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In this paper we characterize those hemirings for which each k-ideal is idempotent. We also characterize those hemirings for which each fuzzy k-ideal is idempotent. The space of prime k-ideals (fuzzy k-prime k-ideals) is topologized.

The notion of semiring, introduced by H. S. Vandiver in 1934 [

Ideals play an important role in the structure theory of hemirings and are useful for many purposes. But they do not coincide with usual ring ideals. For this reason many results in ring theory have no analogues in semirings using only ideals. Henriksen defined in [

The concept of fuzzy set was introduced by Zadeh in 1965 [

In Section 2, we summarize some basic concepts which will be use throughout this paper; these concepts are related to hemirings and fuzzy sets. In Section 3, k-product and k-sum of fuzzy sets in a hemiring are given. It is shown that k-product (k-sum) of fuzzy k-ideals of a hemiring is a k-ideal. Characterization of k-hemiregular hemiring in terms of fuzzy left k-ideal and fuzzy right k-ideal is also given in this section. Section 4 is about idempotent fuzzy k-ideals of a hemiring. Different characterization of hemirings in which each fuzzy k-ideal is idempotent is given. In Sections 5 and 6, prime, semiprime, irreducible fuzzy k-ideals are studied. In last section, the space of prime k-ideals (fuzzy k-prime k-ideals) is topologized.

A semiring is an algebraic system consisting of a non-empty set R together with two binary operations called addition “+” and multiplication “·” such that and are semigroups and connecting the two algebraic structures are the distributive laws:

for all.

A semiring is called a hemiring if “+” is commutative and has a zero element 0, such that and for all. An element (if it exists) is called an identity element of if for all. If a hemiring contains an identity element then it is called a hemiring with identity. A hemiring is called a commutative hemiring if “” is commutative in R.

A non-empty subset A of a hemiring R is called a subhemiring of R if A itself is a hemiring with respect to the induced operations of R. A non-empty subset I of a hemiring R is called a left (right) ideal of R if 1) for all and 2) for all,. Obviously for any left (right) ideal I of R. A non-empty subset I of a hemiring R is called an ideal of R if it is both a left and a right ideal of R. A left (right) ideal I of a hemiring R is called a left (right) k-ideal of R if for any and from it follows.

By k-closure of a non-empty subset A of a hemiring R we mean the set

It is clear that if A is a left (right) ideal of R, then A is the smallest left (right) k-ideal of R containing A. So, for all left (right) k-ideals of R. Obviously

for each non-empty. Also for all.

The intersection of any family of left (right) k-ideals of a hemiring R is a left (right) k-ideal of R.

for any subsets A, B of a hemiring R.

[

[

[

A fuzzy subset of a non empty set X is a function. denotes the set of all values of. A fuzzy subset is non-empty if there exist at least one such that. For any fuzzy subsets and of X we define

for all.

More generally, if is a collection of fuzzy subsets of, then by the intersection and the union of this collection we mean the fuzzy subsets

respectively.

A fuzzy subset of a semiring R is called a fuzzy left (right) ideal of R if for all we have 1)2).

Note that for all.

[

for all.

Let be a fuzzy subset of a universe X and. Then the subset is called the level subset of.

Let A be a non-empty subset of a hemiring R. Then a fuzzy set defined by

where, is a fuzzy left (right) k-ideal of R if and only if A is a left (right) k-ideal of R.

Proof. Straightforward. □

[

A fuzzy subset of a hemiring R is a fuzzy left (right) k-ideal of R if and only if each non-empty level subset of R is a left (right) k-ideal of R.

Proof. Suppose is a fuzzy left k-ideal of R and such that. Let, then and. As, so. Hence. For, so. This implies. Hence is a left ideal of. Now let for some, then and. Since, so. Hence . Thus is a left k-ideal of.

Conversely, assume that each non-empty subset of R is a left k-ideal of R. Let such that. Take such that, then but , a contradiction. Hence .

Similarly we can show that.

Let such that. If possible let. Take such that , then but, a contradiction. Hence. Thus is a fuzzy left k-ideal of R. □

The set with operations addition and multiplication given by the following Cayley tables:

is a hemiring. Ideals in are, , ,. All ideals are k-ideals. Let such that.

