The Decomposition Theorems of AG-Neutrosophic Extended Triplet Loops and Strong AG-(l, l)-Loops

In this paper, some new properties of Abel Grassmann‘s Neutrosophic Extended Triplet Loop (AG-NET-Loop) were further studied. The following important results were proved: (1) an AG-NET-Loop is weakly commutative if, and only if, it is a commutative neutrosophic extended triplet (NETG); (2) every AG-NET-Loop is the disjoint union of its maximal subgroups. At the same time, the new notion of Abel Grassmann’s (l, l)-Loop (AG-(l, l)-Loop), which is the AbelGrassmann’s groupoid with the local left identity and local left inverse, were introduced. The strong AG-(l, l)-Loops were systematically analyzed, and the following decomposition theorem was proved: every strong AG-(l, l)-Loop is the disjoint union of its maximal sub-AG-groups.


Introduction
The theory of group and semigroup [1][2][3][4][5][6][7][8][9][10][11] are the basic abstract algebraic structure and they all have an associative binary relation. As a generalization of a commutative semigroup, the notion of an Abel Grassmann's groupoid was introduced by Kazim and Naseeruddin [12] in 1972 and this structure is known as the left almost semigroup (LA-semigroup). An AG-groupoid is a nonassociative algebraic structure and many features of the AG-groupoid can be studied in [13]. In [14][15][16][17][18][19][20][21], some properties and connections of AG-groupoid, with some classes of algebraic structures, have been investigated. An AG-groupoid is called an AG-group if the left identity and inverse exists, while further research on the AG-group can be found in [22].
As a mathematical tool for dealing with uncertain information, the fuzzy set and the intuitionistic fuzzy set theories are widely used in many fields, such as engineering technology and management science. For example, fuzzy sets can be applied in multi-criteria decision-making (MCDM), and the characteristic objects method (COMET) was developed to solve the problem of MCDM (more information about this topic can be found here: www.comet.edu.pl). As an extension of fuzzy set and intuitionistic fuzzy set, the new concept of neutrosophic logic and neutrosophic set was first proposed by Smarandache in [23], and some new theoretical studies were developed [24][25][26]. Moreover, the theory of neutrosophic set has been applied in many domains, such as [27], which applies neutrosophic set to the decision-making, proposed a new model for the selection of transport service providers, and the model was tested on a hypothetical example of the evaluation of five transport service providers.
For a neutrosophic set over the universe, let T, I, F be real functions from U to [0,1], an element x from U is noted with respect to (T(x), I(x), F(x)). Then T(x), I(x), F(x) are called neutrosophic components. Recently, the new concepts of the neutrosophic triplet group (NTG) and neutrosophic extended triplet group (NETG) were proposed by Smarandache and Ali in [28,29] as an application for neutrosophic sets. For a neutrosophic triplet group (N, *), for any a in N, having its own neutral element (denoted by neut(a)) and at least one opposite element (denoted by anti(a)) in N relative to neut(a) satisfying the condition: a*neut(a) = neut(a)*a = a, a*anti(a) = anti(a)*a = neut(a).
The contrast between the neutrosophic set and the neutrosophic triplet group are as shown in Figure 1. And in [30], sorts of general neutrosophic triplet structures were pointed out, and their basic properties were investigated. For the structure of NETG, some research papers are published with a series of results [31][32][33][34][35].
In [35], the concept of Abel Grassmann's neutrosophic extended triplet loop (AG-NET-loop) was introduced, which is both an AG-groupoid and a neutrosophic extended triplet loop (NET-loop). In this paper, we investigated the Abel Grassmann's neutrosophic extended triplet loop (AG-NET-Loop) further, and introduced the new concept of AG-(l, l)-Loop, which is defined as an Abel Grassmann's groupoid with the local left identity and local left inverse. We analyzed the decomposition theorems of AG-NET-Loop and AG-(l, l)-Loop. The differences between the contents and results of several related papers are described intuitively in Table 1. We also describe the development of groupoids and the relationship with AG-(l, l)-Loop, AG-NET-Loop, and NETG in Figure 2, where the symbol "A→B" means that "A includes B." Table 1. The differences between the contents and results of several related papers.

