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k-Fuzzy Ideals of Ternary Semirings

Sathinee Malee and Ronnason Chinram

Abstract—The notion of k-fuzzy ideals of semirings was intro-

duced by Kim and Park in 1996. In 2003, Dutta and Kar introduced

a notion of ternary semirings. This structure is a generalization of

ternary rings and semirings. The main purpose of this paper is to

introduce and study k-fuzzy ideals in ternary semirings analogous to

k-fuzzy ideals in semirings considered by Kim and Park.

Keywords—k-ideals, k-fuzzy ideals, fuzzy k-ideals, ternary

semirings

I. INTRODUCTION

The notion of ternary algebraic system was introduced by

Lehmer [15] in 1932. He investigated certain ternary algebraic

systems called triplexes. In 1971, Lister [16] characterized

additive semigroups of rings which are closed under the triple

ring product and he called this algebraic system a ternary ring.

Dutta and Kar [3] introduced a notion of ternary semirings

which is a generalization of ternary rings and semirings, and

they studied some properties of ternary semirings ([3], [4], [5],

[6], [7] and [11], etc.).

The theory of fuzzy sets was first studied by Zadeh [17]

in 1965. Many papers on fuzzy sets appeared showing the

importance of the concept and its applications to logic, set

theory, group theory, ring theory, real analysis, topology,

measure theory, etc. Fuzzy ideals of semirings were studied

by some authors ([1], [2], [8], [9], [10] and [14], etc.). The

notion of k-fuzzy ideals of semirings was introduced by

Kim and Park [14]. Recently, Kavikumar, Khamis and Jun

studied fuzzy ideals, fuzzy bi-ideals and fuzzy quasi-ideals

in ternary semirings in [12] and [13]. The fuzzy ideal of

ternary semirings is a good tool for us to study the fuzzy

algebraic structure. The main purpose of this paper is to study

k-fuzzy ideals in ternary semirings analogous to k-fuzzy ideals

in semirings considered by Kim and Park.

II. PRELIMINARIES

In this section, we refer to some elementary aspects of the

theory of semirings and ternary semirings and fuzzy algebraic

systems that are necessary for this paper.

Definition 2.1. A nonempty set S together with two associative

binary operations called addition and multiplication (denoted

Sathinee Malee is with the Department of Mathematics, Faculty of Science,

Prince of Songkla University, Hat Yai, Songkhla, 90110 THAILAND, e-

mail:sathine e@hotmail.com

Ronnason Chinram is with the Department of Mathematics, Faculty of

Science, Prince of Songkla University, Hat Yai, Songkhla, 90110 THAILAND

and Centre of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok

10400, THAILAND, e-mail:ronnason.c@psu.ac.th.

Most of the work in this paper is a part of the Master thesis written by

Miss Satinee Malee under the supervision of Assistant Professor Dr.Ronnason

Chinram.

This research is supported by the Centre of Excellence in Mathematics, the

Commission on Higher Education, Thailand.

by + and ·, respectively) is called a semiring if (S,+) is a

commutative semigroup, (S,·) is a semigroup and multiplica-

tive distributes over addition both from the left and from the

right, i.e., a(b + c) = ab + ac and (a + b)c = ac + bc for all

a,b,c ∈ S.

Definition 2.2. A nonempty set S together with a binary

operation and a ternary operation called addition + and ternary

multiplication, respectively, is said to be a ternary semiring if

(S,+) is a commutative semigroup satisfying the following

conditions: for all a,b,c,d,e ∈ S,

(i) (abc)de = a(bcd)e = ab(cde),

(ii) (a + b)cd = acd + bcd,

(iii) a(b + c)d = abd + acd and

(iv) ab(c + d) = abc + abd.

We can see that any semiring can be reduced to a ternary

semiring. However, a ternary semiring does not necessarily

reduce to a semiring by this example. We consider Z−

the set of all non-positive integers under usual addition and

multiplication, we see that Z−

is closed under the triple multiplication but is not closed under

the binary multiplication. Moreover, Z−

but is not a semiring under usual addition and multiplication.

0,

0is an additive semigroup which

0is a ternary semiring

Definition 2.3. Let S be a ternary semiring. If there exists an

element 0 ∈ S such that 0+x = x = x+0 and 0xy = x0y =

xy0 = 0 for all x,y ∈ S, then 0 is called the zero element or

simply the zero of the ternary semiring S. In this case we say

that S is a ternary semiring with zero.

