Interval-Valued Semiprime Fuzzy Ideals of Semigroups

Abstract

We introduce the notion of (i-v) semiprime (irreducible) fuzzy ideals of semigroups and investigate its different algebraic properties. We study the interrelation among (i-v) prime fuzzy ideals, (i-v) semiprime fuzzy ideals, and (i-v) irreducible fuzzy ideals and characterize regular semigroups by using these (i-v) fuzzy ideals.

1. Introduction

Zadeh [1] first introduced the concept of fuzzy sets in 1965. After that it has become an important research tool in mathematics as well as in other fields. It has many applications in many areas like artificial intelligence, coding theory, computer science, control engineering, logic, information sciences, operations research, robotics, and others. Likewise, an idea of connecting the fuzzy sets and algebraic structures came first in Rosenfeld’s mind. He first introduced the notion of fuzzy subgroup [2] in 1971 and studied many results related to groups. After that fuzzification of any algebraic structures has become a new area of research for the researchers. Some of fuzzy algebraic structures are mentioned in [3–9].

During the progress of the research on fuzzy sets, several types of extensions of fuzzy subsets were introduced. Interval-valued (in short, (i-v)) fuzzy subset is one of such extensions. In 1975, the concept of interval-valued fuzzy subset was introduced by Zadeh [10]. In this concept, the degree of membership of each element is a closed subinterval in [0,1]. Using such concept, it is possible to describe an object in a more precise way. There are many applications of (i-v) fuzzy subsets in different areas: Davvaz [11] on near rings, Hedayati [12] on semirings, Gorzałczany [13] on approximate reasoning, Turksen [14] on multivalued logic, Mendel [15] on intelligent control, Roy and Biswas [16] on medical diagnosis, and so on.

Similar to fuzzy set theory, (i-v) fuzzy set theory gradually developed on different algebraic structures. Biswas [17] defined the (i-v) fuzzy subgroups of Rosenfeld’s nature and investigated some elementary properties. Narayanan and Manikantan [18] introduced the notions of (i-v) fuzzy subsemigroup and various (i-v) fuzzy ideals in semigroups. In [19], Kar et al. introduced the concept of (i-v) prime (completely prime) fuzzy ideal of semigroups and studied their properties. Khan et al. [20] introduced the concept of a quotient semigroup by an interval-valued fuzzy congruence relation on a semigroup. In [21], Thillaigovindan and Chinnadurai introduced the notion of (i-v) fuzzy interior (quasi, bi) ideals of semigroup and studied their properties. However, the concept of (i-v) semiprime (irreducible) fuzzy ideals of semigroups has not been considered so far in the best of our knowledge.

In this paper our main goal is to study the semiprime (completely semiprime) ideal of a semigroup by using (i-v) fuzzy concept and discuss their properties. Also, we prove by an example that every (i-v) semiprime fuzzy ideal may not be (i-v) prime fuzzy ideal, although the converse is true. Finally, we define (i-v) irreducible fuzzy ideal of a semigroup and discuss different relations among (i-v) prime fuzzy ideal, (i-v) semiprime fuzzy ideal, and (i-v) irreducible fuzzy ideal.

2. Preliminaries

In this section we give some basic definitions and results of fuzzy algebra which will be used in this paper.

An interval number on is defined as a closed subinterval of satisfying . Denote as the set of all interval numbers on and , . Let , . Then (i) if and only if and . (ii) if and only if and . (iii) if and only if and . (iv) . (v) . If , then the difference is defined by , whenever, ; , whenever, . If is a family of interval numbers, where , then and . In this paper, we assume that any two interval numbers in are comparable; that is, for any two interval numbers , we have either or .

