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Advances in Fuzzy Systems
Volume 2012 (2012), Article ID 801650, 8 pages
http://dx.doi.org/10.1155/2012/801650
Research Article

Proper Fuzzification of Prime Ideals of a Hemiring

403 Poorvarang Apartments, 13th Lane, Kolhapur 416008, Maharashtra, India

Received 26 April 2012; Accepted 8 October 2012

Academic Editor: Sendren Sheng-Dong Xu

Copyright © 2012 H. V. Kumbhojkar. 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.

Abstract

Prime fuzzy ideals, prime fuzzy k-ideals, and prime fuzzy h-ideals are roped in one condition. It is shown that this way better fuzzification is achieved. Other major results of the paper are: every fuzzy ideal (resp., k-ideal, h-ideal) is contained in a prime fuzzy ideal (resp., k-ideal, h-ideal). Prime radicals and nil radicals of a fuzzy ideal are defined; their relationship is established. The nil radical of a fuzzy k-ideal (resp., an h-ideal) is proved to be a fuzzy k-ideal (resp., h-ideal). The correspondence theorems for different types of fuzzy ideals of hemirings are established. The concept of primary fuzzy ideal is introduced. Minimum imperative for proper fuzzification is suggested and it is shown that the fuzzifications introduced in this paper are proper fuzzifications.

1. Introduction

This paper is, in some sense, an extended version of the article “On Fuzzification of Prime Ideals with Special Reference to Semirings” in SciTopics and something more.

Several attempts have been made to fuzzify the concepts of prime ideals/k-ideals/h-ideals of a semiring [17], prime ideals of a ring [815], and prime ideals of a semigroup [1618]. We have discussed elsewhere [6], in detail, the deficiencies in the definition of a prime fuzzy h-ideal proposed in [7]. The definition suffers from three major drawbacks. First, it is very restrictive in the sense that the fuzzy h-ideals, which are prime according to the definition, are 2-valued function. Secondly, since one of the two values is always 1 (the greatest element of the lattice), the function is determined by only one value, thus, severely curtailing its fuzziness. Third, when the zero element of the valuation lattice is not a prime element (and this happens in many important lattices), even the characteristic function of a prime ideal fails to be a prime fuzzy ideal. The technique adopted for the fuzzification by Zhan and Dudek in [7] and by others in [13, 5] is identical. Therefore, their prime fuzzy ideals inherit the same drawbacks. In [6] we have redefined prime fuzzy left h-ideal so that these deficiencies are completely removed. (It should be thankfully mentioned that one of the referees of the present paper has pointed out that in [4] two similar definitions of prime fuzzy ideal are stated. However, while proving major results of the paper, only 2-valued prime fuzzy ideals are used.) In this paper, we show that the problem of fuzzification of left ideal, left k-ideal, and left h-ideal need not be tackled separately. One single condition governs all the three. We also “refine” our definitions so that they look more compact, elegant, and easy for application. We prove that every proper fuzzy ideal (resp., k-ideal, h-ideal) is contained in a prime fuzzy ideal (resp., k-ideal, h-ideal). We introduce the concepts of fuzzy prime radical (or to be more precise, prime radical of a fuzzy ideal) and fuzzy nil radical (or nil radical of a fuzzy ideal), and fuzzy primary ideal. The prime and the nil radicals of a fuzzy k-ideal coincide when the valuation lattice is linearly ordered (e.g., when it is ). An analogous result holds for fuzzy h-ideals. We establish a correspondence between fuzzy ideals (resp., k-ideals, h-ideals) of a hemiring and those of its homomorphic image. The correspondence preserves prime, semiprime, and primary fuzzy ideals/k-ideals/h-ideals. Fuzzifications introduced in this paper can be labeled as “proper fuzzifications”.

