Fuzzy semirings

Fuzzy Sets and Systems 60 (1993) 309-320 North-Holland

309

Fuzzy semirings J. A h s a n ,

K. Saifullah

Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

M. Farid Khan Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan Received March 1993 Revised May 1993

Abstract: In this paper we initiate the study of fuzzy semirings and fuzzy A-semimodules where A is a semiring and A-semimodules are representations of A. In particular, semirings all of whose ideals are idempotent, called fully idempotent semirings, are investigated in a fuzzy context. It is proved, among other results, that a semiring A is fully idempotent if and only if the lattice of fuzzy ideals of A is distributive under the sum and product of fuzzy ideals. It is also shown that the set of proper fuzzy prime ideals of a fully idempotent semiring A admits the structure of a topological space, called the fuzzy prime spectrum of A.

Keywords: Fuzzy semirings; fuzzy subsemimodules; fuzzy prime ideals; fuzzy irreducible ideals; fuzzy prime spectrum; fully idempotent semirings; von Neumann regular semirings.

Introduction The fundamental concept of a fuzzy set, introduced by Zadeh in his classic paper [16] of 1965, has been applied by many authors to generalize some of the basic notions of algebra (see, for example, [3, 7, 8, 10, 11, 13, 14[). The aim of this paper is to initiate the study of fuzzy semirings and fuzzy A-semimodules, where A is a semiring and A-semimodules are representations of A. Recall that a semiring is a set A together with two binary operations ÷ (addition) and • (multiplication) such that (A, + ) is a commutative semigroup, and (A, .) is a (generally) noncommutative monoid with 1 as its identity element: connecting the two algebraic structures are the distributive laws, a.(b+c)--a.b+a.cand (a+b).c=a.c+b.c, for a l l a , b, c ~ A . We shall always assume that (A, + , .) has an absorbing zero 0, that is, a ÷ 0 = 0 ÷ a = a and 0. a = a • 0-- 0 hold for all a E A (cf. [4]). Thus, all rings with identity elements are semirings. Moreover, if (L, v , ^ ) is a distributive lattice with 0 and 1, then L is a semiring with + = v , and . = A. In particular, the unit interval [0, 1] of real numbers is a semiring with + = max and • = min or, with + = min and • = max or, even with + = max and • = usual product of real numbers. Furthermore, as noted in [4, p. 20], for any nonempty set A, we have the semiring of all fuzzy subsets of A. Semirings have proven to be useful in studying automata and formal languages (cf. [1, 4, 5, 12]). On the other hand, the notions of automata and formal languages have been generalized and extensively studied in a fuzzy framework (cf. [9, 12, 15]). Thus one may expect that fuzzy semirings may prove useful in studying fuzzy automata and fuzzy formal languages. In this paper, however, we pursue an algebraic approach to investigate the concept of fuzzy semirings and related notions in order to set the ground for future work. In Section 1 we provide basic definitions and establish some preliminary results. In Section 2, we investigate fully idempotent semirings, that is, semirings all of whose ideals are idempotent. It is proved that such semirings are characterized by the property that each proper fuzzy Correspondence to: Dr. J. Ahsan, Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia. 0165-0114/93/$06.00 © 1993--Elsevier Science Publishers B.V. All rights reserved

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ideal is the intersection of fuzzy prime ideals containing it. In Section 3, we construct the fuzzy prime spectrum of fully idempotent semirings in a manner analogous to the construction of the prime spectrum in classical ring theory.

