Parastrophe of Some Inverse Properties in Quasigroups

Abstract

This work investigates the relationship between the parastrophes of some notion of inverses in quasigroups. Our findings reveal that, of the 5 parastrophes of LIP quasigroup, (23) parastrophe is a LIP quasigroup, (12) and (132) parastrophes are RIP quasigroups, while (13) and (132) parastrophes are an anti-commutative quasigroup. Similarly, the (12) and (132) parastrophes of a RIP quasigroup are LIP quasigroups; the (13) parastrophe of a RIP quasigroup is a RIP quasigroup, while the (23) and (123) parastrophes are anti-commutative quasigroups. As for the CIP quasigroup, only the (12) parastrophe is a CIP quasigroup; other parastrophes are symmetric quasigroups of order two. Finally, the (12) parastrophe of the WIP quasigroup is an IP quasigroup, the (13), (23), and (132) parastrophes of the WIP quasigroup are CIP quasigroups, while the (123) parastrophe of the WIP quasigroup is a WIP quasigroup.

1. Introduction

Quasigroup G is a set having a binary multiplication $x·y$ usually written as $xy$ that satisfies the condition that, for any $a,b$ in G, the equations $a·x=b$ and $y·a=b$ have unique solutions for $x,y\in G$. Suppose G is a non-empty set defined on a binary operation $(·)$ such that $x,y\in G$ for all $x,y$ in G; then, $(G,·)$ is called a groupoid. Alternatively, a quasigroup can be defined in terms of a translational map. For every x in G, define a mapping $R_x$ and $L_x$ of G into itself by $y{R}_x=y·x$ and $y{L}_x=x·y$. Then, $(G,·)$ is a quasigroup if and only if $R_x$ and $L_x$ are bijective for all x in G. The mappings $L_x$ and $R_x$ are called the left and right translation maps. If a quasigroup G contains an element e such that $e·x=x$ for x in G, then e is called the left identity element of G. Similarly, if $x·e=x$ for x in G, then e is called the right identity element. If G contains both left and right identity elements, then these elements must be the same, and G contains a (two-sided) identity element; G is therefore called a loop.

A quasigroup $(G,·,\setminus ,/)$ is a set G together with three binary operations $(·,\setminus ,/)$ such that

$$a·(a\setminus b)=b,(b/a)·a=b\forall a,b\in G.$$

$$a\setminus (a·b)=b,(b·a)/a=b,a\setminus a=b/b,\forall a,b\in G.$$

Standard classical references on quasigroups and loops that can be consulted for further reading are [1,2,3,4,5,6,7]. The study of parastrophes in quasigroups can be traced back to the works of Sade [8] and Artzy [9]. Jaiyeola [10] gave some necessary and sufficient conditions for the parastrophic invariance of associative law in quasigroups using the holomorph of the respective parastrophe of the quasigroup. In [11], the author established a connection between different pairs of conjugates (another name for parastrophe) and described all six possible conjugate sets with regard to the equality (“assembling”) of conjugates. In [12], Dudek studied the idempotent of k-translatable quasigroups and their parastrophes. Osoba et al. [13] gave an algebraic characterization of a generalized middle Bol loop using the concept of the parastrophe and holomorph of loops.

Recently, there has been a surge in the study of parastrophes of some inverse property quasigroups. For instance, ref. [14] found minimal identities that define CIP quasigroups by investigating the dependencies between the invertibility functions. Ref. [15] studied parastrophe orbits of (r,s,t)-inverse quasigroups in general, while [16] specifically studied (following [15]) parastrophe orbits of WIP quasigroups using a permutation arising from a constructed Cayley table of a WIP quasigroup.

