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Structure properties of Q-anti fuzzy left H-ideals in a hemi rings.

Introduction

Ideals of hemi rings play a central role in the structure theory and are very useful for many purposes. However, they do not in general coincide with the usual ring ideals. Many results in rings apparently have no analogues in hemi rings using only ideals. Henriksen defined in [4] a more restricted class of ideals in semi rings, which is called the class of k-ideals, with the property that if the semi ring R is a ring then a complex in R is a k-ideal if and only if it is a ring ideal. Another more restricted, but very important class of ideals, called h-ideals, has been given and investigated by Izuka [5] and La Torre [7] other important results connected with fuzzy ideals in hemi rings were obtained in [6].The concept of Q-fuzzy subgroups can be obtained in [11] [12] [13] [14]. In this paper, we introduced the notion of Q-anti fuzzy left h-ideals in terms of hemi rings and investigate their properties.

Preliminaries

In this section, we review some elementary aspects that are necessary for this paper.

Definition 2.1: An algebra (R, +, *) is said to be a semi ring if it satisfies the following conditions

(R, +) is a semi group

(R, *) is a semi group

a.(b + c) = a.b + a.c and (b + c).a = b.a + c.a [for all] a, b, c[member of] R.

Definition 2.2: A semi ring (R, +, *) is called a hemi ring

[H.sub.1]: + is commutative and

[H.sub.2]: there exists an element 0[right arrow]R such that 0 is the identity of (R, +) and the zero element of (R, *) i.e, 0.a = a.0 = 0 [for all] a[member of]R. A subset I of a semi ring R is called a left ideal of R if I is closed under addition and RI [subset or equal to] I. A left ideal of R is called a left K-ideal of R if y, z [member of] I and x[member of] R, x + y = z implies x[member of] I.

A left h ideal of a hemi ring R is defined to be a left ideal A of R such that (x + a + z = b + z [right arrow] x[member of] A, [for all](x, z[member of]R), ([for all]a, b[member of] A)).

Right h-ideals are defined similarity.

Definition 2.3: A mapping f: [R.sub.1] [right arrow] [R.sub.2] is said to be hemi ring homomorphism of [R.sub.1] is to [R.sub.2] if f(x + y) = f(x) + f(y) and f(xy) = f(x).f(y) for all x, y [member of] R.

Definition 2.4: A mapping [m,u]: X [right arrow] [0,1] where X is an arbitrary non-empty set is called a fuzzy set is X. For any fuzzy set [mu] is X and any X [member of] [0,1] we defined the set L([mu]: [alpha]) = {x[member of] X/[mu](x) [subset or equal to] [alpha]} which is called lower level cut of [mu].

Definition 2.5: Let Q and G be a set and a group respectively. A mapping [mu]: G x Q [right arrow] [0,1] is called a Q-fuzzy set.

Definition 2.6: A fuzzy subset is of a semi ring R is said to be Q-fuzzy left h- ideal of R if [mu](x + y, q) [greater than or equal to] min{[mu](x, q), [mu](y, q)} [for all]x, y[member of] R, q[member of]Q [mu](xy, q) [greater than or equal to] [mu](y,q) [for all]x, y[member of] R, q[member of] Q

Note that if [mu] is a Q-fuzzy left h-ideal if a hemi ring R, then [mu](0,q) [greater than or equal to] [mu](x,q) [for all] x[member of] R.

Definition 2.7: A fuzzy subset [mu] of a hemi ring R is said to be an Q-anti fuzzy left h-ideal of R if 1. [mu](x + y, q) [less than or equal to] max{[mu](x, q), [mu](y, q)} [for all]x, y[member of] R, q[member of] Q

2. [mu](xy, q) [less than or equal to] [mu](y, q) [for all] x, y[member of] R, q[member of] Q

3. x + a + z = b + z [right arrow] [mu](x, q) [less than or equal to] max{[mu](a, q), [mu](b, q)}

Example: Let R = {0, 1, 2, 3, 4} be a hemi ring with zero multiplication and addition defined by the following table
 0 1 2 3 4

0 0 1 2 3 4
1 1 1 4 4 4
2 2 4 4 4 4
3 3 4 4 4 4
4 4 4 4 4 4


We define a fuzzy set [mu]: R [right arrow] [0,1] by letting [mu](0) = [t.sub.1] and [mu](x) = [t.sub.2] [for all]x [not equal to] 0, [t.sub.1] [subset or equal to] [t.sub.2]. By routine computations, we can also easily check that [mu] is an anti fuzzy left h-ideal of hemi ring R.