1. Preliminary definitions and lemmas Throughout, A will denote a semiring with an identity element 1 and an absorbing zero 0. An additively written commutative semigroup M with a neutral element 0 is called a right A-semimodule, MA, if A is a semiring and there is a function a :M x A ~ M such that if a (m, a) is denoted by ma, then the following conditions hold:

(i) ( m + m ' ) a = m a + m ' a, (ii) m(a + a') = ma + ma', (iii) m(aa') = (ma)a', (iv) m . l = m , (v) 0. a = m • 0 = 0, for all a, a' c A and m, m' e M (cf. [4, p. 138]). One can similarly define a left A-semimodule AM. A semiring A is a right semimodule over itself which will be denoted by A a. A subsemimodule N of a right A-semimodule M is a subsemigroup of M such that na • N for all n • N and a • A. By right (left) ideals of A, we mean subsemimodules of AA (AA). The word ideal will always mean a two-sided ideal of A, that is, an ideal which is both a right and a left ideal of A. An ideal generated by an element x will be denoted by (x). The sum and product of ideals of semirings are defined as in rings. A semiring A will be called fully idempotent if each (two-sided) ideal of A is idempotent (an ideal I is idempotent if I z= I). All von Neumann regular semirings (that is, semirings A for which x E xAx for each x in A (cf. [4, p. 4])) are fully idempotent. In the sequel, L will denote a complete lattice, that is, a set L with a partial order on it such that, for any subset X of L, infimum and supremum of X, denoted by Ax~xX and V x ~ x x , respectively, always exist. We also assume that L is a totally ordered distributive lattice with a least element 0 and a greatest element 1, in which the infinite meet distributive law:

holds for any X ~_ L and y • L. If S denotes the set of discourse, then all fuzzy subsets of S in the sequel are L-fuzzy subsets in the sense of Goguen [2], that is, a function from S into L. For convenience, however, we shall write fuzzy subset, instead of L-fuzzy subset. If L is the unit interval [0, 1] of real numbers, L-fuzzy subsets are fuzzy subsets in the usual sense [16]. A fuzzy subset A :S ~ L is nonempty if it is not the constant map which assumes the value 0 of L. For any fuzzy subsets A and/~ of S, A ~
(a • S),

(A v/~)(a) = A(a)v/x(a),

(a E S).

More generally, if {Ai: i • I} is a family of fuzzy subsets of S, then follows:

(AAi)(a)=6(Ai(a))

and

Ai/~i and

Vi Ai are defined as

(y,x,)(a)=y(,xi(a));

and will be called the intersection and union of the family {Ai: i • I} of fuzzy subsets of S. Generalizing the notion of a fuzzy module (cf. [3, 12, 13]) we formulate the following definition.

Definition 1.1. Let M be a right A-semimodule. A function h :M--~ L is called a fuzzy subsemimodule of MA, if the following conditions hold:

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(i) A(OM) 1, =

(ii) A(m + m')>~ A(m)^A(m'), for all m, rn' e M, (iii) A(ma)/> A(m), for all m E M and a E A. Thus, if A is a fuzzy subsemimodule of the right semimodule AA, then A(0A) = 1. In the sequel, fuzzy subsemimodules of AA are called fuzzy right ideals of the semiring A. Fuzzy left ideals of A are defined analogously. By a fuzzy ideal of A we mean a fuzzy subset of A which is both a fuzzy right and a fuzzy left ideal of A. Let A be a fuzzy subsemimodule of a right A-semimodule M, # a fuzzy ideal of A and x any element of M. For any expression form ]~_~ y~z~ of x, where the y/s are elements of M, z/s are elements of A and p e N (N denotes the set of natural numbers), construct the quantity A l~i~p [A(y~)A/~ (Zi)] which is an element of L. Corresponding to each expression form of x, we thus obtain an element of L in this fashion. "We define (A/z)(x) to be the supremum of all these elements of L. Hence we have the following definition. Definition 1.2. Let A be a fuzzy subsemimodule of a right semimodule MA and /z a fuzzy ideal of A. Then the fuzzy subset A/~ of M is defined by (A/X)(X)

V

=

wherexeM,

[ <~6

[A(yi)

A"(Zi)]]

y~M,z~Aandp~N.

Proposition 1.3. If A is" a fuzzy subsemimodule of MA and tx a fuzzy ideal of A, then AIX is a fuzzy subsemimodule of M. Proof. We have [A(y,) A Iu,(Zi)]] ~ ()=Y.~\ lYiZ,

1

~(0M)/~.