An (i-v) fuzzy subset of a nonempty set is a mapping , where is the set of all closed subintervals of . An (i-v) characteristic function of is an (i-v) fuzzy subset of a nonempty set , defined by . If is an i-v fuzzy subset of a set and , then a level subset of , denoted by , is defined by . It would be noted that for every i-v fuzzy subset of a nonempty set , there correspond two fuzzy subsets and of such that for every and vice versa. If and are two (i-v) fuzzy subsets of a set , then is said to be subset of , denoted by , if for all . For given two (i-v) fuzzy subsets and of , and for all . If is an (i-v) fuzzy subset of a nonempty set , then complement of , denoted by , is defined by , where and . If and are two (i-v) fuzzy subsets of a semigroup , then the product of and is an (i-v) fuzzy subset of , defined by , whenever for some , otherwise. If and are two nonempty subsets of a semigroup , then () if and only if , (ii) , and (iii) . A nonempty (i-v) fuzzy subset is said to be an (i-v) fuzzy left (right, two-sided, interior, bi-) ideal of a semigroup if for any , (resp., and and and . A nonempty subset of a semigroup is a left (right, two-sided) ideal of if and only if is an (i-v) fuzzy left (resp., right, two-sided) ideal of . A nonempty (i-v) fuzzy subset of a semigroup is an (i-v) fuzzy left (right, two-sided) ideal of if and only if (resp., and ). An (i-v) fuzzy point of a set is an (i-v) fuzzy subset of , defined by if if ; for fixed and , where . An (i-v) fuzzy point of is said to be contained in or to belong to a nonempty (i-v) fuzzy subset , denoted by , if . We denote as the set of all (i-v) fuzzy points of a semigroup . If , then . If , then (i-v) fuzzy left (right, two-sided) ideal generated by the fuzzy point is (resp., , .

3. (i-v) Semiprime Fuzzy Ideal of Semigroups

In this section we define (i-v) semiprime fuzzy ideals as a generalization of semiprime ideals of a semigroup and discuss its different algebraic properties.

Definition 1. A proper ideal of a semigroup is said to be semiprime if for any ideal of    implies .

Proposition 2. In a semigroup , an ideal of is a semiprime ideal of if and only if implies .

Definition 3. A nonconstant (i-v) fuzzy ideal of a semigroup is called an (i-v) semiprime fuzzy ideal of if for any (i-v) fuzzy ideal of    implies .

Theorem 4. Let be a nonempty proper subset of a semigroup . Then is a semiprime ideal of if and only if the (i-v) characteristic function of is an (i-v) semiprime fuzzy ideal of .

Proof. Let be a semiprime ideal of . Then it is easy to check that is a nonconstant (i-v) fuzzy ideal of . Consider to be an (i-v) fuzzy ideal of such that . Let us assume that . Then there exists such that . Since any two interval numbers in are comparable, . This implies . Since is a semiprime ideal of , by Proposition 2 it follows that for some ; that is, .
Again, which contradict the fact that . Therefore, and, hence, it follows that is an (i-v) semiprime fuzzy ideal of .
Conversely, let be an (i-v) semiprime fuzzy ideal of . Then is a nonconstant (i-v) fuzzy ideal of and hence is a proper ideal of . Let be an ideal of such that . Then is an (i-v) fuzzy ideal of and . Therefore, by our hypothesis, ;  that is,. Thus, is a semiprime ideal of .

Proposition 5. A nonconstant (i-v) fuzzy ideal of a semigroup is an (i-v) semiprime fuzzy ideal of if and only if a level ideal is a semiprime ideal of for every .

Lemma 6. Let be a semiprime ideal of a semigroup and an (i-v) fuzzy subset of defined by where . Then is an (i-v) semiprime fuzzy ideal of .

Proof. Since is a proper ideal of , it is easy to verify that is an (i-v) fuzzy ideal of . Let for any (i-v) fuzzy ideal of . If possible, let . Then for some . Due to comparability condition of interval numbers, we can write . Therefore, which implies implies . Since is a semiprime ideal of , for some ; that is, .
Now, a contradiction. Thus, it follows that and hence is an (i-v) semiprime fuzzy ideal of .

Note. Lemma 6 is an example of an (i-v) semiprime fuzzy ideal of a semigroup. Now, in the following we give an example of an (i-v) semiprime fuzzy ideal which is not an (i-v) prime fuzzy ideal, although every (i-v) prime fuzzy ideal is an (i-v) semiprime fuzzy ideal.

Example 7. Let , set of nonnegative integers. Then, forms a semigroup with respect to usual multiplication. Define an (i-v) fuzzy subset of by where . Then is an (i-v) semiprime fuzzy ideal of . But, is not an (i-v) prime fuzzy ideal of , because (see [19, Theorem 3.8]).

In the following theorem we try to extend Proposition 2 and characterize an (i-v) semiprime fuzzy ideal.

Theorem 8. If is a nonconstant (i-v) fuzzy ideal of a semigroup , then the following conditions are equivalent.(i) is an (i-v) semiprime fuzzy ideal of .(ii)For any (i-v) fuzzy point of , implies .(iii)For any (i-v) fuzzy point of , implies .

