A Novel of Ideals and Fuzzy Ideals of Gamma-Semigroups

Abstract

In this paper, we define new types of ideals, fuzzy ideals, almost ideals, and fuzzy almost ideals of -semigroups by using the elements of . We investigate properties of them.

1. Introduction and Preliminaries

Ideal theory in semigroups, like all other algebraic structures, plays an important role in studying them. Bi-ideals in semigroups were introduced by Good and Hughes [1] in 1952. Steinfeld [2] gave a notion and studied quasi-ideals in semigroups in 1956. In 1965, Zadeh introduced the concept of fuzzy subsets in [3]. Since then, fuzzy subsets are now applied in various fields. Kuroki [4] first studied fuzzy ideals in semigroups. Almost ideals (A-ideals) in semigroups were studied by Grošek and Satko [5–7] in 1980-1981. Next year, Bogdanović [8] studied almost bi-ideals in semigroups by using concepts of almost ideals and bi-ideals in semigroups. Sen [9] introduced -semigroups where is a set of their operations. This algebraic structure is generalization of semigroups. Many results in semigroup theory were generalized to the results in -semigroup theory. In 2006, Chinram [10] studied quasi--ideals in -semigroups, and Jirotkul joined with Chinram [11] to study bi--ideals in -semigroups in 2007. Moreover, Iampan [12] gave remarkable notes on bi--ideals (or bi-ideals) in -semigroups in 2009. Recently, left almost ideals (left A-ideals) in -semigroups were studied by Wattanatripop and Changphas [13] in 2017. Wattanatripop et al. [14] studied fuzzy almost bi-ideals, almost quasi-ideals, and fuzzy almost ideals in semigroups in 2018. So, all of these give the inspiration to study about new types of ideals and fuzzy ideals in -semigroups in this paper.

First, we recall the definition of -semigroups which was defined by Sen and Saha [15].

Definition 1 (see [15]). Let and be nonempty sets. Then, is called a -semigroup if there exists a mapping written as satisfying the axiom for all and .

Remark 1. (1)In case , the definition of -semigroup is a semigroup(2)Every semigroup can be considered to be a -semigroup where (3)If is a -semigroup, then for each , () is a semigroupLet be a -semigroup. For nonempty subsets of , letIf and , we let , , and .

Definition 2. Let be a -semigroup. A nonempty subset of is called(1)A sub--semigroup of if (2)A left ideal of if (3)A right ideal of if (4)An ideal of if it is both a left ideal and a right ideal of (5)A quasi-ideal of if (6)A bi-ideal of if is a sub--semigroup of and Recently, Wattanatripop and Changphas [13] defined the concepts of left almost ideals and right almost ideals of -semigroups. A -semigroup containing no proper left (respectively, right) almost ideals was characterized.
Now, we recall the definitions and some notations of fuzzy subsets. A fuzzy subset of a set is a function from into the closed interval . For any two fuzzy subsets and of a set ,(1) is a fuzzy subset of defined by(2) is a fuzzy subset of defined by(3) is a fuzzy subset of defined by(4) if for all .For a fuzzy subset of a set , the support of is defined byThe characteristic mapping of a subset of is a fuzzy subset of defined byThe definition of fuzzy points was given by Pu and Liu [16]. For and , a fuzzy point of a set is a fuzzy subset of defined bySome basic concepts of fuzzy semigroup theory can be seen in [17].
For a -semigroup , let be the set of all fuzzy subsets of . For each , define a binary operation on byLet . Then, is a -semigroup.
In 2017, Wattanatripop and Changphas [13] defined the concepts of left A-ideals and right A-ideals (almost left ideals and almost right ideals) of a -semigroup as follows.
A nonempty subset [] of a -semigroup is called a left (right) A-ideal of ifIn 1981, Bogdanović [8] gave the definition of almost bi-ideals of semigroups as follows.
A nonempty subset of a semigroup is called an almost bi-ideal of ifIn 2018, Wattanatripop et al. [14, 18] introduced the notions of almost quasi-ideals, fuzzy almost bi-ideals, fuzzy almost left (right) ideals, and fuzzy almost quasi-ideals in semigroups as follows.
A nonempty subset of a semigroup is called an almost quasi-ideal of ifLet be a fuzzy subset of a semigroup such that . Then, is called(1)A fuzzy almost bi-ideal of if for all ,(2)A fuzzy almost left ideal of if for all ,(3)A fuzzy almost right ideal of if for all ,(4)A fuzzy almost quasi-ideal of if for all ,In 1981, Kuroki [4] introduced the notion of fuzzy ideals of semigroups as follows: A fuzzy subset of a semigroup is called:(1)A fuzzy left ideal of if for all .(2)A fuzzy right ideal of if for all .(3)A fuzzy ideal of if it is both a fuzzy left ideal and a fuzzy right ideal of .The aim of this paper is to define new types of ideals and fuzzy ideals of a -semigroup by using elements in . In Section 2, we consider new types of ideals of . In Section 3, we study new types of fuzzy ideals of . In Section 4, we consider new types of almost ideals of . In Section 5, we study new types of fuzzy almost ideals of .

