In this article, some algebraic properties of the Wilson loop have been investigated in a broad manner. These properties include identities, autotopisms, and implications. We use some equivalent conditions to study the behavior of holomorphism of this loop. Under the shadow of this holomorphism, we are able to observe coincident loops.

1. Introduction and Definitions

A nonempty set is defined to be a groupoid relative to the operation if and only if to each ordered pair of elements of , there exists a uniquely defined element (or simply ). If is a fixed element, then there exist two single-valued mappings , of into itself by

An element is said to be unit of if for each , we have the identity law  =  = . A unit if exists is unique. An element is said to be left nonsingular if is a permutation of and similarly an element of is said to be right nonsingular if is a permutation of . A system is called a quasigroup provided each of , , and is a groupoid and

For all, . For a quasigroup , we can associate a permutation defined by . A quasigroup which admits identity law is called a loop. For each , the elements  = ,  =  such that  = ,  =  are called the right and left inverses of under right and left inverse permutations on , respectively. A loop satisfying Wilson identity in [1] is called Wilson loop. From Table 1 of Wilson loop, we have which shows 10 is the right inverse of 11 with right multiplicative identity 0. For a loop , the right and left inverses of an element of are different in general but right and left identities are always same. It is not easy to control this identity algebraically due to the fact that there are three variables appearing at the right and two at the left side. A loop with weak associativity is called Moufang loop for all . Nucleus of the loop is the intersection of the left , middle , and right nuclei of denoted by . An ordered triple of permutations of loop is in the group of autotopisms, , of if and only if . If these bijections are coincident, then is automorphism in group . To every loop with , there corresponds another set . If we define on such that for all , , and , then is a loop and called the holomorph of .

Inspired by the work on Moufang loop in [1], Goodaire and Robinson [2] introduced a new understanding of Wilson loop with the help of weak inverse property loop and conjugacy closed loop. It is shown in [1] that a Moufang loop is Wilson loop if and only if for all . In [3], the concept of holomorph of a loop is given by Bruck. An extension work on holomorphism of a loop is done in [4].

In the present paper, it is shown that we can obtain loop isotopes of Wilson loop under different autotopisms. In this way, we are able to give new forms of Wilson loops by opposite loops, and finally, we see conditions to prove that holomorph of a Wilson loop is a Wilson loop.

We finish this section by giving an example of a noncommutative and nonassociative Wilson loop of order 12 in the form of Table 1.

2. Results and Identities

A permutation of a loop is known as pseudoautotopism of if and only if there exists an element such that is an autotopism of . The element is called a companion of . The pseudoautotopisms form a group denoted by . The following proposition portrays that the pseudoautotopism is helpful to find a loop which is an isotope of the Wilson loop.

Proposition 1. Let be a Moufang loop with the property for all and with companion . Then, we have () as an autotopism of the Wilson loop .

Proof. Because with companion , soConsider  =  =  =  =  which shows that is also a companion of . By [1], is the Wilson loop, so  =  = . With the given property, for all . So, is a companion of , and in Moufang loop , for all is well-defined. Therefore,  =  (, , ) is an autotopism of the Wilson loop .

We are able to find a class of loop isotopes of the Wilson loop with respect to different natural numbers. A loop has the isotopy-isomorphy property provided each loop isotopic to is isomorphic to . Such a loop is called -loop, and in [1], it is proved that the Wilson loop is an example of the -loop.

Lemma 1. If is a Wilson loop and is the right pseudoautotopism on with companion , then is in the nucleus of as well as in the nucleus of the loop isotope of . Moreover, has companions (not all distinct) for natural number .

Proof. Let be isomorphic to a loop isotope under the isomorphism defined by for all where is a companion of . By [1], we have . But, in Wilson loop, three nuclei are coincident, so . Because is an identity of and is a subgroup of , . Since has companions as well as for all , therefore are the companions (not all distinct) of . Thus, has companions for natural number .

Note that the powers of companions play an important role as well as the left and the right inverse permutations are also important under an arbitrary autotopism.

Lemma 2. If is an autotopism of the Wilson loop , then (, ) and are also autotopisms on for all .

Proof. Let and be autotopisms on , so , , , .
Also, , , , . In [5], is also autotopism of . Therefore, we obtain . Hence, we have .

We have autotopisms in the form of the left and right translations for the special subclass and Wilson loops of exponents 2 without depending upon the given arbitrary autotopism.

Proposition 2. If is the Wilson loop of exponent 2, then the following are autotopisms on :(1)(2)(3)(4)

Proof. Wilson identity in the form of left inverse is and serves for every . Replacing by and by in the later identity  = , = . It implies that is an autotopism on Wilson loop . By using the same behavior, we can show that is another autotopism for this . Due to the exponent 2, and become identity mapping. Therefore, and are the autotopisms of . So, and are the members of . Equations (3) and (4) are trivial by definition of .