~.L ( 0 )

= 1.

p

Thus (A~)(0M)= 1. Again,

(a )(m ) =

V

[ A

. . . . 'y~;, , y/z i L1 ~ j ~ q

[A(y/)^~(zj)]]

and (A/z)(m') =

V m ' = Z rk= l y'd z'~

[,_
where m, m' E M. Thus (A/z)(m) A (A/~)(m ') m

=

~I ~v~

v

. "!Z'! tit '=N~, %1,r : I~¢l, k

1 I q

l<~k<~r

I A [A(y/)Aff(z/)lJ['-Z~k---r[A(yDA'~(z;3]jjl 1 r[,,,'='~-,Y,:' 1 ;~: r^

.... z;, ,yj~,,-, ~J-~,,

(using the infinite meet distributive law) rn=~l

lyi~/tn'=~_.~=ly'~z' ~

1

q

l~k~-r

(using the infinite meet distributive law) [A(y~) A/x (z~)]]

V

m +m'

=Z)

[A,

-~,~. [A(Y;")Al~(z")]]=Alz(m+m').

.v~"zT'

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Note that for an arbitrary expression form of m + m', say, Y~=ly;"z;" it is not necessarily possible to write m as ~=1 yj"z;"and m' as El=t+1 y;"z;". Also,

V

(A/z)(m) =

[l
m = z q _ I Y:Z:

V

"~

[I<
m = Eq_ 1y:z~

<~"

V

[l
= A/x (ma). Thus (A/x)(ma)/> (A/z)(m) for m E M and a c A. Hence A/z is a fuzzy subsemimodule of M.

Corollary. If h and t* are ideals of A then At* is a fuzzy ideal of A, called the product of h and tz. Remark. If A and /x are fuzzy ideals of A, then A ^ / , is clearly a fuzzy ideal of A. In general, A^/~ ~Ap~. Definition 1.4. Let A and/x be fuzzy ideals of A. The fuzzy subset A +/x of A is defined by (a+/z)(x)=

V

[a(y)atz(Z)]

forxeA.

x=y+z

Proposition 1.5. For fuzzy ideals a and tz of A, A + Ix is a fuzzy ideal of A (called the sum o f ^ and tz ). Proof. For any x, x' e A (~ + ~ ) ( x ) ^ ( a + ~)(x')

(using the infinite meet distributive law) =

V

[[,~(y)^t~(z)]^[,~(y')^~(z')]]

x--y+z x'=y'+Z'

=

V x-y+z x'=y'+z

<-

[[My)^A(y')lA[ix(z)^tx(z')]] '

V

[a(y + y ' ) ^ t x ( z + z')]

x +x'~(y+y')+(z

+ z')

~(A + ~)(x +x'). Again,

(;~ + ~)(x) = X ~ yV+ Z [A(y)^y(z)] <~

V

[A(Ya)^lx(za)]

V

[A(y') ^ tx(z')]

xa =ya + za

<~

xa=y' +z '

= (,~ + I.~)(xa). Hence A +/z is a fuzzy ideal of A.

(where a is any element of A)

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2. Fully idempotent semirings A semiring A (in the crisp sense) is called fully idempotent if each (two-sided) ideal of A is idempotent (an ideal I is idempotent if 12-- I). Concerning these semirings, we prove the following characterization theorem.

Theorem 2.1. The following assertions for a semiring A are equivalent: (1) A is fully idempotent, (2) each fuzzy ideal o f A is idempotent (a fuzzy ideal A o f A is called idempotent if AA = A2 = A), (3) for each pair of fuzzy ideals A and tx of A, A AIx = AIx.

If A is assumed to be commutative (that is, xy = yx for all x, y E A), then the above assertions are equivalent to: (4) A is yon Neumann regular. Proof. ( 1 ) ~ ( 2 ) . Let 8 be a fuzzy ideal of A. For any x • A,

82(X) = (88)(X)

v

1

x =W.~L~y,z,

:

~<

V [ /~ (8(yiZi) AS(YiZi))] ~-=~{~lytzt l<~i<~p

=x=~?jy,z [ [l<6~pS(YiZi)]A[l<6
V

x- E~' ~y,zi

I

[8(x)^8(x)l=a(x).

