Abstract
In this study, we introduce a new concept called “anti-involution” in relation to ordered LA-semihypergroups. An anti-involution is basically an involuntary automorphism, which is just a fancy term for a mathematical function that can be reversed. We looked at several fundamental results before introducing anti-involution hyperideals. We studied the anti-involution hyperideals of ordered anti-involution LA-semihypergroups using the rough set theory. In an ordered anti-involution LA-semihypergroup, the -upper and -lower rough approximations of anti-involution hyperideals are an anti-involution hyperideals.
1. Introduction
One of the reasons for studying hyperstructures is to understand biological inheritance and physical phenomena such as nuclear fission. Chemical and redox processes are another source of inspiration for the research of hyperstructures. Classical algebraic structures can be generalized with algebraic superstructures. A composition of two elements is an element of a classical algebraic structure, and a composition of two elements is a set of algebraic superstructures. The term “hyperstructure” originally came from Marty [1], a French mathematician who pioneered hyperstructure theory. He extended the notion of classical binary operation to binary hyperoperation in his study. Several scholars directed their studies in this way, resulting in an excess of novel notions. Many applications of hyperstructures in geometry, codes, and cryptography are described in the book produced by Corsini and Leoreanu-Fotea [2] and Vougiouklis [3].
Kazim and Naseeruddin [4], as the originators of the notion of almost semigroups, enlarged the notion of associative law to left invertive law in his study. Hila and Dine [5] provided the notion of left almost semihypergroups. Yaqoob et al. [6] also contributed to the idea of left almost semihypergroups. Yaqoob and Gulistan [7] introduced the notion of partially ordered left almost semihypergroups.
Pawlak [8] established the notion of rough sets in 1982. Biswas and Nanda [9] applied the rough sets theory to groups. Jun [10] investigated the lower and upper approximations of -subsemigroups/ideals in -semigroups. Qurashi and Shabir [11, 12] generalized rough fuzzy ideals in quantales and roughness in quantale modules. Shabir and Irshad [13] studied roughness in ordered semigroups. Several writers have applied the rough set theory to various algebraic hyperstructures, such as Ameri et al. [14] applied roughness to bi-hyperideals of semihypergroups. Anvariyeh et al. [15] examined the roughness of -semihypergroups. The rough set theory was applied to hyperrings by Davvaz [16], -semihyperrings by Dehkordi and Davvaz [17], hyperlattices by He et al. [18], hypergroups by Leoreanu-Fotea [19], and nonassociative po-semihypergroups by Zhan et al. [20].
An involution is a transformation [21] (or unary operation) that satisfies the following 3 axioms:
Axiom 1. . An involution is its own inverse.
Axiom 2. An involution is linear: and , where is a real constant.
Axiom 3. .
An anti-involution is a self-inverse transformation similar to an involution. It satisfies the Axiom 1, 2, and Axiom 4:
Axiom 4. .
Foulis [22] submitted his Ph.D thesis in 1958, in which he studied unary operations in semigroups and provided some results for the theory of involution semigroups. Following that, Scheiblich and Nordahl [23] established the concept of regular -semigroups. They labeled a semigroup with a unary operation a regular -semigroup if the following identities were satisfied:Following the introduction of this notion, various writers studied natural structures on semigroups with involution [24–26] and analyzed some findings on regular semigroups with involution [27]. Reilly [28] introduced a new class of ordinary -semigroups. Wu [29] recently studied intraregular ordered -semigroups. Aburawash [30] introduced the definition of involution group rings. Baxter [31] investigated rings with proper involution. Herstein [32, 33] added various results for involution rings.
In hyperstructures, the theory of involutions is seldom studied. Feng et al. [34] investigated regular equivalence relations on ordered -semihypergroups. Yaqoob et al. [35] investigated the structures of involution -semihypergroups. Tang and Yaqoob [36] introduced a fuzzy set theory to hyperideals of ordered -semihypergroups.