Definition 2.4. An additive subsemigroup T of S is called a

ternary subsemiring of S if t1t2t3∈ T for all t1,t2,t3∈ T.

Definition 2.5. An additive subsemigroup I of S is called a

left [resp. right, lateral] ideal of S if s1s2i ∈ I [resp. is1s2∈

I,s1is2∈ I] for all s1,s2∈ S and i ∈ I. If I is a left, right

and lateral ideal of S, then I is called an ideal of S.

It is obvious that every ideal of a ternary semiring with zero

contains a zero element.

Definition 2.6. Let S and R be ternary semirings. A mapping

ϕ : S → R is said to be a homomorphism if ϕ(x + y) =

ϕ(x)+ϕ(y) and ϕ(xyz) = ϕ(x)ϕ(y)ϕ(z) for all x,y,z ∈ S.

Let ϕ : S → R be an onto homomorphism of ternary

semirings. Note that if I is an ideal of S, then ϕ(I) is an

ideal of R. If S and R be ternary semirings with zero 0, then

ϕ(0) = 0.

Definition 2.7. Let S be a non-empty set. A mapping f : S →

[0,1] is called a fuzzy subset of S.

Definition 2.8. Let A be a subset of a non-empty set S. The

characteristic function χAof A is a fuzzy subset of S defined

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as follows:

χA(x) =

?

1

0

if x ∈ A,

if x / ∈ A.

Definition 2.9. Let f be a fuzzy subset of a non-empty subset

S. For t ∈ [0,1], the set ft= {x ∈ S | f(x) ≥ t} is called a

level subset of S with respect to f.

III. MAIN RESULT

Definition 3.1. An ideal I of a ternary semiring S is said to

be a k-ideal if for x,y ∈ S,x + y,y ∈ I ⇒ x ∈ I.

Example 3.1. Consider the ternary semiring Z−

addition and ternary multiplication, let I

{−5,−6,−7,...}. It is easy to prove that I is an ideal of

Z−

−2 ?∈ I.

Example 3.2. Consider the ternary semiring Z−

addition and ternary multiplication, let I = {−3k | k ∈ N ∪

{0}}. It is easy to show that I is a k-ideal of Z−

Definition 3.2. For each ideal I of a ternary semiring S, the

k-closure I of I is defined by

0under usual

= {0,−3} ∪

0but not a k-ideal of Z−

0because −3,(−2)+(−3) ∈ I but

0under usual

0.

I = {x ∈ S | a + x = b for some a,b ∈ I}.

The next theorem holds.

Theorem 3.1. Let I be an ideal of a ternary semiring S with

zero. Then I is a k-ideal of S if and only if I = I.

Definition 3.3. A fuzzy subset f of a ternary semiring S is

called a fuzzy ideal of S if for all x,y,z ∈ S,

(i) f(x + y) ≥ min{f(x),f(y)} and

(ii) f(xyz) ≥ max{f(x),f(y),f(z)}.

By the definitions of ideals and fuzzy ideals of ternary

semirings, the following lemma holds.

Lemma 3.2. Let I be a non-empty subset of a ternary semiring

S. Then I is an ideal of S if and only if the characteristic

function χIis a fuzzy ideal of S.

Lemma 3.3. Let f be a fuzzy ideal of a ternary semiring S

with zero 0. Then f(x) ≤ f(0) for all x ∈ S.

Proof. For any x ∈ S,f(0) = f(00x) ≥ max{f(0),f(x)} ≥

f(x).

?

Definition 3.4. A fuzzy ideal f of a ternary semiring S with

zero 0 is said to be a k-fuzzy ideal of S if

f(x + y) = f(0) and f(y) = f(0) ⇒ f(x) = f(0)

for all x,y ∈ S.

Example 3.3. Consider the ternary semiring Z−

addition and ternary multiplication. Define a fuzzy subset f

on Z−

?

0.5

0under usual

0by

f(x) =

0

if x = −1,

otherwise.

It is easy to prove that f is a fuzzy ideal of Z−

f is not a k-fuzzy ideal of Z−

0. However,

0because f((−1) + (−2)) =

f(−3) = 0.5 = f(0) and f(−2) = 0.5 = f(0) but f(−1) =

0 ?= 0.5 = f(0).