Since each ideal of A is idempotent, therefore, (x) -- (x) 2 for each x • A. Since x • (x) it follows that x • (x) 2 = A x A A x A . Hence, x = •q=l aixa[bixb; where ai, a;, hi, b~ • A and q • N. Now,

8(x) = 8 (x) A 8(X) <~8(aixa;) A 8(bixb;)

(1 <~ i ~< q).

Therefore,

[8(aixa[)Aa(bixb[)]

8(x) ~< A

<~ [6(aixai) .v=~q=latxa[bixb, 1 <-'<-q

V

< =

A

8(bixb

[,Ar[(a(yj),',a(zj)l 1

(aS)(x)

=

82(x).

Thus 82= 8. (2) ~ (1). Let 1 be an ideal of A. Thus 81, the characteristic function of L is a fuzzy ideal of A. Hence, (Sz) 2= 81. Therefore, 8181 = ,51. Hence, 812 = 8/. It follows that 12= L Hence (2)<=>(1). (1) ~ (3). Let a and IX be any pair of fuzzy ideals of A. For any x • A

(AIX)(X) =

V x = ~,~' I y , z ,

[ ~<6< (A(yi)ml~(Zi))] 1

p

<~ V [ ~6<- (A(YiZi)AIX(YiZi))I x=~_,f_ly,z' 1 p

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=x=YY,y,z,[[l<6
V [;~(x),,ix(x)] x =El_~yizi

= A(X),,Ix(x) = (,~ ,,Ix)(X). Again, since A is fully idempotent, (x) = (x) 2, for any x E A. Hence, as argued in the first part of the proof of this theorem, we have (~ ,, Ix)(x) = A(x) ,, Ix (x)

<~ V [I<.i/\<~p[A(YiAIx(Zi)]] ) X = ~ { % I YiZi

= (Aix)(x). Thus A/x Ix = AIx. ( 3 ) ~ ( 1 ) . Let A and Ix be any pair of fuzzy ideals of A. We have, A^Ix = A/x. Take Ix =A. Thus, A ^ A = A2, that is, A = A2, where A is any fuzzy ideal of A. Hence, (3) ~ (2). Since we already proved that (1) and (2) are equivalent, hence it follows that ( 3 ) ~ ( 1 ) and so (1)¢:>(3). This establishes (1)¢:> (2)¢:> (3). Finally, if A is commutative then it is easy to verify that (1) ¢:~(4). Next, we prove another characterization theorem for fully idempotent semirings.

Theorem 2.2. The following assertions for a semiring A are equivalent: (1) A is fully idempotent, (2) the set of all f u z z y ideals of A (ordered by <~) forms a distributive lattice ~m under the sum and intersection o f f u z z y ideals with A ^IX = AIX, for each pair o f f u z z y ideals A, tx of A.

Proof. (1) ~ (2). The set ~ z of all fuzzy ideals of A (ordered by ~<) is clearly a lattice under the sum and intersection of fuzzy ideals. Moreover, since A is a fully idempotent semiring, it follows from Theorem 2.1(3) that A ^/x = AIX for each pair of fuzzy ideals A, IX of A. We now show that Le4 is a distributive lattice, that is, for fuzzy ideals A, 8 and ~: of A, we have [(A ^ 6) + ~] = [(A + ~ ) ^ ( 8 + ~)]. For any x e A,

[(A^6)+~:](x)= V [(AA6)(y)^~(z)] x~y+z

=

V

[A(y)Aa(y)^~(z)]

x=y+z

=

V

[A(y)^8(y)^~(z)^~(z)]

x=y+z

=

V

[A(y)^~(z)l^[6(y)^~(z)]

x~-y+z

-< V

[(A + ~)(x),,(8 + ~)(x)]

x=y+z

because, for y + z = x, A(y) ^ ~(z) ~<(A + ~)(x) and, similarly, 6 ( y ) ^ ~(z) ~ (6 + ~:)(x) = (a + ~ ) ( x ) ^ (8 + ~ ) ( x ) = [(,~ + ~ ) / , (8 + ¢ ) ] ( x )

