1. Introduction
The notion of semiring, introduced by H. S. Vandiver in 1934 [1] is a common generalization of rings and distributive lattices. Semirings play an important role in the development of automata theory, formal languages, optimization theory and other branches of applied mathematics (see for example [2-8]). Hemirings, which are semirings with commutative addition and zero element are also very important in theoretical computer science (see for instance [3,6,7]). Some other applications of semirings with references can be found in [5-7,9]. On the other hand, the notions of automata and formal languages have been generalized and extensively studied in a fuzzy frame work (cf. [8-10]).
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 [11] a more restricted class of ideals in semirings, which is called the class of k-ideals. These ideals have the property that if the semiring R is a ring then a subset of R is a k-ideal if and only if it is a ring ideal. Another class of ideals is defined by Iizuka [12], which is called the class of h-ideals. In [13] La Torre studied these ideals, thoroughly.
The concept of fuzzy set was introduced by Zadeh in 1965 [14]. Many researchers used this concept to generalized different notions of algebra. Fuzzy semirings were first studied by Ahsan et al. [15] (see also [16]). Fuzzy k-ideals are studied in [17-22]. Fuzzy h-ideals are studied in [23-29]. In this paper we characterize those hemirnigs for which each k-ideal is idempotent and also those hemirings for which each fuzzy k-ideal is idempotent. The rest of this is organized as follows.
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.
2. Basic Results on Hemirings
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:
![](https://www.scirp.org/html/3-7401286\ec52978b-d3e4-4a4d-bbc5-563e7bf7a0e6.jpg)
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) ![](https://www.scirp.org/html/3-7401286\90f6b076-14ce-4a2c-80f8-30b9158e56ce.jpg)
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
![](https://www.scirp.org/html/3-7401286\5426e499-0cb6-40ec-9e86-9351368864db.jpg)
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
.
2.1. Lemma
The intersection of any family of left (right) k-ideals of a hemiring R is a left (right) k-ideal of R.
2.2. Lemma
for any subsets A, B of a hemiring R.
2.3. Lemma
[30] If A and B are, respectively, right and left k-ideals of a hemiring R, then
![](https://www.scirp.org/html/3-7401286\41f039f7-0d69-4a1c-9d4d-1a68486676e3.jpg)
2.4. Definition
[30] A hemiring R is said to be k-hemiregular if for each
, there exist
such that
.
2.5. Lemma
[30] A hemiring R is k-hemiregular if and only if for any right k-ideal A and any left k-ideal B, we have
![](https://www.scirp.org/html/3-7401286\e567740d-3f67-476a-868f-12f264395c9f.jpg)
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
![](https://www.scirp.org/html/3-7401286\2fea0825-475d-4b56-be59-e5c69e557f55.jpg)
![](https://www.scirp.org/html/3-7401286\b2054328-bc5c-456d-8bc1-3cff1c0f2319.jpg)
![](https://www.scirp.org/html/3-7401286\4db619cd-b082-482b-b0d4-05959dc042af.jpg)
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
![](https://www.scirp.org/html/3-7401286\d7f2593a-04d6-4443-b21a-e6ec616eea40.jpg)
![](https://www.scirp.org/html/3-7401286\b781b2ad-0e50-4a00-990f-d8dda1529992.jpg)
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
.
2.6. Definition
[21] A fuzzy left (right) ideal
of a hemiring R is called a fuzzy left (right) k-ideal if
for all
.
2.7. Definition
Let
be a fuzzy subset of a universe X and
. Then the subset
is called the level subset of
.
2.8. Proposition
Let A be a non-empty subset of a hemiring R. Then a fuzzy set
defined by
![](https://www.scirp.org/html/3-7401286\314cb91f-78cd-479f-8fa3-35539647fb36.jpg)
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. □
2.9. Proposition
[23] If
are subsets of a hemiring
such that
then 1)
2)
.
2.10. Proposition
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. □
2.11. Example
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
![](https://www.scirp.org/html/3-7401286\7900226c-3bda-4717-a615-8032128885e9.jpg)
![](https://www.scirp.org/html/3-7401286\52139b54-0703-4ec5-8a51-6bf3c771b602.jpg)
![](https://www.scirp.org/html/3-7401286\eee92531-ad7a-4105-901b-e3c6c562fb20.jpg)
![](https://www.scirp.org/html/3-7401286\29c3696b-4a13-45ca-b8af-fb185354444a.jpg)
Then
![](https://www.scirp.org/html/3-7401286\97fa8a31-4a7b-4b85-8d25-fd096025ef2b.jpg)
Thus by Proposition 2.10,
is a fuzzy k-ideal of R.
