C om mun.Fac.Sci.U niv.A nk.Series A 1 Volum e 66, N umb er 2, Pages 340–348 (2017) D O I: 10.1501/C om mua1_ 0000000824 ISSN 1303–5991
http://com munications.science.ankara.edu.tr/index.php?series= A 1
N -FUZZY IDEALS OF LATTICES YILDIRAY ÇELIK
Abstract. In this paper, the new concepts of N -fuzzy ideals and N -fuzzy prime ideals of lattices have been introduced. Also, some of theirs basic prop-erties are investigated. Hence, some results about the homomorphic N -image and pre-image of N -fuzzy ideals of lattices are established.
1. Introduction
The concept of fuzzy sets was …rstly introduced by Zadeh [16]. Rosenfeld [12] used this concept to formulate the notion of fuzzy groups. Since then many other fuzzy algebraic concepts had been studied by several authors. Yuan and Wu [15] introduced the concepts of fuzzy sublattice and fuzzy ideals of a lattice. Biswas [1] introduced the concept of anti fuzzy subgroups of groups. Shabir and Nawaz [13] introduced the concept of anti fuzzy ideals in semigroups. Khan and Asif [6] characterized di¤erent classes of semigroups by the properties of their anti fuzzy ideals. Lekkoksung [10] introduced the concept of an anti fuzzy bi-ideal of ordered -semigroups. Kim and Jun [7] studied the notion of anti fuzzy ideals of a near-ring. Datta [2] introduced the concept of anti fuzzy bi-ideals in rings. Anti fuzzy ideals of -rings were studied by Zhou et al. [17]. Srinivas et al. [14] introduced the concept of anti fuzzy ideals of -near-ring. Dheena and Mohanraaj [3] introduced the notion of anti fuzzy right ideal, anti fuzzy right k-ideal and intuitionistic fuzzy right k-ideal in semiring. Hong and Jun [5] introduced the notion of anti fuzzy ideals of BCK algebras. In this paper, we introduce the concepts of N -fuzzy ideals and N -fuzzy prime ideals of lattices and investigate some related properties. Also, we give some results about the homomorphic N -image and pre-image of N -fuzzy ideals of lattices.
2. Preliminaries
In this section, let X denotes a bounded lattice with the least element 0 and the greatest element 1 unless otherwise speci…ed.
Received by the editors: May 31, 2016, Accepted: March 02, 2017.
2010 Mathematics Subject Classi…cation. Primary 06D72; Secondary 20M12. Key words and phrases. Fuzzy ideal, N -fuzzy ideal, fuzzy lattice, N -fuzzy sublattice.
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De…nition 1. [4] A non-empty subset I of X is called an ideal of X if, for any a; b 2 I, a _ b 2 I; a ^ b 2 I and, for any a 2 I and x 2 X, x ^ a = x implies x 2 I: A non-empty subset D of X is called a dual ideal of X if, for any a; b 2 D, a ^ b 2 D, a _ b 2 D and, for a 2 D and x 2 X, x _ a = x implies x 2 D: An ideal P of X is called a proper ideal if P 6= X. A proper ideal P of X is called a prime ideal of X if, for any a; b 2 X, a ^ b 2 P implies a 2 P or b 2 P:
De…nition 2. [11] Let X, Y be two sets and f be a mapping from X to Y . A fuzzy set of X (see [16]) is a map from X to [0; 1]. If F(X) is the family of all fuzzy sets of X, then, for all ; 2 F(X), ! 2 F(Y ) and x 2 X; y 2 Y , the following operations are de…ned:
( _ )(x) = maxf (x); (x)g ( ^ )(x) = minf (x); (x)g ( !)(x; y) = minf (x); !(y)g
f ( )(y) = W
f (a)=y (a) if f 1(y) 6= ;,
0 otherwise,
f 1(!)(x) = !(f (x))
where f ( ) and f 1(!) are called, respectively, the image of under f and the pre-image of ! under f .
We denote if and only if (x) (x) for every x 2 X. For T X,
T 2 F(X) is called characteristic function of T , and de…ned by T(x) = 1 if
x 2 T and T(x) = 0 otherwise for all x 2 X.
