Normal L-fuzzy ideals in semirings

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sets and systems ELSEVIER

Fuzzy Sets and Systems 82 (1996) 383-386

Normal L-fuzzy ideals in semirings Young Bae Jun a, j. Neggers b,,, Hee Sik Kim c aDepartment of Mathematics Education, Gyeongsang National University, Chinju 660-701, South Korea bDepartment of Mathematics, College of Arts & Science, University of Alabama, 345 Gordon Palmer Hall, Tusealoosa, AL 35476-0350, USA CDepartment of Mathematics Education, Chungbuk National University, Chongju 360-763, ,South Korea

Received March 1995; revised July 1995

Abstract

In this paper we introduce the notion of a normal L-fuzzy left (resp. right) ideal in a semiring, and investigate some properties. Keywords: Semiring, (normal) L-fuzzy ideal

Following the introduction of fuzzy sets by Zadeh [ 10], the fuzzy set theory developed by Zadeh himself and others have found many applications in the domain of mathematics and elsewhere. In 1982, Liu [6] defined and studied fuzzy subrings as well as fuzzy ideals in rings. Subsequently, Mukherjee and Sen [7], Swamy and Swamy [8], and Zhang Yue [9] fuzzified certain standard concepts/results on rings and ideals. The theory of semirings has been studied by many authors [1-3, 5]. This paper is a continuation of [4]. In [4] we fuzzified the concept of left (resp. right) ideals in semirings. In this paper we introduce the notion of normal L-fuzzy left (resp. right) ideals in semirings, and investigate some interesting results. By a semiring we mean a set R endowed with two associative binary operations called addition and multiplication (denoted by + and., respectively) satisfying the following conditions: (i) addition is a commutative operation,

* Corresponding author.

(ii) there exists 0 E R such that x + 0 = x and x0 = 0x = 0 for each x E R, and (iii) multiplication distributes over addition both from the left and from the right. For the rest of the paper R is taken to be a semiring unless otherwise specified. A non-empty subset A of R is a left (resp. right) ideal if x, y E A and r E R imply that x + y E A and r x E A (resp. xr E A).

Throughout this paper L = (L,~<,A,V) will be a completely distributive lattice, which has the least and the greatest elements, say 0 and 1, respectively. Let X be a non-empty (usual) set. An L-fuzzy set in X is a map # : X L, and ~ ( X ) will denote the set of all L-fuzzy sets in X. If #,v E ~-(X), then #C_v if and only if # ( x ) ~ v ( x ) for all x E X, and p Cv if and only if # c v and # ~ v. It is easily seen that ~ ( X ) = ( ~ ( X ) , C , A , V ) is a completely distributive lattice, which has the least and the greatest elements, say 0 and 1, respectively in natural manner, where 0(x)=0, l ( x ) = l for all xEX.

0165-0114/96/$15.00 Copyright(~) 1996 Elsevier ScienceB.V. All rights reserved SSDI 0165-01 14(95)00275-8

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Y.B. Jun et al./Fuzzy Sets and Systems 82 (1996) 383-386

Definition 1. Let # E o~(R). Then # is called an Lfuzzy left (resp. right) ideal of R if for all x, y E R, (i) # is an L-fuzzy subsemigroup of (R, +); that is, #(x + y) ~> min{p(x), #(y)}, (ii) #(xy) >>.#(y) (resp. #(xy) >~#(x)). The following theorem shows that the concept of an L-fuzzy left (resp. right) ideal of R is an extension of a left (resp. right) ideal of R.

Theorem 5. Let # be an L-fuzzy left (resp. right)

ideal of R and let #+ be an L-fuzzy set in R defined by #+(x) = #(x) + 1 - #(o)

for all x E R. Then #+ is a normal L-fuzzy left (resp. right) ideal of R containing #. Proof. Let x, y E R. Then

min{p+(x), #+(y)} = min{#(x) + 1 - #(0), Theorem 2. Let A be a non-empty subset of R and let # be an L-fuzzy set in R such that # is into {0, 1 },

# ( y ) + 1 - #(0)} = m i n { # ( x ) , # ( y ) } + 1 - #(0)

so that # is the characteristic function of A. Then # is an L-fuzzy left (resp. right) ideal of R if and only if A is a left (resp. right) ideal of R.