3. k-Product of Fuzzy Subsets
To avoid repetitions from now R will always mean a hemiring
.
3.1. Definition
The k-product of two fuzzy subsets
and
of R is defined by
![](https://www.scirp.org/html/3-7401286\3ea3b2b4-8922-43af-9505-88728995ba0b.jpg)
and
if x can not be expressed as
.
By direct calculations we obtain the following result.
3.2. Proposition
Let
be fuzzy subsets of R. Then
and
.
For any subset A in a hemiring R,
will denote the characteristic function of A.
3.3. Lemma
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
![](https://www.scirp.org/html/3-7401286\2e2887bc-0cc4-42f6-b673-86e7fe1b286e.jpg)
and so
![](https://www.scirp.org/html/3-7401286\3e071887-4c08-4ef8-9b64-a74ff64a2161.jpg)
If
then
. If possible, let
Then
![](https://www.scirp.org/html/3-7401286\eca88abd-a6e6-467f-abb2-2dfa9be0b17a.jpg)
Hence there exist
such that
![](https://www.scirp.org/html/3-7401286\7331a416-c405-4837-a208-d689e8b8ddc3.jpg)
and
![](https://www.scirp.org/html/3-7401286\df75095c-f4ee-4778-b983-8e8a267d10c5.jpg)
that is
![](https://www.scirp.org/html/3-7401286\dd78d08b-41eb-447d-b61b-7b6375ccd8a0.jpg)
hence
and
, and so
which is a contradiction. Thus we have
.
Hence in any case, we have
. □
3.4. Theorem
If
are fuzzy
-ideals of
, then
is a fuzzy
-ideal of
and
.
Proof. Let
be fuzzy
-ideals of
. Let
, then
![](https://www.scirp.org/html/3-7401286\ac5d5838-23b4-4d44-8183-9c73ba9fbabb.jpg)
and
![](https://www.scirp.org/html/3-7401286\d945a2ca-df6b-401d-ab90-a64eb260cc6b.jpg)
Thus
![](https://www.scirp.org/html/3-7401286\51acadb5-b455-49ef-b93e-844fa9d08d31.jpg)
Since for each expression
and
we have
so we have
![](https://www.scirp.org/html/3-7401286\fb08544f-f4f9-4c4a-bac2-034dca4a82b5.jpg)
Similarly,
![](https://www.scirp.org/html/3-7401286\4ea19dc5-735d-41a0-ac1b-49df84a35e99.jpg)
Analogously we can verify that
for all
. This means that
is a fuzzy ideal of
.
To prove that
implies
![](https://www.scirp.org/html/3-7401286\746e54ab-38d3-4eab-bd0b-919f0776b465.jpg)
observe that
(1)
together with
, gives
. Thus
![](https://www.scirp.org/html/3-7401286\658faec6-c1ab-4b41-9ce3-5bc0d81db0f7.jpg)
and, consequently,
![](https://www.scirp.org/html/3-7401286\4aaf15f0-1f39-46ce-a453-8e6c5169d303.jpg)
Therefore
(2)
Now, we have
![](https://www.scirp.org/html/3-7401286\8460036b-ee26-4dba-9657-032621f39e56.jpg)
Thus
.
Hence
is a fuzzy k-ideal of R.
By simple calculations we can prove that
. □
3.5. Definition
The k-sum
of fuzzy subsets
and
of R is defined by
![](https://www.scirp.org/html/3-7401286\ecd96c5e-9602-40e7-936d-a69a374ae0b4.jpg)
where
.
3.6. Theorem
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
![](https://www.scirp.org/html/3-7401286\4e08b8b8-a6d2-455c-8ed0-a403ea13f0dc.jpg)
Similarly,
![](https://www.scirp.org/html/3-7401286\faa158e3-7069-4572-942c-2cf24ae59c36.jpg)
Similarly
This proves that
is a fuzzy ideal of
.