Define by

Then

Thus by Proposition 2.10, is a fuzzy k-ideal of R.

To avoid repetitions from now R will always mean a hemiring.

The k-product of two fuzzy subsets and of R is defined by

and if x can not be expressed as

.

By direct calculations we obtain the following result.

Let be fuzzy subsets of R. Then and.

For any subset A in a hemiring R, will denote the characteristic function of A.

Let R be a hemiring and. Then we have 1) if and only if.

2).

3).

Proof. 1) and 2) are obvious. For 3) let. If, then and for some and. Thus we have

and so

If then. If possible, let Then

Hence there exist such that

and

that is

hence and, and so which is a contradiction. Thus we have

.

Hence in any case, we have

. □

If are fuzzy -ideals of, then is a fuzzy -ideal of and.

Proof. Let be fuzzy -ideals of. Let, then

and

Thus

Since for each expression and we have

so we have

Similarly,

Analogously we can verify that

for all. This means that is a fuzzy ideal of.

To prove that implies

observe that

together with, gives. Thus

and, consequently,

Therefore

Now, we have

Thus

.

Hence is a fuzzy k-ideal of R.

By simple calculations we can prove that

. □

The k-sum of fuzzy subsets and of R is defined by

where.

The k-sum of fuzzy k-ideals of R is also a fuzzy k-ideal of R.

Proof. Let be fuzzy k-ideals of R. Then for we have

Similarly,

Similarly This proves that is a fuzzy ideal of.

Now we show that implies

. For this let and Then,

whence

and

Then

Thus

Therefore

Thus is a fuzzy k-ideal of. □

If is a fuzzy subset of a hemiring R, then the following are equivalent:

1) satisfies a) and b)2).

Proof. 1) ® 2) Let, then

Thus.

2) ® 1) First we show that for all.

Thus for all.

Now

Again

If then and so

□

A fuzzy subset in a hemiring R is a fuzzy left (right) k-ideal if and only if 1)2).

Proof. Let be a fuzzy left k-ideal of R. By Theorem 3.7, satisfies 1). Now we prove condition 2). Let. If, then

. Otherwise, there exist elements

such that. Then we have

This implies that.

Conversely, assume that the given conditions hold. In order to show that is a fuzzy left k-ideal of R it is sufficient to show that the condition holds. Let. Then we have

since, so and is a fuzzy left k-ideal of R. □

For k-hemiregular hemirings we have stronger result.

A hemiring R is k-hemiregular if and only if for any fuzzy right k-ideal and any fuzzy left k-ideal of R we have.

Proof. Let R be a k-hemiregular hemiring and be fuzzy right k-ideal and fuzzy left k-ideal of R, respectively. Then by Lemma 3.8, we have and. Thus. To show the converse inclusion, let. Since R is k-hemiregular, so there exist such that. Then we have

This implies that. Therefore .

Conversely, let C, D be any right k-ideal and any left k-ideal of R, respectively. Then the characteristic functions, of C, D are fuzzy right k-ideal and fuzzy left k-ideal of R, respectively. Now, by the assumption and Lemma 3.3, we have

So,. Hence by Lemma 2.5, R is khemiregular hemiring. □

From Lemma 2.5 it follows that in a k-hemiregular hemiring every k-ideal A is k-idempotent, that is. On the other hand, in such hemirings we have for all fuzzy k-ideals. Fuzzy k-ideal with this property will be called idempotent.

The following statements are equivalent for a hemiring R:

1) Each k-ideal of R is idempotent.

2) for each pair of k-ideals A, B of R.

3) for every.

4) for every non empty subset X of R.

5) for every k-ideal A of R.

If R is commutative, then the above assertions are equivalent to 6) R is k-hemiregular.

Proof. 1) ® 2) Assume that each k-ideal of R is idempotent and A, B are k-ideals of R. By Lemma 2.3,

. Since is a k-ideal of R, so by 1)

. Thus.

2) ® 1) Obvious.

1) ® 3) Let. The smallest k-ideal containing x has the form, where is the set of whole numbers. By hypothesis

. Thus

3) ® 4) This is obvious.