Example 3.4. Let f be a fuzzy subset of a ternary semiring

Z−

?

0.5

It is easy to show that f is a fuzzy ideal of Z−

such that f(x+y) = f(0) and f(x) = f(0). So f(x+y) = 0.5

and f(y) = 0.5. Thus x+y and y are even. Hence x is even,

this implies f(x) = 0.5 = f(0). Therefore f is a k-fuzzy ideal

of Z−

From the condition of Definition 3.4 and Lemma 3.2, the

following theorem holds.

Theorem 3.4. Let S be a ternary semiring with zero 0 and I

is a non-empty subset of S. Then I is a k-ideal of S if and

only if the characteristic function χIis a k-fuzzy ideal of S.

Proof. Assume I is a k-ideal of S. By Lemma 3.2, χI is

a fuzzy ideal of S. Next, let x,y ∈ S and assume χI(x +

y) = χI(0) and χI(y) = χI(0). Since I is an ideal of S,

0 ∈ I. Thus χI(0) = 1, this implies χI(x + y) = 1 and

χI(y) = 1. Then x + y,y ∈ I. Since I is a k-ideal of S,

x ∈ I. Hence χI(x) = 1 = χI(0). Therefore χI is a k-

fuzzy ideal of S. Conversely, assume characteristic function

χI is a k-fuzzy ideal of S. By Lemma 3.2, I is an ideal of

S. So 0 ∈ I, this implies χI(0) = 1. Let x,y ∈ S such that

x+y,y ∈ I. So χI(x+y) = χI(0) and χI(y) = χI(0). Then

χI(x) = χI(0) = 1. So x ∈ I. Hence I is a k-ideal of S. ?

Theorem 3.5. Let f be a fuzzy subset of a ternary semiring

S. Then f is a fuzzy ideal of S if and only if for any t ∈ [0,1]

such that ft?= ∅,ftis an ideal of S.

Proof. Let f be a fuzzy ideal of S. Let t ∈ [0,1] such that

ft?= ∅. Let x,y ∈ ft. Then f(x),f(y) ≥ t. Then f(x + y) ≥

min{f(x),f(y)} ≥ t. Next, let x,y ∈ S and a ∈ ft. We

have f(xya) ≥ max{f(x),f(y),f(a)} ≥ f(a) ≥ t. Thus

xya ∈ ft. Similarly, xay,axy ∈ ft. Therefore ftis an ideal

of S. Conversely, let x,y,z ∈ S and t = min{f(x),f(y)}.

Then f(x),f(y) ≥ t. Thus x,y ∈ ft. By assumption,

x + y ∈ ft. So f(x + y) ≥ t = min{f(x),f(y)}. Next,

let s = max{f(x),f(y),f(z)}. Then f(x) = s or f(y) = s

or f(z) = s. Thus x ∈ fsor y ∈ fsor z ∈ fs. By assumption,

xyz ∈ fs. So f(xyz) ≥ s = max{f(x),f(y),f(z)}.

Therefore f is a fuzzy ideal of S.

0under usual addition and ternary multiplication defined by

0.3

f(x) =

if x is odd

if x is even.

0. Let x,y ∈ Z−

0

0.

?

However, it is not true in general that f is a fuzzy ideal of

a ternary semiring S with zero 0, then for any t ∈ [0,1] such

that ft?= ∅,ftis a k-ideal of S. We can see this example.

Example 3.5. Consider the ternary semiring Z−

addition and ternary multiplication. Define a fuzzy subset f

on Z−

?

0.5

Then f is a fuzzy ideal of Z−

ideal of Z−

but −1 ?∈ f0.5.

0under usual

0by

f(x) =

0

if x = −1,

otherwise.

0but f0.5= Z−

0\{−1} is not a k-

0because (−1)+(−2) = −3 ∈ f0.5and −2 ∈ f0.5

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Theorem 3.6. Let f be a fuzzy subset of a ternary semiring

S with zero 0. If for any t ∈ [0,1] such that ft?= ∅,ftis a

k-ideal of S, then f is a k-fuzzy ideal of S.

