Fuzzy Ideals in Pseudo-Hoop Algebras

Alemayehu, Teferi Getachew

doi:https://doi.org/10.1155/2022/4643252

International Journal of Mathematics and Mathematical Sciences

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 4643252 | https://doi.org/10.1155/2022/4643252

Fuzzy Ideals in Pseudo-Hoop Algebras

Teferi Getachew Alemayehu¹

Academic Editor: Maria A. Lledo

Received16 Jun 2022

Accepted21 Jul 2022

Published23 Sept 2022

Abstract

In this study, fuzzy ideals in pseudo-hoop algebras are presented. Also, we investigate fuzzy congruences relations on pseudo-hoop algebras induced by fuzzy ideals. By using fuzzy ideals, we create the fuzzy quotient pseudo-hoop algebras and identify and demonstrate the one-to-one relationship between the set of all normal fuzzy ideals of a pseudo-hoop algebra with conditions and the set of all fuzzy congruences relation on . Additionally, we investigate the relationship between fuzzy ideals and fuzzy filters in pseudo-hoop algebras by considering the set of complement elements of pseudo-hoop algebras. Lastly, we study fuzzy prime ideals and fuzzy -prime ideals and their relation.

1. Introduction

Bosbach [1, 2] introduced Hoop-algebras which are ordered commutative residualed integral monoids. Georgescu et al. [3] introduce the concepts of pseudo-hoop algebras, which is non-commutative generalizations of hoop algebras, following after Georgescu and Iorgulescu [4] introduce the notions of pseudo-MV algebras, also Nola et al. [5] present pseudo-BL algebras. Pseudo-hoop algebras are weaker structures than Pseudo-MV algebras and pseudo-BL algebras. They are particular cases of pseudo-hoop algebras. In recent years, the investigation of hoop algebras and pseudo-hoop algebras has made great progress. Many researchers study in filter and ideals of pseudo-hoop algebras in [6–9] and ideals of pseudo-hoop algebras in [10].

Fuzzy set theory has been studied by many scholars in many directions. Rosenfeld [11] introduced fuzzy subgroups and his work led to a new field in fuzzy mathematics. Since then many mathematicians have been used this idea and applied in different algebras such as in group [12–14], ring [15–17], Universal algebra [18–22], lattice [23–26], Ockham algebra [27, 28], MS-algebra [29–32], BL-algebra [33–37], BE-algebra [38, 39] etc.

In this study, fuzzy ideals in pseudo-hoop algebras are presented. Also, we investigate fuzzy congruences relations on pseudo-hoop algebras induced by fuzzy ideals. By using fuzzy ideals, we create the fuzzy quotient pseudo-hoop algebras and identify and demonstrate the one-to-one relatonship between the set of all normal fuzzy ideals of a pseudo-hoop algebra with conditions and the set of all fuzzy congruences relation on . For a pseudo-hoop algebra, the pre image of the (normal) fuzzy ideals is the (normal) fuzzy ideals under homomorphism, and the image is the (normal) fuzzy ideals under isomorphism.

Finally, by taking into account the set of complement elements of pseudo-hoop algebras, we investigate the connection between fuzzy ideals and fuzzy filters in pseudo-hoop algebras. Additionally, we investigate the relationship between fuzzy prime ideals and -prime ideals.

2. Preliminaries

We retain significant results and fundamental terminology from this area that we will require in the follow-up. To learn more, go to [1–3, 10, 40, 41].

Definition 1 (see [1]). A structure is a partially ordered monoid if(1) is a monoid, i.e., is associative and for all ,(2) is a partial order on ,(3)for all , and .The largest element (under ) of the set , if it exists, is called the left-residual of relative to , and is denoted by . Thus can be defined by the condition .

Definition 2. Reference [1] is left-residuated if exists for all .
In this event the enriched structure is called a left-residuated partially ordered monoid.
A left-residuated partially ordered monoid can be thought of as an algebra , since the partial order can be retrieved via if and only if . The inverse right divisibility relation on a monoid is defined by .

