# Study of Two Kinds of Quasi AG-Neutrosophic Extended Triplet Loops

Authors: Xiaogang An; Mingming Chen
Journal: Journal of Mathematics
 (2021)

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## 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.


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## Body

## 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 Section2, 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 groupoidG,∗ is called an AG-groupoid if it holds the left invertive law, that is, for all x,y,z∈G, x∗y∗z=z∗y∗x. In an AG-groupoid G,∗ the medial law holds, for all x1,x2,x3,x4∈G, x1∗x2∗x3∗x4=x1∗x3∗x2∗x4. An AG-groupoid G,∗ is called locally associative if for all x∈G,x∗x∗x=x∗x∗x. In an AG-groupoid G,∗, for all x∈G, k∈Z+, xk is defined as follows: x1=x,x2=x∗x,x3=x2∗x,x4=x3∗x,…,xk=xk−1∗x.Definition 1 (see [21]).
LetG be a nonempty set together with a binary operation ∗. Then, G is called a neutrosophic extended triplet set if, for all x∈G, there exist a neutral of ”x” and an opposite of ”x” (denoted by neutx and antix, respectively), such that neutx,antix∈G, and neutx∗x=x∗neutx=x,antix∗x=x∗antix=neutx. The triplet x,neutx,antix is called a neutrosophic extended triplet (NET).Definition 2 (see [18]).
An NET setG,∗ is called an NET-Loop, if, for all x,y∈G, one has x∗y∈G.Definition 3 (see [18]).
An AG-groupoidG,∗ is called an AG-NET-Loop if it is an NET-Loop.
An AG-NET-LoopG is called a commutative AG-NET-Loop if for all x,y∈G,x∗y=y∗x.Theorem 1 (see [18]).
LetG,∗ be an AG-NET-Loop. Then,(1)
For allx∈G, neutx is unique(2)
For allx∈G, neutx2=neutxDefinition 4 (see [2]).
AG-groupoidG,∗ is called regular if, for all a∈G, there exists m∈G, a=a∗m∗a.Definition 5 (see [20]).
LetG,∗ be an AG-groupoid. Then, G is called an AG-(l,l)-Loop if, for all a∈G, there exist a local (l,l)-neutral element of”a” and a local (l,l)-opposite element of”a” (denoted by nlla and olla, respectively), such that nlla∈G,olla∈G, and nlla∗a=a and olla∗a=nlla.Definition 6 (see [20]).
LetG,∗ be an AG-groupoid. Then, G is called an AG-(r,r)-Loop if, for all a∈G, there exist a local (r,r)-neutral element of “a” and a local (r,r)-opposite element of “a” (denoted by nrra and orra, respectively), such that nrra∈G,orra∈G, and a∗nrra=a and a∗orra=nrra.Definition 7.
LetG,∗ be an AG-groupoid. Then, G is called an AG-(l,r)-Loop if, for all a∈G, there exist a local (l,r)-neutral element of “a” and a local (l,r)-opposite element of ”a” (denoted by nlra and olra, respectively), such that nlra∈G,olra∈G, and nlra∗a=a and a∗olra=nlra.Remark 1.
For quasi AG-NET-Loop, we will use the notations such as AG-NET-Loop. Ifnlra and olra are not unique, then the set of all local (l,r)-neutral elements of”a” and the set of all local (l,r)-opposite elements of “a” are denoted by nlra and olra, respectively.Definition 8.
LetG,∗ be an AG-groupoid. Then, G is called an AG-(r,l)-Loop if, for all a∈G, there exist a local (r,l)-neutral element of ”a” and a local (r,l)-opposite element of”a” (denoted by nrla and orla, respectively), such that nrla∈G,orla∈G, and a∗nrla=a and orla∗a=nrla.Definition 9.
LetG,∗ be an AG-(l,r)-Loop. Then, G is called an AG-(l,lr)-Loop if, for all a∈G, olra∗a=a∗olra=nlra.Definition 10 (see [22]).
An AG-groupoidG with a left identity is called an AG-group if each a∈G has an inverse element a′.


## 3. AG-(l,r)-Loop and Strong AG-(l,r)-Loop

Theorem 2.