2. Preliminaries and Basic Definitions
Some basic definitions are provided in this section on ordered LA-semihypergroups.
Definition 1. (see [5, 6]). A hypergroupoid is said to be an LA-semihypergroup if for all ,Equation 2 is known as the left invertive law.
Definition 2. (see [7]). Let be a non-empty set and “” be an ordered relation on . Then, is called an ordered LA-semihypergroup if(1) is an LA-semihypergroup;(2) is a partially ordered set;(3)for every , implies and , where means that for every there exists such that .
Definition 3. (see [7]). If is a nonempty subset of , then is the subset of defined as follows:
Definition 4. (see [7]). A non-empty subset of an ordered LA-semihypergroup is called an LA-subsemihypergroup of if .
An anti-involution is an automorphism that does not change the order of the elements. The distinction between involutions and anti-involutions can exist only in algebra. The involution property does not work in left invertive structures where associativity and commutativity both do not hold.
3. Ordered Anti-InvolutionLA-Semihypergroups
In this section, we define the concept of an ordered anti-involution LA-semihypergroup and provide some results.
Definition 5. An ordered LA-semihypergroup with an unary operation is called an ordered anti-involution LA-semihypergroup if it satisfies(1)(2)
If , then . For any nonempty subset of an ordered anti-involution LA-semihypergroup ,
If for any with , we have , then is called an order preserving anti-involution.
Example 1. Consider a set with the following hyperoperation “” and the order “” (Table1):
We give the covering relation “” as . The figure of is shown in Figure 1.
Then, is an ordered LA-semihypergroup. Now, we define the anti-involution by , , and . Then, it is easy to check that is an ordered anti-involution LA-semihypergroup with order preserving anti-involution .
Lemma 1. Suppose is an ordered anti-involution LA-semihypergroup. Then, we have the following:(i) for any (ii) for any with (iii) for all (iv) for all (v)For any right (left, two-sided) hyperideal of , (vi)If and are hyperideals of , then and are also hyperideals of
Proposition 1. If and are nonempty subsets of an ordered anti-involution LA-semihypergroup, then the following hold:(1) if and only if ;(2);(3).
Proof. (i)Consider . Also, consider Therefore, . Conversely, consider . Also, consider Therefore, .(ii)Consider Hence, we obtain .(iii)The proof is similar to (ii).
Proposition 2. Let be an ordered anti-involution LA-semihypergroup. Then,(1), for any,(2), for any.
Proof. (1)Let . By definition , for some . This implies that Therefore, , that is, and we get that . On the other hand, if , then for some , we have . This implies that So, , that is, . Therefore, . Consequently, .(2)The proof is similar to (1).
Definition 6. A nonempty subset of an ordered anti-involution LA-semihypergroup is called an ordered sub anti-involution LA-semihypergroup of if and .
Example 2. Consider a set with the following hyperoperation “” and the order “” (Table 2):
The figure of is shown in Figure 2.
Then, is an ordered LA-semihypergroup. Now, we define the anti-involution by , , , , and . Then, it is easy to check that is an ordered anti-involution LA-semihypergroup with order preserving anti-involution . Here, and are sub anti-involution LA-semihypergroups of . One can see that but , so is not a sub anti-involution LA-semihypergroup of .
Definition 7. A nonempty subset of an ordered anti-involution LA-semihypergroup is called a right (resp., left) anti-involution hyperideal of if(1) (resp., )(2)If and , then for every (3)
Definition 8. A nonempty subset of an ordered anti-involution LA-semihypergroup is called an anti-involution hyperideal of if it is both a right and a left anti-involution hyperideal of .
Definition 9. A nonempty subset of an ordered anti-involution LA-semihypergroup is called an anti-involution bi-hyperideal of if(1),(2),(3)If and , then for every ,(4).
Example 3. Consider a set with the following hyperoperation “” and the order “” (Table 3):
The figure of is shown in Figure 3.