Definition 3 (see [1]). An algebra is a left-complemented monoid if is a partially ordered monoid with left-residuation .
The notions of right-residual, right-residuated a partially ordered monoid, inverse left divisibility relation , and right-complemented monoid are defined similarly.

Lemma 1 (see [1, 2]). (1)If is a left-complemented monoid, then is a meet-semilattice order, where for all (2)If is a right-complemented monoid, then is a meet-semilattice order, where for all A variety of left- (and right-) complemented monoids exist, and they have a straightforward equational characterization.

Theorem 1 (see [1, 2]). An algebra is a left-complemented monoid iff the following identities hold:(1),(2),(3),(4).

Theorem 2 (see [3]). An algebra is a right-complemented monoid iff the following identities hold:(1),(2),(3),(4).

Definition 4. (see [3]). If an algebra of type satisfies the following requirements or any , it is known to be a pseudo-hoop:(1) is a left-complemented monoid,(2) is a right-complemented monoid,(3).We will agree in the next section that has priority towards the operations . The inverse left and right divisibility relations are coincident in a pseudo-hoop by (3) and are indicated by the symbol .
It follows that for any , .
We specify that for any natural integer on , and .
A partial order on H is represented by the relation denoted by s t s t s t = 1 s t = 1. If is commutative or, alternatively, if = H, then H is referred to as a hoop algebra. Additionally, H is said to be bounded if 0 s for any s H.
The relation described by is a partial order on . If is commutative or alternatively is said to be a hoop algebra. Additionally, is said to be bounded if for any In this instance, and on are defined. If for all then the bounded pseudo-hoop algebra is called good (see [42]). In a bounded pseudo-hoop algebra , if for all then is called satisfying the condition (see [42]). A good pseudo-hoop algebra is known as normal if it satisfies for each .

Theorem 3 (see [42]). Let’s consider the pseudo-hoop algebr . For any , the subsequent characteristics are true:(1),(2)if , then and ,(3) is a monoid,(4),(5), then and ,(6), then and ,(7) and .

Theorem 4 (see [42]). Let be a pseudo-hoop algebra with bounds. The following statements are valid for every :(1),(2),(3),(4),(5),(6)if is good, then and ,(7)if is good, then and .

Definition 5 (see [3]). A filter of a pseudo-hoop algebra is a nonempty subset of which satisfies(1),(2)for any if and , then .

Definition 6 (see [3]). A filter of a pseudo-hoop algebra is known as normal if .

Theorem 5 (see [10]). Let be pseudo-hoop algebra with bounds. For every , we describe left addition and right addition as follows: and .

Theorem 6 (see [10]). Let be a pseudo-hoop algebra. For all , if and , then and .

Theorem 7 (see [10]). Let be a pseudo-hoop algebra. If is normal, then left addition and right addition are associative.

Theorem 8 (see [10]). Let be a nonempty subset of a bounded pseudo-hoop algebra . Then is called a left ideal of if:(1),(2)for any and . Similarly, is known as a right ideal of if:(3),(4)for any and .If is a left ideal and a right ideal of , we say to be an ideal of .
For every ideal of , we get . For each , we get .
An ideal of is known as proper if . An ideal of is known as normal if for every .
Let and be pseudo-hoop algebras. In [10], a map is known as a pseudo-hoop homomorphism if preserves the operations , and The pseudo-hoop homomorphism is called a bounded pseudo-hoop homomorphism if , are bounded and .

3. Fuzzy Ideals in Pseudo-hoop Algebras

In this subsection, we discuse the principles of fuzzy ideals in pseudo-hoop algebras and two types of fuzzy ideals (left and right fuzzy ideals). We provide several similar characterizations of fuzzy ideals of good pseudo-hoop algebras.

Definition 7. Let be a fuzzy subset of a bounded pseudo-hoop algebra . Then is called the left fuzzy ideal of , if(1), for every ,(2), for every . Similarly, is called the right fuzzy ideal of , if.(3), for every ,(4), for every .If is a left fuzzy ideal and a right fuzzy ideal of , we say to be a fuzzy ideal of .