Properties of Q-anti fuzzy left h-ideals

Proposition 3.1 let 'R' be a hemi ring and '[mu]' be a Q-fuzzy set in R. Then [mu] is an Q-anti fuzzy left h-ideal in R if and only if [[mu].sup.c] is a Q-fuzzy left h- ideal in R.

Proof: Let [mu] be an Q-anti fuzzy left h-ideal in R. For x, y [member of] R, We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let x, z, a, b [member of] R be such that x + a + z = b + z

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence [[mu].sup.c] is a Q-fuzzy left h-ideal of R.

Conversly, [[mu].sup.c] is a Q-fuzzy left h-ideal of R.

For x, y [member of] R, q[member of] Q, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let x, z, a, b [member of] R be such that x + a + z = b + z,

Then [mu](x, q) = 1 - [[mu].sup.c](x, q)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence [mu] is a Q-anti fuzzy left h-ideal of R.

Proposition 3.2 let '[mu]' be Q-anti fuzzy left h-ideal in a hemi ring R such that L([mu]: [alpha]) is a left h-ideal of R for each [alpha][member of] [I.sub.m]([mu]), [alpha][member of] [0,1].Then [mu] is an Q-anti fuzzy left h-ideal in R.

Proof: Let x, y [member of] R, q [member of] Q be such that [mu](x, q) = [[alpha].sub.1], [mu](y, q) = [[alpha].sub.2] then x + y [member of] L([mu]: [alpha]) without loss of generality, we may assume that [[alpha].sub.1] > [[alpha].sub.2]. It follows that L([mu]; [[alpha].sub.2]) [subset or equal to] L([mu]; [[alpha].sub.1]) so that x [member of] L([mu]; [[alpha].sub.1]) and y [member of] L([mu]; [[alpha].sub.2]). Since L([mu]; [[alpha].sub.1]) is a left is a left h-ideal of R. We have x + y [member of] L([mu]; [[alpha].sub.1]). Thus

[mu](x + y, q) [less than or equal to] [[alpha].sub.1] = max {[mu](x, q), [mu](y, q)} [mu](xy, q) [less than or equal to] [[alpha].sub.1] = [mu](y, q)

Let x, z, a, b [member of] R be such that x + a + z = b + z, then

[mu](x, q) [less than or equal to] [[alpha].sub.1] = max {[mu](a, q), [mu](b, q)}.

This shows that [mu] is an Q-anti fuzzy left h-ideal in R.

Corollorry 3.3: Let [mu] be Q-anti fuzzy left h-ideal in R then [mu] is an Q- anti fuzzy left h-ideal in R if and only if L([mu]; [alpha]) is a left h-ideal in R for every [alpha][member of] [[mu],(0,q),1] with [alpha][member of] [0,1].

Proposition 3.4: let '[mu]' be Q-anti fuzzy set in a hemi ring R then two lower level subsets L([mu]; [t.sub.1]) and L([mu]; [t.sub.2]), ([t.sub.1] < [t.sub.2]) are equal iff there is no x[member of] R such that [t.sub.1] [less than or equal to] [mu](x, q) [less than or equal to] [t.sub.2].