LetG,∗ be a groupoid. Then, G is an AG-(l,r)-Loop iff it is a regular AG-groupoid.Proof.
Necessity: ifG is an AG-(l,r)-Loop, from Definition 7, for all a∈G, there exist nlra,olra∈G,nlra∗a=a, and a∗olra=nlra. We have a∗olra∗a=a. By Definition 4, G is a regular AG-groupoid.
Sufficiency: ifG is a regular AG-groupoid, from Definition 4, for all a∈G, there exists m∈G and a=a∗m∗a. Set nlra=a∗m, by Definition 7, G is an AG-(l,r)-Loop.
Example1 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.
LetG=1,2,3,4,5,6,7,8, and the definition of operation ∗ on G is shown in Table 1. There is no oll2,olr2,orr2, and orl2 in G. That is, the element “2” in G 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, G 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.
Example2 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.Table 1
Table of Example1.
∗12345678111111111211121111311311111412141111511115111611111688711111878811111888Example 2.
LetG=1,2,3,4,5,6,7, and the definition of operation ∗ on G is shown in Table 2. From Definition 7, G is an AG-(l,r)-Loop. However, there is no oll2,nrr2, and nrl2 in G. From Definitions 5, 6, and 8, G is neither an AG-(l,l)-Loop nor an AG-(r,r)-Loop nor an AG-(r,l)-Loop.Table 2
Table of Example2.
∗123456711111111211144113113133741211211512345376113136771171777Definition 11.
An AG-(l,r)-Loop G,∗ is called a strong AG-(l,r)-Loop if, for all a∈G,nlra2=nlra.
Example3 illustrates that an AG-(l,r)-Loop is not always a strong AG-(l,r)-Loop.Example 3.
LetG=1,2,3,4,5,6,7, and the definition of operation ∗ on G is shown in Table 3. From Definition 7, G is an AG-(l,r)-Loop. However, nlr2=3,3∗3=1; thus, G is not a strong AG-(l,r)-Loop.
Example4 illustrates that a strong AG-(l,r)-Loop is not always an AG-NET-Loop.Table 3
Table of Example3.
∗123456711111111211331333121212241234567511155116123616671237167Example 4.
LetG=1,2,3,4,5,6,7, and the definition of operation ∗ on G is shown in Table 4. By Definition 11, G is a strong AG-(l,r)-Loop. However, since 1∗4≠4∗1, G is not an AG-NET-Loop.Table 4
Table of Example4.
∗123456711134111212341113441344443341333511345116113416671134167Theorem 3.
LetG,∗ be a strong AG-(l,r)-Loop. Then,(1)
For alla∈G, nlra is unique(2)
For alla∈G, nlrnlra=nlra(3)
For alla∈G and for any r∈olra,nlra∗r∈olra(4)
For alla,b∈G, nlra∗b=nlra∗nlrbProof.
(1)
IfG,∗ is a strong AG-(l,r)-Loop, suppose a∈G, there exist nlr1,nlr2∈nlra. By Definition 11, nlr1∗a=a,nlr2∗a=a, nlr1∗nlr1=nlr1,andnlr2∗nlr2=nlr2, and there exist olr1,olr2∈G which satisfy a∗olr1=nlr1anda∗olr2=nlr2. We have(1)nlr1∗nlr2=nlr1∗nlr1∗nlr2=nlr2∗nlr1∗nlr1=nlr2∗nlr1∗a∗olr1=nlr2∗a∗nlr1∗olr1by the medial law=nlr1∗a∗nlr1∗olr1=nlr1∗nlr1∗a∗olr1by the medial law=nlr1∗nlr1=nlr1,nlr2∗nlr1=nlr2∗nlr2∗nlr1=nlr1∗nlr2∗nlr2=nlr1∗nlr2∗a∗olr2=nlr1∗a∗nlr2∗olr2by the medial law=nlr2∗a∗nlr2∗olr2=nlr2∗nlr2∗a∗olr2by the medial law=nlr2∗nlr2=nlr2,nlr2=nlr2∗nlr1=nlr2∗nlr2∗nlr1=nlr1∗nlr2∗nlr2=nlr1∗nlr2=nlr1.