Lemma 2. Let be every fuzzy ideal of a bounded pseudo-hoop . Then(1),(2) for every .

Proof. (1)Since for every . Then by (2) by Definition 7. (2) and (4).(2). Conversely, since . Then . Thus . Similarly, .Obviously, are fuzzy ideals of .
The set of all fuzzy ideals of is denoted by . is known as a proper fuzzy ideal if is a nonconstant fuzzy ideal of .

Definition 8. Let be bounded pseudo-hoop algebra. A fuzzy ideal of is called fuzzy normal ideal if for any .

Example 1. Let . Define the operations and on as follows:
Then is a bounded hoop algebra. It is easy to see that and.
Define a fuzzy subset as , and , a fuzzy subset as , and as are fuzzy ideals of .

Theorem 9. Let be a fuzzy subset of . Then is a fuzzy ideal if and only if for any , is an ideal of .

Proof. The proof is straight forward.

Corollary 1. is an ideal of iff is a fuzzy ideal.

Definition 9. Let be a bounded pseudo-hoop and be a fuzzy subset of . The fuzzy ideal generated by means the smallest fuzzy ideal inluding , which is denoted by .

Theorem 10. Let is a fuzzy ideal of pseuo-hoop algebra . Then is a fuzzy ieal of .

Theorem 11. Let is a fuzzy subset of . Then .

Proof. The proof is clear.

Theorem 12. Let be any subset of . Then .

Theorem 13. Let be a nonconstant fuzzy subset of a good pseudo-hoop algebra such that for any . Then the following conditions are equivalent:(1) is a fuzzy ideal of ,(2) for every ,(3) for every .

Proof. Assume that is a fuzzy ideal of . Since , then we have . Now, . This implies holds.
Assume that holds. Let such that . Then we have and so .
by (2) and hypothesis.
AS , then . Since good pseudo-hoop, and . Also we know that .
Thus is fuzzy ideal of . Hence holds. Similarly, we show that . □

Theorem 14. Let be a nonconstant fuzzy subset of a good pseudo-hoop algebra such that for any . Then the following conditions are equivalent:(1) is a fuzzy ideal of ,(2) for every ,(3) for every .

Proof. Assume that is a fuzzy ideal of . Since and .
by Theorem 11 (2) and Definition 7 (1) and (3).
Suppose that holds. Since .
by (2).
Now, By (2). Hence is a fuzzy ideal of by (2).
Hence holds. Similarly, we show that .

Theorem 15. Let be a bounded and normal pseudo-hoop algebra and be a non constant fuzzy subset of . Then

Proof. Put . We prove that is the smallest ideal containing . Clearly .This implies by Theorem 15, is fuzzy ideals of .
Suppose that is any fuzzy ideal of containing . Now,Hence . With similar procider, we have .

4. Fuzzy Ideals and Fuzzy Congruences Relations

In this subsection, we introuce fuzzy congruences relation on pseudo-hoop algebras induced by fuzzy ideals. We construct the fuzzy quotient pseudo-hoop algebras via fuzzy ideals and prove that there is a one-to-one correspondence between the set of all normal fuzzy ideals of a pseudo-hoop algebra with condition and the set of all fuzzy congruences relation on .

In pseudo-hoop algebra’s, pre image of (normal) fuzzy ideals is (normal) fuzzy ideals under homomorphism, while image is (normal) fuzzy ideals under isomorphism.

Definition 10. Let be a pseudo-hoop algebra and be a fuzzy equivalence relation on . Then(1)The fuzzy equivalence relation is Known as a left fuzzy congruence relation if for every .(2)The fuzzy equivalence relation is Known as a right fuzzy congruence relation if for every .(3)The fuzzy equivalence relation is known as a fuzzy congruence relation if for every .

Example 2. Let be a hoop algebra of eExample 4. Define a fuzzy relation on as . It is easily check that is a fuzzy congruence relation.