Given a quasigroup $(G,·)$, there exist five other associated quasigroups, which are called parastrophes. In associative binary systems, the concept of an inverse element or inverse property is only meaningful if the system has an identity element. In a group, $a·a^{-1}=a^{-1}·a=e$. An inverse property (IP) quasigroup is a set G and a binary operation, where G contains an identity e such that $a·e=a=e·a$ for all $a\in G$ and where $x\in G$ has a two-sided inverse $x^{-1}$ such that for all $y\in G$.

$$x^{-1}·\left(xy\right)=y=\left(yx\right)·y^{-1}.$$

Such IP quasigroups are regarded as loops, which are not the focus of this study. The class of some inverse properties quasigroups formed the basis of this study investigating how parastrophes relate to some notions of inverses in quasigroups. The aim in this study was to answer the following: are the parastrophes of LIP, RIP, CIP, and WIP quasigroups parastrophically invariant? It is our belief that the results and techniques contained in this study offer insights into a similar investigation on m-inverse quasigroups and $(r,s,t)$-inverse quasigroups.

2. Basic Concepts

In this section, we give definitions of terminologies used throughout this paper and some previous results used in this work.

Definition 1. 

A groupoid is a non-empty set together with a binary operation $(G,·)$ for all $x,y\in G, x·y\in G$.

Definition 2. 

A groupoid $(G,·)$ is called a quasigroup if the maps $L\left(x\right):G\to G$ and $R\left(x\right):G\to G$ are bijections for all $x\in G$.

Definition 3. 

Let $(G,·)$ be a groupoid, and let a be any fixed element in G. Then, the translation maps $L\left(x\right)$ and $R\left(x\right)$ are defined as $yL\left(x\right)=x·y$ and $yR\left(x\right)=y·x$ for all $y\in G$.

Definition 4. 

A quasigroup $(G,·)$ is said to be of exponent two if, for all $x\in G$, we have $x^2=e$; that is, $x^{-1}=x$.

Definition 5. 

A quasigroup $(G,·)$ is a LIP quasigroup if there exists a bijection $J_{\lambda }:a\to a^{\lambda }$ on G such that $a^{\lambda }(a·x)=x$ for every $x\in G$.

Definition 6 

([4]). A quasigroup $(G,·)$ has a right inverse property (RIP) if there exists an a bijection $J_{\rho }:a⟶a^{\rho }$ on G such that

$$(x·a)a^{\rho }=x$$

for every $x\in G$

Parastrophe of Quasigroups (Loops)

Let $(G,·)$ be a quasigroup. If, given any two of $x,y,z$ as elements in G, the third can be uniquely obtained in G so that, if

$$x·y=z,$$

we have the left and right divisors $y=x\setminus z$ and $x=z/y$. The binary product $x·y=z$ can be expressed in six ways by permuting the order in which the symbols appear.

Some authors used functional notation for operations on set G; instead of writing $a·b=c$, one writes $F(a,b)=c$. In this case, the quasigroup $(G,·)$ is denoted by $G\left(F\right)$. For operations $(\setminus )$ and (/), one uses symbols $F^{-1}$ and ⁻¹F, i.e., if $F(a,b)=c$, then $F^{-1}(c,b)=a$ and ⁻¹F$(a,c)=b$. One can now determine three other conjugate operations on G associated with the operation F, namely ⁻¹(F⁻¹), (⁻¹F)⁻¹ and (⁻¹(F⁻¹))⁻¹. The six conjugate quasigroups F, F⁻¹,⁻¹F, ⁻¹(F⁻¹), (⁻¹(F⁻¹)⁻¹ and (⁻¹(F⁻¹))⁻¹ are called parastrophe.

According to Pflugfelder [4], we can obtain new quasigroups from existing quasigroups as noted in a 3-web. If in a 3-web, one permutes 3 pencils, and a new 3-web is produced, which in turn gives rise to new quasigroups. If, for instance, the permutation

$$\pi =\left(\begin{array}{ccc}1\hfill & 2\hfill & 3\hfill \\ 2\hfill & 3\hfill & 1\hfill \end{array}\right)$$

is performed on a 3-web W, and if, as a result, the lines are mapped so that

$$w_1⟷w_2^{\prime },w_2⟼w_3^{\prime },w_3\mapsto w_1^1$$

for all $W\in \sum$, then the quasigroup $(G,\circ )$ in which one has, say, $x\circ z=y$, goes into a quasigroup $(G,\ast )$, in which $y\ast x=z$. A quasigroup produced in this way is called a parastrophe and, in particular, a $\pi$ parastrophe if it is based on the permutation $\pi$. $(G,\ast )$ is said to be $\pi$-parastrophic to $(G,\circ )$. Parastrophes of quasigroups have been studied in different contexts by different authors, among which are [8,9,17,18,19,20,21,22].

