If x does not have any reduction in S, then the result is trivial. Otherwise, ( ∗T )(x) = 6
x= x=
sup
P
ÿnite
sup
P
ÿnite
ai bi ai bi
T {(a1 ); : : : ; (am ); (b1 ); : : : ; (bm )} T {(a1 ); : : : ; (am ); (b1 ); : : : ; (bm )}
= ( ∗T )(x): The other case follows similarly. Theorem 3.4. Let S be an additive monoid in which addition is commutative and multiplication is distributive over addition. If ; ; ∈ F(S); such that Im(); Im(); Im() ⊆ DT where DT = {x | xTx = x} and T is an inÿnitely distributive T -norm; then (1) ∗T ( +T ) ⊆ ∗T +T ∗T . (2) If max((x); (x)) 6 (0) = (0); then equality holds in (1). (3) ( +T ) ∗T ⊆ ∗T +T ∗T . (4) If max((x); (x)) 6 (0) = (0); then equality holds in (3). Proof. (1) Let x ∈ R. Suppose that x has a reduction x = a1 b1 +a2 b2 + · · · + am bm in R. Let bj = uj + vj , 1 6 j 6 m. We then have T {(a1 ); : : : ; (am ); T {(u1 ); (v1 )}; T {(u2 ); (v2 )}; : : : ; T {(um ); (vm )}} = T {T {(a1 ); (a2 ); : : : ; (am ); (u1 ); (u2 ); : : : ; (um )}; T {(a1 ); (a2 ); : : : ; (am ); (v1 ); (v2 ); : : : ; (vm )}} 6 T {( ∗T )(a1 u1 + a2 u2 + · · · + am um ); ( ∗T )(a1 v1 + a2 v2 + · · · + am vm )} 6 (( ∗T ) +T ( ∗T ))(x): Now, ( ∗T ( +T ))(x) =
x=
sup
P
ÿnite
ai bi
T {(a1 ); : : : ; (am ); ( +T )(b1 ); ( +T )(b2 ); : : : ; ( +T )(bm )}
(1)
K.C. Gupta, M.K. Kantroo / Fuzzy Sets and Systems 117 (2001) 419–429
=
x=
sup P ÿnite
sup
b1 =u1 +v1
ai bi
sup
b2 =u2 +v2
423
···
sup
bm =um +vm
T {(a1 ); : : : ; (am ); T {(u1 ); (v1 )};
T {(u2 ); (v2 )}; : : : ; T {(um ); (vm )}} · · ·
6 (( ∗T ) +T ( ∗T ))(x)
using Eq: (1):
If x has no reductions in S, then ( ∗T ( +T ))(x) = 0. Consequently, ∗T ( +T ) ⊆ ∗T +T ∗T . (2) For equality let us assume that max((x); (x)) 6 (0) = (0) for all x in S. Let x ∈ S, and let x = a + b. If neither a nor b has any reduction in S, then of course we get T {( ∗T )(a); ( ∗T )(b)} = 0 6 ( ∗T ( +T ))(x): So, suppose that a and b has any reduction in S. Let a = u1 v1 + · · · + um vm , and b = p1 q1 + · · · + pn qn be any reductions in S. Now, we obtain T {T {(u1 ); : : : ; (um ); (v1 ); : : : ; (vm )}; T {(p1 ); : : : ; (pn ); (q1 ); : : : ; (qn )}} = T {(u1 ); : : : ; (um ); (p1 ); : : : ; (pn ); T {(v1 ); (0)}; : : : ; T {(vm ); (0)}; T {(0); (q1 )}; : : : ; T {(0); (qn )}} 6 T {(u1 ); : : : ; (um ); (p1 ); : : : ; (pn ); ( +T )(v1 ); : : : ; ( +T )(vm ); ( +T )(q1 ); : : : ; ( +T )(qn )} 6 ( ∗T ( +T ))(u1 v1 + · · · + um vm + p1 q1 + · · · + pn qn ) = ( ∗T ( +T ))(x): Consequently, we obtain T {( ∗T )(a); ( ∗T )(b)} = sup P x=
ui vi
T {(u1 ); : : : ; (um ); (v1 ); : : : ; (vm )}; T {(p1 ); : : : ; (pn ); (q1 ); : : : ; (qn )} sup P
x=
pi qi
6 ( ∗T ( +T ))(x): Taking the supremum over all the sums of x = a + b of x we get ( ∗T +T ∗T )(x) 6 ( ∗T ( +T ))(x): This completes (2). One can apply a similar technique to prove (3) and (4). Lemma 3.5. If is a fuzzy subset of S; then the following two statements are equivalent: (a) ∗T ⊆ ; (b) T {(a1 ); : : : ; (am ); (b1 ); : : : ; (bm )}6 (a1 b1 + · · · + am bm ) for all ai ; bi ∈S. Proof. (a) ⇒ (b) Let ai ; bi ∈ S; 16i6m, we have T {(a1 ); : : : ; (am ); (b1 ); : : : ; (bm )} 6
x=
sup
P
ÿnite
!