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Again, [(,~ + ~) ^ (8 + ~)](x) = [(~ + ~ ) ( ~ + ~)](x)

] [(A + ~:)(y,) ^ (fi + sC)(zi)]| x = ~5[f I yiz,

d

1

=x__~y,y+z [l~i~p[ [v,=rVs,(A(ri)A'(Si)']m[z+=tV+u,(8(ti)A'(Ui')]

]]

(using the infinite meet distributive law)

=x=?zVyiz,[l~Ai~p[y~_t[Vs,[('~l'(ri)A'(Si)A~(Si)At~(ti)A~(Ui))]]] <~ V x=~p

<-

[ A ly, Z,

l~i~p

[ V

[(A(r/,)^6(r/i)^~(s/,)^~(s,ui)^~(riui)]]]

yi=rs+si LZi=fi+lli

Viy,z, 1As vi=r,+si x =~,~'. Zi=ti+lti

x:Z~%y,z, 1

V

~< x

:

[(a^8)+¢](x)

S,~'t >',z~

[(a ^~) +

~](x).

Thus [(A + ~ ) ^ ( 6 + ~:)] = [(A ^ 6 ) + ~]. (2) ~ (1). Suppose that the set, ~fA, of all the fuzzy ideals of A (ordered by ~<) is a distributive lattice under the sum and intersection of fuzzy ideals with A ^/x = A/x for each pair of fuzzy ideals A, /x of A. Then for any fuzzy ideal A of A, we have A2 = A • A = A ^A = g.l.b, of {A, A} = A. Hence A is fully idempotent.

fuzzy prime ideal of A if for fuzzy ideals A ~ is called fuzzy irreducible if for fuzzy ideals A, /~ of A,

Definition 2.3. A fuzzy ideal ~ of a semiring A is called a

and /~ of A, A/~ ~ A ~ < ~ A^Iz = ~ : ~ A = ~ or/~ = ~:.

or /~ ~ ;

Let A be a fully idempotent semiring. For a fuzzy ideal ~ of A, the following conditions are equivalent: (1) ~ is a fuzzy prime ideal, (2) ~ is a fuzzy irreducible ideal. T h e o r e m 2.4.

Proof. (1) Assume that ~: is a fuzzy prime ideal. We show that ~ is fuzzy irreducible, that is, for fuzzy ideals A, /~ of A, A/x/~ = £ ~ A = £ or /~ = £. Since A is a fully idempotent semiring, the set of fuzzy ideals of A (ordered by ~<) is a distributive lattice under the sum and intersection of fuzzy ideals by

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F u z z y semirings

Theorem 2.2. This implies that ~ = g.l.b, of {h, Ix}, since ~ = h ^Ix. Thus it follows that ~ ~
V l~i(X) = y (t~i(X))~ V (Ai(xr))=

y

hi(xr ).

Similarly, Vi hi(x) <~ Vi M(rx). Finally, we show that Vi hi(x ÷ y) ~ Vi hi(x) ^ Vi Ai(y), for any x, y e A. Consider

V hi(x) A iV~ l h,(y)= iVe l hi(X)a jVE I hi(y)

iel

= [ V hi(x)] ^ y [h,(y)] (by the infinite meet distributive law) (by the infinite meet distributive law)

= V IV

-
where hi = max{h/, hi}; note that hi ~ {hi:i e I}

/[hl(x + y)] I,J

<~Vi [Ai(x + y)] = V

i

+ y).

