Abstract
Abel-Grassmann’s groupoid and neutrosophic extended triplet loop are two important algebraic structures that describe two kinds of generalized symmetries. In this paper, we investigate quasi AG-neutrosophic extended triplet loop, which is a fusion structure of the two kinds of algebraic structures mentioned above. We propose new notions of AG-(l,r)-Loop and AG-(r,l)-Loop, deeply study their basic properties and structural characteristics, and prove strictly the following statements: (1) each strong AG-(l,r)-Loop can be represented as the union of its disjoint sub-AG-groups, (2) the concepts of strong AG-(l,r)-Loop, strong AG-(l,l)-Loop, and AG-(l,lr)-Loop are equivalent, and (3) the concepts of strong AG-(r,l)-Loop and strong AG-(r,r)-Loop are equivalent.
1. Introduction
The so-called left almost semigroup (LA-semigroup) was actually the concept of an Abel-Grassmann’s groupoid (AG-groupoid), which was put forward by Kazim and Naseeruddin [1] at the first time in 1972. Different classes of AG-groupoids and their concerned characteristics have been studied in [2–5].
Neutrosophic set (NS) was first put forward by Smarandache in [6]. Then, it has been growing promptly over the previous 15 years. Nowadays, NS theory is widely used in a couple of sectors such as professional selection [7], integrated speech and text sentiment analysis [8], finite automata [9], clustering methods [10], and deep learning [11]. Besides, more new theoretical studies on NS in [12–17] have been conducted and a few significant results have been gained.
The concept of Abel-Grassmann’s neutrosophic extended triplet loop (AG-NET-Loop), which plays a significant role in neutrosophic triplet algebraic structures, was proposed in [18], that is, an AG-NET-Loop is both an AG-groupoid and a neutrosophic extended triplet loop (NET-Loop). In [19], the concept of neutrosophic triplet elements (NT-elements) and quasi neutrosophic triplet loops were introduced. In [20], two kinds of quasi AG-NET-Loops (AG-(l,l)-Loop and AG-(r,r)-Loop) were proposed and their basic properties were investigated. As a continuation of [20], we propose two other kinds of quasi AG-NET-Loops, which are the AG-(l,r)-Loop and the AG-(r,l)-Loop. We study their properties and analyze their relationship.
The rest of this paper is arranged as follows. In Section 2, some definitions and properties on quasi AG-NET-Loop are given. Some properties and structures about the AG-(l,r)-Loop are discussed in Section 3. The relations among four kinds of quasi AG-NET-Loops are analyzed in Section 4. Some properties about the alternative quasi AG-NET-Loops are discussed in Section 5. Lastly, Section 6 presents the summary and the direction of future efforts.
2. Preliminaries
A groupoid is called an AG-groupoid if it holds the left invertive law, that is, for all , . In an AG-groupoid the medial law holds, for all , . An AG-groupoid is called locally associative if for all . In an AG-groupoid , for all , , is defined as follows: .
Definition 1 (see [21]). Let be a nonempty set together with a binary operation . Then, is called a neutrosophic extended triplet set if, for all , there exist a neutral of ”” and an opposite of ”” (denoted by and , respectively), such that , and . The triplet is called a neutrosophic extended triplet (NET).
Definition 2 (see [18]). An NET set is called an NET-Loop, if, for all , one has .
Definition 3 (see [18]). An AG-groupoid is called an AG-NET-Loop if it is an NET-Loop.
An AG-NET-Loop is called a commutative AG-NET-Loop if for all .
Theorem 1 (see [18]). Let be an AG-NET-Loop. Then,(1)For all , is unique(2)For all ,
Definition 4 (see [2]). AG-groupoid is called regular if, for all , there exists ,
Definition 5 (see [20]). Let be an AG-groupoid. Then, is called an AG-(l,l)-Loop if, for all , there exist a local (l,l)-neutral element of”” and a local (l,l)-opposite element of”” (denoted by and , respectively), such that , and and .