We know thatnlr2=nlr1, and nlra is unique.(2)
IfG,∗ is a strong AG-(l,r)-Loop, from Definition 11, we have, for all a∈G, nlra2=nlra. Thus, nlrnlra=nlra.(3)
Supposer∈olra; then,(2)a∗nlra∗r=nlra∗a∗nlra∗r=nlra∗nlra∗a∗rby the medial law=nlra∗nlra=nlra.
So, we getnlra∗r∈olra.(4)
From Definition11, we have, for all a,b∈G,(3)a∗b=nlra∗a∗nlrb∗b=nlra∗nlrb∗a∗b,nlra∗nlrb=a∗olra∗b∗olrb=a∗b∗olra∗olrb.
Therefore,nlra∗b=nlra∗nlrb.Example 5.
LetG=1,2,3,4,5,6,7, and the definition of operation ∗ on G is shown in Table 5. It is a strong AG-(l,r)-Loop. We have (corresponding to the results of Theorem 3)(1)
For alla∈G, we can verify that nlra is unique.(2)
Beingnlrnlr1=nlr1, nlrnlr2=nlr2, nlrnlr3=nlr3, nlrnlr4=nlr4, nlrnlr5=nlr5, nlrnlr6=nlr6, and nlrnlr7=nlr7, that is, for all a∈G, nlrnlra=nlra.(3)
For anya∈G, let a=1, and we can get nlr1=1 and olr1=1,2,5,6,7. Being 1∗1=1∗2=1∗5=1∗6=1∗7=1∈olr1, that is, nlr1∗olr1∈olr1, let a=3, and we can get nlr3=1, olr3=3. Being 1∗3=3=olr3, that is, nlr3∗olr3∈olr3, we can verify other cases; thus, nlra∗r∈olra.(4)
For anya,b∈G, without loss of generality, let a=1andb=3; we can get nlr1∗3=nlr1∗nlr3. We can verify other cases; thus, nlra∗b=nlra∗nlrb.Table 5
Table of Example5.
∗123456711134111212341123441344443341333511345156113416171234517Theorem 4.
LetG,∗ be a strong AG-(l,r)-Loop. A binary ≈ on G is introduced as follows:(4)for alla,b∈G,a≈b⇔nlra=nlrb.
Then,(1)
The binary≈ on G is an equivalence relation, and the equivalent class contained x is denoted by x≈(2)
For allx∈G, x≈ is a sub-AG-group(3)
G=∪x∈Gx≈, that is, each strong AG-(l,r)-Loop can be represented as the union of its disjoint sub-AG-groupsProof.
(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 alla∈x≈, let nlrx=ex, and we have nlra=nlrx=ex. From Theorem 3 (2), nlrex=ex, and we have ex∈x≈:(i)
By Definition11, we have ex∗a=nlra∗a=a; thus, ex is a left identity of x≈.(ii)
For alla,b,c∈x≈, the left invertive law holds directly.(iii)
For alla,b∈x≈, nlra=nlrb=ex; from Theorem 3 (4), nlra∗b=nlra∗nlrb=ex; thus, a∗b∈x≈.(iv)
For alla∈x≈, let nlra=ex, and suppose p∈olra,q=nlra∗p; by Theorem 3 (3), we have q∈olra,a∗q=nlra=ex, and(5)nlrq=nlrnlra∗p=nlrnlra∗nlrpby Theorem34=nlra∗nlrpby Theorem32=nlra∗pby Theorem34=nlrnlra=nlraby Theorem32=ex.(v)
q∗a=nlrq∗q∗a=ex∗q∗a=a∗q∗ex=ex. Thus, q∈x≈ and q is an inverse element of a. From Definition 10, x≈ is a sub-AG-group of G.(3)
By Theorem3 (1), for all a∈x≈, nlra is unique. Then, G=∪x∈Gx≈.Example 6.
LetG=1,2,3,4,5,6,7,8, and the definition of operation ∗ on G is shown in Table 6. 1≈=1,2,3,4 and 5≈=5,6,7,8. G=1≈∪5≈, and 1≈ and 5≈ are sub-AG-groups of G.
LetG be an AG-groupoid; then, a is an idempotent in G if a∈G, a2=a. The set of all idempotents in G is denoted by EG. An AG-groupoid G is called an AG-band if G=EG.