Proof. By Theorem 3.5, f is a fuzzy ideal of S. Next, let

x,y ∈ S such that f(x+y) = f(0) and f(y) = f(0). Then x+

y,y ∈ ff(0). By assumption, x ∈ ff(0). Hence f(x) ≥ f(0).

Since f is a fuzzy ideal of S, by Lemma 3.3, f(x) = f(0).

Therefore f is a k-fuzzy ideal of S.

However, the converse of Theorem 3.6 does not hold. We

can see this example.

Example 3.6. Consider the ternary semiring Z−

addition and ternary multiplication. Let f be a fuzzy subset

of Z−

⎧

⎪

Then f is a fuzzy ideal of Z−

y) = f(0) and f(x) = f(0). So f(x + y) = 1 and f(y) = 1.

Thus x + y and y are even. Hence x is even, this implies

f(x) = 1 = f(0). Therefore f is a k-fuzzy ideal of Z−

However, f0.5= Z−

(−1) + (−2) = −3 ∈ f0.5and −2 ∈ f0.5but −1 ?∈ f0.5.

Definition 3.5. Let S be a ternary semiring with zero 0 and

f a fuzzy ideal of S. the k-fuzzy closure f of f is defined by

?

f(0)

The next theorem holds.

Theorem 3.7. Let S be a ternary semiring with zero 0 and f

a fuzzy ideal of S. Then f is a k-fuzzy ideal of S if and only

if f = f.

Proof. Assume f is a k-fuzzy ideal of S and let x ∈ ff(0).

Then f(x) = f(0). Since x ∈ ff(0), there exist a,b ∈ ff(0)

such that a + x = b. Thus f(a) = f(0) and f(x + a) =

f(b) = f(0). Then f(x) = f(0). Then f = f. Conversely,

assume f = f. So ff(0)= ff(0), by Theorem 3.1, ff(0)is a

k-ideal of S. Let x,y ∈ S such that f(x+y) = f(0) = f(y).

So x + y,y ∈ ff(0). Then x ∈ ff(0). So f(x) = f(0). Hence

f is a k-fuzzy ideal of S.

Definition 3.6. Let ϕ : S → R be a homomorphism of ternary

semirings. Let f be a fuzzy subset of R. We define a fuzzy

subset ϕ−1(f) of S by

ϕ−1(f)(x) = f(ϕ(x)) for all x ∈ S.

We call ϕ−1(f) the preimage of f under ϕ.

Theorem 3.8. Let ϕ : S → R be an onto homomorphism of

ternary semirings. If f be a fuzzy ideal of R, then ϕ−1(f) is

a fuzzy ideal of S.

Proof. Let f be a fuzzy ideal of R. Then for any x,y,z ∈ S,

ϕ−1(f)(x + y) = f(ϕ(x + y))

= f(ϕ(x) + ϕ(y))

≥ min{f(ϕ(x)),f(ϕ(y))}

= min{ϕ−1(f)(x),ϕ−1(f)(y)}

?

0under usual

0defined by

f(x) =

⎪

⎩

⎨

1

0

0.5

if x is even,

if x = −1,

otherwise.

0. Let x,y ∈ Z−

0such that f(x+

0.

0\ {−1} is not a k-ideal of Z−

0because

f(x) =

f(x)

if x / ∈ ff(0),

if x ∈ ff(0).

and

ϕ−1(f)(xyz) = f(ϕ(xyz))

= f(ϕ(x)ϕ(y)ϕ(z))

≥ max{f(ϕ(x)),f(ϕ(y)),f(ϕ(z))}

= max{ϕ−1(f)(x),ϕ−1(f)(y),ϕ−1(f)(z)}.

This shows that ϕ−1(f) is a fuzzy ideal of S.

?

Theorem 3.9. Let S and R be ternary semirings with zero 0

and ϕ : S → R an onto homomorphism. Let f be a fuzzy

ideal of R. Then f is a k-fuzzy ideal of R if and only if

ϕ−1(f) is a k-fuzzy ideal of S.

Proof. Suppose that f is a k-fuzzy ideal of R. Let x,y ∈ S.

Assume ϕ−1(f)(x + y) = ϕ−1(f)(0) and ϕ−1(f)(y) =

ϕ−1(f)(0). Then f(ϕ(x + y)) = f(ϕ(0)) = f(0) and

f(ϕ(y)) = f(ϕ(0)) = f(0). Since f is a k-fuzzy ideal of R,

f(ϕ(x)) = f(0) = f(ϕ(0)). Thus ϕ−1(f)(x) = ϕ−1(f)(0).