Definition 6 (see [20]). Let be an AG-groupoid. Then, is called an AG-(r,r)-Loop if, for all , there exist a local (r,r)-neutral element of “” and a local (r,r)-opposite element of “” (denoted by and , respectively), such that , and and .
Definition 7. Let be an AG-groupoid. Then, is called an AG-(l,r)-Loop if, for all , there exist a local (l,r)-neutral element of “” and a local (l,r)-opposite element of ”” (denoted by and , respectively), such that , and and .
Remark 1. For quasi AG-NET-Loop, we will use the notations such as AG-NET-Loop. If and are not unique, then the set of all local (l,r)-neutral elements of”” and the set of all local (l,r)-opposite elements of “” are denoted by and , respectively.
Definition 8. Let be an AG-groupoid. Then, is called an AG-(r,l)-Loop if, for all , there exist a local (r,l)-neutral element of ”” and a local (r,l)-opposite element of”” (denoted by and , respectively), such that , and and .
Definition 9. Let be an AG-(l,r)-Loop. Then, is called an AG-(l,lr)-Loop if, for all , .
Definition 10 (see [22]). An AG-groupoid with a left identity is called an AG-group if each has an inverse element .
3. AG-(l,r)-Loop and Strong AG-(l,r)-Loop
Theorem 2. Let be a groupoid. Then, is an AG-(l,r)-Loop iff it is a regular AG-groupoid.
Proof. Necessity: if is an AG-(l,r)-Loop, from Definition 7, for all , there exist , and . We have . By Definition 4, is a regular AG-groupoid.
Sufficiency: if is a regular AG-groupoid, from Definition 4, for all , there exists and . Set , by Definition 7, is an AG-(l,r)-Loop.
Example 1 illustrates that an AG-groupoid may be neither an AG-(l,l)-Loop nor an AG-(l,r)-Loop nor an AG-(r,r)-Loop nor an AG-(r,l)-Loop.
Example 1. Let , and the definition of operation on is shown in Table 1. There is no , and in . That is, the element “2” in has no local (l,l)-opposite element, no local (l,r)-opposite element, no local (r,r)-opposite element, and no local (r,l)-opposite element. From Definitions 5–8, is neither an AG-(l,l)-Loop nor an AG-(l,r)-Loop nor an AG-(r,r)-Loop nor an AG-(r,l)-Loop.
Example 2 illustrates that an AG-(l,r)-Loop may be neither an AG-(l,l)-Loop nor an AG-(r,r)-Loop nor an AG-(r,l)-Loop.
Example 2. Let , and the definition of operation on is shown in Table 2. From Definition 7, is an AG-(l,r)-Loop. However, there is no , and in . From Definitions 5, 6, and 8, is neither an AG-(l,l)-Loop nor an AG-(r,r)-Loop nor an AG-(r,l)-Loop.
Definition 11. An AG-(l,r)-Loop is called a strong AG-(l,r)-Loop if, for all .
Example 3 illustrates that an AG-(l,r)-Loop is not always a strong AG-(l,r)-Loop.
Example 3. Let , and the definition of operation on is shown in Table 3. From Definition 7, is an AG-(l,r)-Loop. However, ; thus, is not a strong AG-(l,r)-Loop.
Example 4 illustrates that a strong AG-(l,r)-Loop is not always an AG-NET-Loop.
Example 4. Let , and the definition of operation on is shown in Table 4. By Definition 11, is a strong AG-(l,r)-Loop. However, since , is not an AG-NET-Loop.
Theorem 3. Let be a strong AG-(l,r)-Loop. Then,(1)For all , is unique(2)For all , (3)For all and for any (4)For all ,
Proof. (1)If is a strong AG-(l,r)-Loop, suppose , there exist By Definition 11, , , and there exist which satisfy . We haveWe know that , and is unique.(2)If is a strong AG-(l,r)-Loop, from Definition 11, we have, for all , . Thus, .(3)Suppose ; then,So, we get .(4)From Definition 11, we have, for all ,Therefore, .