De…nition 3. [8] Let be a fuzzy set of X: Then the complement of ; denoted by
c; is the fuzzy set of X given by c(x) = 1 (x) for all x 2 X: For t 2 [0; 1]; the set t = fx 2 Xj (x) tg is called a lower t-level cut of and +t = fx 2 Xj (x) tg
is called an upper t-level cut of : It is clearly seen that t = c +
1 t for all t 2 [0; 1]:
De…nition 4. [8] Let f iji 2 g be a family of fuzzy sets in X; then the union
(_ i)i2 is de…ned by (_ i)i2 (x) = supf i(x)ji 2 g for each x 2 X:
De…nition 5. [4] A fuzzy set of X is proper if it is a non constant function. De…nition 6. [9] A fuzzy set of X is called a fuzzy sublattice of X if (x ^ y) minf (x); (y)g and (x _ y) minf (x); (y)g for all x; y 2 X:
De…nition 7. [9] A fuzzy sublattice of X is called a fuzzy ideal of X if (x _y) = minf (x); (y)g for all x; y 2 X:
De…nition 8. [9] A proper fuzzy ideal of X is called fuzzy prime ideal of X if (x _ y) maxf (x); (y)g for all x; y 2 X:
Figure 1. .
Theorem 1. [9] A nonempty subset P of X is a prime ideal of X if and only if
p is a fuzzy prime ideal of X:
3. N -Fuzzy Ideals of Lattices
De…nition 9. A fuzzy set of X is called an N -fuzzy sublattice of X if (x ^ y) maxf (x); (y)g and (x _ y) maxf (x); (y)g for all x; y 2 X:
Example 1. Consider the lattice X = f0; ; ; 1g given by as follows
Let de…ne a fuzzy set of X by (0) = 0:1; ( ) = 0:2; ( ) = 0:3 and (1) = 0:3: Then is an N -fuzzy sublattice of X:
Theorem 2. A fuzzy set of X is an N -fuzzy sublattice of X if and only if c is
a fuzzy sublattice of X:
Proof. Let be an N -fuzzy sublattice of X: Then for x; y 2 X;
c (x ^ y) = 1 (x ^ y) 1 maxf (x); (y)g = minf1 (x); 1 (y)g = minf c(x); c(y)g c (x _ y) = 1 (x _ y) 1 maxf (x); (y)g = minf1 (x); 1 (y)g = minf c(x); c(y)g
Hence c is a fuzzy sublattice of X. Similarly, the converse can be proved.
De…nition 10. Let be an N -fuzzy sublattice of X: Then is an N -fuzzy ideal of X if (x _ y) = maxf (x); (y)g for all x; y 2 X:
Example 2. Let consider the lattice X = f0; ; ; 1g given by in the Figure 1. Let de…ne a fuzzy set of X by (0) = 0:1; ( ) = 0:2; ( ) = 0:3 and (1) = 0:3: Then is an N -fuzzy ideal of X:
Every N -fuzzy sublattice of X need not be N -fuzzy ideal of X.
Example 3. Let consider the lattice X = f0; ; ; 1g given by in the Figure 1. Let de…ne a fuzzy set of X by (0) = 0; ( ) = 0:2; ( ) = 0:3 and (1) = 0:2: Then is an N -fuzzy sublattice of X; but is not an N -fuzzy ideal of X as ( _ ) = (1) 6= maxf ( ); ( )g:
Remark 1. Let be an N -fuzzy ideal of X: As (x) = (x_0) = maxf (x); (0)g, then we get (0) (x) for any x 2 X:
Theorem 3. A fuzzy set of X is an N -fuzzy ideal of X if and only if c is a
fuzzy ideal of X:
Proof. Let be an N -fuzzy ideal of X: By Theorem 2, c is a fuzzy sublattice of
X. For x; y 2 X, c (x _ y) = 1 (x _ y) = 1 maxf (x); (y)g = minf1 (x); 1 (y)g = minf c(x); c(y)g
Thus, c is a fuzzy ideal of X: Similarly, the converse can be proved.