2. Preliminaries

2.1. Ideals of a Semiring

In the following discussion, stands for a semiring. That is, is a commutative monoid having identity element 0 and is a semigroup satisfying the following identities: , and . A commutative semiring with unity is a semiring such that is a commutative monoid. We denote the identity element of by 1. With abuse of notation, we denote by . A left ideal of is a nonempty set which is closed under the addition of and is such that, for all and we have . A left ideal of is called a left k-ideal, if for all , and . It is called a left -ideal, if for all , and . A right ideal (resp., k-ideal, h-ideal) is similarly defined. Whenever a statement is made about left ideals, it is to be understood that the analogous statement is made about right ideals. An ideal is one, which is both right and left ideal. A left ideal is called prime left ideal, if it satisfies the following conditions:(i) and(ii)for all left ideals of , we have It is natural to call a k-prime (resp., h-prime) left ideal, if the condition holds for left k-ideals (resp., h-ideals) and .

Clearly, every prime left ideal is k-prime and every k-prime left ideal is h-prime. However, as will be seen in Example 1, the reverse implications, in general, are not true.

Example 1. (a) If is the Boolean lattice of four elements, then 0 is not a k-prime ideal, as the condition fails for k-ideals and . However, being the only h-ideal of is h-prime. Clearly, 0 is neither prime nor an h-ideal.
(b) Consider the semiring , where the binary operations and are defined as follows: and . One can easily see that has only three proper ideals, namely, , and . Since we have and is not a prime ideal. However, 0 and being the only k-ideals of , one can see that is a k-prime ideal. Again, is neither prime nor a k-ideal.
We shall soon see that the concepts of primeness and k-primeness (resp., h-primeness) coincide for k-ideals (resp., h-ideals).

Proposition 2 (see [5, 7]). If is a semiring and and are left ideals of , then and , where and , respectively, denote k-closure and h-closure of .

Using Proposition 2 we get the following.

Theorem 3. Let be a proper left k-ideal (resp., h-ideal) of a semiring . The following statements are equivalent.(a) is prime.(b) is k-prime (resp., h-prime).(c)For all implies .

Proof. We prove the statement: “(b) implies (c)”, for h-ideals.
Suppose is a proper h-prime left h-ideal such that for . Clearly, we have . Our first claim is that for and , where stands for the h-closure of a left ideal of , we have . Suppose we have and . Then for some , in , we have and and, therefore, the equalities: , and . As are elements of and is an h-ideal, is in . Therefore, and consequently, are in . It, then, follows that and, being h-prime, we have either or . Suppose . If is the left ideal generated by , then we have and hence, . Since is h-prime, we have and consequently, .

Using Zorn’s Lemma one can prove the following.

Theorem 4. Every proper ideal (resp., k-ideal, h-ideal) of a commutative hemiring with unity is contained in a prime ideal (resp., k-ideal, h-ideal) of .

Theorem 5. If is a multiplicatively closed set in a commutative hemiring with unity, disjoint from an ideal (resp., k-ideal, h-ideal) of , then there exists a prime ideal (resp., k-ideal, h-ideal) of such that and .

2.2. Prime Ideals of N

In the hemiring of nonnegative integers, obviously, an ideal is a k-ideal if and only if it is an h-ideal. Moreover, is a k-ideal if and only if for some . The prime k-ideals of are either where is a prime number in or the zero ideal. For each prime the ideal is a maximal k-ideal [19]. Clearly, is not a maximal ideal of .

Proposition 6. Let be a prime integer in . There is no prime ideal such that .

Proof. We first prove that the proposition holds for . Assuming the contrary, let be a prime ideal such that . Let be the smallest element of . Then for some positive integer . Let . Since and are in and is closed under addition, we have . Clearly, if , then . Therefore, we have and . Consider . For sufficiently large value of , we have is in and hence, in . Since is a prime ideal, we have . This contradicts the assumption that is the smallest element of . Therefore, is not a prime ideal.
Consider a prime integer and a prime ideal such that . Let be the smallest element of . Then, for some and . Consider . Clearly, and thus, we have . Let . Observe that for all we have . Therefore . However, contains and therefore, . Now set . Since we assume to be a prime ideal, we get . This contradicts the choice of as the smallest element in . Therefore, . Consider for . Then, clearly, we have . We claim that . If , then . Obviously, we have . On the other hand, if , then, we get the absurd result that for . Now set, as before, to get the contradiction to the assumption that is the smallest element of and complete the proof.