Example 5. Let , and the definition of operation on is shown in Table 5. It is a strong AG-(l,r)-Loop. We have (corresponding to the results of Theorem 3)(1)For all , we can verify that is unique.(2)Being , , , , , , and , that is, for all , .(3)For any , let , and we can get and . Being , that is, , let , and we can get , . Being , that is, , we can verify other cases; thus, .(4)For any , without loss of generality, let ; we can get . We can verify other cases; thus, .
Theorem 4. Let be a strong AG-(l,r)-Loop. A binary on is introduced as follows:Then,(1)The binary on is an equivalence relation, and the equivalent class contained is denoted by (2)For all , is a sub-AG-group(3), that is, each strong AG-(l,r)-Loop can be represented as the union of its disjoint sub-AG-groups
Proof. (1)From the binary definition, it is easy to verify that has the properties of reflexive, symmetric, and transitive. Thus, it is an equivalence relation.(2)For all , let , and we have . From Theorem 3 (2), , and we have :(i)By Definition 11, we have ; thus, is a left identity of .(ii)For all , the left invertive law holds directly.(iii)For all , ; from Theorem 3 (4), ; thus, (iv)For all , let , and suppose ; by Theorem 3 (3), we have , and(v). Thus, and is an inverse element of . From Definition 10, is a sub-AG-group of .(3)By Theorem 3 (1), for all , is unique. Then, .
Example 6. Let , and the definition of operation on is shown in Table 6. and . , and and are sub-AG-groups of .
Let be an AG-groupoid; then, is an idempotent in if , . The set of all idempotents in is denoted by . An AG-groupoid is called an AG-band if .
From now on, we assume that is a strong AG-(l,r)-Loop, which is the same as Theorem 4. Let be an AG-band, , and for any , the equivalent class , which is defined in Theorem 4, will be denoted by , and the elements of will be denoted by , .
Theorem 5. Let be a groupoid, be an AG-band, . , is a strong AG-(l,r)-Loop with a left identity for each , and , . If, for all , for all , , then is a strong AG-(l,r)-Loop.
Proof. Suppose is the groupoid, is an AG-band, for each , and is a strong AG-(l,r)-Loop with a left identity and if in .
We first prove that is an AG-groupoid. Let , , and be arbitrary elements. Since , and are strong AG-(l,r)-Loops, we havewhere Since is a strong AG-(l,r)-Loop, the left invertive law holds directly for elements . Thus, is an AG-groupoid.
For any , we have . Let , we denote is the left identity in , , and . Being if in , we can get and .
Depending on , we have three cases to discuss.
case 1. , . Being if in , we can get . That is, there is no element such that .
case 2. , . Being if in , we can get . That is, there is no element such that , and there exists .
case 3. when ; when . That is, there is no element such that , and there exists .
From all the above cases, has a unique and . Consequently, is a strong AG-(l,r)-Loop.
Example 7. Let , and the definition of operation on is shown in Table 7. An AG-band and , , and . For any , without losing generality, let , and we have , and . The other cases can be verified; thus, is an AG-groupoid.
Let , ; for all , there is no element such that . That is, the element “10” has a unique and .
Let , ; for all , ; thus, there is no element such that there exists . That is, the element “3” has a unique and .
Let , when ; when . That is, there is no element such that , and there exists . The element “7” has a unique and .
The other cases can be verified; thus, is a strong AG-(l,r)-Loop.
Theorem 6. Let be a groupoid, be an AG-band, . , be a strong AG-(l,r)-Loop with a left identity for each , and , . If, for all , for all , , then is a strong AG-(l,r)-Loop.
Proof. Theorem 6 is proved similarly to Theorem 5.
The strong AG-(l,r)-Loop constructed by Theorem 5 is not isomorphic to the strong AG-(l,r)-Loop constructed by Theorem 6.
Definition 12 (see [20]). An AG-(l,l)-Loop is called a strong AG-(l,l)-Loop if for all .
