Applied Mathematics
Vol. 4 No. 5 (2013) , Article ID: 31225 , 16 pages DOI:10.4236/am.2013.45104
Characterizations of Hemirings by the Properties of Their k-Ideals
Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
Email: mshabirbhatti@yahoo.co.uk
Copyright © 2013 Muhammad Shabir, Rukhshanda Anjum. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received November 26, 2012; revised March 22, 2013; accepted March 30, 2013
Keywords: Hemiring; Fuzzy k-Ideal; Idempotent k-Ideals; Prime Ideals; Semiprime Ideals
ABSTRACT
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.
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:
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
.
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
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
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
.
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
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
Then
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
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
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
. □
3.4. Theorem
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
(1)
together with, gives
. Thus
and, consequently,
Therefore
(2)
Now, we have
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
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
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
. □
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
Thus.
2) ® 1) First we show that for all
.
Thus for all
.
Now
Again
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)
.
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.
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
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. □
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
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
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
. □
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
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
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
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. □
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
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
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.
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
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
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
. □
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.
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