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.
[
[
, there exist 
such that
.
[
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
.
[
of a hemiring R is called a fuzzy left (right) k-ideal if
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. □
[
are subsets of a hemiring 
such that 
then 1)
2)
.
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.
.
and 
of R is defined by
if x can not be expressed as
.
be fuzzy subsets of R. Then 
and
.
will denote the characteristic function of A.
. Then we have 1) 
if and only if
.
.
.
. If
, then 
and 
for some 
and
. Thus we have
then
. If possible, let 
Then
such that
and
, and so 
which is a contradiction. Thus we have
.
. □
are fuzzy 
-ideals of
, then 
is a fuzzy 
-ideal of 
and
.
be fuzzy 
-ideals of
. Let
, then
and 
we have
so we have
for all
. This means that 
is a fuzzy ideal of
.
implies
, gives
. Thus
.
is a fuzzy k-ideal of R.
. □
of fuzzy subsets 
and 
of R is defined by
.
be fuzzy k-ideals of R. Then for 
we have
This proves that 
is a fuzzy ideal of
.
implies
. For this let 
and 
Then,
is a fuzzy k-ideal of
. □
is a fuzzy subset of a hemiring R, then the following are equivalent:
satisfies a) 
and b)
2)
.
, then
.
for all
.
for all
.
then 
and so
□
in a hemiring R is a fuzzy left (right) k-ideal if and only if 1)
2)
.
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
.
is a fuzzy left k-ideal of R it is sufficient to show that the condition 
holds. Let
. Then we have
, so 
and 
is a fuzzy left k-ideal of R. □
and any fuzzy left k-ideal 
of R we have
.
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
. Therefore 
.
, 
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
. Hence by Lemma 2.5, R is khemiregular hemiring. □
. On the other hand, in such hemirings we have 
for all fuzzy k-ideals
. Fuzzy k-ideal with this property will be called idempotent.
for each pair of k-ideals A, B of R.
for every
.
for every non empty subset X of R.
for every k-ideal A of R.
. Since 
is a k-ideal of R, so by 1)
. Thus
.
. The smallest k-ideal containing x has the form
, where 
is the set of whole numbers. By hypothesis
. Thus
. Hence
.
. □
for all fuzzy k-ideals of R.
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
.
. □
for each pair of k-ideals A, B of R.
for all fuzzy k-ideals of R.
By Proposition 4.1.
By Proposition 4.2.
. The smallest k-ideal of R containing x has the form
. By hypothesis, we have
. Thus
, this implies
.
and 
for each
, so
.
. By Theorem 3.4,
. Thus
.
of A is a fuzzy k-ideal of R. Hence by hypothesis
. Thus
. □
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 
.
is a Brouwerian lattice, that is, for any 
the set 
contains a greatest element.
contains a maximal element M. Since each k-ideal of R is idempotent, so 
and
. Thus
. Consequently,
.
, for every 
there exist 
such that
. Thus 
for any
. As 
we have
, which implies
.
. This means that
, i.e., 
whence 
because M is maximal in
. Therefore 
for every
.
of all k-ideal of R is distributive.
.
. 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.
for all
. Let
, then
is a distributive lattice.
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
.
from 
it follows 
or
.
for some
. Obviously, 
and 
are left k-ideals of R generated by a and b, respectively. So, 
and consequently 
or
. If
, then
. If
, then
.
from 
it follows 
or
.
from 
it follows 
or
.
of R is called prime (in the second sense) if for all 
and 
the following condition is satisfied:
for every 
then 
or
.
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.
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.
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.
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.
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.
of a commutative hemiring R with identity is prime if and only if 
for all
.
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,
.
for all
. If 
for every
, then replacing 
by the identity of R, we obtain
. Thus
, i.e., 
or
, which means that 
is prime. □
. 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
. □
for some k-ideals A, B of R, then by Proposition 4.1,
. Thus
. Since 
is a distributive lattice, so
.
or
, that is either 
or
.
for some
, then
. Thus 
or
. But 
and
. Hence 
or
. □
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.
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. □
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
.
, 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
. □
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
.
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
.
is a fuzzy k-ideal of R.
we have
. Thus 
is a fuzzy ideal.
, where
. Then
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
.
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
or
. Thus 
is an irreducible fuzzy k-ideal of R. By Theorem 5.12, 
is k-prime. □
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
.
be a fuzzy k-ideal of R then 
, so 
where
are k-prime fuzzy k-ideals of R. Thus each 
contains
, and hence
. So
, but
always. Hence
. □
implies
. Similarly, a non-constant fuzzy k-ideal 
of R is called semiprime if for any fuzzy k-ideal 
of R, 
implies
.
from 
it follows
.
from 
it follows
.
. Then
. By hypothesis
, so
. Hence A is semiprime.
is a k-ideal of R. Also
. Hence by hypothesis
. But
always. Hence
. □
of R we have 
. If each fuzzy k-ideal of 
is semiprime, then 
implies
. Hence 
.
of R is called semiprime (in the second sense) if for all 
and 
the following condition is satisfied:
for every 
then
.
of R is semiprime in the second sense if and only if each its proper level set 
is a semiprime k-ideal of R.
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.
of a commutative hemiring R with identity is semiprime if and only if 
for every
.
defined by the following tables:
and R. Obviously these k-ideals are idempotent.
of R and any 
we have
. Indeed,
.
. Consequently,
.
for every fuzzy k-ideal of this hemiring.
always, so we have to show that
. Obviously, for every 
we have
implies
.
implies
. Similarly 
implies
,
implies
.
it follows
.
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.
. Thus
but neither 
nor
, that is 
is not a k-prime fuzzy k-ideal.
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 
.
forms a topology on the set
.
, where 
is the usual empty set, because 0 belongs to each k-ideal. So empty set belongs to
.
, because 
is the set of all proper prime k-ideals of R. Thus 
belongs to
.
where I1 and I2 are in
. Then
.
.
. So 
belongs to
.
be an arbitrary family of members of
. Then
is the k-ideal generated by
.
is a topology on
. □
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
.
of a hemiring R is a k-prime fuzzy k-ideal of 
if and only if 1) 
is a prime k-ideal of R.
contains exactly two elements.
.
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
.
of R is called proper if
, where 
is the fuzzy k-ideal of R defined by
,
.
forms a topology on the set
.
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
.
. This is true, since 
is the set of proper k-prime fuzzy k-ideals of R. So 
is an element of
.
with
.
. Since each fuzzy k-ideal of R is idempotent, this implies 
. Thus
.
of fuzzy k-ideals of R. Since
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
.
. This implies 
that is
.
for some 
implies
.
then there exists an element 
such that
.
, then we have
.
being the corresponding breakup of x, where only a finite number of 
and 
are not zero).
.
it follows that 
for some
.
implies that 
for some
.
and 2)
for some 
are equivalent.
is also a fuzzy k-ideal of R.
. Hence it follows that 
forms a topology on the set
. □