Example 8 illustrates that an AG-(l,l)-Loop is not always a strong AG-(l,l)-Loop.
Example 8. Let , and the definition of operation on is shown in Table 8. From Definitions 5 and 7, is both an AG-(l,l)-Loop and an AG-(l,r)-Loop. However, ; thus, it is neither a strong AG-(l,l)-Loop nor a strong AG-(l,r)-Loop.
Theorem 7. Let be an AG-groupoid. Then, the following three statements are equivalent:(1) is a strong AG-(l,r)-Loop(2) is a strong AG-(l,l)-Loop(3) is an AG-(l,lr)-Loop
Proof. (1) (2). Suppose is a strong AG-(l,r)-Loop; from Definition 11, for all , there exist , , , and . Let , and we have . From Definition 12, is a strong AG-(l,l)-Loop.(2) (3). Suppose is a strong AG-(l,l)-Loop; from Definition 12, for all , there exist , , , and . So, . By Definition 9, is an AG-(l,lr)-Loop.(3) (1). If is an AG-(l,lr)-Loop, from Definition 9, for all , there exist , , and . So, . By Definition 11, is a strong AG-(l,r)-Loop.Figure 1 shows the relationships among AG-(l,l)-Loop and AG-(l,r)-Loop. Here, A stands for AG-NET-Loop, B stands for strong AG-(l,r)-Loop shown in Example 4 rather than AG-NET-Loop, C stands for AG-(l,r)-Loop and AG-(l,l)-Loop shown in Example 8, which is, however, not strong AG-(l,r)-Loop, D stands for AG-(l,l)-Loop rather than AG-(l,r)-Loop, E stands for AG-(l,r)-Loop shown in Example 2 rather than AG-(l,l)-Loop, and F stands for AG-groupoid shown in Example 1, which is, however, not either AG-(l,l)-Loop or AG-(l,r)-Loop. A + B stands for strong AG-(l,r)-Loop, A + B + C + D stands for AG-(l,l)-Loop, A + B + C + E stands for AG-(l,r)-Loop, and A + B + C + D + E + F stands for AG-groupoid.
4. AG-(r,r)-Loop and AG-(r,l)-Loop
Theorem 8. Let be an AG-(r,r)-Loop. Then,(1) is an AG-(r,l)-Loop(2) is an AG-(l,l)-Loop
Proof. (1)Suppose is an AG-(r,r)-Loop; from Definition 6, for all , there exist , , and . Let , and we have . By Definition 8, is an AG-(r,l)-Loop.(2)Suppose is an AG-(r,r)-Loop; from Definition 6, for all , there exist , , and . Let , and we have .By Definition 5, is an AG-(l,l)-Loop.
Definition 13. An AG-(r,r)-Loop is called a strong AG-(r,r)-Loop if for all .
Definition 14. An AG-(r,l)-Loop is called a strong AG-(r,l)-Loop if for all .
Example 9 illustrates that an AG-(r,r)-Loop is not always a strong AG-(r,r)-Loop and an AG-(r,l)-Loop is not always a strong AG-(r,l)-Loop.
Example 9. Let , and the definition of operation on is shown in Table 9. From Definitions 6, 8, 5, and 7, is both an AG-(r,r)-Loop and an AG-(r,l)-Loop and an AG-(l,l)-Loop and AG-(l,r)-Loop. However, ; . Thus, is neither a strong AG-(r,r)-Loop nor a strong AG-(r,l)-Loop nor a strong AG-(l,l)-Loop nor a strong AG-(l,r)-Loop.