Theorem 7. is the only prime ideal of which is not a k- ideal (resp., an h-ideal).

Proof. One easily observes that is a prime ideal and is not a k-deal. Let be any other ideal of , which is not a k-ideal. Clearly, then, we have . Therefore, there exist such that . Let be the prime factorization of . If is prime, there is at least one prime integer in . Therefore, we have . As is not a k-ideal we have . On the other hand, by Proposition 6 we cannot have a prime ideal such that . Therefore, is not a prime ideal.

2.3. Fuzzy Ideals of a Semiring

Throughout this paper stands for a complete Heyting algebra, that is, a complete lattice such that for all subsets of and all and . An L-fuzzy subset (or simply an L-fuzzy set) of a set is a function ; a fuzzy set is an L-fuzzy set when is the unit interval . If , then the set is called -level cut or in short -cut of and is denoted by . The strict -level cut of is the set . An L-fuzzy left ideal of is an L-fuzzy set such that for all the following conditions are satisfied: (i) , (ii) . An L-fuzzy left ideal of is called an L-fuzzy left k-ideal, if the following condition is satisfied: . It is an L-fuzzy left h-ideal, if . An L-fuzzy right ideal (resp., k-ideal, h-ideal) is similarly defined. Whenever a statement is made about L-fuzzy left ideals, it is to be understood that the analogous statement is made about an L-fuzzy right ideals. An L-fuzzy ideal is one, which is both L-fuzzy right and L-fuzzy left ideal.

3. Prime Fuzzy Ideals

We defined L-fuzzy prime h-ideal in [6]. We extend the definition to L-fuzzy ideals and k-ideals.

Definition 8. An L-fuzzy left ideal (resp., k-ideal, h-ideal) of is called a prime L-fuzzy left ideal (resp., k-ideal, h-ideal), if it is nonconstant and, for all and , the following condition is satisfied:

Proposition 9. A nonconstant L-fuzzy left ideal (resp., k-ideal, h-ideal) of is prime if and only if its every nonempty level cut of is either a prime left ideal (resp., k-ideal, h-ideal) of or itself.

Corollary 10. A left ideal (resp., k-ideal, h-ideal) of is prime if and only if its characteristic function is an L-fuzzy prime left ideal (resp., k-ideal, h-ideal) for every complete Heyting algebra L.

Proposition 9 is proved for L-fuzzy left h-ideal in [6].

Let be an L-fuzzy prime (two sided) ideal of . Then and, therefore, . On the other hand, for , we have: Therefore, .

Let, further, . Then, .

Thus, is totally ordered.

Conversely, let be totally ordered and . Then, Therefore, is a prime L-fuzzy ideal.

This leads to the following elegant characterizations of prime fuzzy ideals.

Proposition 11. Let be a nonconstant L-fuzzy ideal (resp., k- ideal, h- ideal) of , and .(1) is prime if and only if is totally ordered.(2)Let be commutative hemiring with unity. is prime if and only if is totally ordered.(3)A nonconstant fuzzy ideal (resp., k- ideal, h- ideal) is prime if and only if Inf .(4)Let be commutative hemiring with unity. A nonconstant fuzzy ideal (resp., k- ideal, h- ideal) is prime if and only if .

The following example shows that the condition that is totally ordered is necessary for to be prime.

Example 12. Let be the Boolean algebra of four elements. Consider the L-fuzzy ideal defined as follows: Clearly, the L-fuzzy h-ideal is not prime, though holds for all .

Remark 13. While fuzzifying the condition of “primeness” stated in 2.1 three types of products of fuzzy left ideals and of , are used in the literature: namely, [13, 5, 7]. They are defined as follows: This was needed, because the problem of fuzzification of left ideals, left k-ideals, and left h-ideals were treated as three separate problems. Theorem 3 allows us to rope all the three in one and leads us to a compact characterization of primeness given in Proposition 11.
A semiprime fuzzy ideal, now defines itself.