Papers
The Algebra Structures Involved The Associative Law is Satisfied or Not

Preliminaries
A groupoid (S, *) is called an Abel Grassmann's groupoid (AG-groupoid) [18,19] if it holds the left invertive law, that is, for all a, b, c ∈ S, (a*b)*c = (c*b)*a. In an AG-groupoid the medial law holds, for all a, b, c, d ∈ S, (a*b)*(c*d) = (a*c)*(b*d). An AG-groupoid (S, *) is called locally associative if it satisfies (a*a)*a = a*(a*a), for all a in S. If an AG-groupoid(S,*) with left identity, then it holds a*(b*c) = b*(a*c) and An AG-groupoid is a non-associative algebraic structure midway between a groupoid and a commutative semigroup, because if an AG-groupoid contains a right identity then it becomes a commutative semigroup. An AG-group is a generalization of the abelian group and a special case of quasigroup, which is not commutative or associative in general. But if one of them is allowed, an AGgroup becomes an abelian group.
Theorem 4. ( [19]) Let (N, *) be an AG-group. Then N has exactly one idempotent, which is the left identity.

Then, we call N a neutrosophic extended triplet set. Thus, a neutrosophic extend triplet is (a, neut(a), anti(a)), where neut(a) is extend neutral of "a"(not necessarily the identity element), and anti(a) is the opposite of "a."
In the following, we use the notations {neut(a)} and {anti(a)} to represent the sets of neut(a) and anti(a); we also use neut(a) and anti(a) to represent any certain one of neut(a) and anti(a).

Definition 4. ([32]) Assume that (N, *) is a NETG. When for all a, b∈ N, neut(a) * b = b * neut(a), N is said to be a weak commutative neutrosophic extended triplet group (WCNETG).
Proposition 2. ( [32]) Let (N, *) be a weak commutative NETG with respect to * and for any a, b ∈ N, Remark 1. In [30,32], the name of neutrosophic triplet loop is used. To be more rigorous with and echo the neutrosophic extended triplet group (NETG), the name of the neutrosophic extended triplet loop (NET-loop) is used in this paper.

∀x,y ∈ N, x ≈ y ⟺ neut(x) = neut(y)
Then (1) The binary ≈ is a congruence relation on N, and we denote the equivalent class contained x by [ ] ≈ ,

that is, every AG-NET-Loop is the disjoint union of its maximal subgroups.
Proof.

neut (c) = b, anti (c) = c; neut (d) = a, anti (d) = d; neut (e) = a, anti (e) = d. Let S is the set of neutral element "a" and H is the set of neutral element "b". Then S = {a,d,e} and H = {b,c}.
It is easy to verify that N = S∪H, both S and H are subgroups of N.
Therefore, neut(l, l) (a) is unique. Moreover, by (i) and (iii) we can get that p*a = x implies a*p=x.

Conclusions
We have investigated the structure and properties of NETG and AG-NET-Loop in [35]. In the paper, we studied the structure of the AG-NET-Loop further and introduced the AG-(l, l)-Loop (which is the Abel Grassmann's groupoid with local left identity and local left inverse), gave some infinite examples of them, and obtained some important results. We proved that every weak commutative AG-NET-Loop (or weak commutative AG-(l, l)-Loop) is commutative NETG (CNETG) and every AG-NET-loop is the disjoint union of its maximal subgroups. Moreover, we introduced the new notion of an AG-(l, l)-Loop and investigated the decomposition theorem of a strong AG-(l, l)-Loop. The main results of this paper are described in Figure 3. As the next research direction, we will explore the structure of the combination of the neutrosophic set, fuzzy set, soft set, and algebra systems (see [36][37][38]). Author Contributions: X.Z. and X.W. proposed the idea of the research, and X.W. wrote the paper.