Proof. (i)(ii). Let be an (i-v) semiprime fuzzy ideal of and for an (i-v) fuzzy point of . Then is an (i-v) fuzzy ideal of and . Therefore, by our assumption, . Again, , which implies .
(ii)(iii). Let (ii) hold and . Then . Therefore, by (ii), it follows that .
(iii)(i). Let hold and for an (i-v) fuzzy ideal of . Let us assume that . Then for some implies . Now, we can choose an interval number such that . This implies . Again, . By , it follows that , a contradiction. Consequently, it shows that our assumption, that is, , is not true. Therefore, and hence is an (i-v) semiprime fuzzy ideal of .

Now, we investigate the nature of image and preimage of an (i-v) semiprime fuzzy ideal of a semigroup under homomorphism. For this reason, we first give the following definition of image and preimage of an (i-v) fuzzy set and then prove Proposition 10.

Definition 9 (see [2]). Let and be two nonempty sets and a function. Let and be the (i-v) fuzzy subsets of and , respectively. Then image of under the function is an (i-v) fuzzy subset of defined by where and .
Preimage of under the function is an (i-v) fuzzy subset of defined by for any .

Proposition 10. Let and be two semigroups and an epimorphism. If is an (i-v) fuzzy ideal of and an (i-v) fuzzy ideal of , then (i) is an (i-v) fuzzy ideal of ;(ii) is an (i-v) fuzzy ideal of .

Proof. Since is nonempty, there exists an element such that . Again, since is surjective, ; say, . Therefore, ; that is, . Hence, is nonempty. Let . Then (since is homomorphism) (by assumption) . Similarly, we find that . Therefore, is an (i-v) fuzzy ideal of .
(ii) Since is nonempty, there exists an element such that . Let . Then and . Hence, is nonempty. Let . Since is surjective, there exist such that and . Therefore, and hence . Now, = = ≥ (since is an (i-v) fuzzy ideal of ) . Similarly, we get . Hence, it follows that is an (i-v) fuzzy ideal of .

Theorem 11. Let be an epimorphism from a semigroup to another semigroup . If is an (i-v) semiprime fuzzy ideal of , then homomorphic preimage is an (i-v) semiprime fuzzy ideal of .

Proof. Since is a nonconstant (i-v) fuzzy ideal of , by Proposition 10, is an (i-v) fuzzy ideal of and there are two elements such that . Since is surjective, there exist such that and . Therefore, . Thus, is nonconstant. Now consider an (i-v) fuzzy point of such that . Let us assume that . Then ; that is, . This implies that . Again, since , for any (i-v) fuzzy point of , ⇒⇒ (since is homomorphism) ⇒. Since is arbitrary, is also arbitrary in (since is surjective). Therefore, is an arbitrary (i-v) fuzzy point in . Thus, . Since is an (i-v) semiprime fuzzy ideal of , by Theorem 8 it follows that , that is, , which is an inconsistent result. Therefore, and hence is an (i-v) semiprime fuzzy ideal of .

Theorem 12. Let be an epimorphism from a semigroup to another semigroup . If is an (i-v) semiprime fuzzy ideal of and also -invariant i.e., for any , then homomorphic image of is an (i-v) semiprime fuzzy ideal of .

Proof. Since is nonconstant, there are two elements such that . Again, from Proposition 10, it follows that is an (i-v) fuzzy ideal of . Let and where . Therefore, and . Again, since is -invariant, for every implies that . Also, for every implies . But . It shows that is nonconstant. Now, consider an (i-v) fuzzy point of such that . Then for any (i-v) fuzzy point of , ⇒—(i). Since is surjective, there exist elements such that and . Therefore, for any ( is homomorphism). Since is -invariant, . Thus, from (i), it implies that ≥ for any (i-v) fuzzy point of . Therefore, . Since is an (i-v) semiprime fuzzy ideal of , by Theorem 8, , that is, , and (since is -invariant) . This implies that and hence, by Theorem 8, is an (i-v) semiprime fuzzy ideal of .

Theorem 13. Let and be two semigroups and an epimorphism. Then there is a one-to-one correspondence between the -invariant (i-v) semiprime fuzzy ideals of and (i-v) semiprime fuzzy ideals of .