4) ® 5) Let A be a k-ideal of R. Then

. Hence.

5) ® 1) This is obvious.

If R is commutative then by Lemma 2.5,. □

The following statements are equivalent for a hemiring R.

1) Each fuzzy k-ideal of R is idempotent.

2) for all fuzzy k-ideals of R.

If R is commutative, then the above assertions are equivalent to 3) R is k-hemiregular.

Proof. 1) ® 2) Let and be fuzzy k-ideals of R. By Proposition 3.2,. Since is a fuzzy k-ideal of R, so by hypothesis is idempotent. Thus. By Theorem 3.4,. Thus.

2) ® 1) Obvious.

If R is commutative then by Theorem 3.9,. □

Let R be a hemiring with identity 1, then the following assertions are equivalent:

1) Each k-ideal of R is idempotent.

2) for each pair of k-ideals A, B of R.

3) Each fuzzy k-ideal of R is idempotent.

4) for all fuzzy k-ideals of R.

Proof. By Proposition 4.1.

By Proposition 4.2.

1) ® 3) Let. The smallest k-ideal of R containing x has the form. By hypothesis, we have

. Thus

, this implies

for some.

As and for each, so

Therefore.

Similarly

Therefore

Hence. By Theorem 3.4,. Thus.

3) ® 1) Let A be a k-ideal of R, then the characteristic function of A is a fuzzy k-ideal of R. Hence by hypothesis. Thus. □

If each k-ideal of R is idempotent, then the collection of all k-ideals of R is a complete Brouwerian lattice.

Proof. Let be the collection of all k-ideals of R, then is a poset under the inclusion of sets. It is not difficult to see that is a complete lattice under the operations, defined as and .

We now show that is a Brouwerian lattice, that is, for any the set contains a greatest element.

By Zorn’s Lemma the set contains a maximal element M. Since each k-ideal of R is idempotent, so and. Thus

. Consequently,.

Since, for every there exist such that

. Thus for any. As we have, which implies

.

Hence. This means that

, i.e., whence because M is maximal in. Therefore for every.

□

If each k-ideal of R is idempotent, then the lattice of all k-ideal of R is distributive.

Proof. Each complete Brouwerian lattice is distributive (cf. [

Each fuzzy k-ideal of R is idempotent if and only if the set of all fuzzy k-ideal of R (ordered by ≤) forms a distributive lattice under the k-sum and k-product of fuzzy k-ideals with.

Proof. Suppose that each fuzzy k-ideal of R is idempotent. Then by Proposition 4.2,. Let be the collection of all fuzzy k-ideals of R. Then is a lattice (ordered by ≤) under the k-sum and k-product of fuzzy k-ideals.

We show that for all. Let, then

So, is a distributive lattice.

The converse is obvious.

A proper (left, right) k-ideal P of R is called prime if for any (left, right) k-ideals A, B of R, implies or. A proper (left, right) k-ideal P of R is called irreducible if for any (left, right) k-ideals A, B of R, implies or. By analogy a non-constant fuzzy k-ideal of R is called prime (in the first sense) if for any fuzzy k-ideals, of R, implies or, and irreducible if implies or.

A left (right) k-ideal P of a hemiring R with identity is prime if and only if for all from it follows or.

Proof. Assume that P is a prime left k-ideal of R and

for some. Obviously, and are left k-ideals of R generated by a and b, respectively. So, and consequently or. If, then. If, then.

The converse is obvious. □

A k-ideal P of a hemiring R with identity is prime if and only if for all from it follows or.

A k-ideal P of a commutative hemiring R with identity is prime if and only if for all from it follows or.

The result expressed by Corollary 5.3, suggests the following definition of prime fuzzy k-ideals.

A non-constant fuzzy k-ideal of R is called prime (in the second sense) if for all and the following condition is satisfied:

if for every then or.

In other words, a non-constant fuzzy k-ideal is prime if from the fact that for every it follows or. It is clear that any fuzzy k-ideal is prime in the first sense is prime in the second sense. The converse is not true.