Proof: From definition of L([mu]; [alpha]) it follows that L([mu]; t) = [[mu].sup.-1]([[mu](0,q);t]) for t[member of] [0,1]. Let [t.sub.1], [t.sub.2][member of] [0,1] be such that [t.sub.1] < [t.sub.2] then L([mu]; [t.sub.1]) = L([mu]; [t.sub.2])

[??] [[mu].sup.-1]([[mu](0,q); [t.sub.1]]) = [[mu].sup.-1]([[mu](0,q); [t.sub.2]])

[??] [[mu].sup.-1]([t.sub.1], [t.sub.2]) = [phi]

[??] There is no x [member of] R such that [t.sub.1] [less than or equal to] [mu](x, q) [less than or equal to] [t.sub.2].

This completes the proof.

Definition 3.5: A left h-ideal A of hemi ring R is said to be characteristic iff f(A) = A, for all f [member of] Aut (R), Where Aut (R) is the set of all automorphisms of R. Q-anti fuzzy left h-ideal [mu] of hemi ring R is said to be Q-anti fuzzy characteristic if [[mu].sup.f](x, q) = [mu](x, q) for all and f [member of] Aut (R).

Lemma3.6: Let [mu] be an Q-anti fuzzy left h-ideal of a hemi ring R and let x [member of] R then [mu](x, q) = s iff x [member of] L([mu]; s) and x [not member of] L([mu]; t) for all s > t.

Proof: Straight forward.

Proposition 3.7: let '[mu]' be an Q-anti fuzzy left h-ideal of a hemi ring R then each level left h-ideal of [mu] is characteristic iff [mu] is an Q-anti fuzzy Characteristic.

Proof: Suppose that [mu] is an Q-anti fuzzy Characteristic and let S [member of] [I.sub.m]([mu]), f [member of] Aut (R) and x [member of] L([mu]; s) then

[[mu].sup.f] (x, q) = [mu](x, q) [less than or equal to] s [??] [mu](f(x, q)) [greater than or equal to] s [??] f(x, q) [member of] L([mu]; s)

Thus

f(L([mu]; s)) and y [member of] R such that f(y, q) = (x, q)

then

[mu](y, q) = [[mu].sup.f](y, q) = [mu](f(y, q)) = (x, q) [less than or equal to] s [??] y [member of] L([mu]: s) so that (x, q) = f(y, q) [member of] L([mu]; s).

Consequently, L([mu]; s) [subset or equal to] f (L([mu]; s)).

Hence f (L([mu]; s)) = L([mu]; s)

i.e. L([mu]; s) is characteristic. Conversely, suppose that each level h-ideal of [mu] is characteristic and let x [member of] R, f [member of] Aut (R) and [mu](x, q) = s. then by virtue Lemma, x[member of] L([mu]; s) and x [not member of] L([mu]; t) for all s > t. It follows from the assumption that

f(x, q) [member of] f (L([mu]; s))= L([mu]) so that [[mu].sup.f](x, q) = [mu](f(x, q)) [less than or equal to] s.

Let t = [[mu].sup.f](x, q) and assume that s > t. then f(x, q) [member of] L([mu]; t)= f(L([mu]; t)) which implies from the injectivity of f that x[member of] L([mu]: t), a contradiction.

Hence [[mu].sup.f](x, q) = [mu](f(x, q)) = s = [mu](x, q) showing that [mu] is an Q-anti fuzzy characteristic.

Proposition 3.8: Let f: [R.sub.1] [right arrow] [R.sub.2] be an epimorphism of hemi rings. If V is an Q-anti fuzzy left h-ideal of [R.sub.2] and [mu] is the pre-image of V under f then [mu] is an Q-anti fuzzy left h-ideal of [R.sub.1].

Proof: For any x, y [member of] [R.sub.1] and q[member of] Q, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let x, z, a, b [member of] R be such that x + a + z = b + z, then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence [mu] is an Q-anti fuzzy left h-ideal of [R.sub.1].

Definition 3.9: Let [R.sub.1] and [R.sub.2] be two hemi rings and f be a function of [R.sub.1] into [R.sub.2]. If [mu] is a Q-anti fuzzy in [R.sub.2] then the Pre-image of [mu] under f then [mu] is the Q-anti fuzzy in [R.sub.1] defined by [f.sup.-1]([mu])(x, q) = [mu](f(x, q)) [for all] x[member of] [R.sub.1], q[member of] Q.