Theorem 16. A fuzzy relation on a pseudo-hoop algebra is a fuzzy congruence relation iff it is a left and a right fuzzy congruence relation.

Proof. The proof of it is clear.
Let be a fuzzy ideal of a bounded pseudo-hoop algebra . Then we expres a fuzzy relation on as follows:for every .

Theorem 17. Let be a bounded pseudo-hoop algebra and a fuzzy ideal of . Then is a fuzzy equivalence relation on .

Proof. Since and , for any . Hence is fuzzy reflexive.
Clearly is fuzzy symmetric. Next, we show that is fuzzy transitive.
Since , andThis implies is transitive. Hence is fuzzy equivalence relation on . □

Theorem 18. Let be a good pseudo-hoop algebra and a fuzzy normal ideal of . Then is a fuzzy congruence relation on .

Proof. is an equvalence relation, by Theorem 15.
As ,
an is a fuzzy normal. Then we haveSimilarly, .
Since and .Similarly, .
For any ,Similarly, , and . Hence is a fuzzy congruence relation on .
Let be a good pseudo-hoop algebra and be a fuzzy normal ideal of . We define , where . For every , we expres the operation on by .

Theorem 19. Let be a good pseudo-hoop algebra and be a fuzzy normal ideal of . Then is a bounded pseudo hoop algebra with condition .

Theorem 20. .Let be a good pseudo-hoop algebra and is a fuzzy subset of . Then(1)Let be a fuzzy congruence relation on . Then is a normal fuzzy ideal of and is a fuzzy congruence relation on . Let satisfies the condition . Then concide with ,(2)Let be a fuzzy normal ideal of . Then is a fuzzy congruence relation on . Additionaly, is a fuzzy normal ideal of and coincides with ,(3)Let satisfies the condition . Then there is a one-to-one correspondence between the set of fuzzy congruence relations on and the set of fuzzy normal ideals of .

Proof. (1)Firest, we want to show that is a normal fuzzy ideal of . For every , Similarly, . Also, let such that . Now, Thus is a fuzzy ideal of . Since is a good pseudo-hoop, we have . For every , Similarly, . Hence and is normal fuzzy ideal. By Theorem 16, is a fuzzy congruence relation on . Assume that satisfies condition. Next, we want to show that . For any , . With similar procider, , and . This implies . Conversely, Similarly . This implies . Hence .(2)By Theorem 20 is a fuzzy congruence relation on . By (1), is normal fuzzy ideal of . Next, we want to show that . Since . Then we have . Hence .(3)It is clear by (1) and (2).

Theorem 21. Let be two pseudo-hoop algebras with boundes and a bounded pseudo-hoop homomorphism. Then we have the following results:(1)Let be a (normal) fuzzy ideal of J. Then is a (normal) fuzzy ideal of ,(2)Let be a isomorphism and be a fuzzy (normal) ideal of . Then is a fuzzy (normal) ideal of .

Proof. (1)Assume that is a fuzzy ideal of such that . Then we have that . As is a bounded pseudo-hoop homomorphism, . Hence . Now, . Next, for any , . With similarly, . This implies is a fuzzy ideal of . Assume that is a normal fuzzy ideal of . Let , . This implies is a normal fuzzy ideal of .(2)Assume that is a fuzzy ideal of . We want to show that is a fuzzy ideal of . Let such that . As is bounded pseudo-homomorphism, there exist such that and .This implies and also .
Now, . Additionaly, . With similar procidure, .
This implies is a fuzzy ideal of .
Assume that is a normal fuzzy ideal of . Let , . Thus implies normal fuzzy ideal of .

5. Fuzzy Ideals and Fuzzy Filters

The set of complement elements of pseudo-hoop algebras is taken into consideration in this part as we discuse the connection between fuzzy ideals and fuzzy filters in pseudo-hoop algebras. In addition, we investigate the relationship between fuzzy prime ideals and fuzzy -prime ideals.

Let be a nonempty fuzzy subset of a bounded pseudo-hoop algebra . We represents and .