In an ordinary hemiring of natural numbers the set of even numbers forms a k-ideal. A fuzzy set

is a fuzzy k-ideal of this hemiring. It is prime in the second sense but it is not prime in the first sense.

A non-constant fuzzy k-ideal of a hemiring R with identity is prime in the second sense if and only if each its proper level set is a prime k-ideal of R.

Proof. Suppose is a prime fuzzy k-ideal of R in the second sense and let be its arbitrary proper level set, i.e.,. If, then for every. Hence or , i.e., or, which, by Corollary 5.3, means that is a prime k-ideal of R.

To prove the converse, consider a non-constant fuzzy k-ideal of R. If it is not prime then there exist a, such that for all, but and. Thus, , but and. Therefore is not prime, which is a contradiction. Hence is a prime fuzzy k-ideal in the second sense.

The fuzzy set defined in Proposition 2.8, is a prime fuzzy k-ideal of R (with identity) in the second sense if and only if A is a prime k-ideal of R.

In view of the Transfer Principle the second definition of prime fuzzy k-ideal is better. Therefore fuzzy k-ideals which are prime in the first sense will be called k-prime.

A non-constant fuzzy k-ideal of a commutative hemiring R with identity is prime if and only if for all.

Proof. Let be a non-constant fuzzy k-ideal of a commutative hemiring R with identity. If, then for every, we have

. Thus for every, which implies or. If, then, whence . If, then, as in the previous case,. So,.

Conversely, assume that for all. If for every, then replacing by the identity of R, we obtain. Thus, i.e., or, which means that is prime. □

Every proper k-ideal of a hemiring R is contained in some proper irreducible k-ideal of R.

Proof. Let P be a proper k-ideal of R such that. Let be a family of all proper k-ideals of R containing P and not containing a. By Zorn’s Lemma, this family contains a maximal element, say M. This maximal element is an irreducible k-ideal. Indeed, let for some k-ideals of R. If M is a proper subset of and, then, according to the maximality of M, we have and. Hence, which is impossible. Thus, either

or. □

If all k-ideals of R are idempotent, then a k-ideal P of R is irreducible if and only if it is prime.

Proof. Assume that all k-ideals of R are idempotent. Let P be a fixed irreducible k-ideal. If for some k-ideals A, B of R, then by Proposition 4.1,

. Thus. Since is a distributive lattice, so

.

So either or, that is either or.

Conversely, if a k-ideal P is prime and for some, then. Thus or. But and. Hence or. □

Let R be a hemiring in which all k-ideals are idempotent. Then each proper k-ideal of R is contained in some proper prime k-ideal.

Let R be a hemiring in which all fuzzy k-ideals are idempotent. Then a fuzzy k-ideal of R is irreducible if and only if it is k-prime.

Proof. Assume that all fuzzy k-ideals of R are idempotent and let be an arbitrary irreducible fuzzy k-ideal of R. We prove that it is k-prime. If for some fuzzy k-ideals of R then also. Since the set of all fuzzy k-ideals of R is a distributive lattice, we have . Thus or. Thus or. This proves that is k-prime.

Conversely, if is a k-prime fuzzy k-ideal of R and for some, then, which implies or. Since, so we have also and. Thus or. So, is irreducible. □

The following assertions for a hemiring R are equivalent:

1) Each k-ideal of R is idempotent.

2) Each proper k-ideal P of R is the intersection of all prime k-ideals of R which contain P.

Proof. 1) ® 2) Let P be a proper k-ideal of R and let be the family of all prime k-ideals of R which contain P. Theorem 5.9, guarantees the existance of such ideals. Clearly. If then by Theorem 5.9, there exists an irreducible k-ideal such that and. By Theorem 5.10, is prime. So there exists a prime k-ideal such that and. Hence. Thus.

2) ® 1) Assume that each k-ideal of R is the intersection of all prime k-ideals of R which contain it. Let A be a k-ideal of R. If, then we have, which means that A is idempotent. If then is a proper k-ideal of R and so it is the intersection of all prime k-ideals of R containing. Let. Then for each. Since is prime, we have. Thus. But. Hence. □

Let R be a hemiring in which each fuzzy k-ideal is idempotent. If is a fuzzy k-ideal of R with, where a is any element of R and, then there exists an irreducible k-prime fuzzy k-ideal of R such that and.

