Characterizations of Hemirings by the Properties of Their k-Ideals

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.


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][3][4][5][6][7][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][6][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][9][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.
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.

Basic Results on Hemirings
A semiring is an algebraic system   , , R   consisting of a non-empty set R together with two binary operations called addition "+" and multiplication "•" such that   , R  and   , R  are semigroups and connecting the two algebraic structures are the distributive laws: commutative and R has a zero element 0, such that 0 0 a a a     and 0 0 0 a a     for all a R  .An element 1 R   (if it exists) is called an identity element of R if 1 1 a a a     for all a R  .If a hemiring contains an identity element then it is called a hemiring with identity.A hemiring   , , R   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.

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

Lemma
AB AB  for any subsets A, B of a hemiring R. [30] If A and B are, respectively, right and left k-ideals of a hemiring R, then .AB A B   2.4.Definition [30] A hemiring R is said to be k-hemiregular if for each a R  , there exist , x y R  such that a axa aya   .

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 .

AB A B
  A fuzzy subset  of a non empty set X is a function . Im  denotes the set of all values of  .
A fuzzy subset For any fuzzy subsets  and  of X we define More generally, if   is a collection of fuzzy subsets of X , then by the intersection and the union of this collection we mean the fuzzy subsets

Definition
[21] A fuzzy left (right x y z R  .

Definition
Let  be a fuzzy subset of a universe X and the level subset of  .

Proposition
Let A be a non-empty subset of a hemiring R. Then a fuzzy set A  defined by Proof.Straightforward.□ 2.9.Proposition [23] If , A B are subsets of a hemiring R such that Im Im

Proposition
A fuzzy subset  of a hemiring R is a fuzzy left (right) Similarly we can show that    

Example
The set

 
0,1, 2,3 R  with operations addition and multiplication given by the following Cayley tables: Thus by Proposition 2.10,  is a fuzzy k-ideal of R.

k-Product of Fuzzy Subsets
To avoid repetitions from now R will always mean a hemiring   , , R   .

Definition
The k-product of two fuzzy subsets  and  of R is defined by By direct calculations we obtain the following result.

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

Lemma
Let R be a hemiring and , A B R  .Then we have 1) Hence there exist , , , q q B   , and so x AB  which is a contradiction.Thus we have Hence in any case, we have and , together with x a b   , gives and, consequently, Now, we have

Definition
The , , , , x a b a b R  .

Theorem
The .

Theorem
If  is a fuzzy subset of a hemiring R, then the following are equivalent: by ( 1) by ( 2)

Lemma
A fuzzy subset  in a hemiring R is a fuzzy left (right) k-ideal if and only if 1) Proof.Let  be a fuzzy left k-ideal of R. By Theorem 3.7,  satisfies 1).Now we prove condition 2).Let . Otherwise, there exist elements , , , . 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 and  is a fuzzy left k-ideal of R. □ For k-hemiregular hemirings we have stronger result.

Theorem
A hemiring R is k-hemiregular if and only if for any fuzzy right k-ideal  and any fuzzy left k-ideal  of 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 Hence by Lemma 2.5, R is k- hemiregular hemiring.□

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

Proposition
The following statements are equivalent for a hemiring R:  , where 0  is the set of whole numbers.By hypothesis  .□

Proposition
The following statements are equivalent for a hemiring R.
Proof. 1)  2) Let  and  be fuzzy k-ideals of R.

Theorem
Let R be a hemiring with identity 1, then the following assertions are equivalent: for all fuzzy k-ideals of R. Proof. 1)  2)  By Proposition 4.1.

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 R  be the collection of all k-ideals of R, then R  is a poset under the inclusion of sets.It is not difficult to see that R  is a complete lattice under the operations  ,  defined as Proof.Each complete Brouwerian lattice is distributive (cf.[31], 11.11).□

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 We show that  The converse is obvious.

Definition
In other words, a non-constant fuzzy k-ideal  is prime if from the fact that . 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.

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.

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 To prove the converse, consider a non-constant fuzzy  is not prime, which is a contradiction.Hence  is a prime fuzzy k-ideal in the second sense.

Corollary
The fuzzy set A  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.