From now on, we assume thatG is a strong AG-(l,r)-Loop, which is the same as Theorem 4. Let Y be an AG-band, Y⊂G, and for any α∈Y, the equivalent class α≈, which is defined in Theorem 4, will be denoted by Sα, and the elements of Sα will be denoted by aα, bα,…,.Table 6
Table of Example6.
∗12345678112341111221432222343214444434123333512345678612346587712348765812347856Theorem 5.
LetG,∗ be a groupoid, Y be an AG-band, Y⊂G. G=∪α∈YSα, Sα,∗ is a strong AG-(l,r)-Loop with a left identity eα for each α∈Y, and Sα∩Sβ=∅, α,β∈Yandα≠β. If, for all aα∈Sα, for all bβ∈Sβ, aα∗bβ=aα∗eα,andbβ∗aα=aα, then G is a strong AG-(l,r)-Loop.Proof.
SupposeG=∪α∈YSα is the groupoid, Y is an AG-band, for each α∈Y, and Sα is a strong AG-(l,r)-Loop with a left identity eα and Sα∩Sβ=∅ if α≠β in Y.
We first prove thatG is an AG-groupoid. Let aα∈Sα, bβ∈Sβ, and cγ∈Sγ be arbitrary elements. Since Sα,Sβ, and Sγ are strong AG-(l,r)-Loops, we have(6)aα∗bβ∗cγ=aα∗eα∗cγ=aα∗eα∗eα=eα∗eα∗aαby the left invertive law=eα∗aα=aα,where cγ∗bβ∗aα=bβ∗aα=aα=aα∗bβ∗cγ. Since Sα is a strong AG-(l,r)-Loop, the left invertive law holds directly for elements aα,bα,cα∈Sα. Thus, G is an AG-groupoid.
For anybβ∈Sβ, we have nlrbβ=eβandolrbβ∗bβ=bβ∗olrbβ=eβ. Let x∈G−Sβ, we denote ex is the left identity in x≈, LSβ=x|x∗bβ=x∗ex,bβ∗x=x,x∈G−Sβ, and RSβ=x|x∗bβ=bβ,bβ∗x=bβ∗eβ,x∈G−Sβ. Being Sα∩Sβ=∅ if α≠β in Y, we can get LSβ∩Sβ∩RSβ=∅ and LSβ⋃Sβ⋃RSβ=G.
Depending onSβ, we have three cases to discuss.case 1.
LSβ=G−Sβ,RSβ=∅,x∈LSβ, x∗bβ=x∗ex,andbβ∗x=x. Being Sα∩Sβ=∅ if α≠β in Y, we can get x∗ex∈x≈,x∗bβ∉Sβ. That is, there is no element x∉Sβ such that x∗bβ=bβ.case 2.
LSβ=∅,RSβ=G−Sβ,x∈RSβ, x∗bβ=bβ,andbβ∗x=bβ∗eβ. Being Sα∩Sβ=∅ if α≠β in Y, we can get bβ∗x=bβ∗eβ∈Sβ. That is, there is no element x∉Sβ such that x∗bβ=bβandbβ∗y=x, and there exists y∈G−Sβ.case 3.
LSβ≠∅andRSβ≠∅, when x∈LSβ,x∗bβ=x∗ex∉Sβ,andbβ∗x=x∉RSβ; when x∈RSβ,x∗bβ=bβ,bβ∗x=bβ∗eβ∉RSβ. That is, there is no element x∉Sβ such that x∗bβ=bβandbβ∗y=x, and there exists y∈G−Sβ.
From all the above cases,bβ has a unique nlrbβ=eβ and olrbβ⊆Sβ. Consequently, G is a strong AG-(l,r)-Loop.Example 7.
LetG=1,2,3,4,5,6,7,8,9,10,11,12,13, and the definition of operation ∗ on G is shown in Table 7. An AG-band Y=1,5,9 and S1=1,2,3,4,e1=1, S5=5,6,7,8,e5=5, and S9=9,10,11,12,13,e9=9. For any a1∈S1,b5∈S5,andc9∈S9, without losing generality, let a1=3,b5=7,andc9=10, and we have 3∗7=3∗1and7∗3=3,3∗10=3∗1and10∗3=3,7∗10=7∗5and10∗7=7, and 3∗7∗10=10∗7∗3. The other cases can be verified; thus, G is an AG-groupoid.
