Scientiae Mathematicae
, No. 3(1999), 411-413
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Vol. 2
A NOTE ON FUZZY SETS IN SEMIGROUPS
NIOVI KEHAYOPULU AND MICHAEL TSINGELIS
Received October 4, 1999
Abstract. We introduce the concept of the complement fuzzy set in semigroups and
we give some properties which let to the conclusion that a fuzzy subset f of a semigroup
S is an ideal of S if and only if the complement f of f is a lter of S .
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Denition 1. Let S be a semigroup and f a fuzzy subset of S . We consider the mapping
f : S ! [0; 1] j f (x) := 1 ; f (x).
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The mapping f is a fuzzy subset of S (a fuzzy set in S ) called the complement of f (in S ).
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Remark. If S is a semigroup and f a fuzzy subset of S , then we have:
f (x) f (y) , f (x) f (y) (x; y 2 S ).
f (x) = f (y) , f (x) = f (y) (x; y 2 S ).
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(f ) = f .
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In the following, we denote by R the set of real numbers.
Proposition 1. Let S be a semigroup and f a fuzzy subset of S . Then, we have
1 ; minff (x); f (y)g = maxff (x); f (y)g 8 x; y 2 S:
Proof. Let x; y 2 S . Since f (x); f (y) 2 [0; 1] R, we have
f (x) f (y) or f (y) f (x).
Let f (x) f (y). Then minff (x); f (y)g = f (x), and
1 ; minff (x); f (y)g = 1 ; f (x) := f (x):::::::()
On the other hand, f (x) f (y). So maxff (x); f (y)g = f (x):::::::()
By (*) and (**), we have 1 ; minff (x); f (y)g = maxff (x); f (y)g.
In case f (y) f (x), the proof is similar.
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Corollary 1. Let S be a semigroup and f a fuzzy subset of S . Then
1 ; minff (x); f (y)g = maxff (x); f (y)g 8 x; y 2 S .
Proof. Let x; y 2 S . Since f is a fuzzy subset of S , by Proposition 1, we have
1 ; minff (x); f (y)g = maxf(f ) (x); (f ) (y)g = maxff (x); f (y)g: 0
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By Proposition 1 and Corollary 1, we have
Corollary 2. If S is a semigroup and f a fuzzy subset of S , then we have
A) 1 ; maxff (x); f (y)g = minff (x); f (y)g 8 x; y 2 S .
B) 1 ; maxff (x); f (y)g = minff (x); f (y)g 8 x; y 2 S .
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1991 Mathematics Subject Classication. 20M10.
Key words and phrases. Fuzzy subsets, prime fuzzy subsets, fuzzy lters in semigroups.
412
N. KEHAYOPULU AND M. TSINGELIS
Corollary 3. Let S be a semigroup and f a fuzzy subset of S . The following are
equivalent:
1) f (xy) = minff (x); f (y)g 8 x; y 2 S .
2) f (xy) = maxff (x); f (y)g 8 x; y 2 S .
Proof. 1) ) 2). Let x; y 2 S . By 1), we have f (xy) = minff (x); f (y)g. Then, by
Proposition 1, f (xy) := 1 ; f (xy) = maxff (x); f (y)g.
2) ) 1). Let x; y 2 S . By 2), we have f (xy) = maxff (x); f (y)g. Then, by Proposition
1, f (xy) := 1 ; f (xy) = minff (x); f (y)g.
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Denition 2. Let S be a semigroup. A fuzzy subset f of S is called a fuzzy lter of S if
f (xy) = minff (x); f (y)g for every x; y 2 S: A fuzzy subset f of S is called a fuzzy ideal of S if f (xy) f (x) and f (xy) f (y) for all
x; y 2 S [1].
Equivalent Denition: f (xy) maxff (x); f (y)g for all x; y 2 S [1].
A fuzzy subset f of a semigroup S is called prime if f (xy) maxff (x); f (y)g for all
x; y 2 S [2]: By Corollary 3, we have
Corollary 4. Let S be a semigroup. A fuzzy subset f of S is a fuzzy lter of S if and
only if the fuzzy subset f of S is a prime fuzzy ideal of S . 0
Proposition 1 is generalized for an arbitrary non-empty subset A of S , as follows:
Lemma. Let ; 6= A R : 9 inf A 2 R. Let b 2 R. Then
b ; inf A = supfb ; a j a 2 Ag.
Proof. Let c := inf A. Then b ; c = supfb ; a j a 2 Ag. In fact:
We have c a 8 a 2 A, then ;c ;a 8 a 2 A, and b ; c b ; a 8 a 2 A. Let now
d 2 R such that d b ; a 8 a 2 A. Since a b ; d 8 a 2 A, we have inf A b ; d. Then
c b ; d, and b ; c d: Concerning the Lemma, we note that: If for a given ; 6= A R, the inf A does not exist
in R, then the supfb ; a j a 2 Ag does not exist in R, as well. More generally, the set
fb ; a j a 2 Ag has no upper bounds in R. Indeed: Let x 2 R; x b ; a 8 a 2 A. Then
b ; x 2 R and b ; x a 8 a 2 A. Since A is a non-empty subset of R having a lower
bound, there exists the inf A in R. Contradiction.
Proposition 2. Let S be a semigroup and f a fuzzy subset of S . Let ; 6= A S . Then
1 ; inf ff (a) j a 2 Ag = supff (a) j a 2 Ag.
Proof. Since f is a fuzzy subset of S and ; =
6 A S , we have ; 6= f (A) [0; 1]. Since
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f (A) is a non-empty subset of R which has a lower bound, there exists the inf f (A) in R.
By the Lemma,
1 ; inf f (A) = supf1 ; x j x 2 f (A)g
= supf1 ; f (a) j a 2 Ag
= supff (a) j a 2 Ag:
Thus we have
1 ; inf ff (a) j a 2 Ag = supff (a) j a 2 Ag.
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FUZZY SETS IN SEMIGROUPS
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Corollary 5. Let S be a semigroup and f a fuzzy subset of S . Let ; 6= A S . Then we
have
1 ; inf ff (a) j a 2 Ag = supff (a) j a 2 Ag.
Proof. Since f is a fuzzy subset of S , by Proposition 2, we have
1 ; inf ff (a) j a 2 Ag = supf(f ) (a) j a 2 Ag = supff (a) j a 2 Ag.
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REFERENCES.
(1) N. Kuroki, On fuzzy ideals and fuzzy bi-ideals in semigroups, Fuzzy Sets and Systems 5 (1981), 203{215.
(2) N. Kehayopulu, Xie Xiang-Yun, Michael Tsingelis, A characterization of prime and
semiprime ideals of semigroups in terms of fuzzy subsets, Soochow Journ. Math.,
submitted.
University of Athens, Department of Mathematics
Mailing (home) address: Niovi Kehayopulu, Nikomidias 18, 161 22 Kesariani, Greece
e-mail address: nkehayop@cc.uoa.gr