Theorem 1. Let be a Wilson loop and , , . The following identities hold:(1) = (2) = (3) = (4) = (5)  = (6)  = 

Proof. (1) In [2], it is shown that is conjugacy closed loop (CCL) and weak inverse property loop (WIPL). We have  =   =   =   =  =  and  = . Thus, in case of Wilson loop, we have  = .
For (3), since by CCL, we have  = ,  =  ,  = ,  =  ,  = . By WIPL,  = . Thus,  = .
For (5),  =  ,  =  ,  =  ,  =  . Finally,  = .

We have the following immediate corollary of the above theorem.

Corollary 1. Let be a commutative Wilson loop with . Then, for all , , , we have(1) = (2) = (3) = The Wilson loop is not ready to accept squaring property unless we select only those which are flexible. The next lemma indicates this notion.

Lemma 3. Let be the flexible Wilson loop. Then,  =  for all ,  L. Moreover,  =  and show that is an autotopism on .

Proof. In [2], the Wilson loop is an extra loop that satisfies the square property  = . So,  =  =  = . entertains the autotopism .

Lemma 4. If is a Wilson loop, then need not to be a Wilson loop.

Proof. Since is a Wilson loop, so for all , , , we have  =  =  = . It implies that , i.e., is not a Wilson loop.

Theorem 2. The homomorphic image of a Wilson loop is a Wilson loop.

Theorem 3. The direct product of Wilson loops is again a Wilson loop.

Wilson loop can be defined by another way of opposite loop in case of nonassociative finite invertible loop (NAFIL). Thus, we have the following consequence of the above discussion.

Theorem 4. Let be any NAFIL. A loop is a Wilson loop if and only if is a Wilson loop, where is opposite loop of . Thus, we have  =  for all , .

Proof. Let be a Wilson loop, so  = , for all , , and . ConsiderIt shows that is Wilson loop.
Conversely, now we suppose that is a Wilson loop,It completes the proof.

We finish this section with the following inclusions.

Lemma 5. Let be a Wilson loop. Then, the following statements hold:(1){Wilson loop}  {Osborn loop}(2){flexible Wilson loop}  {S-loop}(3){Wilson loop}  {automorphic inverse property loop (AIPL)}(4){extra loop}  {Wilson loop}

Proof. First two results are obvious from [2, 6]. For (3), the Wilson loop  the Osborn loop  the Moufang loop  inverse property loop (IPL)  cross inverse property loop (CIPL)  AIPL.

3. Holomorphism of Wilson Loop

In this section, we describe the conditions under which the holomorph of Wilson loop is Wilson loop. Here is the first result.

Lemma 6. If holomorph of a loop is Wilson loop, then is a Wilson loop.

Proof. Since is a normal subloop of , moreover is isomorphic to , so is a normal subloop of . Since is a Wilson loop, so does .

Theorem 5. If is a Wilson loop, then for all , , , , , and , , we have .

Proof. Necessary computations are given below

Thus, we have the following immediate corollary.

Corollary 1. The holomorph of a loop is a Wilson loop if and only if is a Wilson loop and is trivial group.

Lemma 7. If is a Wilson loop of a commutative loop , then  =  = .

Proof. With the aid of the last two results, loop is commutative . Therefore, .

Lemma 8. Let  =  and  =  be two loops and is a nonempty set. If and are isomorphic Wilson loops, then binary operations are coincident.

Proof. Isomorphism of and gives  =  for all , , , , and . Put , then we have  = ( )  =  for all , . Hence, we have two identical loops and .

Similarly, we state the following obvious result.

Lemma 9. Let be any loop with holomorph . is Wilson CIPL if and only if is Wilson CIPL and  = .

Theorem 6. Let be a loop with holomorph . is Wilson loop if and only if for all , , we have

Proof. For all , ,Given hypothesis isThus, equation (9) becomesWe summarize the above discussion in the following manner.

Theorem 7. Let be a loop with holomorph . Then, the following are equivalent:(1) is Wilson loop(2) is Wilson loop with trivial group (3) for all

Lemma 10. Let be the holomorph of the loop . Then, has right action on but not left.

Proof. We define  =  for all .For all , . Moreover,So, has right action on . Clearly, . has not left action on .

Remark 1. has both the left and right actions on Wilson loop .

Proposition 3. If has the Wilson loop, then , , where is the group of weak inverse permutations of .

Proof. Since is Wilson loop, so is trivial. By Wilson identity,  =  for all , , and . Put  = , then  = , is WIPL. is an automorphism of this loop by [5], therefore is an identity mapping because is trivial. To show , we verify  = .It implies that is the weak right inverse permutation, so . It can be shown easilyTherefore, . Similarly, we can prove .

Remark 2. Let be the NAFIL with trivial group  = . is Wilson loop if and only if is Wilson loop.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

Han Li wrote the final version of the paper and arranged funding for this paper, M. Nadeem proved the main results, M. A. Ali wrote the first version of the paper, and H. Mutee ur Rehman proposed the problem and supervised this work.