Thus Vi Ai is a fuzzy ideal of A. Clearly A ~< V~ A~and Vi Ai(a) = V~ A/(a) = a. Thus Vi A~is the 1.u.b. of

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317

Hence, by Zorn's Lemma, there exists a fuzzy ideal ~: of A which is maximal with respect to the property that A ~< ~: and ~ ( a ) = a. We now show that ~: is a fuzzy irreducible ideal of A. Suppose ~: = ~5~^82, where 6~ and 62 are fuzzy ideals of A. Since A is assumed to be a fully idempotent semiring, so by Theorem 2.2, the set of fuzzy ideals of A (ordered by ~< ) is a (distributive) lattice under the sum and intersection of fuzzy ideals. Hence ~ = 61 ^ 62 = g.l.b.{6~, 62}. This implies that ~ ~< 6x and <~ 62. We claim that either {: = 6~ or ~ = 62. Suppose, on the contrary, ~ ~ 6~ and {: g a2. Since ~ is maximal with respect to the property that ~ ( a ) - - a and since ~ 6 ~ and ~:~62, it follows that 6~(a) ~ a and a2(a) ~ a. Hence a = ~:(a) = (61 ^ 62)(a) = {61(a)/x 62(a)} ~ a, which is absurd. Hence either ~ = 6~ or ~: = ~52. This proves that ~ is a fuzzy irreducible ideal. Hence by Theorem 2.4, ~ is a fuzzy prime ideal. We are now ready to prove our main characterization theorem for fully idempotent semirings. Theorem 2.6. The following assertions for a semiring A are equivalent:

(1) (2) fuzzy (3) If, in (4)

A is fully idempotent, the lattice of all fuzzy ideals of A (ordered by <~) is distributive under the sum and intersection of ideals with A ^ Ix = A/x, for each pair of fuzzy ideals A, Ix of A, each fuzzy ideal is the intersection of those fuzzy prime ideals of A which contain it. addition, A is assumed to be commutative, then the above assertions are equivalent to: A is yon Neumann regular.

Proof. ( 1 ) ~ (2). This is Theorem 2.2. (2) ~ (3). Let A be a fuzzy ideal of A and let {A,: a e £2} be the family of all fuzzy prime ideals of A which contain A. Obviously, A < ~ / ~ ~n A~. We now prove t h a t / ~ n A ~ ~< A. Let a be any element of A. By Lemma 2.5, there exists a fuzzy prime ideal A~ (say) such that A ~< A¢ and A(a) = Ao(a). Thus A~ e {A~: a e ~2}. Hence / ~ , 2 A ~ ~(4). This completes the proof of the theorem. We close this section with the following fuzzy theoretic characterization of von Neumann regular semirings. First we recall the following definition. Definition 2.7. Let A and Ix be fuzzy subsets of a semiring A. Then the fuzzy subset h o/z is defined by

(A oix)(x) = Vx=yz [A(y)^ix(z)], for all x c A. Theorem 2.8. The following assertions for a semiring A are equivalent:

(1) A is yon Neumann regular, (2) for any right ideal R and any left ideal L of A, R fq L = RL, (3) for any fuzzy right ideal A and any fuzzy left ideal tx of A, A A Ix = A o Ix. Proof. For (1)<=>(2), we refer to Golan [4, Proposition 5.27, p. 63]. So we prove (1)<::>(3) only. Suppose that A is von Neumann regular. Let 6 be any fuzzy right ideal a n d / z any fuzzy left ideal of A.

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We show that A ^/x = A o/x. Let x e A. Then

(Ao/~)(x)= V [A(y)^/x(z)] x =yz

<~

V

[A(YZ)^tz(YZ)] =

x=yz =

V [A(x)^/z(x)]

x=yz =

Thus h o/z ~
(A ^/z)(x) = (A(x)^/z(x)) ~<(A(xa)^tz(x)) ~< V (A(y)^/z(Z)) = (A o/x)(x). x =yz

Hence h ^/z = A o/z. Conversely, assume that A ^/z = h o/x for any fuzzy right ideal h and any fuzzy left ideal/z of A. We show that A is von Neumann regular. Let x e A. x A and A x are the principal right and left ideals of A, respectively, which are generated by x. Thus, if 6xA and 6Ax denote, respectively, the characteristic functions of x A and A x , then clearly GA and 6Ax are fuzzy right and fuzzy left ideals of A. Hence, by the assumption 6xA ^ 6 A x = 6xA ° 6Ax. This implies that x A ~ A x = x A A x . Thus x • x A fq A x = x A A x ~_ xAx. Hence, there exists a • A such that x = xax, thus showing that A is von Neumann regular.