Letc9=10, LS9=G−S9=1,2,3,4,5,6,7,8andRS9=∅; for all x∈LS9, there is no element x such that x∗10=10. That is, the element “10” has a unique nlr10=9 and olr10=13⊆S9.
Leta1=3, LS1=∅,RS1=G−S1=5,6,7,8,9,10,11,12,13; for all x∈RS1, 3∗x=3∗e1=3∗1=4∉RS1; thus, there is no element x such that there exists y∈RS1,x∗3=3,3∗y=x. That is, the element “3” has a unique nlr3=1 and olr3=4⊆S1.
Letb5=7, LS5=1,2,3,4,andRS5=9,10,11,12,13, when x∈LS5,x∗7=x∗ex∉S5,7∗x=x∉RS5; when x∈RS5,x∗7=7,7∗x=7∗e5=7∗5=8∉RS5. That is, there is no element x∉S5 such that x∗7=7,7∗y=x, and there exists y∈G−S5. The element “7” has a unique nlr7=5 and olr7=7⊆S5.
The other cases can be verified; thus,G is a strong AG-(l,r)-Loop.Table 7
Table of Example7.
∗1234567891011121311234111111111221432222222223432144444444443412333333333512345678555556123465876666671234875688888812347865777779123456789101112131012345678101112139111234567811121391012123456781213910111312345678139101112Theorem 6.
LetG,∗ be a groupoid, Y be an AG-band, Y⊂G. G=∪α∈YSα, Sα,∗ be a strong AG-(l,r)-Loop with a left identity eα for each α∈Y, and Sα∩Sβ=∅, α,β∈Y,α≠β. If, for all aα∈Sα, for all bβ∈Sβ, aα∗bβ=bβ,bβ∗aα=bβ∗eβ, then G is a strong AG-(l,r)-Loop.Proof.
Theorem6 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 G,∗ is called a strong AG-(l,l)-Loop if for all a∈G,nlla2=nlla.
Example8 illustrates that an AG-(l,l)-Loop is not always a strong AG-(l,l)-Loop.Example 8.
LetG=1,2,3,4,5,6,7,8, and the definition of operation ∗ on G is shown in Table 8. From Definitions 5 and 7, G is both an AG-(l,l)-Loop and an AG-(l,r)-Loop. However, nll1=nlr1=3,3∗3=4≠3; thus, it is neither a strong AG-(l,l)-Loop nor a strong AG-(l,r)-Loop.Table 8
Table of Example8.
∗12345678124317568231246875313428657442135786586576875657867568775685786868758657Theorem 7.
LetG,∗ be an AG-groupoid. Then, the following three statements are equivalent:(1)
G is a strong AG-(l,r)-Loop(2)
G is a strong AG-(l,l)-Loop(3)
G is an AG-(l,lr)-LoopProof.
(1)
⟹ (2). Suppose G is a strong AG-(l,r)-Loop; from Definition 11, for all a∈G, there exist nlra,olra∈G, nlra∗a=a, a∗olra=nlra, and nlra2=nlra. Let d=nlra∗olra, and we have d∗a=nlra∗olra∗a=a∗olra∗nlra=nlra2=nlra. From Definition 12, G is a strong AG-(l,l)-Loop.(2)
⟹ (3). Suppose G is a strong AG-(l,l)-Loop; from Definition 12, for all a∈G, there exist nlla,olla∈G, nlla∗a=a, olla∗a=nlla, and nlla2=nlla. So, a∗olla=nlla∗a∗olla=olla∗a∗nlla=nlla2=nlla. By Definition 9, G is an AG-(l,lr)-Loop.(3)
⟹ (1). If G is an AG-(l,lr)-Loop, from Definition 9, for all a∈G, there exist nlra,olra∈G, nlra∗a=a, and olra∗a=a∗olra=nlra. So, nlra∗nlra=olra∗a∗nlra=nlra∗a∗olra=a∗olra=nlra. By Definition 11, G is a strong AG-(l,r)-Loop.