Theorem 4. A fuzzy set of X is an N -fuzzy ideal of X if and only if t is an ideal of X for each t 2 [ (0); 1]:
Proof. Let be an N -fuzzy ideal of X and t 2 [ (0); 1]: Then by Theorem 3, c is
a fuzzy ideal of X. Hence t = c +
1 tis an ideal of X.
Conversely, ; t is an ideal of X for each t 2 [ (0); 1] and s 2 [0; 1 (0)] = [0; c(0)]: Then 1 s 2 [ (0); 1] and c +
s = 1 s is an ideal of X: Hence c +s
is an ideal of X for all s 2 [0; c(0)]; and c is a fuzzy ideal of X. This shows that
is an N -fuzzy ideal of X:
Theorem 5. If I is an ideal of X; then for each t 2 [0; 1], there exists an N -fuzzy ideal of X such that t = I:
Proof. Let I be an ideal of X and t 2 [0; 1]: Let de…ne a fuzzy set of X by (x) =
(
t; if x 2 I 1; if x 62 I
for each x 2 X: Then s = I for any s 2 [t; 1) = [ (0); 1); and 1 = X: Thus s
is an ideal of X for all s 2 [ (0); 1]: Hence, from Theorem 4, is N -fuzzy ideal of X and t = I:
Let be a fuzzy set of X and let de…ne X = fx 2 Xj (x) = (0)g: We then get the following theorem.
Theorem 6. If is an N -fuzzy ideal of X; then X is an ideal of X:
Proof. Let be an N -fuzzy ideal of X and x; y 2 X . Then (x) = (0) and (y) = (0): So (x _ y) = maxf (x); (y)g = (0): Hence x _ y 2 X : Now let x a; x 2 X and a 2 X : Then x _ a = a and (a) = (0): As is an N -fuzzy ideal of X; (x _ a) = maxf (x) (a)g: Thus (a) = maxf (x); (a)g: Therefore (x) (a) = (0): Also by Remark 1, (0) (x): So we get (x) = (0): Hence x 2 X : This shows that X is an ideal of X:
Theorem 7. If f iji 2 g a family of N -fuzzy ideals of X, then so is (_ i)i2 :
Proof. Let f i; j i 2 g be a family of N -fuzzy ideals of X: Let x; y 2 X:
(_ i)i2 (x ^ y) = supf i(x ^ y)ji 2 g
sup maxf i(x); i(y)g
= max supf i(x)g; supf i(y)g = maxf_ i)i2 (x); _ i)i2 (y)g
(_ i)i2 (x _ y) = supf i(x _ y)ji 2 g
sup maxf i(x); i(y)g
= max supf i(x)g; supf i(y)g = maxf_ i)i2 (x); _ i)i2 (y)g
Hence (_ i)i2 is an N -fuzzy sublattice of X: Also,
(_ i)i2 (x _ y) = supf i(x _ y)ji 2 g
= sup maxf i(x); i(y)g = max supf i(x)g; supf i(y)g = maxf(_ i)i2 (x); (_ i)i2 (y)g
Hence (_ i)i2 is an N -fuzzy ideal of X:
Theorem 8. Let f : X ! Y be a lattice homomorphism where Y is a bounded lattice. Let be an N -fuzzy ideal of Y: Then f 1( ) is an N -fuzzy ideal of X:
Proof. Let x; y 2 X. Then
f 1( )(x ^ y) = (f (x ^ y)) = (f (x) ^ f(y))
maxf (f(x)); (f(y))g = maxff 1( )(x); f 1( )(y)g Thus f 1( )(x ^ y) maxff 1( )(x); f 1( )(y)g:
Similarly we can prove f 1( )(x _ y) maxff 1( )(x); f 1( )(y)g: Hence
f 1( ) is an N -fuzzy sublattice of X: Also for x; y 2 X
f 1( )(x _ y) = (f (x _ y)) = (f (x) _ f(y))
= maxf (f(x)); (f(y))g = maxff 1( )(x); f 1( )(y)g
Thus f 1( )(x _ y) = maxff 1( )(x); f 1( )(y)g: This shows that f 1( ) is an
N -fuzzy ideal of X:
De…nition 11. Let f : X ! Y be a mapping where Y is a non-empty set. Let be a fuzzy set of X: Then N -image of under f is a fuzzy set f ( ) of Y de…ned by f ( )(y) = inf f (x)jx 2 X and f(x) = yg for all y 2 Y:
Theorem 9. Let f : X ! Y be an onto lattice homomorphism where Y is a bounded lattice. Let be N -fuzzy ideal of X: Then f( ) is an N -fuzzy ideal of Y: Proof. Let be an N -fuzzy ideal of X and a; b 2 Y: As f is onto, there exist p; q 2 X such that f(p) = a and f(q) = b: Also a ^ b = f(p) ^ f(q) = f(p ^ q) and
f ( )(a ^ b) = inff (z)jz 2 X and f(z) = a ^ bg inf f (p ^ q)jf(p) = a and f(q) = bg inf maxf (p); (q)gjf(p) = a and f(q) = b = max inf f (p)=f(p) = ag; inff (q)jf(q) = bg = maxff( )(a); f( )(b)g
Thus f ( )(a^b) maxff( )(a); f( )(b)g: Similarly we can prove that f( )(a_b) maxff( )(a); f( )(b)g: Hence f( ) is an N -fuzzy sublattice of Y:
Again let x; y 2 Y: As f is onto, there exist r; s 2 X such that f(r) = x and f (s) = y: Also x _ y = f(r) _ f(s) = f(r _ s) and
f ( )(x _ y) = inff (z)jz 2 X and f(z) = x _ yg = inf f (r _ s)jf(r) = x and f(s) = yg = inf maxf (r); (s)gjf(r) = x and f(y) = s = max inf f (r)jf(r) = xg; inff (s)jf(s) = yg = maxff( )(x); f( )(y)g
Thus f ( )(x _ y) = maxff( )(x); f( )(y)g: This shows that f( ) is an N -fuzzy ideal of Y:
Theorem 10. Every N -fuzzy ideal of X is order preserving.
Proof. Let be an N -fuzzy ideal of X: Let x; y 2 X such that x y: Then (y) = (x _ y) = maxf (x); (y)g: Thus (x) (y):
De…nition 12. Let and be fuzzy sets of X: The N Cartesian product : X X ! [0; 1] is de…ned by (x; y) = maxf (x); (y)g for all x; y 2 X: Theorem 11. If and are N -fuzzy ideals of X, then is N -fuzzy ideal of
X X:
Proof. Let (x1; y1) and (x2; y2) 2 X X: Then
((x1; y1) ^ (x2; y2)) = (x1^ x2; y1^ y2)g
= maxf (x1^ x2); (y1; y2)g
max maxf (x1); (x2)g; fmaxf (y1); (y2)g
= max maxf (x1); (y1)g; fmaxf (x2); (y2)g
= maxf (x1; y1); (x2; y2)g
((x1; y1) _ (x2; y2)) = (x1_ x2; y1_ y2)g
= maxf (x1_ x2); (y1_ y2)
max maxf (x1); (x2)g; maxf (y1); (y2)g
= max maxf (x1); (y1)g; maxf (x2); (y2)g
= maxf (x1; y1); (x2; y2)g
Thus is an N -fuzzy sublattice of X X: Also, ((x1; y1) _ (x2; y2)) = (x1_ x2; y1_ y2)
= maxf (x1_ x2); (y1_ y2)
= max maxf (x1); (x2)g; maxf (y1); (y2)g
= max maxf (x1); (y1)g; maxf (x2); (y2)g
= maxf (x1; y1); (x2; y2)g
Hence is an N -fuzzy ideal of X X:
De…nition 13. Let be an N -fuzzy ideal of X. Then is called an N -fuzzy prime ideal of X if (x ^ y) minf (x); (y)g for all x; y 2 X:
Example 4. Consider the lattice X = f0; ; ; 1g given by in the Figure 1. Let de…ne a fuzzy set of X by (0) = 0:2; ( ) = 0:2; ( ) = 0:3 and (1) = 0:3: Then is an N -fuzzy prime ideal of X:
Every N -fuzzy ideal of X need not be N -fuzzy prime ideal of X:
Example 5. Consider the lattice X = f0; ; ; 1g given by in the Figure 1. Let de…ne a fuzzy set of X by (0) = 0:1; ( ) = 0:2; ( ) = 0:3 and (1) = 0:3: Then is an N -fuzzy ideal of X, but is not an N -fuzzy prime ideal of X as