3. Fuzzy prime spectrum of a fully idempotent semiring In this section A will denote a fully idempotent semiring, ~z will denote the lattice of fuzzy ideals of A, and '-'~A the set of all proper fuzzy prime ideals of A. For any fuzzy ideal A of A, we define O A = {jl~ E ' ~ A : /~ ~ 1"£} and ~ ' ( ~ A ) = {OA'- /~ E "~A}" A fuzzy ideal A of A is called proper if A # ~, where the fuzzy ideal ~d of A is defined by d ( x ) = 1, for all x • A. We prove the following.

Theorem 3.1. The

s e t r(o'~A) f o r m s a (classical) topology on the set ~ A . The assignment A ~ @ is an isomorphism between the lattice ~L#Ao f f u z z y ideals o f A and the lattice o f open subsets o f O%~z.

Proof. First we show that Z ( ~ A ) forms a topology on the set ~ a - Note that O~ = {/x e off~A: q ~ / x } = ~b, where ~b is the usual empty set and q~ is the f u z z y zero ideal of A defined by q~(a) = 0 for all a e A. This follows since q~ is contained in every fuzzy (prime) ideal of A. Thus O~ is the empty subset of Z ( ~ A ) . On the other hand, we have O~ = {/z e ~ z : ~/~P tX} = ~ m . This is true, since O~A is the set of proper fuzzy prime ideals of A. So O~¢ = ff~z is an element of T ( ~ A ) . NOW, let O8,, ~ 2 E r(,~bA) with 61 and 6 2 in (~A" Then Oa, rh O~2= {/z e ~i~A: ~1 ~/~1.£ and 62~/~p}. Since A is fully idempotent, therefore, 6162 = ~}1^ ~2, by Theorem 2.1. Since/z is fuzzy prime, so 6162 ~
U 19~:,= U {jt£ E o'~bA: ~i~/~ /.f} i~l

i~l

= {]d. E ff~A: 3k e I so that ~k ~P/Z}.

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Note that (5,(x,) A 52(G) A 53(%) A ’ . *) c si (x) = v i rel 1 _I(=X,+l~+X~+.~. where only a finite number of the x,‘s are not 0. Thus, since t,(O) = 1, therefore, we are considering the infimum of a finite number of terms, because the l’s are effectively not being considered. Now, if for some k E I, tk $ p, then there exists x E A such that [,&) > P(X). Consider the particular factorization of x for which x k =x and xi = 0 for all i #k. We see that t,(x) is an element of the set whose supremum is defined to be (ZZ,tl 5,) (x). Th us, (St, 5;) (x) 2 &AX)> P(X). Thus (Eisl5,) (x) > P(X). H ence, we have Xi,, 5, $ p. Hence, tl, $ p for some k E I implies that Ertl 6; $ p. there exists an element x E A such that Conversely, suppose that Xi,, 5; $ + Therefore, (Xi,, 5;) (x) > p(x). This means that _~=X,+X,+XI+.,_ V [51(x1) A 52(X*) A 54x3) A.

. .I > CL(x).

Now, if all the elements of the set, whose supremum we are taking, are individually less than or equal to p(x), then we have i ;,ri

1 (x) = _+:,,+...

(~,(x,)A\52(X2)A53(Xj)A’

“>

d P(X)

which does not agree with what we have assumed. Thus, there is at least one element of the set (whose supremum we are taking), say, ~,(X;)A52(X;)A53(X;)A. ” which is greater than p(x) (X=X;+X;+X;+..’ being the corresponding breakup of x, where only a finite number of the xi’s are nonzero). Thus ~,(X;)A~Z(X;)A~~(X;)~’ That is,

..>~(x)b~(x;)~~(X;)~~(x;)~....

5,(XI)A\52(X;)A53(X;)A. That is,

“>~(X:)A~(X;)A~(X;)A’

5,(X;)A52(X;)A53(X;)A.’

“.

‘>P(X;)

where P(X;) = P(X!)AP(x;)AP(X;)A’

”

(P E 1).