The following is obvious in view of the existence of 6 permutations of 3 pencils.

Theorem 1 

([4]). There are 6 quasigroups parastrophic to every quasigroup.

Definition 7 

([4]). The operation in the π parastrophe of the quasigroup $(G,·)$ is denoted by $\left(\pi \right)$, i.e., we write $x\left(\pi \right)z$ instead of $x\circ z$.

If the operation $(·)$ in $(G,·)$ is denoted by F and the operation in the $\pi$ parastrophe is denoted by $\left({\pi }_i\right),i=1,2,3,4,5,6$, then the correspondence is as follows:

$$\begin{array}{cc}& \left({\pi }_1\right)=\left(\begin{array}{ccc}1\hfill & 2\hfill & 3\hfill \\ 1\hfill & 2\hfill & 3\hfill \end{array}\right)=F\mathrm{and}x\left({\pi }_1\right)z=y=F(x,z)\hfill \\ & \left({\overline{\pi }}_2\right)=\left(\begin{array}{ccc}1\hfill & 2\hfill & 3\hfill \\ 2\hfill & 1\hfill & 3\hfill \end{array}\right)=F^{-1}\mathrm{and}y\left({\pi }_2\right)z=x=F^{-1}(y,z)\hfill \\ & \left({\pi }_3\right)=\left(\begin{array}{ccc}1\hfill & 2\hfill & 3\hfill \\ 1\hfill & 3\hfill & 2\hfill \end{array}\right){=}^{-1}F\mathrm{and}x\left({\pi }_3\right)y=z{=}^{-1}F(x,y)\hfill \\ & \left({\pi }_4\right)=\left(\begin{array}{ccc}1\hfill & 2\hfill & 3\hfill \\ 2\hfill & 3\hfill & 1\hfill \end{array}\right){=}^{-1}\left(F^{-1}\right)\mathrm{and}y\left({\pi }_4\right)x=z=-1{(}^{-1})(y,x)\hfill \\ & \left({\pi }_5\right)=\left(\begin{array}{ccc}1\hfill & 2\hfill & 3\hfill \\ 3\hfill & 1\hfill & 2\hfill \end{array}\right)={{(}^{-1}F)}^{-1}\mathrm{and}z\left({\pi }_5\right)y=x={{(}^{-1}F)}^{-1}(z,y)\hfill \\ & \left({\pi }_6\right)=\left(\begin{array}{ccc}1\hfill & 2\hfill & 3\hfill \\ 3\hfill & 2\hfill & 1\hfill \end{array}\right)={{(}^{-1}\left(F^{-1}\right))}^{-1}\mathrm{and}z\left({\pi }_6\right)x=y=\left({{[}^{-1}\left(F^{-1}\right))}^{-1}(z,x)\right.\hfill \end{array}$$

Remark 1. 

If $(G,·)$ is a quasigroup, its conjugates or parastrophes are also quasigroups.

Definition 8 

([4]). A quasigroup $(G,·)$ has a left inverse property (LIP) if there exists a bijection $J_{\lambda }:a\to a^{\lambda }$ on G such that

$$a^{\lambda }·(a·x)=x$$

for every $x\in G$.

Definition 9 

([4]). A quasigroup $(G,·)$ has a right inverse property (RIP) if there exists a bijection $J_{\rho }:a⟶a^{\rho }$ on G such that

$$(x·a)·a^{\rho }=x$$

for every $x\in G$.

Theorem 2 

([4]). If $(G,·)$ is a LIP or a RIP quasigroup, then $J_{\lambda }=J_{\rho }=J(i.e{a}^{\lambda }=a^{\rho }=a^{-1}, wherea·a^{-1}=a^{-1}·a=e)$.

Definition 10 

([4]). A quasigroup $(G,·)$ is called a cross-inverse property (CIP) quasigroup if any two elements $x,y\in L$ satisfy the relation

$$\begin{array}{ccc}\hfill xy·x^{\rho }=& y\hfill & \end{array}$$

(1)

$$\begin{array}{ccc}\hfill x·\left(y{x}^{\rho }\right)=& y\hfill & \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\left(xy\right)}^{\rho }=& x^{\rho }{y}^{\rho }\hfill \end{array}$$

(3)

Definition 11. 