Proof. Let set of all -invariant (i-v) semiprime fuzzy ideals of and set of all (i-v) semiprime fuzzy ideals of .
Define a mapping by , where . Let such that . Since is surjective, for any . Therefore, = . This implies that . Hence, is well defined.
Let for any two . Then for any —(i). Since and are both -invariant, for any , implies and . Hence, it follows from (i) that for every . Since is arbitrary and is surjective, is arbitrary. It implies that for all ; that is, . Hence, is one-one.
Consider and . Since , . This implies that where, . This implies that is onto. Therefore, is bijective and the result follows.

In the following we try to give the definition of (i-v) fuzzy -system and characterize (i-v) semiprime fuzzy ideal using it.

Definition 14. A nonempty subset of a semigroup is called a -system of if for every there exists such that .

Definition 15. A nonempty (i-v) fuzzy subset of a semigroup is called an (i-v) fuzzy -system of if for any and     implies for some .

Theorem 16. A nonempty subset of a semigroup is a p-system of if and only if the characteristic function is an (i-v) fuzzy p-system of .

Proof. Let be a -system of . Then is nonempty. Let and such that . This implies that implies . Therefore, by our assumption, for some ; that is, . Consequently, it follows that is an (i-v) fuzzy -system of .
Conversely, let be an (i-v) fuzzy -system of . Then for any , . Therefore, for any ,  . Now, by our hypothesis, for some . This implies that ; that is, . Thus, it follows that is a -system of .

Theorem 17. A nonconstant (i-v) fuzzy ideal of a semigroup is an (i-v) semiprime fuzzy ideal of if and only if is an (i-v) fuzzy p-system of .

Proof. Let be an (i-v) semiprime fuzzy ideal of . Since is nonconstant, there exists such that and hence . Therefore, is nonempty. Consider an interval number and such that ; that is, . Then implies . Now, using Theorem 8 it implies that for some (i-v) fuzzy point , which implies ; that is, . Consequently, it follows that is an (i-v) fuzzy -system of .
Conversely, let be an (i-v) fuzzy -system of and an (i-v) fuzzy ideal of such that . Let us choose . Then for some ; that is, . Now, we can choose an interval number such that . This implies that ; that is, . Therefore, by our hypothesis, for some ; that is, . Further, , a meaningless result due to wrong assumption. Thus, it follows that ; that is, is an (i-v) semiprime fuzzy ideal of .

The concept of completely semiprime ideal (see Definition 18) is defined in [22], in which it is known as semiprime ideal. Now, we try to generalize this concept using (i-v) fuzzy points and define (i-v) completely semiprime fuzzy ideal. Also, we investigate its various properties.

Definition 18. A proper ideal of a semigroup is said to be completely semiprime if for any     implies .

Definition 19. A nonconstant (i-v) fuzzy ideal of a semigroup is called an (i-v) completely semiprime fuzzy ideal of if for any (i-v) fuzzy point of    implies .

Proposition 20. A proper subset of a semigroup is a completely semiprime ideal of if and only if is an (i-v) completely semiprime fuzzy ideal of .

Proof. Let be a completely semiprime ideal of . Then is a proper ideal of and hence is a nonconstant (i-v) fuzzy ideal of . Consider an (i-v) fuzzy point of such that . This implies that . Therefore, by our assumption, implies implies . Thus, it follows that is an (i-v) completely semiprime fuzzy ideal of .
Conversely, let be an (i-v) completely semiprime fuzzy ideal of . Then is a nonconstant (i-v) fuzzy ideal of and hence is a proper ideal of . Let for any . Then for any interval number . Therefore, . Hence, by our assumption, . This implies that . This shows that is a completely semiprime ideal of .

Proposition 21. A nonconstant (i-v) fuzzy ideal of a semigroup is an (i-v) completely semiprime fuzzy ideal of if and only if a level ideal is completely semiprime ideal of for every .

Proposition 22. A nonconstant (i-v) fuzzy ideal of a semigroup is an (i-v) completely semiprime fuzzy ideal of if and only if for every .

Proof. Let be an (i-v) completely semiprime fuzzy ideal of and . If for some , then implies . Hence, by our assumption, . This implies that (since is an (i-v) fuzzy ideal of ). Thus, .
Conversely, let for all and consider an (i-v) fuzzy point of such that . Therefore, . This shows that is an (i-v) completely semiprime fuzzy ideal of .