Now we show that
implies
. For this let
and
Then,
![](https://www.scirp.org/html/3-7401286\55d07ef2-4669-4936-a833-bb92da7b8f65.jpg)
whence
![](https://www.scirp.org/html/3-7401286\c94fe56f-905b-485c-b574-7830e9775647.jpg)
and
![](https://www.scirp.org/html/3-7401286\5c4bdb59-ebfc-44cc-8c2b-f2f4ab907099.jpg)
Then
![](https://www.scirp.org/html/3-7401286\e0fbb559-a416-42d2-b07d-00e1502be635.jpg)
Thus
![](https://www.scirp.org/html/3-7401286\860ff9f5-42dc-4818-8f66-72534de960d1.jpg)
Therefore
![](https://www.scirp.org/html/3-7401286\80359db0-6993-468a-b929-9dc7b2050ea8.jpg)
Thus
is a fuzzy k-ideal of
. □
3.7. Theorem
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
![](https://www.scirp.org/html/3-7401286\7f5352f2-829f-4b82-8c65-8019253dfda0.jpg)
Thus
.
2) ® 1) First we show that
for all
.
![](https://www.scirp.org/html/3-7401286\005799a2-b20d-4dc1-920a-1fa4feecd3d6.jpg)
Thus
for all
.
Now
![](https://www.scirp.org/html/3-7401286\2c630b9a-e38d-4d87-846e-f3c910566708.jpg)
Again
![](https://www.scirp.org/html/3-7401286\da7c468d-28a7-4d63-8a7a-b528718bcab1.jpg)
If
then
and so
□
3.8. Lemma
A fuzzy subset
in a hemiring R is a fuzzy left (right) k-ideal if and only if 1)
2)![](https://www.scirp.org/html/3-7401286\b64910ad-5b91-4785-9249-3c540724d949.jpg)
.
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
![](https://www.scirp.org/html/3-7401286\b20ec2fc-947e-46f5-b50e-018fc963407b.jpg)
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
![](https://www.scirp.org/html/3-7401286\3f1ba1f0-8a7d-474e-bb64-eccb35b468b1.jpg)
since
, so
and
is a fuzzy left k-ideal of R. □
For k-hemiregular hemirings we have stronger result.
3.9. Theorem
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
![](https://www.scirp.org/html/3-7401286\4a5ba473-1335-4cbb-aafc-93d11bfdf7a7.jpg)
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
![](https://www.scirp.org/html/3-7401286\961c0b90-f8c7-40a3-9914-3246ef2ffa5e.jpg)
So,
. Hence by Lemma 2.5, R is khemiregular hemiring. □
4. Idempotent k-Ideals
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.
4.1. Proposition
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
![](https://www.scirp.org/html/3-7401286\562cbee3-d2d3-4326-a571-f739400cc596.jpg)
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,
. □
4.2. Proposition
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,
. □
4.3. Theorem
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
![](https://www.scirp.org/html/3-7401286\4e51767e-0624-47a6-a516-9c6e6ebca4fb.jpg)
for some
.
As
and
for each
, so
![](https://www.scirp.org/html/3-7401286\115c2f36-55f1-4f89-9efd-f803a5a98843.jpg)
Therefore
.
Similarly
![](https://www.scirp.org/html/3-7401286\d92b6030-d54a-43ab-ab29-c11b601720a0.jpg)
Therefore
![](https://www.scirp.org/html/3-7401286\d18f1711-42f0-49a8-8104-414224a7eafa.jpg)
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
. □
4.4. Theorem
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
.
□
4.5. Corollary
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. [31], 11.11). □
4.6. Theorem
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
![](https://www.scirp.org/html/3-7401286\eec512db-2d29-48a8-8005-65ea49e378b4.jpg)
So,
is a distributive lattice.
The converse is obvious.
5. Prime k-Ideals
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
.
5.1. Theorem
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. □
5.2. Corollary
A k-ideal P of a hemiring R with identity is prime if and only if for all
from
it follows
or
.
5.3. Corollary
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.
5.4. Definition
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.
5.5. Example
In an ordinary hemiring of natural numbers the set of even numbers forms a k-ideal. A fuzzy set
![](https://www.scirp.org/html/3-7401286\5594a2e7-4128-4ad3-a0e4-f0e289f0cf37.jpg)
is a fuzzy k-ideal of this hemiring. It is prime in the second sense but it is not prime in the first sense.
5.6. Theorem
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.
5.7. Corollary
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.
5.8. Proposition
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. □
5.9. Theorem
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
. □
5.10. Theorem
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
. □
5.11. Corollary
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.