2. New Types of Ideals

In this section, we will focus on -ideals, -quasi-ideals, and -bi-ideals of -semigroups for .

2.1. -Ideals

First, we will define -ideals of -semigroups as follows.

Definition 3. Let be a -semigroup, be a nonempty subset of , and . Then, is called(1)A left -ideal of if .(2)A right -ideal of if .(3)An -ideal of if it is both a left -ideal and a right -ideal of .(4)An -ideal of if it is an -ideal of .

Remark 2. (1)Every left ideal of a -semigroup is a left -ideal of for all .(2)Every right ideal of a -semigroup is a right -ideal of for all .(3)Every ideal of a -semigroup is an -ideal of for all .However, the converse of Example 1 is not generally true. We can see in the following example.

Example 1. Let and for all and . Then, is a -semigroup. Let . It is easy to show that is a left 4-ideal but not a left ideal of .

Theorem 1. The following statements are true:(1)If is a left -ideal of a -semigroup , then is a left ideal of a semigroup .(2)If is a right -ideal of a -semigroup , then is a right ideal of a semigroup .(3)If is an -ideal of a -semigroup , then is an ideal of a semigroup .For a nonempty subset of a -semigroup , let , and be the left -ideal, the right -ideal, and the -ideal of generated by , respectively.

Theorem 2. Let be a nonempty subset of a -semigroup . Then,(1).(2).(3).

Proof. (1) Let be a nonempty subset of a -semigroup . Let . Clearly, . Since is a -semigroup, . Therefore, is a left -ideal of . Next, let be any left -ideal of containing . Since is a left -ideal of and , . Therefore, . Hence, is the smallest left -ideal of containing . Therefore, , as required.
The proofs of (2) and (3) are similar to the proof of (1).

Theorem 3. Let be a left -ideal and a right -ideal of a -semigroup . Then, is an -ideal of for all .

Proof. Let and be a left -ideal and a right -ideal of , respectively, and let . Clear that . We have and . Therefore, is an -ideal of .

Theorem 4. Let and be two left -ideals of a -semigroup . The following statements are true:(1) is a left -ideal of .(2) is a left -ideal of , where .

Proof. (1) Let and be two left -ideals of . Clear that . Then, . Hence, is a left -ideal of .
(2) Since , we have . Hence, is a left -ideals of .

Theorem 5. Let and be two right -ideals of a -semigroup . Then,(1) is a right -ideal of .(2) is a right -ideal of , where .

Proof. It is similar to Theorem 4.

Theorem 6. Let and be two -ideals of a -semigroup . Then,(1) is an -ideal of .(2) is an -ideal of , where .

Proof. It follows by Theorems 4 and 5.

2.2. -Quasi-Ideals

We define -quasi-ideals of -semigroups as follows.

Definition 4. Let be a -semigroup. A nonempty subset of is called(1)An -quasi-ideal of if .(2)An -quasi-ideal of if it is an -quasi-ideal of .

Theorem 7. Let be a -semigroup and an -quasi-ideal of for all . If , then is an -quasi-ideal of .

Proof. Let be a -semigroup and an -quasi-ideal of for all . Then, for all , so . Therefore, is an -quasi-ideal of .
Let be a nonempty subset of a -semigroup . Let be the -quasi-ideal of generated by .

Theorem 8. Let be a nonempty subset of a -semigroup . Then,

Proof. Let be a nonempty subset of a -semigroup . Let . Clearly, . We have . Therefore, is an -quasi-ideal of .

Let be any -quasi-ideal of containing . Since is an -quasi-ideal of and , . Therefore, .

Hence, is the smallest -quasi-ideal of containing . Therefore, , as required.

Theorem 9. Let be a -semigroup. Let and be a left -ideal and right -ideal of , respectively. If , then is an -quasi-ideal of .

Proof. Let and be a left -ideal and a right -ideal of a -semigroup , respectively. Then, . Hence, is an -quasi-ideal of .

Corollary 1. Let be a -semigroup. Let and be a left -ideal and right -ideal of , respectively. Then, is an -quasi-ideal of .

Proof. We have ; this implies . By Theorem 9, is an -quasi-ideal of .

Theorem 10. Every -quasi-ideal of a -semigroup is the intersection of a left -ideal and a right -ideal of .

Proof. Let be an -quasi-ideal of a -semigroup . Let and . Then, , and also, . We have that . Therefore, .

Definition 5. A -semigroup is called -quasi-simple if does not contain proper -quasi-ideals.
A -semigroup is called -quasi-simple if is -quasi-simple.

Theorem 11. Let be a -semigroup. Then, is -quasi-simple if and only if for all .

Proof. Assume that is -quasi-simple. Let ; we claim that is an -quasi-ideal of . We have ; this implies . Moreover,  = . Therefore, is an -quasi-ideal of . Since is -quasi-simple, we have .
Conversely, assume that for all . Let be an -quasi-ideal of and . By assumption, . Since is an -quasi-ideal of , . Then, . Therefore, is -quasi-simple.