Proposition 23. Let be a semigroup epimorphism. Then the following statements are true.(i) If is an (i-v) completely semiprime fuzzy ideal of , then homomorphic preimage is an (i-v) completely semiprime fuzzy ideal of .(ii) If is an f-invariant (i-v) completely semiprime fuzzy ideal of , then homomorphic image is an (i-v) completely semiprime fuzzy ideal of .

Proof. (i) It is clear that is an (i-v) fuzzy ideal of . Now, for any , ( is a homomorphism) (by our assumption) . Thus, Proposition 22 implies that is an (i-v) completely semiprime fuzzy ideal of .
(ii) From Proposition 10, it follows that is an (i-v) fuzzy ideal of . Let . Since is onto, . Let . Therefore, (since is homomorphism), which implies . Now, (since is -invariant) (using Proposition 22) . Hence, it follows that is an (i-v) completely semiprime fuzzy ideal of .

Proposition 24. Every (i-v) completely semiprime fuzzy ideal of a semigroup is an (i-v) semiprime fuzzy ideal of .

Proof. Let be an (i-v) completely semiprime fuzzy ideal of and consider for an (i-v) fuzzy ideal of . Let us choose . Then for some ; that is, . Now, we take an interval number such that . This implies that and hence ; that is, . Hence, by our assumption, which implies , an absurd result. Consequently, it follows that and so is an (i-v) semiprime fuzzy ideal of .

Note. But the converse of Proposition 24 is not always true; that is, every (i-v) semiprime fuzzy ideal of a semigroup may not be an (i-v) completely semiprime fuzzy ideal of .

Example 25. Consider . Then is a semigroup with respect to matrix multiplication. Define an (i-v) fuzzy subset of by where such that . Then it is easily shown that is an (i-v) semiprime fuzzy ideal of . But, if we take a matrix such that , then we see that , a null matrix in . Therefore, . Thus, from Proposition 22, it follows that is not an (i-v) completely semiprime fuzzy ideal of .

Proposition 26. In a commutative semigroup , every (i-v) semiprime fuzzy ideal is an (i-v) completely semiprime fuzzy ideal of .

Proof. Let be an (i-v) semiprime fuzzy ideal of and an (i-v) fuzzy point of such that . Then implies . Now, for any , we have If , then . Again, if , then (since is commutative). Thus, for both cases . Thus, it follows from the above that , which implies implies . This shows that is an (i-v) completely semiprime fuzzy ideal of .

Corollary 27. In a commutative semigroup , a nonconstant (i-v) fuzzy ideal of is an (i-v) semiprime fuzzy ideal of if and only if for every .

Theorem 28. A commutative semigroup is regular if and only if every nonconstant (i-v) fuzzy ideal of is an (i-v) semiprime fuzzy ideal of .

Proof. Let be a nonconstant (i-v) fuzzy ideal of a regular semigroup and for any (i-v) fuzzy ideal of . Then (since is regular) . This implies that is an (i-v) semiprime fuzzy ideal of .
Conversely, let every nonconstant (i-v) fuzzy ideal of   be an (i-v) semiprime fuzzy ideal of . Let . Since is a principal ideal of generated by , is an (i-v) fuzzy ideal of and hence, by our assumption, is an (i-v) semiprime fuzzy ideal of . Therefore, by Corollary 27, we can write . This implies (since is commutative). Thus, for both cases there always exists such that and hence is regular.

Proposition 29. If is an (i-v) completely semiprime fuzzy ideal of a semigroup , then for all .

Proof. Since is an (i-v) completely semiprime fuzzy ideal of , for any (since is an (i-v) fuzzy ideal of ) . This implies that and hence the result follows.

Definition 30 (see [19]). A nonconstant (i-v) fuzzy ideal of a semigroup is called an (i-v) completely prime fuzzy ideal of if for any two (i-v) fuzzy points of     implies either or .

Theorem 31 (see [19]). Let be an (i-v) prime fuzzy ideal of a semigroup . Then is an (i-v) completely prime fuzzy ideal of if and only if for any two (i-v) fuzzy points and of    implies .

Theorem 32. Let be an (i-v) prime fuzzy ideal of a semigroup . Then is an (i-v) completely prime fuzzy ideal of if and only if is an (i-v) completely semiprime fuzzy ideal of .