Then, is an ordered LA-semihypergroup. Now, we define the anti-involution by , , , , and . Then, it is easy to check that is an ordered anti-involution LA-semihypergroup with order preserving anti-involution . Here, , , and are anti-involution bi-hyperideals of .
Definition 10. A nonempty subset of an ordered anti-involution LA-semihypergroup is called an anti-involution quasi-hyperideal of if(1),(2),(3).
Proposition 3. Let be an ordered LA-semihypergroup with order preserving anti-involution . Then,(1) is a left (resp., right) anti-involution hyperideal for any left (resp., right) hyperideal of ,(2) is an anti-involution bi-hyperideal for any bi-hyperideal of ,(3) is an anti-involution quasi-hyperideal for any quasi-hyperideal of .
Proof. Assume is a left hyperideal of . Since and , we haveNow, let and , then . Thus, , hence, . Therefore, is a left anti-involution hyperideal of . Similar is the case for right hyperideal of .(2)Similar to equation (1).(3)Let be a quasi-hyperideal of . Since , and By Proposition 1, we haveNow, let and , then . Thus, , hence, . This shows that is an anti-involution quasi-hyperideal of .
Theorem 1. Let be an ordered LA-semihypergroup with order preserving anti-involution . Let be a family of(1)Left (resp., right) anti-involution hyperideals of . Then, the intersection is a left (resp., right) anti-involution hyperideal of .(2)Anti-involution bi-hyperideals of . Then, the intersection is an anti-involution bi-hyperideal of .(3)Anti-involution quasi-hyperideals of . Then, the intersection is an anti-involution quasi-hyperideal of .
Proof. Straightforward.
Proposition 4. Let be an ordered LA-semihypergroup with order preserving anti-involution and be any anti-involution hyperideal of . For any , if , then .
Proof. Let . Then, for some . Let . Then, for some . Similarly, for some . Consequently, we haveHence, . On the other hand, let . Then, we have for some , because . Clearly,Thus, . Hence, .
Definition 11. An ordered anti-involution LA-semihypergroup is called a regular ordered anti-involution LA-semihypergroup, if , for all .
Proposition 5. Let be an ordered LA-semihypergroup with order preserving anti-involution . If is regular and has left identity, thenfor any .
Proof. Let . Since is regular, we have . Then,Thus, . On the other hand, we haveThus, . Hence, .
4. Rough Anti-Involution Hyperideals
In this section, we applied rough set theory to anti-involution hyperideals of ordered anti-involution LA-semihypergroups.
Definition 12. A relation on an ordered anti-involution LA-semihypergroup is called a pseudohyperorder if(1)(2) is transitive, that is, implies for all .(3) is compatible, that is, if then, and for all .(4)for all , we have .
Definition 13. A pseudohyperorder relation on an ordered anti-involution LA-semihypergroup is said to be complete if .
Definition 14. [20] Let be a nonempty set and be a binary relation on . By , we mean the power set of . For all , we define and bywhere . and are called the lower approximation and the upper approximation operations, respectively.
Lemma 2. Let be a pseudohyperorder on an ordered anti-involution LA-semihypergroup. Then, for any , .
Proof. Consider . Then, , this implies that . By definition of pseudohyperorder relation , this implies that . So, . Thus, we get . Conversely, consider , then . By definition of pseudohyperorder relation , this implies that . So, , this implies that . Thus, we obtain . Hence, .
Theorem 2. Let be a pseudohyperorder on an ordered LA-semihypergroup. If and are nonempty subsets of , then
Proof. The proof is straightforward.
Theorem 3. Let be a complete pseudohyperorder on an ordered LA-semihypergroup. If and are nonempty subsets of , then
Proof. The proof is straightforward.
Definition 15. Assume is a pseudohyperorder on an ordered anti-involution LA-semihypergroup. Then, a nonempty subset of is called a -upper rough LA-subsemihypergroup (resp., left anti-involution hyperideal, right anti-involution hyperideal, anti-involution hyperideal, and anti-involution bi-hyperideal) of if is an LA-subsemihypergroup (resp., left anti-involution hyperideal, right anti-involution hyperideal, anti-involution hyperideal, and anti-involution bi-hyperideal) of .