Definition 6. An -quasi-ideal of a -semigroup is called minimal if for all -quasi-ideal of , if , then .

Theorem 12. Let be a -semigroup and an -quasi-ideal of . If is -quasi-simple, then is a minimal -quasi-ideal of .

Proof. Assume that is a -semigroup and an -quasi-ideal of . Let be -quasi-simple. Let be an -quasi-ideal of such that . Then, . Therefore, is an -quasi-ideal of . Since is -quasi-simple, . Hence, is a minimal -quasi-ideal of .

2.3. -Bi-Ideals

We will define -bi-ideals of -semigroups as follows.

Definition 7. Let be a -semigroup and . A nonempty subset of is called(1)An -bi-ideal of if .(2)An -bi-ideal of if it is an -bi-ideal of .

Theorem 13. Every -quasi-ideal of a -semigroup is a -bi-ideal of .

Proof. Let be an -quasi-ideal of . Then,Hence, is a -bi-ideal of .

Theorem 14. Let be a -semigroup and an -bi-ideal of for all . If , then is an -bi-ideal of .

Proof. Let be a -semigroup and an -bi-ideal of for all . Then, for all , so . Therefore, is an -bi-ideal of .
Let be a nonempty subset of a -semigroup , and let denote the -bi-ideal of generated by .

Theorem 15. Let be a nonempty subset of a -semigroup and . Then,

Proof. Let be a nonempty subset of a -semigroup . Let . Clearly, . We have that . Therefore, is an -bi-ideal of .
Let be any -bi-ideal of containing . Since is an -bi-ideal of and , . Therefore, .
Hence, is the smallest -bi-ideal of containing . Therefore, .

Theorem 16. Let be a -semigroup, a nonempty subset of , and an -bi-ideal of . The following statements are true:(1) is an -bi-ideal of .(2) is an -bi-ideal of .

Proof. We have thatThen, is an -bi-ideal of . Similarly, is an -bi-ideal of .

Theorem 17. Let be a -semigroup. Assume that , and are -bi-ideals of . Then, is an -bi-ideal of .

Proof. Since , and are -bi-ideals of , we haveThen,Therefore, is an -bi-ideal of .

Definition 8. A -semigroup is called -bi-simple if does not contain proper -bi-ideals.

Theorem 18. Let be a -semigroup. Then, is -bi-simple if and only if for all .

Proof. Assume that is -bi-simple. Let . We claim that is an -bi-ideal of . We have ; this implies that . Moreover, . Therefore, is an -bi-ideal of . Since is -bi-simple, .
Conversely, assume that for all . Let be an -bi-ideal of and . By assumption, . Since is an -bi-ideal of , . Then, . Therefore, is -bi-simple.

Definition 9. An -bi-ideal of a -semigroup is called minimal if for all -bi-ideal of , if , then .

Theorem 19. Let be a -semigroup and an -bi-ideal of . If is -bi-simple, then is a minimal -bi-ideal of .

Proof. Assume that is a -semigroup and an -bi-ideal of . Let be -bi-simple. Let be an -bi-ideal of such that . Then, . Therefore, is an -bi-ideal of . Since is -bi-simple, . Then, is a minimal -bi-ideal of .

3. New Types of Fuzzy Ideals

3.1. Fuzzy -Ideals

We will define fuzzy -ideals of -semigroups as follows.

Definition 10. Let and be a fuzzy subset of a -semigroup . Then, is called(1)A fuzzy left -ideal of if for all .(2)A fuzzy right -ideal of if for all .(3)A fuzzy -ideal of if it is both a fuzzy left -ideal and a fuzzy right -ideal of .The following theorems show the relationship between -ideals and fuzzy -ideals.

Theorem 20. Let be a nonempty subset of a -semigroup . Then, the following statements are true:(1) is a left -ideal of if and only if is a fuzzy left -ideal of .(2) is a right -ideal of if and only if is a fuzzy right -ideal of .(3) is an -ideal of if and only if is a fuzzy -ideal of .

Proof. (1) Suppose that is a left -ideal of . Let .
If , then . Thus, so that .
If , then .
Therefore, is a fuzzy left -ideal of .
Conversely, assume that is a fuzzy left -ideal of . Let and . Then, . Since , we have . Hence, is a left -ideal of .
(2) is similar to (1).
(3) follows by (1) and (2).

Theorem 21. Let be a fuzzy subset of a -semigroup . Then, the following properties hold:(1) is a fuzzy left -ideal of if and only if .(2) is a fuzzy right -ideal of if and only if .(3) is a fuzzy ()-ideal of if and only if and .

Proof. (1) Assume that is a fuzzy left -ideal of a -semigroup . Let and .
If , then . So, .
If , then there exist such that . Since is a fuzzy left -ideal of , we haveWe conclude that .
Conversely, assume that . Let be such that . Then,Hence, is a fuzzy left -ideal of .
(2) and (3) can be seen in a similar fashion.

Theorem 22. Let and be fuzzy left -ideals of a -semigroup . Then,(1) is a fuzzy left -ideal of .(2) is a fuzzy left -ideal of .