Proof. Let be an (i-v) completely prime fuzzy ideal of . Then it is clear that is an (i-v) completely semiprime fuzzy ideal of .
Conversely, let be an (i-v) completely semiprime fuzzy ideal of and consider two (i-v) fuzzy points and of such that . Then . This implies that (by Proposition 29) . Hence, by Theorem 31, it follows that is an (i-v) completely prime fuzzy ideal of .

Definition 33 (see [22]). A semigroup is called intra-regular if for each element there exist elements such that .

Theorem 34. In a semigroup , the following statements are equivalent.(i) is intra-regular.(ii) is a semilattice of simple semigroups.(iii)Every ideal of is completely semiprime.(iv)Every i-v fuzzy ideal of is an i-v completely semiprime fuzzy ideal of .

Proof. The equivalence conditions (i), (ii), and (iii) follow from [22, Theorem ].
(i)(iv). Let be intra-regular. Then, for any , there exist such that . If is an (i-v) fuzzy ideal of , . This implies that for all . Hence, Proposition 22 implies that is an (i-v) completely semiprime fuzzy ideal of .
(iv)(i). Let (iv) hold and . Since is a principal ideal of generated by , is an (i-v) fuzzy ideal of and hence, by our assumption, is an (i-v) completely semiprime fuzzy ideal of . Now, using Proposition 22, we can write . This implies that . Therefore, for any possible form of , it implies that . Hence, is intra-regular.

Theorem 35. The following conditions are equivalent in a semigroup .(i) is intra-regular.(ii)Every (i-v) fuzzy interior ideal of is an i-v completely semiprime fuzzy ideal of .(iii)For every (i-v) fuzzy interior ideal of and , .

Proof. Since proof is simple, we omit the proof.

4. (i-v) Irreducible Fuzzy Ideal of Semigroups

In this section we have defined (i-v) irreducible fuzzy ideal of a semigroup and we study its several properties.

Definition 36. A proper ideal of a semigroup is called an irreducible ideal of if for any two ideals and of implies either or .

Definition 37. A nonconstant (i-v) fuzzy ideal of a semigroup is called an (i-v) irreducible fuzzy ideal of if for any two (i-v) fuzzy ideals and of    implies either or .

Definition 38. A proper ideal of a semigroup is called a strongly irreducible ideal of if for any two ideals and of implies either or .

Definition 39. A nonconstant (i-v) fuzzy ideal of a semigroup is called an (i-v) strongly irreducible fuzzy ideal of if for any two (i-v) fuzzy ideals , of implies either or .

Proposition 40. Every (i-v) strongly irreducible fuzzy ideal of a semigroup is an (i-v) irreducible fuzzy ideal of . But the converse is not true in general.

Proof. For the converse part, we give a counter example.
Consider a semigroup where and a binary operation “” on is defined by Then ,  ,  ,  , and are the ideals of in which, ,   are the irreducible ideals of , but not the strongly irreducible ideals of . Define an (i-v) fuzzy subset of such that where . Then is an irreducible fuzzy ideal of , but not an (i-v) strongly irreducible fuzzy ideal of .

Proof of Propositions 41–45 is straightforward and so we omit the proof.

Proposition 41. Let be an (i-v) irreducible fuzzy ideal of a semigroup and . If for some fuzzy ideals and of , then and are both irreducible fuzzy ideals of .

Proposition 42. A proper ideal of a semigroup is an irreducible ideal of if and only if the characteristic function is an (i-v) irreducible fuzzy ideal of .

Proposition 43. An (i-v) fuzzy ideal of a semigroup is an (i-v) irreducible fuzzy ideal of if and only if a level ideal is an irreducible ideal of for every .

Proposition 44. If and are two (i-v) irreducible fuzzy ideals of a semigroup , then is an (i-v) irreducible fuzzy ideal of , provided is nonempty.

Proposition 45. Let and be two nonempty sets and a function. If ,   and ,   are the (i-v) fuzzy ideals of and , respectively, then the following statements are true.(i).(ii), provided is -invariant.(iii), provided and are -invariant.(iv). In the following theorem we try to find the homomorphic image and preimage of an (i-v) irreducible fuzzy ideal of a semigroup.

Theorem 46. Let be a semigroup epimorphism and ,   (i-v) irreducible fuzzy ideals of and , respectively. Then(i)homomorphic image is an (i-v) irreducible fuzzy ideal of , provided is -invariant;(ii)homomorphic preimage is an (i-v) irreducible fuzzy ideal of , provided every (i-v) fuzzy ideal of is -invariant.