~< #(x + y ) + 1 - #(0) = #+(x + y)

Proof. Suppose # is an L-fuzzy left (resp. fight) ideal of R. Let x, y E A and r E R. Then #(x + y ) ~> m i n { # ( x ) , p ( y ) } = 1 and #(rx)>>.#(x) = 1 (resp. #(xr)>>.#(x) = 1), which imply that #(x + y) = 1 and #(rx) = 1 (resp. #(xr) = 1). Hence, x + y E A and rx E A (resp. xr E A). Consequently, A is a left (resp. right) ideal of R. Conversely, assume that A is a left (resp. right) ideal of R. Let x, y E R . Ifx, y EA, thenx+yEA a n d x y E A. Thus, #(x + y ) = 1 = min{p(x), # ( y ) } and #(xy) = 1 = # ( y ) (resp. #(xy) = 1 = #(x)). If x, y ¢ A, then #(x) = # ( y ) = 0 and so clearly #(x + y ) 1> min{#(x), #(y)} and #(xy) >~#(y) (resp. #(xy)>Jp(x)). Ifxf[A and y E A then xy E A (resp. yx E A), #(x) = 0 and # ( y ) = 1. Hence, #(x + y)~>0 = m i n { # ( x ) , # ( y ) } and #(xy) = 1 = # ( y ) (resp. #(yx) = 1 = #(y)). A similar argument x E A and y ~ A leads to completion of the proof. []

and

#+(xy) = #(xy) + 1 - #(0) >>.#(y) + 1 - #(O) (resp../>#(x) + 1 - #(0)) = # + ( y ) (resp. = #+(x)). Hence, #+ is an L-fuzzy left (resp. right) ideal of R. Clearly, #+(0) = 1 and # c #+. This completes the proof. [] Noticing that p C # + , we have the following corollary. Corollary 6. l f # is an L-fuzzy left (resp. right) ideal of R satisfying #+(x) = Ofor some x E R, then #(x) = 0 also.

L e m m a 3 (Jun et al [4]). I f # is anL-fuzzy left (resp.

right) ideal of R, then the set

Using Theorem 2 and Definition 4, we obtain the following theorem.

R. = {x E R I #(x) = #(0)}

is a left (resp. right) ideal of R.

Theorem 7. For any left (resp. right) ideal A of R,

the characteristic function 7A of A is a normal L-fuzzy left (resp. right) ideal of R and RzA = A.

For a given L-fuzzy left (resp. fight) ideal of R, we note that #(0) is the largest value o f # . It is often convenient to have #(0) -- 1.

Theorem 8. Let # and v be L-fuzzy left (resp. right) ideals of R. I f # C v and #(0) -- v(0), then R~ CRv.

Definition 4. An L-fuzzy left (resp. fight) ideal # of R is said to be normal if #(0) = 1.

Proof. Assume that # C v and #(0) = v(0). I f x E R~ then v(x)/> #(x) = #(0) = v(0). Noticing that v(x) <~

Y.B. Jun et al./Fuzzy Sets and Systems 82 (1996) 383-386 v(0) for all x ER, we have fix) = v(0), that is, x E Rv. This completes the proof. []

385

Proof. Let x, y E R. Then #f(x + y) = f ( # ( x + y))

Corollary 9. / f # and v are normal L-fuzzy left (resp. right) ideals of R satisfying # C v, then R# C R~.

>/f(min{#(x), #(y)}) = r a i n { f (#(x)), f ( # ( y ) ) }

Theorem 10. An L-fuzzy left (resp. right) ideal # of R is normal if and only if #+ = #.

= min{#f(x), # f ( y ) } and

Proof. The sufficiency is obvious. Assume that # is

#f(xy) = f(#(xy))

a normal L-fuzzy left (resp. right) ideal of R and let x E R. Then #+(x) = #(x) + 1 - #(0) = #(x), and hence #+ = #. E3

Theorem 11. / f # is an L-fuzzy left (resp. right) ideal of R, then (#+)+ = #+. Proof. For any x E R we have (#+)+(x) = / t + ( x ) + 1 - #+(0) = #+(x), completing the proof.

[]

Corollary 12. If # is a normal L-fuzzy left (resp. right) ideal of R, then (#+)+ = #. Theorem 13. Let # be an L-fuzzy left (resp. right) ideal of R. I f there exists an L-fuzzy left (resp. right) ideal v of R satisfying v+ C I~, then # is normal.