A quasigroup $(G,·)$ is called a weak inverse property (WIP) quasigroup if it satisfies the identical relation

$$\begin{array}{cc}\hfill y{\left(xy\right)}^{\rho }=& x^{\rho }\hfill \end{array}$$

(4)

Definition 12. 

A quasigroup $(G,·)$ is said to be anti-commutative if it satisfies any of the following identities:

$$\begin{array}{ccc}\hfill \left(xy\right)=& {\left(yx\right)}^{\rho }\hfill & \end{array}$$

(5)

$$\begin{array}{cc}\hfill \left(xy\right)=& {\left(yx\right)}^{\lambda }\hfill \end{array}$$

(6)

Definition 13. 

A quasigroup $(G,·)$ is said to be symmetric of order 2 if it satisfies the following identities:

$$\begin{array}{ccc}\hfill yx·y=& x\hfill & \end{array}$$

(7)

$$\begin{array}{cc}\hfill x=& x^{\lambda }\mathrm{or}x=x^{\rho }\hfill \end{array}$$

(8)

Theorem 3 

([4]). Every CIP quasigroup has a WIP.

Theorem 4 

([4]). Let $(G,·)$ be a quasigroup, then the following statements are equivalent:

(i) $(G,·)$ is a WIP quasigroup;

(ii) The relation $xy·z=e$ implies $x·yz=e$;

(iii) $(G,·)$ satisfies the identical relation.

$${\left(xy\right)}^{\lambda }·x=y^{\lambda }$$

(9)

3. Results

3.1. Parastrophes of Left Inverse Property (LIP) Quasigroups

Theorem 5. 

Let G be a left inverse property (LIP) quasigroup. Then, the (12) parastrophes of G are a right inverse property (RIP) quasigroup if $a=a^{\rho }$.

Proof. 

$a^{\lambda }\left(ax\right)=x$. Let $ax=p$ then, $a^{\lambda }p=x\stackrel{\left(12\right)}{\Rightarrow }$

$$p{a}^{\lambda }=x.$$

(10)

Also, $ax=p\stackrel{\left(12\right)}{\Rightarrow }$

$$xa=p$$

(11)

Substituting Equation (11) into Equation (10), $p·a^{\lambda }=x\Rightarrow xa·a^{\lambda }=x$. Now, set $a=a^{\rho }$ to obtain $x{a}^{\rho }·a=x$. Interchange the roles of a and $a^{\rho }$ to obtain $xa·a^{\rho }=x$. Thus, the (12) parastrophes of a left inverse property (LIP) quasigroup are a right inverse property (RIP) quasigroup. □

Theorem 6. 

Let G be a left inverse property (LIP) quasigroup; then, the (23) parastrophe of G has a left inverse property (LIP) if $a=a^{\lambda }$.

Proof. 

Suppose $a^{\lambda }\left(ax\right)=x$ and let $ax=p$ then, $a^{\lambda }p=x\stackrel{\left(23\right)}{\Rightarrow }$

$$a^{\lambda }x=p$$

(12)

If $ax=p\stackrel{\left(23\right)}{\Rightarrow }ap=x$

$$p=a\setminus x$$

(13)

Substituting (13) into (12), $a^{\lambda }x=p\Rightarrow a^{\lambda }x=a\setminus x$. Thus, $x=a·a^{\lambda }x$. Interchanging the roles of a and $a^{\lambda }$ obtains $x=a^{\lambda }·ax$. Therefore, the (23) parastrophe of the LIP quasigroup is a LIP quasigroup. □

Theorem 7. 

Let G be a left inverse property (LIP) quasigroup. Then, the (13) parastrophe of G is anti-commutative.

Proof. 