Definition 11. Let be a pseudo-hoop algebra with boundes and be a fuzzy subset of . The fuzzy sets and .

Example 3. Let . We define and on as follows:
Then is a bouneded pseudo-hoop algebra. Define a fuzzy subset as . Then . Also, .
After routine calculation show that is a fuzzy filter of .

Theorem 22. Let be a fuzzy filter of a good pseudo-hoop algebra . Then is a fuzzy ideal generated by and is a fuzzy ideal generated by

Proof. For any ,This implies is a fuzzy ideal of by Theorem 12.
Next, we show that, .
For any , there exist such that and also clearly .. This implies .
Suppose that is a fuzzy ideal such that .
Since .
This implies . Hence is a fuzzy ideal generated by . Similarly, is a fuzzy ideal generated by .

Theorem 23. Let be a bounded pseudo-hoop algebra and a fuzzy ideal of . If is good, then and are fuzzy filters of such that amd .

Proof. For any , if , then and .Also, . This implies is a fuzzy filter of . For any , there exists such that . Now, as by Lemma 2. Thus . This implies . Similarly, we can prove that is fuzzy filter of and .

Theorem 24. Let be a fuzzy ideal of a bounded pseudo-hoop algebra . Then .

Proof. For any , . Hence . Similarly .

Theorem 25. Let be a fuzzy filter of a bounded pseudo-hoop algebra . Then and .

Proof. Since , Now, . Hence . Similarly, .

Theorem 26. Let be a bounded is known asa fuzzy prime ideal if is for every .

Example 4. In example 4, and are fuzzy prime ideal of . As and . This show that . Therefore is not fuzzy prime ideal of .

Theorem 27. Let be two bounded pseudo-hoop algebras and be a bounded pseudo-hoop homomorphism. Then, the following facts are correct:(1)If is a fuzzy prime ideal of , then is a fuzzy prime ideal of .(2)If is a fuzzy prime ideal of and is a bounded pseudo-hoop isomorphism, then is a fuzzy prime ideal of .

Proof. (1) is a fuzzy ideal of by Theorem 21. Next, we want to show that is a fuzzy prime ideal of . Let , . Therefore is a fuzzy prime ideal of .(2)Also, is a fuzzy ideal of , by Theorem 22 and also easily proved that is a fuzzy prime ideal of .

Definition 12. Let be a pseudo-hoop algebra with boundes and a fuzzy ideal of . Then is known as a fuzzy -prime ideal of if for every .

Example 5. In example 4, a fuzzy subset as , is a fuzzy -prime ideal of .

Theorem 28. Let be a pseudo-hoop algebra with bounds. Then every fuzzy -prime ideal of is a fuzzy prime ideal of , but the converse ot it may not be true.

Proof. Assume that is fuzzy -prime ideal of . We prove that is fuzzy prime ideal. Since . Now, . Conversely, . Hence . This implies every fuzzy -prime ideal of is a fuzzy prime ideal of .
In example 4, a fuzzy subset as , . It is simple to demonstrate that is the fuzzy -prime ideal of , but not the fuzzy prime ideal.

6. Conclusions

In this study, fuzzy ideals in pseudo-hoop algebras are discused. Also, we investigate fuzzy congruences relations on pseudo-hoop algebras induced by fuzzy ideals. By using fuzzy ideals, we create the fuzzy quotient pseudo-hoop algebras and identify and demonstrate the one-to-one relationship between the set of all normal fuzzy ideals of a pseudo-hoop algebra with conditions and the set of all fuzzy congruences relation on . For a pseudo-hoop algebra, the pre image of the (normal) fuzzy ideals is the (normal) fuzzy ideals under homomorphism, and the image is the (normal) fuzzy ideals under isomorphism.

Lastly, by taking into account the set of complement elements of pseudo-hoop algebras, we investigate the connection between fuzzy ideals and fuzzy filters in pseudo-hoop algebras. Additionally, we investigate the relationship between fuzzy prime ideals and -prime ideals.

Data Availability

This study is not supported by any data.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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