Hence ϕ−1(f) is a k-fuzzy ideal of S. Conversely, assume

ϕ−1(f) is a k-fuzzy ideal of S. Let x,y ∈ R such that

f(x + y) = f(0) and f(y) = f(0). Since ϕ is onto, there

exist a,b ∈ S such that f(a) = x and f(b) = y. So f(ϕ(a)+

ϕ(b)) = f(ϕ(0)) and f(ϕ(b)) = f(ϕ(0)). Hence ϕ−1(f)(a+

b) = ϕ−1(f)(0) and ϕ−1(f)(b) = ϕ−1(f)(0). Since ϕ−1(f)

is a k-fuzzy ideal of S, ϕ−1(f)(a) = ϕ−1(f)(0), this implies

f(x) = f(ϕ(a)) = f(ϕ(0)) = f(0). Hence f is a k-fuzzy

ideal of R.

?

Definition 3.7. Let ϕ : S → R be a homomorphism of ternary

semirings. Let f be a fuzzy subset of S. We define a fuzzy

subset ϕ(f) of R by

⎧

⎩

The following lemma is case L = [0,1] of Proposition 8 in

[10].

ϕ(f)(y) =

⎨

sup

x∈ϕ−1(y)

0

f(x)

if ϕ−1(y) ?= ∅,

otherwise.

We call ϕ(f) the image of f under ϕ.

Lemma 3.10. ([10]) Let ϕ be a mapping from a set X to a

set Y and f a fuzzy subset of X. Then for every t ∈ (0,1],

(ϕ(f))t=

0<s<t

?

ϕ(ft−s).

Lemma 3.11. The intersection of arbitrary set of ideals of a

ternary semiring S is either empty or an ideal of S.

Theorem 3.12. Let ϕ : S → R be an onto homomorphism of

ternary semirings. If f is a fuzzy ideal of S, then ϕ(f) is a

fuzzy ideal of R.

Proof. By Theorem 3.5, it is sufficient to show that each

nonempty level subset of ϕ(f) is an ideal of R. Let t ∈ [0,1]

such that (ϕ(f))t?= ∅. If t = 0, then (ϕ(f))t= R. Assume

that t ?= 0. By Lemma 3.10,

(ϕ(f))t=

0<s<t

Then ϕ(ft−s) ?= ∅ for all 0 < s < t, and so ft−s ?= ∅ for

all 0 < s < t. By Theorem 3.5, ft−sis an ideal of S for all

?

ϕ(ft−s).

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0 < s < t. Since ϕ is an onto homomorphism, ϕ(ft−s) is

an ideal of R for all 0 < s < t. By Lemma 3.11, (ϕ(f))t=

?

Definition 3.8. Let S and R be any two sets and ϕ : S → R

be any function. A fuzzy subset f of S is called ϕ-invariant

if ϕ(x) = ϕ(y) implies f(x) = f(y) where x,y ∈ S.

Lemma 3.13. Let S and R be ternary semirings and ϕ : S →

R a homomorphism. Let f be a ϕ-invariant fuzzy ideal of S.

If x = ϕ(a), then ϕ(f)(x) = f(a).

Proof. If t ∈ ϕ−1(x), then ϕ(t) = x = ϕ(a). Since f is

ϕ-invariant, f(t) = f(a). This implies

0<s<t

ϕ(ft−s) is an ideal of R.

?

ϕ(f)(x) = sup

t∈ϕ−1(x)

f(t) = f(a).

Hence ϕ(f)(x) = f(a).

?

Theorem 3.14. Let S and R be ternary semirings and ϕ :

S → R an onto homomorphism. If f is a ϕ-invariant fuzzy

ideal of S, then ϕ(f) is a fuzzy ideal of R.

Proof. Let x,y,z ∈ R. Then there exist a,b,c ∈ S such that

ϕ(a) = x,ϕ(b) = y and ϕ(c) = z and then x+y = ϕ(a+b)

and xyz = ϕ(abc). Since f is ϕ-invariant, by Lemma 3.13,

we have

ϕ(f)(x + y) = f(a + b)

≥ min{f(a),f(b)}

= min{ϕ(f)(x),ϕ(f)(y)}

and

ϕ(f)(xyz) = f(abc)

≥ max{f(a),f(b),f(c)}

= max{ϕ(f)(x),ϕ(f)(y),ϕ(f)(z)}.