Suppose $a^{\lambda }\left(ax\right)=x$, and let $ax=p$. Then, $a^{\lambda }p=x\stackrel{\left(13\right)}{\Rightarrow }$

$$xp=a^{\lambda }$$

(14)

If $ax=p\stackrel{\left(13\right)}{\Rightarrow }$ then, $px=a$

$$p=a/x$$

(15)

Substituting (15) into (14) yields

$$xp=a^{\lambda }\Rightarrow x·a/x=a^{\lambda }.$$

Let

$$\overline{a}=a/x\Rightarrow \overline{a}x=a$$

$$x\overline{a}={\left(\overline{a}x\right)}^{\lambda }.$$

Thus, the (13) parastrophe of the LIP quasigroup is anti-commutative. □

Theorem 8. 

For a left inverse property (LIP) quasigroup, the (123) parastrophe of G has a right inverse property if $a=a^{\rho }$.

Proof. 

Suppose $a^{\lambda }\left(ax\right)=x$, and let $ax=p$. Then, $a^{\lambda }p=x\stackrel{\left(123\right)}{\Rightarrow }$

$$x{a}^{\lambda }=p$$

(16)

If $ax=p\stackrel{\left(123\right)}{\Rightarrow }pa=x\Rightarrow$

$$p=x/a.$$

(17)

Substituting (17) into (16), $x{a}^{\lambda }=p\Rightarrow x{a}^{\lambda }=x/a\Rightarrow x{a}^{\lambda }·a=x$. Set $a=a^{\rho }$; then, $xa·a^{\rho }=x$. Thus, the (123) parastrophe of a LIP quasigroup is a RIP quasigroup. □

Theorem 9. 

Let G be a left inverse property (LIP) quasigroup. Then, the (132) parastrophe of G is anti-commutative.

Proof. 

$a^{\lambda }\left(ax\right)=x$, and let $ax=p$. Then, $a^{\lambda }p=x\stackrel{\left(132\right)}{\Rightarrow }$

$$px=a^{\lambda }$$

(18)

Let $ax=p\stackrel{\left(132\right)}{\Rightarrow }xp=a$

$$p=x/a$$

(19)

Substituting (19) into (18), $px=a^{\lambda }\Rightarrow (x\setminus a)·x=a^{\lambda }$. Set $\overline{a}=x\setminus a\Rightarrow x\overline{a}=a\Rightarrow \overline{a}x={\left(x\overline{a}\right)}^{\lambda }$. Thus, the result follows. □

Remark 2. 

The (23) parastrophe is the only parastrophically invariant among the parastrophes of the LIP quasigroup. The (13) and (132) parastrophes are anti-commutative, while the (12) and (123) parastrophes are RIP.

Example 1. 

We give the following example to illustrate the fact that the (23) parastrophe of a LIP is a LIP (Table 1).

Table 1. A left inverse property quasigroup.

·	1	2	3	4	5
1	1	2	3	4	5
2	2	1	4	5	3
3	3	5	1	2	4
4	4	3	5	1	2
5	5	4	2	3	1

The (23)-parastrophe in the above table is given below, and it can be seen that it is LIP (Table 2):

Table 2. The (23) parastrophe of a left inverse property quasigroup.

•	1	2	3	4	5
1	1	2	3	4	5
2	2	1	5	3	4
3	3	4	1	5	2
4	4	5	2	1	3
5	5	3	4	2	1

3.2. Parastrophes of Right Inverse Property (RIP) Quasigroups

Theorem 10. 

Let G be a right Inverse property (RIP) quasigroup. Then, the (12) parastrophe of G is a left inverse property (LIP) quasigroup.

Proof. 

Suppose $\left(xa\right)·a^{\rho }=x$, and let $xa=q$. Then, $q{a}^{\rho }=x\stackrel{\left(12\right)}{\Rightarrow }$

$$a^{\rho }q=x$$

(20)

If $xa=q$, $\stackrel{\left(12\right)}{\Rightarrow }$

$$ax=q$$

(21)

Substituting (21) into (20),

$$a^{\rho }·q=x\Rightarrow a^{\rho }·\left(ax\right)=x.$$

Set $a=a^{\lambda }$; then, ${\left(a^{\lambda }\right)}^{\rho }·a^{\lambda }x=x\Rightarrow a·a^{\lambda }x=x$. Interchanging the roles of a and $a^{\lambda }$ obtains $a^{\lambda }·ax=x$. Thus, the (12) parastrophe of a right inverse property (RIP) quasigroup is a left inverse property (LIP) quasigroup. □

Theorem 11. 