Proposition 3.10: Let f: [R.sub.1] [right arrow] [R.sub.2] be an onto homomorphism of hemi rings. If [mu] is an Q-antifuzzy left h-ideal of [R.sub.2] then [f.sup.-1] ([mu]) is an Q-anti fuzzy left h-ideal of [R.sub.1].

Proof: Let [x.sub.1], [x.sub.2] [member of] [R.sub.1], then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let x, z, a, b [member of] [R.sub.1] be such that x + a + z = b + z, then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence [f.sup.-1]([mu]) is an Q-anti fuzzy left h-ideal of [R.sub.1].

Definition 3.11: Let [R.sub.1] and [R.sub.2] be any sets and let f: [R.sub.1] [right arrow] [R.sub.2] be any function. A Q-fuzzy subset [mu] of [R.sub.1] is called f-invariant if f(x) = f(y) implies [mu](x, q) = [mu](y, q)) [for all] x, y[member of] R, q[member of] Q.

Proposition 3.12: Let f: [R.sub.1] [right arrow] [R.sub.2] be an epimorphism of hemi rings. Let [mu] be an f-invariant Q-anti fuzzy left h-ideal of [R.sub.1], then f([mu]) is an Q-anti fuzzy left h- ideal of [R.sub.2].

Proof: Let x, y[member of] [R.sub.2] then there exists a, b [member of] [R.sub.1], such that f(a) = x and f(b) = y then x + y = f (a + b) and xy = f(ab). Since [mu] is invariant,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let x, z, a, b [member of] [R.sub.2] be such that x + a + z = b + z, then there exist [bar.x], [bar.z], [bar.a], [bar.b] such that f([bar.x]) = x, f([bar.y]) = y, f([bar.a]) = a and f([bar.b] = b. Since [mu] is f- invariant,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence f([mu]) is an Q-anti fuzzy left h-ideal of [R.sub.2].

Definition 3.13: A Q-fuzzy left h-ideal [mu] of a hemi ring R is said to be normal if there exist x [member of] R such that [mu](x, q) = 1. Note that if [mu] is a normal Q-anti fuzzy left h-ideal of [R.sub.1] then [mu](0, q) = 1 and hence [mu] is normal if and only if [mu](0, q) = 1.

Proposition 3.14: Let [mu] be an Q-anti fuzzy left h-ideal of a hemi ring R. Let [[mu].sup.+] be a Q-fuzzy set in R defined by [[mu].sup.+] (x, q) = [mu](x, q) = 1 - [mu](0, q) for all x[member of] R then [[mu].sup.+] is a normal Q-anti fuzzy left h-ideal of R which contains [mu].

Proof: For any x, y[member of] R, we have [[mu].sup.+](x, q) = [mu](0, q) + 1 - [mu](0,q) = 1

And [[mu].sup.+] (x + y, q) = [mu](x + y, q) + 1 - [mu](0,q)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This Shows that [[mu].sup.+] is an q-anti fuzzy left H-ideal of R. Let a, b, x, z, [member of] R be such that x + a + z = b + z, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence [[mu].sup.+] is a normal Q-anti fuzzy left h-ideal of hemi ring of R. Clearly [mu] [less than or equal to] [[mu].sup.+].

Definition 3.15: Let N(R) denote the set of all normal Q-anti fuzzy left h- ideals of R. Note that N(R) is a Poset under the set inclusion. A Q-fuzzy set [mu] in a hemi ring R is called a Maximal Q-anti fuzzy left h-ideal of R if it is non-constant and [[mu].sup.+] is a maximal element of (N(R), [subset or equal to]).

Proposition 3.16: Let [mu] [member of]N(R) be non-constant such that it is a maximal element of (N(R), [subset or equal to]) then it takes only two values {0,1}.