Theorem 9. Let be an AG-groupoid. Then, the following three statements are equivalent:(1) is a strong AG-(r,r)-Loop(2) is a strong AG-(r,l)-Loop(3) is an AG-NET-Loop
Proof. (1)(2). Suppose is a strong AG-(r,r)-Loop; from Definition 13, for all , there exist , , , and . Let , and we have . By Definition 14, is a strong AG-(r,l)-Loop.(2)(3). Suppose is a strong AG-(r,l)-Loop; from Definition 14, for all , there exist , , , and . So, and . By Definition 3, is an AG-NET-Loop.(3)(1). It is obvious that an AG-NET-Loop is a strong AG-(r,r)-Loop.Figure 2 shows the relationships among AG-(r,l)-Loop and AG-(l,r)-Loop. Here, A stands for AG-NET-Loop, B stands for AG-(r,l)-Loop and strong AG-(l,r)-Loop shown in Example 4, which is, however, not AG-NET-Loop, C stands for AG-(r,l)-Loop and AG-(l,r)-Loop shown in Example 9, which is, however, not strong AG-(l,r)-Loop, D stands for AG-(r,l)-Loop rather than AG-(l,r)-Loop, E stands for strong AG-(l,r)-Loop rather than AG-(r,l)-Loop, F stands for AG-(l,r)-Loop shown in Example 2, which is, however, not either AG-(r,l)-Loop or strong AG-(l,r)-Loop, and G stands for AG-groupoid shown in Example 1, which is, however, not either AG-(l,r)-Loop or AG-(r,l)-Loop. A + B + E stands for strong AG-(l,r)-Loop, A + B + C + D stands for AG-(r,l)-Loop, A + B + C + E + F stands for AG-(l,r)-Loop, and A + B + C + D + E + F + G stands for AG-groupoid.
Figure 3 shows the relationships among AG-(r,l)-Loop and AG-(l,l)-Loop. Here, A stands for AG-NET-Loop, B stands for AG-(r,r)-Loop and strong AG-(l,l)-Loop shown in Example 4, which is, however, not AG-NET-Loop, C stands for AG-(r,r)-Loop shown in Example 9 rather than strong AG-(l,l)-Loop, D stands for AG-(r,l)-Loop and AG-(l,l)-Loop rather than AG-(r,r)-Loop, E stands for AG-(r,l)-Loop rather than AG-(l,l)-Loop, F stands for strong AG-(l,l)-Loop rather than AG-(r,l)-Loop, and G stands for AG-(l,l)-Loop, which is, however, not either AG-(r,l)-Loop or a strong AG-(l,l)-Loop. A + B + C stands for AG-(r,r)-Loop, A + B + F stands for strong AG-(l,l)-Loop, A + B + C + D + E stands for AG-(r,l)-Loop, and A + B + C + D + F + G stands for AG-(l,l)-Loop.
5. Alternative Quasi AG-NET-Loop
Definition 15. Let be an AG-NET-Loop (AG-(l,l)-Loop, AG-(l,r)-Loop, AG-(r,r)-Loop, and AG-(r,l)-Loop). Then, is called a right alternative AG-NET-Loop (AG-(l,l)-Loop, AG-(l,r)-Loop, AG-(r,r)-Loop, and AG-(r,l)-Loop) if , for all .
Definition 16. Let be an AG-NET-Loop (AG-(l,l)-Loop, AG-(l,r)-Loop, AG-(r,r)-Loop, and AG-(r,l)-Loop). Then, is called an alternative AG-NET-Loop (AG-(l,l)-Loop, AG-(l,r)-Loop, AG-(r,r)-Loop, and AG-(r,l)-Loop), if for all , , .
Example 10 illustrates that an AG-NET-Loop is not always an alternative AG-NET-Loop.
Example 10. Let , and the definition of operation on is shown in Table 10. By Definition 3, is an AG-NET-Loop. However, is not an alternative AG-NET-Loop because .
Theorem 10. Let be an AG-NET-Loop. Then, the following three statements are equivalent:(1) is a right alternative AG-NET-Loop(2) is a commutative AG-NET-Loop(3) is an alternative AG-NET-Loop
Proof. (1)(2). Suppose is a right alternative AG-NET-Loop; from Definition 15, for all , soConsequently, is a commutative AG-NET-Loop.(2)(3). If is a commutative AG-NET-Loop, for all , and . By Definition 16, is an alternative AG-NET-Loop.(3)(1). It is obvious that an alternative AG-NET-Loop is a right alternative AG-NET-Loop.