Proof. (i) Let be -invariant (i-v) irreducible fuzzy ideal of . Since is a nonconstant (i-v) fuzzy ideal of , by Proposition 10,    is a nonconstant (i-v) fuzzy ideal of . Let and be two (i-v) fuzzy ideals of such that . Since is -invariant, by Proposition 45, . Since and are (i-v) fuzzy ideals of (by Proposition 10) and is an (i-v) irreducible fuzzy ideal of , either or . This follows that either or . Hence, is an (i-v) irreducible fuzzy ideal of .
(ii) Let every (i-v) fuzzy ideal of be -invariant. Since is a nonconstant (i-v) fuzzy ideal of , by Proposition 10   is a nonconstant (i-v) fuzzy ideal of . Let and be two (i-v) fuzzy ideals of such that . Then, by Proposition 45, . Since is an (i-v) irreducible fuzzy ideal of , either or . This follows that either or . Hence, is an (i-v) irreducible fuzzy ideal of .

Theorem 47. If is a semigroup epimorphism, then there is a one-to-one correspondence between the set of -invariant (i-v) irreducible fuzzy ideals of and the set of all (i-v) irreducible fuzzy ideals of .

Proof. Let be the set of all -invariant (i-v) irreducible fuzzy ideals of and the set of all (i-v) irreducible fuzzy ideals of . Now, we define a mapping by . Then, it is a well-defined map. Let , where . This implies that = for all for all (since and are -invariant). Since is onto and is arbitrary, is also arbitrary and hence . Therefore, it follows that is one-to-one.

Theorem 48. Every (i-v) prime fuzzy ideal of a semigroup is an (i-v) irreducible fuzzy ideal of .

Proof. Let be an (i-v) prime fuzzy ideal of . Consider and to be two (i-v) fuzzy ideals of such that . Then and . Again, since and are (i-v) fuzzy ideals of , . Therefore, by our assumption, either or . This implies that either or . Hence, is an (i-v) irreducible fuzzy ideal of .

For converse part, we set an example as follows.

Consider a semigroup with usual multiplication and an ideal of . Then is an irreducible ideal of , but not a prime ideal of . Now, if we define an (i-v) fuzzy subset of by where , then it is easily shown that is an (i-v) irreducible fuzzy ideal of , but not an (i-v) prime fuzzy ideal of .

Remark 49. But the converse of Theorem 48 is true in a regular semigroup.

Theorem 50. Every (i-v) irreducible fuzzy ideal of a regular semigroup is an (i-v) prime fuzzy ideal of .

Proof. Let be an (i-v) irreducible fuzzy ideal of and consider and to be two (i-v) fuzzy ideals of such that . Since is regular, for every , there exists such that . Therefore, which implies . But . Thus, by our assumption, either or . This implies either or . Hence, is an (i-v) prime fuzzy ideal of .

Theorem 51. A nonconstant (i-v) fuzzy ideal of a semigroup is an (i-v) prime fuzzy ideal of if and only if is an (i-v) irreducible fuzzy ideal as well as an (i-v) semiprime fuzzy ideal of .

Proof. Let be an (i-v) prime fuzzy ideal of . Then is an (i-v) semiprime fuzzy ideal of and, by Theorem 48, is also an (i-v) irreducible fuzzy ideal of .
Conversely, let be an (i-v) irreducible fuzzy ideal as well as an (i-v) semiprime fuzzy ideal of . Consider and to be two (i-v) fuzzy ideals of such that . Then, . But is an (i-v) semiprime fuzzy ideal of . Therefore, . Now, which implies either or implies either or . Hence, is an (i-v) prime fuzzy ideal of .

Definition 52. A set of (i-v) fuzzy ideals of a semigroup forms an -chain if or for any two (i-v) fuzzy ideals . If such a set exists in , then we say has an -chain.

Definition 53. An (i-v) fuzzy ideal of a semigroup is called maximal if, for any (i-v) fuzzy ideal of , implies .

Theorem 54 shows the existence of an irreducible ideal corresponds to an ideal of a semigroup. In Theorem 55, we try to generalize this result in terms of interval-valued fuzzy concept.

Theorem 54. Let be an ideal of a semigroup and such that . Then, there exists an irreducible ideal of such that and .