Theorem 4. Let be a pseudohyperorder on an ordered anti-involution LA-semihypergroup and an LA-subsemihypergroup (resp., left anti-involution hyperideal, right anti-involution hyperideal, anti-involution hyperideal, and anti-involution bi-hyperideal) of . Then, is an LA-subsemihypergroup (resp., left anti-involution hyperideal, right anti-involution hyperideal, anti-involution hyperideal, and anti-involution bi-hyperideal) of .
Proof. Let be an anti-involution bi-hyperideal of .(1)By Theorem 2, we have(2)By Theorem 2, we have(3)Let and such that . Then, there exists , such that and . Since is transitive, so implies .(4)ConsiderThus, . Hence, is an anti-involution bi-hyperideal of . The other cases can be seen in a similar way.
Definition 16. Assume is a complete pseudohyperorder on an ordered anti-involution LA-semihypergroup. Then, a nonempty subset of is called a -lower rough LA-subsemihypergroup (resp., left anti-involution hyperideal, right anti-involution hyperideal, anti-involution hyperideal, and anti-involution bi-hyperideal) of if is an LA-subsemihypergroup (resp., left anti-involution hyperideal, right anti-involution hyperideal, anti-involution hyperideal, and anti-involution bi-hyperideal) of .
Theorem 5. Assume is a complete pseudohyperorder on an ordered anti-involution LA-semihypergroup and an LA-subsemihypergroup (resp., left anti-involution hyperideal, right anti-involution hyperideal, anti-involution hyperideal, and anti-involution bi-hyperideal) of . Then, is an LA-subsemihypergroup (resp., left anti-involution hyperideal, right anti-involution hyperideal, anti-involution hyperideal, and anti-involution bi-hyperideal) of .
Proof. Let be an anti-involution bi-hyperideal of .(1)By Theorem 3, we have(2)By Theorem 3, we have(3)Let and such that . Then, there exists , such that and . Since is transitive, so implies .(4)ConsiderThus, . Hence, is an anti-involution bi-hyperideal of . The other cases can be seen in a similar way.
The following example shows that the converse of Theorems 4 and 5, for the case of anti-involution bi-hyperideal, but does not hold in general.
Example 4. Let be an ordered LA-semihypergroup with the following multiplication table and the order “.” (Table 4)
The figure of is shown in Figure 4.
Clearly, is an ordered LA-semihypergroup. We define the anti-involution by , , , , and . Then, it is easy to check that is an ordered anti-involutionLA-semihypergroup with order preserving anti-involution . Now, letbe a complete pseudohyperorder on , such thatNow, for ,It is clear that and are both anti-involution bi-hyperideals of but is not an anti-involution bi-hyperideal of .
5. Conclusion
Involutions can be applied to associative algebraic structures. This theory fails for noncommutative and nonassociative structures (left invertive structures). Here, we chose a noncommutative and nonassociative structure (ordered left almost semihypergroup) and then applied the concept of involutions. However, the involution theory failed on the ordered LA-semihypergroup because of its noncommutative and nonassociative nature. Then, we applied anti-involutions to an ordered LA-semihypergroup and provided some results and examples. We constructed some results on roughness in an ordered anti-involution LA-semihypergroup.
In the future, the following topics might be considered for further studies:(i)Regular and intraregular ordered anti-involution LA-semihypergroups(ii)Prime and weakly prime hyperideals in ordered anti-involution LA-semihypergroups(iii)Fuzzy ordered anti-involution LA-semihypergroups(iv)Soft ordered anti-involution LA-semihypergroups
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research project was funded by the Deanship of Scientific Research, Princess Nourah Bint Abdulrahman University, through the Program of Research Project Funding After Publication, grant no. (43-PRFA-P-12).