>~ f ( # ( y ) ) (resp. >ff ( p ( x ) ) ) = #f(y)

(resp. = #f(x)).

Hence, # f is an L-fuzzy left (resp. right) ideal of R. If f ( # ( 0 ) ) = 1 then clearly # f is normal. Assume that f(t)>~t for all t E [0,#(0)]. Then #f(x) = f(p(x))>t#(x) tbr all x E R, which proves that #c#f.

Theorem 16. Let # be a non-constant normal Lfuzzy left (resp. right) ideal of R, which is maximal in the poset of normal L-fuzzy left (resp. right) ideals under set inclusion. Then # takes only the values 0 and 1. Proofi Note that #(0) = 1. Let x E R be such that

Proof. Suppose there exists an L-fuzzy left (resp. right) ideal v of R such that v+ C #. Then 1 = v+(0) ~<#(0), whence ~ ( 0 ) = 1. The proof is complete. []

#(x) 7£ 1. It is sufficient to show that #(x) = 0. Assume that there exists a E R such that 0 < /ffa) < 1. Define an L-fuzzy set v : R ~ L by v(x) = ½{#(x) + #(a)} for all x E R. Then clearly v is well-defined. Let x, y E R. Then

By using Theorem 10, we have the following corollary.

v(x + y) = ½{#(x + y) + #(a)} ~> ½{min{#(x),#(y)} + #(a)}

Corollary 14. Let # be an L-fuzzy left (resp. right) ideal of R. I f there exists an L-fuzzy left (resp. right) ideal v of R satisfying v+ C #, then i~+ = #.

= min{½{#(x) +/~(a)}, ½{#(y) + #(a)}} = min{v(x), v(y)},

Theorem 15. Let # be an L-fuzzy left (resp. right) ideal of R and let f : [0,#(0)] ~ L be an increasing function. Define an L-fuzzy set #f : R ---4 L by #f(x) = f ( # ( x ) ) for all x E R. Then #f is an L-fuzzy left (resp. right)ideal of R. ln particular, if f ( # ( 0 ) ) = 1 then #f is normal, and iff(t)>>,tfor all t E [0,#(0)] then # C # f .

and

v(xy) = ½{#(xy) + #(a)} t> ½{/~(y) + #(a)} (resp. ~> ½{#(x) + #(a)})

= v(y) (resp. = v(x)).

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Y.B. Jun et al./ Fuzzy Sets and Systems 82 (1996) 383-386

Hence, v is an L-fuzzy left (resp. right) ideal of R. Now we have

v+(x) = v(x) + 1 - v(O) =

:

+

½{,(x)

+ 1 -

+

+ 1},

and so v+(0) = ½{/~(0)+ 1} = 1. Thus, v+ is a normal L-fuzzy left (resp. right) ideal of R. Noticing that v + ( 0 ) = 1 > v + ( a ) = ½{l~(a)+ 1} > g(a), we know that v+ is non-constant. From v+(a) > #(a) it follows that /~ is not maximal. This proves the theorem.

References [1] J. Ahsan and M. Shabir, Semirings with projective ideals, Math. Japonica 38 (1993) 271276. [2] P.J. Allen, A fundamental theorem of homomorphisms for semirings, Proc. Amer. Math. Soc. 21 (1969) 412-416 [3] L. Dale, Direct sums of semirings and the KrulI-Schmidt theorem, Kyungpook Math. J. 17 (1977) 135-141. [4] Y.B. Jun, J. Neggers and H.S. Kim, On L-fuzzy ideals in semirings, submitted. [5] H.S. Kim, On quotient semiring and extension of quotient halfring, Comm. Korean Math. Soc. 4 (1989) 17-22.

[6] Wang-jin Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 (1982) 133 139. [7] T.K, Mukherjee and M.K. Sen, On fuzzy ideals of a ring 1, Fuzz)' Sets and Systems 21 (1987) 99-104. [8] K.L.N. Swarny and U.M. Swamy, Fuzzy prime ideals of rings, J. Math. Anal. Appl. 134 (1988) 94-103. [9] Zhang Yue, Prime L-fuzzy ideals and primary L-fuzzy ideals, Fuzz)" Sets and Systems 27 (1988) 345-350. [I0] L.A. Zadeh, Fuzzy sets, InJorm. and Control 8 (1965) 338-353.