Definition 14. An L-fuzzy left h-ideal of is called semiprime, if is nonconstant and, for all and , the following condition is satisfied: It follows that a nonconstant L-fuzzy ideal (resp., k-ideal, h-ideal) of is semiprime if and only if for all . In case is commutative hemiring with unity, the above equation is further simplified to . Analogues of Proposition 9 and Corollary 10 can easily be proved.

Theorem 15. Every nonconstant fuzzy ideal (resp., k-ideal, h-ideal) of a commutative ring with unity is contained in a minimal prime fuzzy ideal (resp., k-ideal, h-ideal).

Proof. As usual we prove the result for fuzzy h-ideals. Let be a nonconstant fuzzy h-ideal of a commutative ring with unity and . Let be a prime h-ideal containing . Define a fuzzy ideal by Clearly, is a prime fuzzy h-ideal containing and, thus, the class C of all prime fuzzy h-ideals containing is non-empty. We partially order C by reverse containment, that is, we define if and only if for all , and consider a totally ordered subset of C. Then, the set of the -level cuts of is a totally ordered set consisting of prime h-ideals (and possibly of ) for each . Therefore, is either a prime h-ideal of or itself. By Proposition 9, is a prime fuzzy h-ideal containing . Since is an upper bound of the family , C has a maximal element which, clearly, is a minimal prime fuzzy h-ideal containing .

Remark 16. Example 12 will testify that Theorem 15 is not valid when , in general.

4. Prime Radicals of a Fuzzy Ideal

In this section, we assume to be a commutative hemiring with unity.

Definition 17. If is an L-fuzzy ideal of , then the intersection of all prime L-fuzzy ideals (resp., k-ideals, h-ideals) of containing is called the prime (resp., k-prime, h-prime) radical of . We denote it by (resp., , ). If the set of prime L-fuzzy ideals (resp., k-ideals, h-ideals) of containing is empty, we define (resp., , ) to be .
Note that , (resp., , ) is a semiprime fuzzy ideal (resp., k-ideal, h-ideal) containing . Clearly . However, the following examples show that strict containment holds.

Example 18. Let be a prime integer. Consider and . Define a fuzzy set by By Proposition 9, is a prime fuzzy ideal (also k-ideal and h-ideal), for all . We will call the fuzzy ideal a prime fuzzy k-ideal induced by the prime number and denote it by .

Example 19. Suppose and . Define a fuzzy set by By Proposition 9, is a prime fuzzy ideal which is neither a fuzzy k-ideal nor a fuzzy h-ideal, for all . We will call the fuzzy ideal a prime fuzzy ideal induced by the prime integer and denote it by . Note that, in the light of Theorem 7, these are the only prime fuzzy ideals of which are not fuzzy k-ideals.

Example 20. Consider a fuzzy ideal defined by : Let and be the fuzzy k-ideal defined by : Let is prime, and , is prime}. Clearly, is the set of all prime fuzzy k-ideals of containing and is the set of all those prime fuzzy ideals containing , which are not fuzzy k-ideals. Since and , it is mundane to verify that and is the fuzzy ideal defined by : Clearly, .

Example 21. Let be the Boolean algebra of four elements. Consider the prime fuzzy ideal defined by and . Then, being a prime fuzzy k-ideal . Since all the fuzzy ideals of are fuzzy k-ideals, we have . Since the set of fuzzy h-ideals of is empty, .
Clearly, .

5. Nil Radicals of a Fuzzy Ideal

In this section, we assume to be a commutative hemiring with unity.

Recall that if is an ideal of , then its radical (also called nil radical) is defined as , for some integer .

We define the fuzzy analogue of nil radical as follows.

Definition 22. If is an L-fuzzy ideal of , then the L-fuzzy set defined by is called the L-fuzzy (nil) radical of .
Through series of propositions we prove that, when is totally ordered and is a fuzzy k-ideal (resp., h-ideal) of , so is .
The following results are the direct consequences of Definition 22.