Proof. Let . Then,Hence, is a fuzzy left -ideal of . Next,Hence, is a fuzzy left -ideal of .

Theorem 23. Let and be fuzzy right -ideals of a -semigroup . Then,(1) is a fuzzy right -ideal of .(2) is a fuzzy right -ideal of .

Proof. It is similar to Theorem 22.

Theorem 24. Let and be fuzzy -ideals of a -semigroup . Then,(1) is a fuzzy -ideal of .(2) is a fuzzy -ideal of .

Proof. It follows by Theorems 22 and 23.

Theorem 25. Let be a nonzero fuzzy subset of a -semigroup and . The following statements are true:(1) is a fuzzy left -ideal of if and only if for all , if is a nonempty set, then is a left -ideal of .(2) is a fuzzy right -ideal of if and only if for all , if is a nonempty set, then is a right -ideal of .(3) is a fuzzy -ideal of if and only if for all , if is a nonempty set, then is an -ideal of .

Proof. (1) Suppose that is a fuzzy left -ideal of . Then, for all . Let be such that . Let and . Since , . Thus, . Hence, is a left -ideal of .
Conversely, assume that is a left -ideal of if and . Let and . Since , we have so that . Thus, is a left -ideal of . Since and , we have . Then, . Hence, is a fuzzy left -ideal of .
(2) is similar to (1).
(3) follows by (1) and (2).

3.2. Fuzzy -Quasi-Ideals

We will define fuzzy -quasi-ideals of -semigroups as follows.

Definition 11. Let and be a fuzzy subset of a -semigroup . Then, is called a fuzzy -quasi-ideal of if .
A fuzzy subset of is called a fuzzy -quasi-ideal of if is a fuzzy -quasi-ideal of .

Theorem 26. Let be a -semigroup. Let and be a fuzzy left -ideal and a fuzzy right -ideal of , respectively. Then, is a fuzzy -quasi-ideal of .

Proof. Let and be a fuzzy left -ideal and a fuzzy right -ideal of a -semigroup , respectively. We have ; this implies . Then, . Hence, is a fuzzy -quasi-ideal of .

Theorem 27. Every fuzzy ()-quasi-ideal of a -semigroup is the intersection of a fuzzy left -ideal and a fuzzy right -ideal of .

Proof. Let be a fuzzy ()-quasi-ideal of a -semigroup . Let and . Then, , and also, . Thus, and are a fuzzy left -ideal and a fuzzy right -ideal of , respectively. We claim that . That is, , and conversely, . Therefore, .

Theorem 28. Let be a nonempty subset of a -semigroup . Then, is an -quasi-ideal of if and only if is a fuzzy -quasi-ideal of .

Proof. Assume that is an -quasi-ideal of a -semigroup . If , . Let . Then, . So . Hence, . Therefore, is a fuzzy -quasi-ideal of .
Conversely, assume that is a fuzzy -quasi-ideal of . Then, . Let . Then, , and this implies that . So, . Consequently, is an -quasi-ideal of .

3.3. Fuzzy -Bi-Ideals

Definition 12. Let and be a fuzzy subset of a -semigroup . Then, is called a fuzzy -bi-ideal of if .

Theorem 29. Let be a -semigroup, a fuzzy subset of , and a fuzzy -bi-ideal of . Then, the following statements are true:(1) is a fuzzy -bi-ideal of .(2) is a fuzzy -bi-ideal of .

Proof. (1) Let be a fuzzy point of . Then,Hence, is a fuzzy -bi-ideal of .
(2) is similar to (1).

Theorem 30. Let be a -semigroup. If and are fuzzy -bi-ideals of , then is a fuzzy bi-()-ideal of .

Proof. Let be a fuzzy point of . Then,Hence, is a fuzzy -bi-ideal of .

Theorem 31. Let be a -semigroup and a nonempty subset of . Then, is an -bi-ideal of if and only if is a fuzzy -bi-ideal of .

Proof. Assume that is an -bi-ideal of . Then, . If , we have . Let . Then, . Thus, ; then, . This implies that . Hence, is a fuzzy -bi-ideal of .
Conversely, assume that is a fuzzy -bi-ideal of . Then, . Let . Then, ; thus, . Hence, so that . Thus, is an -bi-ideal of .

Theorem 32. Let and be two fuzzy ()-bi-ideals of a -semigroup . If , then is a fuzzy ()-bi-ideal of .

Proof. Let and be fuzzy ()-bi-ideals of a -semigroup . Then, . Hence, is a fuzzy -bi-ideal of .

4. New Types of Almost Ideals

4.1. Almost -Ideals

Definition 13. Let be a -semigroup and , and be nonempty subsets of . Let . Then,(1) is called an almost left -ideal of if .(2) is called an almost right -ideal of if .(3) is called an almost -ideal of if it is both an almost left -ideal and an almost right -ideal of .

Theorem 33. Let be a -semigroup. If is a left -ideal of , then is an almost left -ideal of .

Proof. Let be a left -ideal of . Then, so that . Thus, is an almost left -ideal of .