Proof: Since [mu] is normal, [mu](0,q) = 1. We claim that [mu](x, q) = 0. If not, then there exists [x.sub.0] [member of] R such that 0 [less than or equal to] [mu]([x.sub.0], q) < 1. Define on R a Q-fuzzy set V by putting V(x, q) = 1/2 {[mu](x, q) + [x.sub.0], q)} for each x [member of]R then clearly V is well defined and for all x, y [member of] R. we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus V is an Q-anti fuzzy left h-ideal of R.

Let a, b, x, z, [member of] R be such that x + a + z = b + z, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence V is an Q-anti fuzzy left h-ideal of R. By theorem 3.16, [V.sup.+] is a maximal Q-anti fuzzy left h-ideal of R. Note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [V.sup.+]([x.sub.0], q) [less than or equal to] 1 = [V.sup.+](0,q).

Hence [V.sup.+] is a non-constant and [mu] is not a maximal non-constant and [mu] is not a maximal element of N(R). This is a contradiction.

Conclusion

Y.B.Jun[6] introduced the concept on fuzzy h-ideals in hemi rings and [14] investigated the idea of anti fuzzy left h-ideals in hemi rings. In this paper, we established the some structure properties of Q-anti fuzzy left h-ideals in hemi rings. One can obtain similar results, by using the intuitionistic anti fuzzy ideals in a hemi rings.

Acknowledgements

The authors are highly greatful to the referees for their valuable comments and suggestions for improving the paper. This research work was supported by J J Educational & Charitable Trust, Tirchirappalli.

References

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[3] T.K.Dutta and B.K.Biswas, Fuzzy k-ideals of semi rings, Bull.Calcutta Math.Soc.87(1995), 91-96.

[4] M.Henriksen, Ideals in Semi rings with commutative addition, Am.Math.Soc Notices, 6(1958) 3-21

[5] K.Izuka, On the Jacobson radical of a semi ring, Tohoku, Math. J.,11(2)(1959), 409-421.

[6] Y.B.Jun, M.A.Ozturk and S.Z.Song, On fuzzy h-ideals in hemi rings, info. Scien.162(2004), 211-226.

[7] D.R.LaTorre, On h-ideals and k-ideals in hemi rings, Publ.Math.Debrecen, 12(1965), 219-226.

[8] D.M.Olson, A note on the homomorphism theorem for hemi rings, IJMMS, 1(1978), 439-439.

[9] A.Rosenfeld, Fuzzy groups, J.Math.Anal.Appl.35(1971), 512-517.

[10] A.Solairaju and R.Nagarajan, Q-fuzzy left R-subgroups of near rings with respect to T-norms, Antarctica Journal of mathematics, 5(2)(2008), 59-63.

[11] A.Solairaju and R.Nagarajan, A New Structure and Constructions of Q-Fuzzy groups, Advances in Fuzzy mathematics, 4(1)(2009), 23-29.

[12] A.Solairaju and R.Nagarajan, Lattice valued Q-fuzzy sub modules of near rings with respect to To norms, Advances in Fuzzy mathematics, 4(2)(2009), 137-145.

[13] A.Solairaju and R.Nagarajan, Q-fuzzy subgroups of Beta fuzzy congruence relations on a group, Accepted for publications in International Journal of Computer Science, Network and Security(IJCSNS), 2010.

[14] A.Solairaju and R.Nagarajan, characterization of interval valued anti fuzzy left h-ideals over hemi rings, Advances in Fuzzy Mathematics, 4(2)(2009), 129-136.

[15] L.A.Zadeh, Fuzzy sets, Information control, 8(1965), 338-353.

R. Nagarajan (1) and M. Muruga Ganesan (2)

(1) Assistant Professor, (2) Senior Lecturer

(1,2) Dept. of Mathematics, JJ College of Engineering & Technology Trichirappalli--620 009, India

E-mail: (1) nagalogesh@yahoo.co.in, (2) murugaganesan.m@gmail.com
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Author:Nagarajan, R.; Ganesan, M. Muruga
Publication:International Journal of Computational and Applied Mathematics
Date:Jul 1, 2010
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