ai bi
T {(a1 ); : : : ; (am ); (b1 ); : : : ; (bm )}
424
K.C. Gupta, M.K. Kantroo / Fuzzy Sets and Systems 117 (2001) 419–429
= ( ∗T )(x);
where x = a1 b1 + a2 b2 + · · · + am bm
6 (x) = (a1 b1 + · · · + am bm ): (b) ⇒ (a) Let x ∈ S. If x has no reductions in S, then ( ∗T )(x) = 0; so ( ∗T )(x) 6 (x): Suppose x has a reduction x = a1 b1 + a2 b2 + · · · + am bm in S. Then T {(a1 ); : : : ; (am ); (b1 ); : : : ; (bm )}6 (a1 b1 + · · · + am bm ) = (x): Taking the supremum over all reductions of x in S, we get ( ∗T )(x) 6 (x);
consequently ( ∗T )(x) ⊆ (x):
4. T-fuzzy subrings, ideals and T-intrinsic product In this section, we deÿne T -fuzzy subrings and ideals and analyze their T -intrinsic product. Deÿnition 4.1. Let ∈ F(R). We say that is a T-fuzzy subring of R, if (1) T ((x); (y)) 6 (x − y), (2) T ((x); (y)) 6 (xy) for all x; y in R, (3) Im ⊆ DT where DT = {x | xTx = x}. If is a T -fuzzy subring of a ring R, then for x ∈ R (x) = T ((x); (x)) = T ((x); (−x)) 6 (x − x) = (0): Deÿnition 4.2. Let ∈ F(R). We say that is a T-fuzzy left ideal of R, if (1) T ((x); (y)) 6 (x − y), (2) (y) 6 (xy) for all x; y in R, (3) Im ⊆ DT where DT = {x | xTx = x}. Similarly, is called a T-fuzzy ideal of R, if (1) T ((x); (y)) 6 (x − y), (2) (x) 6 (xy) for all x; y in R, (3) Im ⊆ DT where DT = {x | xTx = x}. is called a T-fuzzy ideal of R if it is both T -fuzzy left and right ideal of R, i.e. (1) T ((x); (y)) 6 (x − y), (2) max((x); (y)) 6 (xy) for all x; y in R, (3) Im ⊆ DT where DT = {x | xTx = x}. Since T ((x); (y)) 6 min((x); (y)) 6 (x); (y) for all x; y in R. Therefore if is either a T -fuzzy left ideal or a T -fuzzy right ideal of R, then is a T -fuzzy subring of R. Deÿnition 4.3. ∈ F(G), is called T-fuzzy subgroup of a group G if (1) T ((x); (y)) 6 (x − y) for all x; y in G. (2) Im ⊆ DT where DT = {x | xTx = x}. We note that from Deÿnitions 4.1–4.3 (x) 6 (0) follows quite naturally.
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425
Theorem 4.4. If is a fuzzy subset of a ring R; then the following two statements are equivalent: (a) is a T-fuzzy subring of R. (b) is a T-fuzzy subgroup of the additive group (R; +) and ∗T ⊆ . Proof. (a) ⇒ (b) If is a T -fuzzy subring of R, then clearly is a T -fuzzy subgroup of (R, +). Secondly if ai ; bi ∈ R; 1 6 i 6 m, we have T {(a1 ); : : : ; (am ); (b1 ); : : : ; (bm )} = T {T ((a1 ); (b1 )); : : : ; T ((am ); (bm ))} 6 T {(a1 b1 ); : : : ; (am bm )} 6 (a1 b1 + · · · + am bm ): By Lemma 3.5 this implies ∗T ⊆ . (b) ⇒ (a) It follows by using the deÿnition of a T -fuzzy subring and Lemma 3.5. Theorem 4.5. If ; are T-fuzzy subrings of a ring R such that ∗T = ∗T ; where T is an inÿnitely distributive t-norm; then ∗T is a T-fuzzy subring of R. Proof. Let x; y ∈ R. If either x or y has no reduction in R; then either ( ∗T )(x) = 0
or
( ∗T )(y) = 0:
Therefore, T (( ∗T )(x); ( ∗T )(y)) = 0 6 ( ∗T )(x − y): So we assume that x and y have reductions in R. Let x=
m X
ai bi ;
y=
i=1
n X
ci di :
i=1
P P Hence x − y = ai bi + (− ci )di . We then have T {(a1 ); : : : ; (am ); (− c1 ); : : : ; (− cn ); (b1 ); : : : ; (bm ); (d1 ); : : : ; (dn )} 6 ( ∗T )(x − y): Therefore, T (( ∗T )(x); ( ∗T )(y)) ! =
x=
sup
P
ÿnite ai bi
x=
sup
P
ÿnite ci di
6 ( ∗T )(x − y):