Theorem 34. Let be a -semigroup. If is a right -ideal of , then is an almost right -ideal of .

Proof. It is similar to Theorem 33.

Theorem 35. Let be a -semigroup. If is an ()-ideal of , then is an almost ()-ideal of .

Proof. It follows by Theorems 33 and 34.

Theorem 36. Let be a -semigroup. If is an almost left -ideal of and , then is an almost left -ideal of .

Proof. Let be an almost left -ideal of and . Since and , we have . Therefore, is an almost left -ideal of .

Theorem 37. Let be a -semigroup. If is an almost right -ideal of and , then is an almost right -ideal of .

Proof. It is similar to Theorem 36.

Theorem 38. Let be a -semigroup. If is an almost ()-ideal of and , then is an almost ()-ideal of .

Proof. It follows by Theorems 36 and 37.

Corollary 2. Let be a -semigroup. If and are almost left -ideals of , then is an almost left -ideal of .

Corollary 3. Let be a -semigroup. If and are almost right -ideals of , then is an almost right -ideal of .

Corollary 4. Let be a -semigroup. If and are almost ()-ideals of , then is an almost ()-ideal of .

Example 2. Consider a -semigroup with and
We have and are almost left -ideals of . However, is not an almost left -ideal of .

Remark 3. The intersection of two almost left -ideals of a -semigroup need not be an almost left -ideal of .

Remark 4. The intersection of two almost right -ideals of a -semigroup need not be an almost right -ideal of .

Remark 5. The intersection of two almost ()-ideals of a -semigroup need not be an almost ()-ideal of .

Definition 14. A -semigroup is called(1)Almost left -simple if does not contain proper almost left -ideals.(2)Almost right -simple if does not contain proper almost right -ideals.(3)Almost ()-simple if does not contain proper almost ()-ideals.

Theorem 39. Let be a -semigroup. Then, is almost left -simple if and only if for each there exists such that .

Proof. Assume that is almost left -simple. Then, has no proper almost left -ideals. Let . Then, is not an almost left -ideal of . Thus, there exists such that ; this implies that .
Conversely, for each , there exists such that . Assume that is a proper almost left -ideal of , and let . Since , we have is an almost left -ideal of by Theorem 36. Thus, ; we get a contradiction. Hence, has no proper left almost -ideals. Therefore, is almost left -simple.

Theorem 40. Let be a -semigroup. Then, is almost right -simple if and only if for each , there exists such that .

Proof. It is similar to Theorem 39.

4.2. Almost -Quasi-Ideals

Definition 15. A nonempty subset of a -semigroup is called an almost -quasi-ideal of if for all .

Proposition 1. Every -quasi-ideal of a -semigroup is either for some or an almost -quasi-ideal of .

Proof. Assume that is an -quasi-ideal of a -semigroup . Assume that for all . Let . Then, . That is, . Hence, is an almost -quasi-ideal of .

Theorem 41. Every almost -quasi-ideal of a -semigroup is an almost left -ideal of .

Proof. Assume that is an almost -quasi-ideal of a -semigroup . Let . Then, . Hence, is an almost left -ideal of .
Similarly, every almost -quasi-ideal of a -semigroup is an almost right -ideal of , but the converse is not true.

Theorem 42. If is an almost -quasi-ideal of a -semigroup and , then is an almost -quasi-ideal of .

Proof. Assume that is an almost -quasi-ideal of a -semigroup and . Let . Then, . Therefore, is an almost -quasi-ideal of .

Corollary 5. The union of two almost -quasi-ideals of a -semigroup is an almost -quasi-ideal of .

Example 3. Consider a -semigroup with and
We have and are almost -quasi-ideals of .

Remark 6. The intersection of two almost -quasi-ideals of a -semigroup need not be an almost -quasi-ideal of .

Definition 16. A -semigroup is called almost -quasi-simple if does not contain proper almost -quasi-ideals.

Theorem 43. A -semigroup is almost -quasi-simple if and only if for any , there exists such that .

Proof. Assume that a -semigroup is almost -quasi-simple and is not an almost -quasi-ideal of . Then, there exists such that . Therefore, .
Conversely, assume that, for any , there exists such that . Then, . Hence, is not an almost -quasi-ideal of . Let be a proper almost -quasi-ideal of . Then, for some ; this is a contradiction. Therefore, has no proper almost -quasi-ideals. Hence, is almost -quasi-simple.

4.3. Almost -Bi-Ideals

Definition 17. A nonempty subset of a -semigroup is called an almost -bi-ideal of if for all .

Theorem 44. If is an almost -bi-ideal of a -semigroup and , then is an almost -bi-ideal of .

Proof. Let be an almost -bi-ideal of a -semigroup and . Since for all and , we have . Therefore, is an almost -bi-ideal of .

Corollary 6. The union of two almost -bi-ideals of a -semigroup is also an almost -bi-ideal of .

Example 4. Consider a -semigroup with and
We have and are almost -bi-ideals of .

Remark 7. The intersection of two almost -bi-ideals of a -semigroup need not be an almost -bi-ideal of .