5.12. Theorem
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. □
5.13. Theorem
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
. □
5.14. Lemma
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
![](https://www.scirp.org/html/3-7401286\04509bd0-b9ba-447e-b0d7-b87d6e3476fa.jpg)
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
![](https://www.scirp.org/html/3-7401286\9d640df4-87d9-4ae8-a361-c0b51dcfc453.jpg)
Similarly
![](https://www.scirp.org/html/3-7401286\820262f5-88f6-4060-b91e-edaf5910606d.jpg)
and
![](https://www.scirp.org/html/3-7401286\7fef5e09-7c4d-4b7c-9d07-379a0e0fc319.jpg)
for all
. Thus
is a fuzzy ideal.
Now, let
, where
. Then
![](https://www.scirp.org/html/3-7401286\d6ae49c4-2a4e-4137-9a74-dd5e5a072afc.jpg)
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
![](https://www.scirp.org/html/3-7401286\10788ae1-84fd-49f6-90b8-16ddb6eb82c1.jpg)
which is impossible. Hence
or
. Thus
is an irreducible fuzzy k-ideal of R. By Theorem 5.12,
is k-prime. □
5.15. Theorem
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 ![](https://www.scirp.org/html/3-7401286\d13d0f8f-c769-46eb-8f4f-deece5782605.jpg)
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
. □
6. Semiprime k-Ideals
6.1. Definition
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
.
6.2. Theorem
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. □
6.3. Corollary
A k-ideal P of a commutative hemiring R with identity is semiprime if and only if for all
from
it follows
.
6.4. Theorem
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
. □
6.5. Theorem
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.
6.6. Definition
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
.
6.7. Theorem
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. □
6.8. Corollary
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.
6.9. Proposition
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.
6.10. Example
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
![](https://www.scirp.org/html/3-7401286\e28829bf-f5ee-485c-b784-de535e0232d0.jpg)
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
![](https://www.scirp.org/html/3-7401286\c338ab26-235b-45fb-b4cb-1254db67b423.jpg)
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:
![](https://www.scirp.org/html/3-7401286\82fa0c49-1a3a-4e42-a391-bc2aec25fe4e.jpg)
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
![](https://www.scirp.org/html/3-7401286\c4a56000-a18f-4844-ae4e-15f80f4ab951.jpg)
and
![](https://www.scirp.org/html/3-7401286\6ac36d01-1186-43a1-bd64-e9342b600a4e.jpg)
So,
but neither
nor
, that is
is not a k-prime fuzzy k-ideal.
7. Prime Spectrum
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
.
7.1. Theorem
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 I1 and I2 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
![](https://www.scirp.org/html/3-7401286\2c3d8783-9bb6-4f89-beea-fd2178888f9f.jpg)
where
is the k-ideal generated by
.
Hence
is a topology on
. □
7.2. Definition
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 [29].
7.3. Theorem
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)
.
7.4. Corollary
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
,
.
7.5. Theorem
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
![](https://www.scirp.org/html/3-7401286\ec6c05b1-4b6e-4644-bbb4-d76b142b93e4.jpg)
Note that
![](https://www.scirp.org/html/3-7401286\ea269fdd-5e2c-4d08-bb5e-cd0c71dfb3bd.jpg)
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
![](https://www.scirp.org/html/3-7401286\941ac21d-658c-4c89-b66a-3744bc5c259a.jpg)
Now, if all the elements of the set (whose supremum we are taking) are individually less than are equal to
, then we have
![](https://www.scirp.org/html/3-7401286\2b187c4c-3a46-453f-8b85-05f6dd0a9894.jpg)
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,
![](https://www.scirp.org/html/3-7401286\fa1c832b-242e-408d-8bd1-cfdbcb8c658a.jpg)
Let
![](https://www.scirp.org/html/3-7401286\4d8db192-ad5b-4744-9e51-e18532a1ad4e.jpg)
and
![](https://www.scirp.org/html/3-7401286\b3702c57-832c-4f69-8cbb-7d63a5ac6bc3.jpg)
where
.
So,
it follows that
for some
.
Hence
implies that
for some
.
Hence the two statements 1)
and 2)
for some
are equivalent.
Hence
![](https://www.scirp.org/html/3-7401286\d9f17cd5-c2be-47d8-8a5e-8a2a32792495.jpg)
because,
is also a fuzzy k-ideal of R.
Thus,
. Hence it follows that
forms a topology on the set
. □
8. Conclusion
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.