Proof. Let be an arbitrary fuzzy k-ideal of R and be fixed. Consider the following collection of fuzzy k-ideals of R

is non-empty since. Let be a totally ordered subset of containing, say.

We claim that is a fuzzy k-ideal of R.

For any we have

Similarly

and

for all. Thus is a fuzzy ideal.

Now, let, where. Then

Thus is a fuzzy k-ideal of R. Clearly and. Thus is the least upper bound of. Hence by Zorn’s lemma there exists a fuzzy k-ideal of R which is maximal with respect to the property that and.

We will show that is an irreducible fuzzy k-ideal of R. Let, where are fuzzy k-ideals of R. Then and. We claim that either or. Suppose and. Since is maximal with respect to the property that and since and, so and. Hence

which is impossible. Hence or. Thus is an irreducible fuzzy k-ideal of R. By Theorem 5.12, is k-prime. □

Each fuzzy k-ideal of R is idempotent if and only if each fuzzy k-ideal of R is the intersection of those k-prime fuzzy k-ideals of R which contain it.

Proof. Suppose each fuzzy k-ideal of R is idempotent. Let be a fuzzy k-ideal of R and let be the family of all k-prime fuzzy k-ideals of R which contain. Obviously. We now show that. Let a be an arbitrary element of R. Thenby Lemma 5.14, there exists an irreducible k-prime fuzzy k-ideal such that and. Hence

and. So,

. Thus. Therefore.

Conversely, assume that each fuzzy k-ideal of R is the intersection of those k-prime fuzzy k-ideals of R which contain it. Let be a fuzzy k-ideal of R then

is also a fuzzy k-ideal of, so where

are k-prime fuzzy k-ideals of R. Thus each contains, and hence. So, but

always. Hence. □

A proper (left, right) k-ideal A of R is called semiprime if for any (left, right) k-ideal B of R, implies. Similarly, a non-constant fuzzy k-ideal of R is called semiprime if for any fuzzy k-ideal of R, implies.

A (left, right) k-ideal P of a hemiring R with identity is semiprime if and only if for every from it follows.

Proof. Proof is similar to the proof of Theorem 5.1. □

A k-ideal P of a commutative hemiring R with identity is semiprime if and only if for all from it follows.

The following assertions for a hemiring R are equivalent:

1) Each k-ideal of R is idempotent.

2) Each k-ideal of R is semiprime.

Proof. Suppose that each k-ideal of R is idempotent. Let A, B be k-ideals of R such that. Then

. By hypothesis, so. Hence A is semiprime.

Conversely, assume that each k-ideal of R is semiprime. Let A be a k-ideal of R, then is a k-ideal of R. Also. Hence by hypothesis. But

always. Hence. □

Each fuzzy k-ideal of R is idempotent if and only if each fuzzy k-ideal of R is semiprime.

Proof. For any fuzzy k-ideal of R we have . If each fuzzy k-ideal of is semiprime, then implies. Hence .

The converse is obvious. □

Theorem 6.2, suggest the following definition of semiprime fuzzy k-ideals.

A non-constant fuzzy k-ideal of R is called semiprime (in the second sense) if for all and the following condition is satisfied:

if for every then.

A non-constant fuzzy k-ideal of R is semiprime in the second sense if and only if each its proper level set is a semiprime k-ideal of R.

Proof. Proof is similar to the proof of Theorem 5.6. □

A fuzzy set defined in Proposition 2.8 is a semiprime fuzzy k-ideal of R in the second sense if and only if A is a semiprime k-ideal of R.

In view of the Transfer Principle the second definition of semiprime fuzzy k-ideal is better. Therefore fuzzy kideals which are semiprime in the first sense should be called k-semiprime.

A non-constant fuzzy k-ideal of a commutative hemiring R with identity is semiprime if and only if for every.

Proof. Proof is similar to the proof of Proposition 5.8. □

Every fuzzy k-prime k-ideal is fuzzy k-semiprime kideal but the converse is not true.