Hence, &,(xL) > I. It follows that &, + p for some Hence, the two statements, that is, (i) Cit15,+~, and (ii) & + p for some p E I

p E N.

Hence, Ej,, &$ p +&, $ p for some

p E N.

are equivalent.

Hence

because, E:,l I 5, is also a fuzzy ideal of A. Thus, lJie, 0c8 E r(S%&). Hence it follows that r(%!Q forms a topology on the set E&. Let 4 :2$, + (SPA) be the mapping defined by h H 0,. It follows from the above that the prescription +(A) = 0, preserves finite intersection and arbitrary union. Thus 4 is a

J. Ahsan et al. / Fuzzy semirings

320

lattice homomorphism. To conclude the proof, we must show that ~b is bijective. In fact, we need to prove the equivalence 6~ = 6 2 if and only if O8, = O82 for 61, 62 in ~(~Z' Suppose that O8, = O82. If 61 ~ 62, then there exists x E A such that 61(x) ~ 62(x). Thus, either 6~(x) > 62(x) o r 62(X) > 61(X ). Suppose that 61(x) > 62(x). Using Lemma 2.5, there exists a fuzzy prime ideal t~ of A such that 32 ~ 62(x) =/x(x). Therefore, 61(x) > ~(x). Thus, /z e O8,. Our assumption is that O8, = O8~. Hence, we have/z e O82. Hence 62 ~/x. This is a contradiction. If, in the beginning, we had assumed that 62(X)>6a(x) then, again, by similar reasoning, we get a contradiction. Thus, O81 = O82 implies that 61 = 62. Conversely, if 61 = 32, then, by definition, it is obvious that O8, = O82. Thus, we have proved that 6~ = 62 if and only if O8, = 082 for 61 and 32 in "~A" This completes the proof of the theorem. The set ~ A will be called the fuzzy prime spectrum of A and the topology r ( ~ A ) constructed in the above theorem will be called the fuzzy spectral topology on ~ A . The associated topological space will be called the fuzzy spectral space of A.

Acknowledgements The authors are grateful to the referees for suggesting improvements. The first two authors would like to acknowledge the support provided by the King Fahd University of Petroleum and Minerals.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

S. Eilenberg, Automata, Languages and Machines, Vol. A (Academic Press, New York, 1974). J.A. Goguen, L-fuzzy sets, J. Math. AnaL AppL 18 (1967) 145-174. J.S. Golan, Making modules fuzzy, Fuzzy Sets and Systems 32 (1989) 91-94. J.S. Golan, The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science, Pitman Monographs and Surveys in Pure and App. Maths., no. 54 (Longman, New York, 1992). W. Kuich and A. Salomaa, Semirings, Automata, Languages (Springer-Verlag, Berlin, 1986). N. Kuroki, Two-sided purity on regular semigroups, Comment. Math. Univ. St. Paul 24 (1975) 21-26. N. Kuroki, Fuzzy bi-ideals in semigroups, Comment. Math. Univ. St. Paul 28 (1979) 17-21. N. Kuroki, On fuzzy ideals and fuzzy bi-ideals in semigroups, Fuzzy Sets and Systems 5 (1981) 203-215. E.T. Lee and L.A. Zadeh, Note on fuzzy languages, Inf. Sci. 1 (1969) 421-434. Wang-Jin Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 (1982) 133-139. Wang-Jin Liu, Operations on fuzzy ideals, Fuzzy Sets and Systems 11 (1983) 31-41. C.V. Negoita and D.A. Ralescu, Applications of Fuzzy Sets to System Analysis (Birkhauser, Basel, 1975). Fu-Zheng Pan, Fuzzy finitely generated modules, Fuzzy Sets and Systems 21 (1987) 105-113. A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517. W.G. Wee and K.S. Fu, A formulation of fuzzy automata and its application as a model of learning systems, IEEE Trans. Systems Sci. Cybernetics SSC-5 (1969) 215-223. L.A. Zadeh, Fuzzy sets, Inf. Contr. 8 (1965) 338-353.