Proposition 23. If is an ideal of , then , where and are the characteristic functions of and , respectively.

Proposition 24. If is a prime L-fuzzy ideal, then .

Proposition 25. If and are L-fuzzy ideals of a hemiring, then, the following statements hold.(a) .(b) .(c) .

Proposition 26. Let be an L- fuzzy ideals of and . Then, the following statements hold.(i) .(ii)If is a totally-ordered set, then , where and are strict level cuts.

Proof. We prove only (ii):

The following example shows that the set inclusion in Proposition 26(i) can be strict.

Example 27. Let be the hemiring of non-negative integers, , and a prime number. Let denote the ideal of generated by .
Define a fuzzy ideal as follows: Since , we have . On the other hand, for each and thus, , for any . Consequently, we have .

Theorem 28. If is totally ordered and an L-fuzzy k- ideal (resp., h- ideal) of , then coincides with (resp., ).

Proof. As usual we restrict our discussion to h-ideals. If is an L-fuzzy prime h-ideal containing , then, by Proposition 26, we have , where is the h-prime radical of the (scrisp) ideal . On the other hand, for any and , we have and therefore, we have .
Suppose . Then, there exists , such that . Let . Since , there exists a prime h-ideal say such that and .
Consider the following prime L-fuzzy h-ideal: Clearly, if , then we have . On the other hand, if , then we have and . Thus, we get and consequently, .
However, this leads to the following contradiction, .
Hence, we have .

Corollary 29. Let be a totally ordered set. If is an L-fuzzy k-ideal (resp., h-ideal) of , then, so is .

6. Correspondence Theorems

In this section, is a homomorphism of hemirings, is an L-fuzzy left ideal of , and is an L-fuzzy left ideal of .

In [20, Proposition 3.11], Zhan claims that if is an L-fuzzy h-ideal with sup property, then is an L-fuzzy h-ideal of . The following example does not substantiate the claim.

Example 30. Let be the hemiring given in Example 1 (b), be the hemiring of non-negative integers, and be the epimorphism given by for all .
Define a mapping by Since for all and for all , it can be verified that , , and .
One can readily see that is an L-fuzzy h-ideal with sup property; but is not an L-fuzzy h-ideal. For, we have and .
Example 30 raises a natural question: What are the sufficient conditions for a homomorphic image of an h-ideal (resp., k-ideal) to be an h-ideal (resp., k-ideal)? In order to answer this question, we introduce the following definition.

Definition 31. Let be a homomorphism of hemirings. An L-fuzzy left ideal of is called f-compatible if, for all .
Recall that is f-invariant, if implies . We leave it to the reader to prove that an f-compatible fuzzy left ideal is f-invariant.

Proposition 32. Let be a homomorphism of hemirings and and L-fuzzy left ideals of and , respectively. Then, the following statements hold.(1) is an f-invariant L-fuzzy left ideal of .(2)If is an L-fuzzy left k-ideal, then so is .(3)If is an L-fuzzy left h-ideal, then is an -compatible L-fuzzy left h-ideal of .(4)If is f-invariant (in particular if is f-compatible), then and therefore, .(5)If is an epimorphism, .

Proof. We prove (4) and (5). If is f-invariant and , then it is obvious that . This proves (4). Moreover, if , then . It, then, follows that if is an epimorphism, then .

This leads to the following correspondence theorem for L-fuzzy left k-ideals and h-ideals.

Theorem 33. Let be an epimorphism of hemirings.(1)There is one-to-one correspondence between the set of L-fuzzy left ideals (resp., k-ideals) of and that of f-invariant L-fuzzy left ideals (resp., k-ideals) of .(2)There is one-to-one correspondence between the set of L-fuzzy left h-ideals of and that of f-compatible L-fuzzy left h-ideals of .The above correspondence preserves prime and semiprime L- fuzzy left ideals (resp., k-ideals, h-ideals).