Consider the hemiring defined by the following tables:

This hemiring has two k-ideals and R. Obviously these k-ideals are idempotent.

For any fuzzy ideal of R and any we have. Indeed,

.

This together with

implies. Consequently,

.

Therefore for every fuzzy k-ideal of this hemiring.

Now we prove that each fuzzy k-ideal of R is idempotent. Since always, so we have to show that. Obviously, for every we have

So, implies

.

Hence implies. Similarly implies

,

implies

.

Analogously, from it follows

.

This proves that for every. Therefore for every fuzzy k-ideal of R, which, by Theorem 6.4, means that each fuzzy k-ideal of R is semiprime.

Consider the following three fuzzy sets:

These three fuzzy sets are idempotent fuzzy k-ideals. Since all fuzzy k-ideal of this hemiring are idempotent, by Proposition 4.1, we have. Thus

and

So, but neither nor, that is is not a k-prime fuzzy k-ideal.

Let R be a hemiring in which each k-ideal is idempotent. Let be the lattice of all k-ideals of R and be the set of all proper prime k-ideals of R. For each k-ideal I of R define and .

The set forms a topology on the set.

Proof. Since, where is the usual empty set, because 0 belongs to each k-ideal. So empty set belongs to.

Also, because is the set of all proper prime k-ideals of R. Thus belongs to.

Suppose where I_{1} and I_{2} are in. Then

.

Since each k-ideal of R is idempotent so.

Thus. So belongs to

.

Let be an arbitrary family of members of

. Then

where is the k-ideal generated by.

Hence is a topology on. □

A fuzzy k-ideal of a hemiring R is said to be normal if there exists such that. If is a normal fuzzy k-ideal of R, then, hence is normal if and only if.

The proof of the following theorem is same as the proof of Theorem 4.4 of [

A fuzzy subset of a hemiring R is a k-prime fuzzy k-ideal of if and only if 1) is a prime k-ideal of R.

2) contains exactly two elements.

3).

Every k-prime fuzzy k-ideal of a hemiring is normal.

Let R be a hemiring in which each fuzzy k-ideal is idempotent, the lattice of fuzzy normal k-ideals of R and the set of all proper fuzzy k-prime k-ideals of R. For any fuzzy normal k-ideal of R, we define and.

A fuzzy k-ideal of R is called proper if, where is the fuzzy k-ideal of R defined by,.

The set forms a topology on the set.

Proof. 1) where is the usual empty set and is the characteristic function of k-ideal. This follows since each k-prime fuzzy k-ideal of R is normal. Thus the empty subset belongs to.

2). This is true, since is the set of proper k-prime fuzzy k-ideals of R. So is an element of.

3) Let with.

Then. Since each fuzzy k-ideal of R is idempotent, this implies . Thus

.

4) Let us consider an arbitrary family of fuzzy k-ideals of R. Since

Note that

where and only a finite number of the and are not zero. Since therefore we are considering the infimum of a finite number of terms because are effectively not being considered. Now, if for some then there exists such that. Consider the particular expression for in which and for all. We see that is an element of the set whose supremum is defined to be

.

Thus. This implies that is.

Hence for some implies.

Conversely, suppose that then there exists an element such that.

This means that

Now, if all the elements of the set (whose supremum we are taking) are individually less than are equal to, then we have

which does not agree with what we have assumed. Thus, there is at least one element of the set (whose supremum we are taking), say,

.

(being the corresponding breakup of x, where only a finite number of and are not zero).

Thus,

Let

and

where.

So, it follows that for some.

Hence implies that for some

.

Hence the two statements 1) and 2)

for some are equivalent.

Hence

because, is also a fuzzy k-ideal of R.

Thus,. Hence it follows that forms a topology on the set. □

In the study of fuzzy algebraic system, the fuzzy ideals with special properties always play an important role. In this paper we study those hemirings for which each fuzzy k-ideal is idempotent. We characterize these hemirings in terms of prime and semiprime fuzzy k-ideals. In the future we want to study those hemirings for which each fuzzy one sided k-ideal is idempotent and also those hemirings for which each fuzzy k-bi-ideal is idempotent.