Theorem 45. A -semigroup has a proper almost -bi-ideal if and only if there exists an element such that for all .

Proof. Assume that a -semigroup contains a proper almost -bi-ideal and . Then, . By Theorem 44, is a proper almost -bi-ideal of , that is, for all .
Conversely, let such that for all . Since , we have is a proper almost -bi-ideal of .

Definition 18. A -semigroup is called almost -bi-simple if does not contain proper almost -bi-ideals.

Theorem 46. A -semigroup is almost -bi-simple if and only if for all , there exists such that .

Proof. Assume that a -semigroup is almost -bi-simple. Then, has no proper almost -bi-ideals, and let . Then, is not an almost -bi-ideal of . Thus, there exists such that . This implies that .
Conversely, suppose that has a proper almost -bi-ideal , that is, . Since , we have is an almost -bi-ideal of by Theorem 44; this is a contradiction. Hence, has no proper almost -bi-ideals. Therefore, is almost -bi-simple.

5. New Types of Fuzzy Almost Ideals

5.1. Fuzzy Almost -Ideals

Definition 19. Let . Let be a fuzzy subset of a -semigroup . Then, is called(1)A fuzzy almost left -ideal of if for all fuzzy point of .(2)A fuzzy almost right -ideal of if for all fuzzy point of .(3)A fuzzy almost -ideal of if it is both a fuzzy almost left -ideal and a fuzzy almost right -ideal of .

Theorem 47. Let be a fuzzy almost left -ideal of a -semigroup and be a fuzzy subset of such that . Then, is a fuzzy almost left -ideal of .

Proof. Assume that is a fuzzy almost left -ideal of a -semigroup and is a fuzzy subset of such that . Then, for each fuzzy point , . We have ; this implies . Therefore, is a fuzzy almost left -ideal of .

Theorem 48. Let be a fuzzy almost right -ideal of a -semigroup and be a fuzzy subset of such that . Then, is a fuzzy almost right -ideal of .

Proof. It is similar to Theorem 47.

Theorem 49. Let be a fuzzy almost ()-ideal of a -semigroup and be a fuzzy subset of such that . Then, is a fuzzy almost ()-ideal of .

Proof. It follows by Theorem 47 and Theorem 48.

Corollary 7. The union of two fuzzy almost left -ideals of a -semigroup is a fuzzy almost left -ideal of .

Corollary 8. The union of two fuzzy almost right -ideals of a -semigroup is a fuzzy almost right -ideal of .

Corollary 9. The union of two fuzzy almost ()-ideals of a -semigroup is a fuzzy almost ()-ideal of .

Theorem 50. Let be a nonempty subset of a -semigroup . Then,(1) is an almost left -ideal of if and only if is a fuzzy almost left -ideal of .(2) is an almost right -ideal of if and only if is a fuzzy almost right -ideal of .(3) is an almost ()-ideal of if and only if is a fuzzy almost ()-ideal of .

Proof. (1) Assume that is an almost left -ideal of a -semigroup . Then, for all . Thus, there exists , and . So, and . Hence, . Therefore, is a fuzzy almost left -ideal of .
Conversely, assume that is a fuzzy almost left -ideal of . Let . Then, . Then, there exists such that . Hence, . So, . Consequently, is an almost left -ideal of .
The proofs of (2) and (3) are similar to the proof of (1).

Theorem 51. Let be a fuzzy subset of a -semigroup . Then,(1) is a fuzzy almost left -ideal of if and only if is an almost left -ideal of .(2) is a fuzzy almost right -ideal of if and only if is an almost right -ideal of .(3) is a fuzzy almost ()-ideal of if and only if is an almost ()-ideal of .

Proof. (1) Assume that is a fuzzy almost left -ideal of a -semigroup . Let . Then, . Hence, there exists such that . So, there exists such that . That is, . Thus, and . Therefore, . Hence, is a fuzzy almost left -ideal of . By Theorem 50, is an almost left -ideal of .
Conversely, assume that is an almost left -ideal of . By Theorem 50, is a fuzzy almost left -ideal of . Then, for all . Then, there exists such that . Hence, and . Then, there exists such that . This means . Therefore, is a fuzzy almost left ideal of .
The proofs of (2) and (3) are similar to the proof of (1).

Definition 20. A fuzzy almost left -ideal of a -semigroup is minimal if for all fuzzy almost left -ideal of such that , we obtain .

Theorem 52. Let be a nonempty subset of a -semigroup . Then,(1) is a minimal almost left -ideal of if and only if is a minimal fuzzy almost left -ideal of .(2) is a minimal almost right -ideal of if and only if is a minimal fuzzy almost right -ideal of .(3) is a minimal almost ()-ideal of if and only if is a minimal fuzzy almost ()-ideal of .

Proof. (1) Assume that is a minimal almost left -ideal of a -semigroup . By Theorem 50 (1), is a fuzzy almost left -ideal of . Let be a fuzzy almost left -ideal of such that . By Theorem 51 (1), is an almost left -ideal of . Then, . Since is minimal, . Therefore, is minimal.
Conversely, assume that is a minimal fuzzy almost left -ideal of . By Theorem 50 (1), is an almost left -ideal of . Let be an almost left -ideal of such that . By Theorem 50 (1), is a fuzzy almost left -ideal of such that . Hence, . Therefore, is minimal.
(2) and (3) can be proved similarly.