Theorem 55. Let be an (i-v) fuzzy ideal of a semigroup and such that . Then, there exists an (i-v) irreducible fuzzy ideal of such that and .

Proof. Let be a set of all (i-v) fuzzy ideals of such that and . Then and, under inclusion, is a poset. Now, if we consider an -chain of , then it is easy to show that the (i-v) fuzzy ideal of is an upper bound of . Therefore, from Zorn’s Lemma, we can say that has a maximal element, say, , which is an (i-v) fuzzy ideal of containing such that . Let and be two (i-v) fuzzy ideals of such that . Then either or . Therefore, by maximality condition, it implies that either or . Hence, is an (i-v) irreducible fuzzy ideal of .

Theorem 56. The following conditions are equivalent in a semigroup .(i) is regular.(ii)Every (i-v) fuzzy ideal of is idempotent.(iii)Every (i-v) fuzzy ideal of is an (i-v) semiprime fuzzy ideal of .(iv)Every (i-v) fuzzy ideal of is an intersection of (i-v) prime fuzzy ideals of containing it.

Proof. (i)(ii). Let be regular and an (i-v) fuzzy ideal of . Then . This shows that is an (i-v) idempotent fuzzy ideal of .
(ii)(i). Let hold and consider and to be two (i-v) fuzzy ideals of . Then and is an (i-v) fuzzy ideal of . Therefore, by , . This implies that and hence is regular.
(ii)(iii). Let every (i-v) fuzzy ideal of be idempotent. Let be an (i-v) fuzzy ideal of and for any (i-v) fuzzy ideal of . Then, by our assumption, ; that is, . Hence, is an (i-v) semiprime fuzzy ideal of .
(iii)(iv). Let hold and consider to be an (i-v) fuzzy ideal of . Then, for any , it follows from Theorem 55 that there exists an (i-v) irreducible fuzzy ideal of such that and . Now, if we consider the set of all those (i-v) irreducible fuzzy ideals of containing , then is the smallest (i-v) irreducible fuzzy ideal of containing . Also, . This implies that for arbitrary . Hence . But implies every () is an (i-v) semiprime fuzzy ideal of . Hence, by Theorem 51, every () is an (i-v) prime fuzzy ideal of . Thus, it follows that is an intersection of (i-v) prime fuzzy ideals of containing .
(iv)(ii). Let hold and consider to be an (i-v) fuzzy ideal of . Then . Again, since is an (i-v) fuzzy ideal of , by , is an (i-v) prime fuzzy ideal of containing . This implies that for every . But each is an (i-v) prime fuzzy ideal of . Hence, . Therefore, ; that is, . Thus, it follows that ; that is, is an (i-v) idempotent fuzzy ideal of .

Theorem 57. Every (i-v) fuzzy ideal of a semigroup is an (i-v) prime fuzzy ideal of if and only if is regular and all (i-v) fuzzy ideals of form an -chain.

Proof. Let every (i-v) fuzzy ideal of be an (i-v) prime fuzzy ideal of . Then each of these ideals is also an (i-v) semiprime fuzzy ideal of and hence, by Theorem 56, is regular. Again, for any two (i-v) fuzzy ideals and of , and is an (i-v) fuzzy ideal of . Therefore, by our assumption, is an (i-v) prime fuzzy ideal of . Thus, it follows that either or ; that is, either or . Thus, the set of all (i-v) fuzzy ideals of forms an -chain.
Conversely, let be regular and all (i-v) fuzzy ideals form an -chain. Let be an (i-v) fuzzy ideal of and for any two (i-v) fuzzy ideals of . Then, from chain property, either or . Again, since is regular, by Theorem 56, and . Now, if , then and if , then . Thus it follows that either or . Hence is an (i-v) prime fuzzy ideal of .

5. Conclusion

Interval-valued fuzzy ideals are new tools to study fuzzy algebra. Interval-valued semiprime fuzzy ideals may be used to further study of fuzzy semigroups and fuzzy semirings and certainly give some important aspects of fuzzy algebra as a whole.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Paltu Sarkar would like to gratefully acknowledge the financial support of the Council of Scientific & Industrial Research (CSIR), India, for carrying out this research work. The authors are highly grateful to the referees and Professor Katsuhiro Honda, Associate Editor, for their valuable comments and suggestions for improving the paper.