T {(a1 ); : : : ; (ar ); (− c1 ); : : : ; (− ct ); (b1 ); : : : ; (br ); (d1 ); : : : ; (dt )}
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K.C. Gupta, M.K. Kantroo / Fuzzy Sets and Systems 117 (2001) 419–429
We now show that Im ⊆ DT ( ∗T )(x)T ( ∗T )(x) ( =T
x=
x=
sup
P
ÿnite ai bi
sup
P
ÿnite cj dj
= sup P x=
T {(a1 ); : : : ; (an ); (b1 ); : : : ; (bn )}; )
T {(c1 ); : : : ; (cm ); (d1 ); : : : ; (dm )}
sup T {T ((a1 ); : : : ; (an ); (b1 ); : : : ; (bn )); T ((c1 ); : : : ; (cm ); (d1 ); : : : ; (dm ))} P
ai bi x=
cj dj
T {T ((a1 ); : : : ; (an ); (b1 ); : : : ; (bn )); T ((a1 ); : : : ; (an ); (b1 ); : : : ; (bn ))} ¿ sup P x=
ai bi
¿ T {(a)T(a1 )T · · · T(an )T(b1 )T(b2 )T · · · T(bn ) T(a1 )T(a2 )T · · · T(an )T(b1 )T(b2 )T · · · T(bn )} = T {(a1 ); (a2 ); : : : ; (an ); (b1 ); (b2 ); : : : ; (bn )}: P Taking the supremum over all reductions x = ÿnite ai bi , we get ( ∗T )(x)T ( ∗T )(x) ¿ ( ∗T )(x): But we know that ( ∗T )(x)T ( ∗T )(x) 6 ( ∗T )(x): Thus ( ∗T )(x)T ( ∗T )(x) = ( ∗T )(x). If x has no reduction in R, then ( ∗T )(x) = 0 = 0T 0 = ( ∗T )(x)T ( ∗T )(x): Hence ∗T is T -fuzzy subgroup of (R; +). Also, ( ∗T ) ∗T ( ∗T ) = ∗T ( ∗T ) ∗T = ∗T ( ∗T ) ∗T = ( ∗T ) ∗T ( ∗T ) ⊆ ∗T : Therefore by Theorem 4.4 ∗T is a T -fuzzy subring of R. Theorem 4.6. If is a T-fuzzy right ideal of R; and a T-fuzzy left ideal of R then ∗T ⊆ T; where T is an inÿnitely distributive T-norm. Proof. Let x ∈ R. If x has no reduction in R, then ( ∗T )(x) = 0 6 (T)(x). Let x = a1 b1 + a2 b2 + · · · + am bm ; ai ; bi ∈ R. Then T {(ai ); (bi )} 6 (ai ) 6 (ai bi ); 1 6 i 6 m.
K.C. Gupta, M.K. Kantroo / Fuzzy Sets and Systems 117 (2001) 419–429
427
We have T {(a1 ); : : : ; (am ); (b1 ); : : : ; (bm )} 6 T {(a1 b1 ); : : : ; (am bm )} 6 (a1 b1 + a2 b2 + · · · + am bm ) = (x): Taking the supremum over all reductions of x in R we get (i) ( ∗T )(x) 6 (x). Since is a fuzzy left ideal of R, we can similarly prove that (ii) ( ∗T )(x) 6 (x). Since Im , Im ⊆ DT , as proved in Theorem 4.5. Therefore from (i) and (ii) we get ( ∗T )(x) = ( ∗T )(x)T ( ∗T )(x) 6 (x)T(x) ∀x ∈ R: Hence ∗T ⊆ T. Theorem 4.7. If is a T-fuzzy left ideal of a ring R and is a T-fuzzy right ideal of R; then ∗T is a T-fuzzy ideal of R; where T is an inÿnitely distributive T-norm. Proof. From the ÿrst part of Theorem 4.5 it follows that ∗T is a fuzzy subgroup of (R; +). Secondly let x; y ∈ R. We show that ( ∗T )(x) 6 ( ∗T )(xy)
and
( ∗T )(y) 6 ( ∗T )(xy):
The above inequalities are obvious if x or y has no reduction in R. We may assume that x and y have reduction in R. Let x = a1 b1 + · · · + am bm ; ai ; bi ∈ R. Then xy = a1 (b1 y) + · · · + am (bm y). We have T {(a1 ); : : : ; (am ); (b1 ); : : : ; (bm )} 6 T {(a1 ); : : : ; (am ); (b1 y); : : : ; (bm y)} 6 ( ∗T )(xy): Therefore, ( ∗T )(x) 6 ( ∗T )(xy). Similarly, ( ∗T )(y) 6 ( ∗T )(xy). Consequently, max{( ∗T )(x); ( ∗T )(y)}6 ( ∗T )(xy): 5. The classiÿcation In this section, we deÿne E(R) a subset of F(R) and isolate its certain subclasses. It is shown that some of these subclasses are ideals of semigroup (E(R); ∗T ). Deÿnition 5.1. We deÿne E(R) as E(R) = { ∈ R: Im ⊆ DT }; where T is an inÿnitely distributive T -norm and DT = {x | xTx = x}. Theorem 5.2. (E(R); ∗T ) is a subsemigroup of the semigroup (F(R); ∗T ).