Figure1 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.Figure 1
The relationships among AG-(l,l)-Loop and AG-(l,r)-Loop.


## 4. AG-(r,r)-Loop and AG-(r,l)-Loop

Theorem 8.
LetG,∗ be an AG-(r,r)-Loop. Then,(1)
G is an AG-(r,l)-Loop(2)
G is an AG-(l,l)-LoopProof.
(1)
SupposeG is an AG-(r,r)-Loop; from Definition 6, for all a∈G, there exist nrra,orra∈G, a∗nrra=a, and a∗orra=nrra. Let q=orra∗nrra, and we have q∗a=orra∗nrra∗a=a∗nrra∗orra=a∗orra=nrra. By Definition 8, G is an AG-(r,l)-Loop.(2)
SupposeG is an AG-(r,r)-Loop; from Definition 6, for all a∈G, there exist nrra,orra∈G, a∗nrra=a, and a∗orra=nrra. Let d=nrra2andq=nrra∗orra, and we have d∗a=nrra∗nrra∗a=a∗nrra∗nrra=a∗nrra=aandq∗a=nrra∗orra∗a=a∗orra∗nrra=nrra∗nrra=d.
By Definition5, G is an AG-(l,l)-Loop.Definition 13.
An AG-(r,r)-Loop G,∗ is called a strong AG-(r,r)-Loop if for all a∈G,nrra2=nrra.Definition 14.
An AG-(r,l)-Loop G,∗ is called a strong AG-(r,l)-Loop if for all a∈G,nrla2=nrla.
Example9 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.
LetG=1,2,3,4,5,6,7,8, and the definition of operation ∗ on G is shown in Table 9. From Definitions 6, 8, 5, and 7, G 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, nrr1=4,nrl1=4,4∗4=3≠4; nll1=3,nlr1=3,3∗3=4≠3. Thus, G 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.Table 9
Table of Example9.
∗12345678124313124231242431313424213442131342513426875642137568724315786831248657Theorem 9.
Let G,∗ be an AG-groupoid. Then, the following three statements are equivalent:(1)
G is a strong AG-(r,r)-Loop(2)
G is a strong AG-(r,l)-Loop(3)
G is an AG-NET-LoopProof.
(1)
⟹(2). Suppose G is a strong AG-(r,r)-Loop; from Definition 13, for all a∈G, there exist nrra,orra∈G, a∗nrra=a, a∗orra=nrra, and nrra2=nrra. Let q=orra∗nrra, and we have q∗a=orra∗nrra∗a=a∗nrra∗orra=a∗orra=nrra. By Definition 14, G is a strong AG-(r,l)-Loop.(2)
⟹(3). Suppose G is a strong AG-(r,l)-Loop; from Definition 14, for all a∈G, there exist nrla,orla∈G, a∗nrla=a, orla∗a=nrla, and nrla2=nrla. So, nrla∗a=nrla∗nrla∗a=a∗nrla∗nrla=a∗nrla=a and a∗orla=nrla∗a∗orla=orla∗a∗nrla=nrla2=nrla. By Definition 3, G is an AG-NET-Loop.(3)
⟹(1). It is obvious that an AG-NET-Loop is a strong AG-(r,r)-Loop.
Figure2 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.
Figure3 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.Figure 2
The relationships among AG-(r,l)-Loop and AG-(l,r)-Loop.Figure 3
The relationships among AG-(r,l)-Loop and AG-(l,l)-Loop.


## 5. Alternative Quasi AG-NET-Loop

Definition 15.
LetG,∗ be an AG-NET-Loop (AG-(l,l)-Loop, AG-(l,r)-Loop, AG-(r,r)-Loop, and AG-(r,l)-Loop). Then, G 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 b∗a∗a=b∗a∗a, for all a,b∈G.Definition 16.
LetG,∗ be an AG-NET-Loop (AG-(l,l)-Loop, AG-(l,r)-Loop, AG-(r,r)-Loop, and AG-(r,l)-Loop). Then, G 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 a,b∈G, a∗a∗b=a∗a∗b, a∗b∗b=a∗b∗b.