Theorem 12. Let P be a non-empty subset of X and r; t 2 [0; 1] such that r < t: Let p be a fuzzy subset of X such that
p(x) =
(
r; if x 2 P
t; if x 62 P
for all x 2 X. Then P is a prime ideal of X if and only if p is an N -fuzzy prime
ideal of X:
Proof. Let P be a prime ideal of X and x; y 2 X: If x ^ y 2 P; then p= (x ^ y) =
r maxf p(x); p(y)g: If x ^ y 62 P; then x 62 P and y 62 P . Then p(x ^ y) = t; p(x) = t and p(y) = t: Hence p(x ^ y) maxf p(x); p(y)g: Therefore we have p(x ^ y) maxf p(x); p(y)g: Similarly we can prove that p(x _ y) maxf p(x); p(y)g: This shows that p is an N -fuzzy sublattice of X:
Now let x; y 2 X: If x _ y 2 P; then x 2 P and y 2 P . Therefore p(x ^ y) =
r; p(x) = r and p(y) = r: Hence p(x _ y) = maxf p(x); p(y)g: If x _ y 62 P; then x 62 P or y 62 P . Therefore p(x ^ y) = t; p(x) = t or p(y) = t: Hence
p(x _ y) = maxf p(x); p(y)g: This shows that p is an N -fuzzy ideal of X:
Again let x; y 2 X: If x ^ y 2 P; then x 2 P or y 2 P (since P is a prime ideal of X). Therefore p(x ^ y) = r, p(x) = r or p(y) = r: Hence p(x _ y)
minf p(x); p(y)g: If x ^ y 62 P; then x 62 P and y 62 P (since P is an ideal
of X). Therefore p(x ^ y) = t, p(x) = t and p(y) = t: Hence p(x ^ y) minf p(x); p(y)g: This shows that p is N -fuzzy prime ideal of X:
Conversely, let p be N -fuzzy prime ideal of X and x; y 2 P: As p is N -fuzzy
sublattice of X; p(x ^ y) maxf p(x); p(y)g = r: Hence x ^ y 2 P: Similarly we can prove that x _ y 2 P: This shows that P is a sublattice of X: Now let a 2 P; x 2 X such that x ^ a = x: Thus x _ a = (x ^ a) _ a = a: Therefore r = p(a) = p(x _ a) = maxf p(x); p(a)g. Hence p(x) = r and so x 2 P: This shows that P is an ideal of X:
Again let x ^ y 2 P: Then p(x ^ y) = r: As p is N -fuzzy prime ideal of
X; p(x ^ y) minf p(x); p(y)g: Therefore p(x) = r or p(y) = r: Hence x 2 P or y 2 P: This shows that P is a prime ideal of X.
4. Conclusion
In this paper we de…ned the notions of N -fuzzy sublattice, N -fuzzy ideal and N fuzzy prime ideal of a bounded lattice. We showed that the complement of N -fuzzy sublattice of a bounded lattice is a -fuzzy sublattice. We also showed that the union a family of N -fuzzy ideals of a bounded lattice is also N -fuzzy ideal of a bounded lattice. We stated how the homomorphic N -images and inverse images of N -fuzzy ideals of a bounded lattice become N -fuzzy ideal of a bounded lattice. We also investigated how the N -Cartesian product of N -fuzzy ideals of a bounded lattice becomes N -fuzzy ideal of a bounded lattice. Our future work will focus on studying the intuitionistic N -fuzzy ideals of a bounded lattice.
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Current address : Y¬ld¬ray Çelik: Department of Mathematics, Ordu University, 52200, Ordu, TURKEY.