Proof. Suppose and are L-fuzzy left ideals of and . By Proposition 32, the correspondence is given by and . We only need to verify that, when is an L-fuzzy left ideal (resp., k-ideals, h-ideal), then so is , under the conditions specified for . A reader may easily prove that, when is f-invariant, is an L-fuzzy left ideal. Let, moreover, be an L-fuzzy k-ideal, , , , and . Then and therefore, we have . Let and consider . Clearly, . Since , we have and being a k-ideal . Therefore, . But, by Proposition 32 (4), this inequality is equivalent to . Thus, is an L-fuzzy left k-ideal.
On similar lines, one can prove that, when is an f-compatible L-fuzzy left h-ideal of , is an L-fuzzy left h-ideal of .

7. Primary Fuzzy Ideals

In this section, we assume that is a commutative hemiring with unity.

Recall that an ideal of a hemiring is primary, if (i) and (ii) or for some positive integer .

We define primary fuzzy ideal as follows.

Definition 34. A nonconstant L-fuzzy ideal/k-ideal/h-ideal of is primary, if or for some positive integer .
The following propositions are immediate consequences of Definition 34.

Proposition 35. A nonconstant L-fuzzy ideal of is a primary L-fuzzy ideal (resp., k-ideal, h-ideal) if and only if each of its nonempty level cuts is either a primary ideal (resp., k-ideal, h-ideal) of or itself.

Proposition 36. Let be an ideal of . The characteristic function is a primary L-fuzzy ideal (resp., k-ideal, h-ideal) of if and only if is primary ideal (resp., k-ideal, h-ideal), for every complete Heyting algebra L.

Proposition 37. Every prime L- fuzzy ideal (resp., k-ideal, h- ideal) is a primary L-fuzzy ideal (resp., k- ideal, h-ideal).
The fuzzy ideal in Example 27 is primary but not prime, as every nonempty, proper level-cut of the fuzzy ideal is primary but not prime.
The proof of the following proposition is straightforward.

Proposition 38. Let be a homomorphism of hemirings and and L-fuzzy ideals of and , respectively.(a)If is primary, then is primary f-invariant.(b)Let be an epimorphism and be f-invariant. If is a primary, then is primary.(c)If is an epimorphism and is an f-compatible primary fuzzy h-ideal, then is a primary L-fuzzy h-ideal.Thus, the correspondence theorems in the previous section preserve primary fuzzy ideals as well.

8. Minimum Imperative for Fuzzification

In this paper, we fuzzified the concepts of prime ideal, semiprime ideal, and primary ideal of a hemiring. Some of these concepts have been fuzzified earlier in different ways. Therefore, it is pertinent to ask: What constitutes “proper fuzzification” of a concept? Our answer is the following:

Suppose is a (crisp) ideal with property of a hemiring and is its fuzzification which inherits property . If it is the best fuzzification, it should satisfy the following properties.(1)For every Heyting algebras L, the characteristic function of satisfies the property if and only if has property .(2) satisfies the property , whenever every nonempty levelcut of different from satisfies the property and conversely.(3)The set has more than two elements. Let be a homomorphism of hemirings.(4)If an L-fuzzy ideal of has property , then , as an L-fuzzy ideal of , has property .(5)If L-fuzzy ideal of has property , then is an L-fuzzy ideal of with property , under some preassigned condition(s). The condition(s) is (are) suggested by the corresponding crisp situation.

The last sentence needs some elaboration. If is a (crisp) ideal of , is an ideal of provided that it satisfies the condition: and implies . The f-invariance of stated above is a “fuzzification” of the condition on the crisp ideal . If is an h-ideal, then is an h-ideal of provided that it satisfies the condition: and implies . The f-compatibility is a “fuzzification” of the condition on the ideal .

As proved earlier, the different types of prime, semiprime, and primary L-fuzzy ideals defined in this paper fulfill the above five conditions and, therefore, they are the best fuzzifications of the concepts.

Acknowledgments

The author expresses his deep sense of gratitude to the Editorial Board in general, and Editor-in-Chief in particular, of the journal “Advances in Fuzzy Systems”. This paper may not have come out in the present form, if the author has not been invited by them to write it within a specified time limit. He is also grateful to the referees for their useful suggestions.

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