Hence ϕ(f) is a fuzzy ideal of R.

?

Theorem 3.15. Let S and R be ternary semirings with zero

0 and ϕ : S → R an onto homomorphism. Let f be a ϕ-

invariant fuzzy ideal of S. Then f is a k-fuzzy ideal S if and

only if ϕ(f) is a k-fuzzy ideal of R.

Proof. Suppose that f is a k-fuzzy ideal of S and let x,y ∈ R

such that ϕ(f)(x + y) = ϕ(f)(0) and ϕ(f)(y) = ϕ(f)(0).

Since ϕ is onto, there exist a,b ∈ S such that ϕ(a) = x and

ϕ(b) = y. By Lemma 3.13, ϕ(f)(0) = f(0),ϕ(f)(x + y) =

f(a + b) and ϕ(f)(y) = f(b). Thus f(a + b) = f(0) and

f(b) = f(0). Since f is a k-fuzzy ideal of S, f(a) = f(0).

By Lemma 3.13, ϕ(f)(x) = f(a) = f(0) = ϕ(f)(0). Hence

ϕ(f) is a k-fuzzy ideal of R. Conversely, if ϕ(f) is a k-fuzzy

ideal of R, then for any x ∈ S,

ϕ−1(ϕ(f))(x) = ϕ(f)(ϕ(x)) = f(x).

So ϕ−1(ϕ(f)) = f. Since ϕ(f) is a k-fuzzy ideal of R, by

Theorem 3.9, f = ϕ−1(ϕ(f)) is a k-fuzzy ideal of S.

?

Next, we define fuzzy k-ideals of ternary semirings analo-

gous to fuzzy k-ideals of semirings.

Definition 3.9. A fuzzy ideal f of a ternary semiring S is said

to be a fuzzy k-ideal of S if

f(x) ≥ min{f(x + y),f(y)}

for all x,y ∈ S.

Example 3.7. Let f be a fuzzy subset of a ternary semiring

Z−

by

?

0.5

0under the usual addition and ternary multiplication defined

f(x) =

0

if x = −1,

otherwise.

Then f is a fuzzy ideal of Z−

because set x = −1 and y = −2, we have f(x) = 0 < 0.5 =

min{f(x + y),f(y)}.

Example 3.8. let f be a fuzzy subset of a ternary semiring

Z−

?

0.5

0but not a fuzzy k-ideal of Z−

0

0under usual addition and ternary multiplication defined by

f(x) =

0.3

if x is odd,

if x is even.

It is easy to show that f is a fuzzy k-ideal of Z−

Lemma 3.16. Let S be a ternary semiring and f a fuzzy ideal

of S. Then f is a fuzzy k-ideal of S if and only if for any

t ∈ [0,1] such that ft?= ∅,ftis a k-ideal of S.

Proof. By Theorem 3.5, ftis an ideal of S. Let x,y ∈ ftand

assume x + y,y ∈ ft. Then f(x + y),f(y) ≥ t. Since f is

a fuzzy k-ideal of S, f(x) ≥ min{f(x + y),f(y)} ≥ t. So

x ∈ ft. Therefore ft is a k-ideal of S. Conversely, assume

for any t ∈ [0,1] such that ft?= ∅,ft is a k-ideal of S. By

Theorem 3.5, f is a fuzzy ideal of S. Next, let x,y ∈ S.

Set t = min{f(x + y),f(y)} Then f(x + y),f(y) ≥ t. So

x+y,y ∈ ft. By assumption, ftis a k-ideal of S, this implies

x ∈ ft. Hence f(x) ≥ t = min{f(x+y),f(y)}. Therefore f

is a fuzzy k-ideal of S.

0.

?

Theorem 3.17. Let S be a ternary semiring with zero 0 and

f a fuzzy ideal of S. If f is a fuzzy k-ideal of S, then f is a

k-fuzzy ideal of S.