Corollary 10. Let be a sub--semigroup of a -semigroup . Then,(1) is almost left -simple if and only if for each fuzzy almost left -ideal of , .(2) is almost right -simple if and only if for each fuzzy almost right -ideal of , .(3) is almost ()-simple if and only if for each fuzzy almost ()-ideal of , .

5.2. Fuzzy Almost -Quasi-Ideals

Definition 21. Let and be a fuzzy subset of a -semigroup . Then, is called a fuzzy almost -quasi-ideal of if for all fuzzy points of .

Theorem 53. Let be a fuzzy almost -quasi-ideal of a -semigroup and a fuzzy subset of such that . Then, is a fuzzy almost -quasi-ideal of .

Proof. Assume that is a fuzzy almost -quasi-ideal of a -semigroup and is a fuzzy subset of such that . Then, for all fuzzy point of , . We have that . This implies that . Therefore, is a fuzzy almost -quasi-ideal of .

Corollary 11. Let and be fuzzy almost -quasi-ideals of a -semigroup . Then, is a fuzzy almost -quasi-ideal of .

Proof. Since , by Theorem 53, is a fuzzy almost -quasi-ideal of .

Example 5. Consider the -semigroup where and , where and . Let defined byand defined byWe have and are fuzzy almost -quasi-ideals of , but is not a fuzzy almost -quasi-ideal of .

Theorem 54. Let be a nonempty subset of a -semigroup . Then, is an almost -quasi-ideal of if and only if is a fuzzy almost -quasi-ideal of .

Proof. Assume that is an almost -quasi-ideal of a -semigroup , and let be a fuzzy point of . Then, . Thus, there exists and . So, and . Hence, . Therefore, is a fuzzy almost -quasi-ideal of .
Conversely, assume that is a fuzzy almost -quasi-ideal of . Let . Then, . Then, there exists such that . Hence, . So, . Consequently, is an almost -quasi-ideal of .

Theorem 55. Let be a fuzzy subset of a -semigroup . Then, is a fuzzy almost -quasi-ideal of if and only if is an almost -quasi-ideal of .

Proof. Assume that is a fuzzy almost -quasi-ideal of a -semigroup . Let and . Then, . Hence, there exists such thatSo, there exist such that , and . That is, . Thus, and . Therefore, . Hence, is a fuzzy almost -quasi-ideal of . By Theorem 54, is an almost -quasi-ideal of .
Conversely, assume that is an almost -quasi-ideal of . By Theorem 54, is a fuzzy almost -quasi-ideal of . Then, for each fuzzy point of , we have . Then, there exists such thatHence, and . Then, there exist such that , and . This means that . Therefore, is a fuzzy almost -quasi-ideal of .
Next, we define minimal fuzzy almost -quasi-ideals in -semigroups and give some relationship between minimal almost -quasi-ideals and minimal fuzzy almost -quasi-ideals of -semigroups.

Definition 22. A fuzzy almost -quasi-ideal of a -semigroup is called minimal if for each fuzzy almost -quasi-ideal of such that , we have .

Theorem 56. Let be a nonempty subset of a -semigroup . Then, is a minimal almost -quasi-ideal of if and only if is a minimal fuzzy almost -quasi-ideal of .

Proof. Assume that is a minimal almost -quasi-ideal of a -semigroup . By Theorem 54, is a fuzzy almost -quasi-ideal of . Let be a fuzzy almost -quasi-ideal of such that . Then, . Since and by Theorem 53, we have is a fuzzy almost -quasi-ideal of . By Theorem 54, is an almost -quasi-ideal of . Since is minimal, . Therefore, is minimal.
Conversely, assume that is a minimal fuzzy almost -quasi-ideal of . Let be an almost -quasi-ideal of such that . By Theorem 54, is a fuzzy almost -quasi-ideal of such that . Since is minimal, . Therefore, is minimal.

Corollary 12. Let be a sub -semigroup of a -semigroup . Then, is almost -quasi-simple if and only if for all fuzzy almost -quasi-ideal of , .

5.3. Fuzzy Almost -Bi-Ideals

Definition 23. Let and be a fuzzy subset of a -semigroup . Then, is called a fuzzy almost -bi-ideal of if for all fuzzy point of .

Theorem 57. Let be a fuzzy almost -bi-ideal of a -semigroup and be a fuzzy subset of such that . Then, is a fuzzy almost -bi-ideal of .

Proof. Assume that is a fuzzy almost -bi-ideal of a -semigroup and is a fuzzy subset of such that . Then, for each fuzzy point , . We have ; this implies . Therefore, is a fuzzy almost -bi-ideal of .

Corollary 13. Let and be fuzzy almost -bi-ideals of a -semigroup . Then, is a fuzzy almost -bi-ideal of .

Example 6. Consider the -semigroup where and , where and . Let defined byand defined byWe have and are fuzzy almost -bi-ideals of .