Theorem 11 (see [23]). Let be a locally associative AG-groupoid. If is finite, then there exists .
Theorem 12. Let be a right alternative AG-(r,l)-Loop. If is finite, then, for all , there exist .
Proof. If is a finite right alternative AG-(r,l)-Loop. Then, for all , there exist , and we have .
When , ,Thus, are all right neutral element.
By Theorem 11, we get that there is an idempotent right neutral element in .
Theorem 13 (see [23]). Let be a finite alternative AG-(l,l)-Loop. Then, is a strong AG-(l,l)-Loop.
Theorem 14. Let be an AG-groupoid. Then, the following three statements are equivalent:(1) is a finite right alternative AG-(r,l)-Loop(2) is a finite alternative AG-NET-Loop(3) is a finite alternative AG-(l,l)-Loop
Proof. (1)(2). If is a finite right alternative AG-(r,l)-Loop, applying Theorem 12, we get that is a strong AG-(r,l)-Loop. From Theorem 9, we get that is a right alternative AG-NET-Loop. Applying Theorem 10, is a finite alternative AG-NET-Loop.(2)(3). It is obvious that a finite alternative AG-NET-Loop is a finite alternative AG-(l,l)-Loop.(3)(1). If is a finite alternative AG-(l,l)-Loop, applying Theorem 13, we get that is a strong AG-(l,l)-Loop. From Definition 12, for all , there exist , , , and . We have By Definition 15, is a finite right alternative AG-(r,l)-Loop.
Example 11. Let , and the definition of operation on is shown in Table 11. We can easily verify that satisfies the alternative law. Being each element in has a neutral element and an opposite element; by Definition 16, is a finite alternative AG-NET-Loop. Obviously, a finite alternative AG-NET-Loop is both a finite right alternative AG-(r,l)-Loop and a finite alternative AG-(l,l)-Loop. Since for all , we have as a commutative AG-NET-Loop.
Figure 4 shows the relationships among alternative AG-NET-Loop and other alternative quasi AG-NET-Loops. In Figure 4, we prove that the right alternative AG-NET-Loop is equivalent to the commutative AG-NET-Loop, and the commutative AG-NET-Loop is equivalent to the alternative AG-NET-Loop. As the finite right alternative AG-(r,l)-Loop is equivalent to the finite alternative AG-(l,l)-Loop, the finite alternative AG-(l,l)-Loop is equivalent to the finite alternative AG-NET-Loop; therefore, they are equivalent to each other.
6. Conclusion
In this paper, the AG-(l,r)-Loop and AG-(r,l)-Loop have been introduced, the structure of the quasi AG-NET-Loops have been studied further, and some important results have been obtained. We prove that the strong AG-(l,r)-Loop, the strong AG-(l,l)-Loop, and the AG-(l,lr)-Loop are equivalent (see Theorem 7); the strong AG-(r,l)-Loop, the strong AG-(r,r)-Loop, and the AG-NET-Loop are equivalent (see Theorem 9); the commutative AG-NET-Loop, the alternative AG-NET-Loop, and the right alternative AG-NET-Loop are equivalent (see Theorem 10). Furthermore, the decomposition theorem of strong AG-(l,r)-Loop (see Theorem 4) and two different ways how to make a strong AG-(l,r)-Loop are obtained (see Theorem 5 and Theorem 6), thus illuminating the structure of strong AG-(l,r)-Loop. Figure 5 shows the main results of this paper. Future efforts will be directed towards discussing the relationship between strong AG-(l,r)-Loop and other related AG-groupoid bands, such as root of band, AG-4-band, and AG-3-band (see [24]).
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research was supported by Scientific Research Program funded by Shaanxi Provincial Education Department (Program No. 20JK0549).