Proof. Let x,y ∈ S such that f(x + y) = f(0) and f(y) =

f(0). Set t = f(0). So x + y,y ∈ ft. By Lemma 3.16, the

level subset ft is a k-ideal of S. So x ∈ ft. This implies

f(x) ≥ t = f(0). By Lemma 3.3, f(x) = f(0).

However, the converse of Theorem 3.17 does not hold. We

can see this example.

?

Example 3.9. Consider the ternary semiring Z−

addition and ternary multiplication. Define a fuzzy subset f

on Z−

⎧

⎪

By Example 3.6, we known that f is a k-fuzzy ideal of Z−

However, f is not a fuzzy k-ideal of Z−

(−2)) = f(−3) = 0.5 and f(−2) = 1 but f(−1) = 0 <

0.5 = min{f((−1) + (−2)),f(−2)}.

0under usual

0by

f(x) =

⎪

⎩

⎨

1

0

0.5

if x is even,

if x = −1,

otherwise.

0.

0because f((−1) +

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Definition 3.10. Let f and g be fuzzy subset of a non-empty

subset S. A fuzzy subset f ∩g of S is defined by (f ∩g)(x) =

min{f(x),g(x)} for all x ∈ S.

Lemma 3.18. Let f and g be fuzzy subset of a ternary semiring

S. If f and g are fuzzy ideals of S, then f ∩g is a fuzzy ideal

of S.

Proof. Let x,y,z ∈ S. We have

(f ∩ g)(x + y) = min{f(x + y),g(x + y)}

≥ min{f(x),f(y),g(x),g(y)}

= min{(f ∩ g)(x),(f ∩ g)(y)}

and

(f ∩ g)(xyz) = min{f(xyz),g(xyz)}

≥ min{max{f(x),f(y),f(z)},max{g(x),g(y),g(z)}}

≥ max{(f ∩ g)(x),(f ∩ g)(y),(f ∩ g)(z)}.

Hence f ∩ g is a fuzzy ideal of S.

Theorem 3.19. Let f and g be fuzzy subset of a ternary

semiring S. If f and g are fuzzy k-ideals of S, then f ∩ g

is a fuzzy k-ideal of S.

Proof. By Lemma 3.18, f ∩g is a fuzzy ideal of S. Let x,y ∈

S. We have

?

(f ∩ g)(x) ≥ min{f(x),g(x)}

≥ min{f(x + y),f(y),g(x + y),g(y)}

= min{(f ∩ g)(x + y),(f ∩ g)(y)}.

Hence f ∩ g is a fuzzy k-ideal of S.

Let f and g be k-fuzzy ideals of a ternary semiring S. In

general, a fuzzy ideal f ∩g need not be a k-fuzzy ideal of S.

See this example.

?

Example 3.10. Consider the ternary semiring Z−

addition and ternary multiplication. Let f and g be fuzzy

subsets on Z−

⎧

⎪

and g(x) = 0.2 for all x ∈ Z−

g are k-fuzzy ideal of Z−

?

0.2

0under usual

0by

f(x) =

⎪

⎩

0. We have

⎨

0.3

0.1

0.2

if x = 0,

if x = −1,

otherwise

0. It is easy to verify that f and

(f ∩ g)(x) =

0.1

if x = −1,

otherwise.

Set x = −1 and y = −2. We have (f ∩ g)(x + y) = 0.2 =

(f ∩g)(0) and (f ∩g)(y) = 0.2 = (f ∩g)(0) but (f ∩g)(x) =

0.1 ?= 0.2 = (f ∩ g)(0). Thus f ∩ g is not k-fuzzy ideal of

Z−

0.

ACKNOWLEDGMENT

This research is (partially) supported by the Centre of Excel-

lence in Mathematics, the Commission on Higher Education,

Thailand.

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Sathinee Malee was born in Yala, Thailand, in 1985. She finished

her B.Sc. from Prince of Songkla University in 2007. Now she

is continuing her graduation in Master degree at same university.

Both of Bachelor and Master level she got a Science Achievement

Scholarship of Thailand to support her education.

Ronnason Chinram was born in Ranong, Thailand, in 1975. He re-

ceived his M.Sc and Ph.D. from Chulalongkorn University, Thailand.

Since 1997, he has been with Prince of Songkla University, Thailand

where now he is an Assistant Professor in Mathematics. His research

interests focus on semigroup theory and algebraic systems.

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