Example10 illustrates that an AG-NET-Loop is not always an alternative AG-NET-Loop.Example 10.
LetG=1,2,3,4,5,6,7, and the definition of operation ∗ on G is shown in Table 10. By Definition 3, G is an AG-NET-Loop. However, G is not an alternative AG-NET-Loop because 3∗4∗4≠3∗4∗4.Table 10
Table of Example10.
∗123456711423312232411343413224342314421523145216142336274132247Theorem 10.
LetG,∗ be an AG-NET-Loop. Then, the following three statements are equivalent:(1)
G is a right alternative AG-NET-Loop(2)
G is a commutative AG-NET-Loop(3)
G is an alternative AG-NET-LoopProof.
(1)
⇒(2). Suppose G is a right alternative AG-NET-Loop; from Definition 15, for all a,b∈G,(7)a∗neutb=a∗neutb∗neutbby Theorem12=a∗neutb∗neutbby the right alternative law=neutb∗neutb∗aby the left invertive law=neutb∗a,so(8)a∗b=neuta∗a∗b∗neutb=neuta∗b∗a∗neutbby the medial law=b∗neuta∗neutb∗a=b∗neutb∗neuta∗a=b∗a.
Consequently,G is a commutative AG-NET-Loop.(2)
⇒(3). If G is a commutative AG-NET-Loop, for all m,n∈G, m∗n∗n=n∗n∗m=m∗n∗n and m∗m∗n=n∗m∗m=m∗n∗m=m∗m∗n. By Definition 16, G 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]).
LetG,∗ be a locally associative AG-groupoid. If G is finite, then there exists a∈G,a2=a.Theorem 12.
LetG,∗ be a right alternative AG-(r,l)-Loop. If G is finite, then, for all a∈G, there exist s,p∈G,a∗s=a,p∗a=s,ands2=s.Proof.
IfG is a finite right alternative AG-(r,l)-Loop. Then, for all a∈G, there exist s,p∈G,a∗s=a,andp∗a=s, and we have a∗s2=a∗s∗s=a∗s∗s=a∗s=a.
Whenk∈Z+, k>2,(9)a∗sk=a∗s∗s2∗sk−2=a∗s2∗s∗sk−2by the medial law=a∗sk−1=⋯⋯=a∗s2=a.
Thus,s,s2,s3,…,sk,… are all right neutral element.
By Theorem11, we get that there is an idempotent right neutral element in G.Theorem 13 (see [23]).
LetG,∗ be a finite alternative AG-(l,l)-Loop. Then, G is a strong AG-(l,l)-Loop.Theorem 14.
LetG,∗ be an AG-groupoid. Then, the following three statements are equivalent:(1)
G is a finite right alternative AG-(r,l)-Loop(2)
G is a finite alternative AG-NET-Loop(3)
G is a finite alternative AG-(l,l)-LoopProof.
(1)
⇒(2). If G is a finite right alternative AG-(r,l)-Loop, applying Theorem 12, we get that G is a strong AG-(r,l)-Loop. From Theorem 9, we get that G is a right alternative AG-NET-Loop. Applying Theorem 10, G 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 G is a finite alternative AG-(l,l)-Loop, applying Theorem 13, we get that G is a strong AG-(l,l)-Loop. From Definition 12, for all a∈G, there exist nlla,olla∈G, nlla∗a=a, olla∗a=nlla, and nlla2=nlla. We have(10)a∗nlla=a∗nlla∗nlla=a∗nlla∗nllaby the right alternative law=nlla∗nlla∗aby the left invertive law=nlla∗a=a.By Definition15, G is a finite right alternative AG-(r,l)-Loop.Example 11.
LetG=1,2,3,4,5,6,7, and the definition of operation ∗ on G is shown in Table 11. We can easily verify that G satisfies the alternative law. Being each element in G has a neutral element and an opposite element; by Definition 16, G 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 a,b∈Ganda∗b=b∗a, we have G as a commutative AG-NET-Loop.
Figure4 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.Table 11
Table of Example11.
∗123456712541311253124223415323341234544534251556123456471234547Figure 4
The relationships among alternative AG-NET-Loop and other alternative quasi AG-NET-Loops.


## 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]).Figure 5
The main results of this paper.



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