Proposition
A non-constant fuzzy k-ideal  of a commutative hemiring R with identity is prime if and only if Proof.Let  be a non-constant fuzzy k-ideal of a commutative hemiring R with identity.If   ab t   , then for every x R  , we have   , then, as in the previous case, x by the identity of R, we obtain   Proof.Let P be a proper k-ideal of R such that a P  .
Let   | P     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 is a proper subset of P  and P  , then, according to the maximality of M, we have a P   and a P   .Hence , which is impossible.Thus, either M P   or M P   .□

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 AB P  for some k-ideals A, B of R, then by Proposition 4.1, So

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.

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.

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

 
| P     be the family of all prime k-ideals of R which contain P. Theorem 5.9, guarantees the existance of such ideals.Clearly for each  .Since P  is prime, we have

Lemma
Let R be a hemiring in which each fuzzy where a is any element of R and Hence by Zorn's lemma there exists a fuzzy k-ideal  of R which is maximal with respect to the property that    and   a    .We will show that  is an irreducible fuzzy k-ideal of R. Let is an irreducible fuzzy k-ideal of R. By Theorem 5.12,  is k-prime.□

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  

Theorem
A (left, right) k-ideal P of a hemiring R with identity is semiprime if and only if for every a R  from aRa P  it follows a P  .
Proof.Proof is similar to the proof of Theorem 5.1.□

Corollary
A k-ideal P of a commutative hemiring R with identity is semiprime if and only if for all a R  from 2 a P  it follows a P  .

Theorem
The following assertions for a hemiring R are equivalent:

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 . The converse is obvious.□ Theorem 6.2, suggest the following definition of semiprime fuzzy k-ideals.

Definition
A non-constant fuzzy k-ideal  of R is called semi- 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.

Proposition
A non-constant fuzzy k-ideal  of a commutative hemiring R with identity is semiprime if and only if 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.This hemiring has two k-ideals   0, c and R. Obviously these k-ideals are idempotent.
For any fuzzy ideal  of R and any x R This together with always, so we have to show that Hence 0 00 00 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

Prime Spectrum
Let R be a hemiring in which each k-ideal is idempotent.Let

Theorem
The set , where  is the usual empty set, because 0 belongs to each k-ideal.So empty set belongs to , where I 1 and : and be an arbitrary family of members of

Definition
A fuzzy k-ideal  of a hemiring R is said to be normal if there exists x R  such that     .The proof of the following theorem is same as the proof of Theorem 4.4 of [29].

Theorem
A fuzzy subset  of a hemiring R is a k-prime fuzzy k-ideal of R if and only if 1) contains exactly two elements.

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, R  the lattice of fuzzy normal k-ideals of R and  is an element of the set whose supremum is defined to be    

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.
A non-empty subset I of a hemiring R is called a left (right) ideal of R if 1)  , r R  .Obviously 0 I  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 , a b I  and x R  from x a b   it follows x I  .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, A A  for all left (right) k-ideals of R. Obviously A A  for each non-empty A R  .Also A B  for all A B R   .
let C, D be any right k-ideal and any left k-ideal of R, respectively.Then the characteristic functions C  , D  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 .

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

.
 be the family of all k-prime fuzzy k-ideals of R which contain  .Obviously Let a be an arbitrary element of R.Then, by Lemma 5.14, there exists an irreducible k-prime fuzzy k-ideal  such that  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 k    is also a fuzzy k-ideal of R , so k are k-prime fuzzy k-ideals of R. Thus each   con- all proper fuzzy k-prime k-ideals of R. For any fuzzy normal k-ideal  of R, we define k-ideal of R is called proper if    ,where  is the fuzzy k-ideal of R defined by the usual empty set and  is the characteristic function of k-ideal   0 .This follows since each k-prime fuzzy k-ideal of R is normal.Thus the empty subset belongs to of proper k-prime fuzzy k-ideals of R.So considering the infimum of a finite number of terms because 1 s  are effectively not being considered.Now, if for some ,

.
all the elements of the set (whose supremum we are taking) are individually less than are equal to a is also a fuzzy k-ideal of R. Hence it follows that   R P  forms a topology on the set R P .□