Remark 8. The intersection of two fuzzy almost ()-bi-ideals of a -semigroup need not be a fuzzy almost ()-bi-ideal of .

Theorem 58. Let be a nonempty subset of a -semigroup . Then, is an almost -bi-ideal of if and only if is a fuzzy almost -bi-ideal of .

Proof. Assume that is an almost -bi-ideal of a -semigroup . Then, for all . Thus, there exists and . So, and . Hence, . Therefore, is a fuzzy almost -bi-ideal of .
Conversely, assume that is a fuzzy almost -bi-ideal of . Let . Then, . Thus, there exists such that . Hence, . So, . Consequently, is an almost -bi-ideal of .

Theorem 59. Let be a fuzzy subset of a -semigroup . Then, is a fuzzy almost -bi-ideal of if and only if is an almost -bi-ideal of .

Proof. Assume that is a fuzzy almost -bi-ideal of a -semigroup . Let . Then, . Hence, there exists such that . So, there exist such that , and . That is, . Thus, and . Therefore, . Hence, is a fuzzy almost -bi-ideal of . By Theorem 58, is an almost -bi-ideal of .
Conversely, assume that is an almost -bi-ideal of . By Theorem 58, is a fuzzy almost -bi-ideal of . Then, for all . Then, there exists such that . Hence, and . Then, there exist such that , and . This means . Therefore, is a fuzzy almost -bi-ideal of .
We define minimal fuzzy almost -bi-ideals in -semigroups and give some relationship between minimal almost -bi-ideals and minimal fuzzy almost -bi-ideals of -semigroups.

Definition 24. A fuzzy almost -bi-ideal of a -semigroup is called minimal if for all fuzzy almost -bi-ideal of such that , we have .

Theorem 60. Let be a nonempty subset of a -semigroup . Then, is a minimal almost -bi-ideal of if and only if is a minimal fuzzy almost -bi-ideal of .

Proof. Assume that is a minimal almost -bi-ideal of a -semigroup . By Theorem 58, is a fuzzy almost -bi-ideal of . Let be a fuzzy almost -bi-ideal of such that . Then, . Since and by Theorem 57, we have is a fuzzy almost bi-ideal of . By Theorem 58, is an almost -bi-ideal of . Since is minimal, . Therefore, is minimal.
Conversely, assume that is a minimal fuzzy almost -bi-ideal of . Let be an almost -bi-ideal of such that . Then, is a fuzzy almost -bi-ideal of such that . Hence, . Therefore, is minimal.

Corollary 14. Let be a sub--semigroup of a -semigroup . Then, is almost -bi-simple if and only if for all fuzzy almost -bi-ideal of , .
Next, we give the relationship between -prime almost -bi-ideals and -prime fuzzy almost -bi-ideals.

Definition 25. Let be a -semigroup and .(1)An almost -bi-ideal of is called -prime if for all implies or .(2)A fuzzy almost -bi-ideal of is called -prime if for all .

Theorem 61. Let be a nonempty subset of a -semigroup . Then, is a -prime almost -bi-ideal of if and only if is a -prime fuzzy almost -bi-ideal of .

Proof. Assume that is a -prime almost -bi-ideal of . By Theorem 58, is a fuzzy almost -bi-ideal of . Let . We consider two cases: Case 1: . So, or . Then, . Case 2: . Then, .Thus, is a -prime fuzzy almost -bi-ideal of .
Conversely, assume that is a -prime fuzzy almost -bi-ideal of . By Theorem 58, is an almost -bi-ideal of . Let be such that . Then, . By assumption, . Therefore, . Hence, or . Thus, is a -prime almost -bi-ideal of .
In this section, we give the relationship between -semiprime almost -bi-ideals and -semiprime fuzzy almost -bi-ideals.

Definition 26. Let be a -semigroup and .(1)An almost -bi-ideal of is called a -semiprime if for all implies .(2)A fuzzy almost -bi-ideal of is called a -semiprime if for all .

Theorem 62. Let be a nonempty subset of . Then, is a -semiprime almost -bi-ideal of if and only if is a -semiprime fuzzy almost -bi-ideal of .

Proof. Assume that is a -semiprime almost -bi-ideal of . By Theorem 58, is a fuzzy almost -bi-ideal of . Let . We consider two cases: Case 1: . Then, . So, . Hence, . Case 2: . Then, .Thus, is a -semiprime fuzzy almost -bi-ideal of .
Conversely, assume that is a -semiprime fuzzy almost -bi-ideal of . By Theorem 58, is an almost -bi-ideal of . Let be such that . Then, . By assumption, . Since , . Hence, . Thus, is a -semiprime almost -bi-ideal of .

6. Discussion and Conclusion

In this paper, we define new types of ideals and fuzzy ideals by using elements in . We show interesting properties of these ideals and fuzzy ideals. Moreover, we show the relationships between these ideals and their fuzzifications.

Data Availability

No data were used to support this research.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Faculty of Science Research Fund, Prince of Songkla University (contract no. 1-2562-02-013).