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Proof. Let ; ∈ E(R), then (x)T(x) = (x) and (x)T(x) = (x) for all x ∈ R. Let x ∈ R. As in Theorem 4.5, we have T {( ∗T )(x); ( ∗T )(x)} = ( ∗T )(x): Hence ∗T ∈ E(R). We now isolate the following subclasses of E(R). Deÿnition 5.3. (1) A−1 (R) = { ∈ E(R): +T ⊇ }: (2) A−s (R) = { ∈ E(R): +T ⊇ ; (− x) = (x) ∀x ∈ R}. (3) As (R) = { ∈ E(R): (− x) = (x) ∀x ∈ R}. (4) A0 (R) = { ∈ E(R): (x) 6 (0) ∀x ∈ R}. (5) A1 (R) = { ∈ E(R): +T ⊆ }. (6) A2 (R) = { ∈ E(R): +T = }. (7) A3 (R) = { ∈ E(R): +T ⊆ ; (x) 6 (0) ∀x ∈ R}. (8) A4 (R) = { ∈ E(R): +T ⊆ ; (− x) = (x) ∀x ∈ R}. Theorem 5.4. (1) A−1 (R) is an ideal of (E(R); ∗T ). (2) A−s (R) is an ideal of (E(R); ∗T ). (3) As (R) is an ideal of (E(R); ∗T ). (4) A0 (R) is an ideal of (E(R); ∗T ). (5) A3 (R) is an ideal of (E(R); ∗T ). (6) A4 (R) is an ideal of (E(R); ∗T ). Proof. (1) Let ∈ E(R); ∈ A−1 (R), then by Theorem 3.4 ∗T ⊆ ∗T ( +T ) ⊆ ∗T + ∗T : Hence ∗T ∈ A−1 (R). Similarly ∗T ∈ A−1 (R). (2) Let ∈ E(R); ∈ A−s (R). Let x ∈ R: x has a reduction in R i − x has a reduction in R. If x has no reduction in R, then ( ∗T )(x) = 0 = ( ∗T )(−x): Let x have a reduction in R. Then for x ∈ R T {(a1 ); : : : ; (an ); (b1 ); : : : ; (bn )} ( ∗T )(x) = sup P x=
ai b i
T {(a1 ); : : : ; (an ); (− b1 ); : : : ; (− bn )} = sup P x=
ai b i
= ( ∗T )(− x): Also from (1) ∗T ∈ A−1 (R). Hence A−s (R) is an ideal of E(R). (3) Obvious. (4) Let ∈ E(R); ∈ A0 (R). Let x ∈ R. If x has no reduction in R, then 0 = ( ∗T )(x) 6 ( ∗T )(0):
K.C. Gupta, M.K. Kantroo / Fuzzy Sets and Systems 117 (2001) 419–429
So let x have a reduction in R. Then ( ∗T )(x) =
sup T {(a1 ); : : : ; (an ); (b1 ); : : : ; (bn )} P
x=
ai bi
T {(a1 ); : : : ; (an ); (0); : : : ; (0)} 6 sup P x=
ai bi
6 ( ∗T )(0): Consequently, ∗T ∈ A0 (R). (5) Let ∈ E(R); ∈ A3 (R), then (x) 6 (0) for all x ∈ R. Therefore by Theorem 3.4 ∗T +T ∗T = ∗T ( +T ) ⊆ ∗T : Also ( ∗T )(x) 6 ( ∗T )(0) from (4). Hence ∗T ∈ A3 (R). (6) Let ∈ E(R); ∈ A4 (R). From (3) ( ∗T )(x) = ( ∗T )(− x); for all x ∈ R. Also, (x) = T ((x); (− x)); 6 ( +T )(x − x) = ( +T )(0) 6 (0)
for all x ∈ R:
Therefore by Theorem 3.4 it follows that ∗T + ∗T = ∗T ( + ) ⊆ ∗T : Consequently, ∗T ∈ A4 (R). Similarly, ∗T ∈ A4 (R). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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