Let G be a right inverse property (RIP) quasigroup. Then, the (23) parastrophe of G is anti-commutative.

Proof. 

Suppose $\left(xa\right)·a^{\rho }=x$, and let $xa=q$. Then, $q·a^{\rho }=x\stackrel{\left(23\right)}{\Rightarrow }$

$$qx=a^{\rho }$$

(22)

If $xa=q\stackrel{\left(23\right)}{\Rightarrow }xq=a$

$$q=x\setminus a$$

(23)

Substituting (23) into (22), $q·x=a^{\rho }\Rightarrow (x\setminus a)·x=a^{\rho }$. Set $\overline{a}=x\setminus a$; then, $x\overline{a}=a\Rightarrow \overline{a}x={\left(x\overline{a}\right)}^p$. Thus, the (23) parastrophe of a RIP quasigroup is anti-commutative. □

Theorem 12. 

Let G be a right inverse property (RIP) quasigroup; then, the (13) parastrophe of G is a right inverse property quasigroup.

Proof. 

Suppose $\left(xa\right)·a^{\rho }=x$, and let $\left(xa\right)=q$. Then, $q{a}^{\rho }=x\stackrel{\left(13\right)}{\Rightarrow }$

$$x{a}^{\rho }=q$$

(24)

Let $xa=q\stackrel{\left(13\right)}{\Rightarrow }qa=x$

$$q=x/a$$

(25)

Substituting (25) into (24),

$$x{a}^{\rho }=q\Rightarrow x{a}^{\rho }=x/a\Rightarrow x{a}^{\rho }·a=x.$$

Interchanging the roles of a and $a^p$ obtains $xa·a^{\rho }=x$. Thus, the (13) parastrophe of a RIP quasigroup is a RIP quasigroup. □

Theorem 13. 

Let G be a right inverse property (RIP) quasigroup. Then, the (123) parastrophe of G is anti-commutative.

Proof. 

Suppose $\left(xa\right)·a^{\rho }=x$ and let $xa=q$.

Then, $q·a^{\rho }=x\stackrel{\left(123\right)}{\Rightarrow }$

$$x·q=a^{\rho }$$

(26)

If $\left(xa\right)=q\stackrel{\left(123\right)}{\Rightarrow },qx=a$

$$q=a/x$$

(27)

Substituting (27) into (26), $x·q=a^{\rho }\Rightarrow x·a/x=a^{\rho }$. Set $\overline{a}=a/x$; then, $\overline{a}x=a\Rightarrow x\overline{a}={\left(\overline{a}x\right)}^{\rho }$. Thus, the $\left(123\right)$ parastrophe of G is anti-commutative. □

Theorem 14. 

Let G be a right inverse property (RIP) quasigroup; then, the (132)- parastrophe of G is a right inverse property quasigroup. However, if $a=a^{\lambda }$, then the (132) parastrophe of G is a left inverse property quasigroup.

Proof. 

Suppose $\left(xa\right)a^{\rho }=x$, and let $xa=q$. Then, $q{a}^{\rho }=x\stackrel{\left(132\right)}{\Rightarrow }$

$$a^{\rho }x=q.$$

(28)

If $xa=q\stackrel{\left(132\right)}{\Rightarrow }a·q=x$

$$q=a\setminus x$$

(29)

Substituting (29) into (28), $a^{\rho }x=q\Rightarrow a^{\rho }x=a\setminus x\Rightarrow a·a^{\rho }x=x$. Set $a=a^{\lambda }$ then, $a^{\lambda }·ax=x$. Thus, the (132) parastrophe of a RIP quasigroup is a LIP quasigroup. □

Remark 3. 

The (13) parastrophe is parastrophically invariant among the parastrophes of a RIP quasigroup. The (12) and (132) parastrophes are LIP, while the (23) and (123) parastrophes are anti-commutative.

3.3. Parastrophes of Cross-Inverse Property (CIP) Quasigroup

Theorem 15. 

Let G be a cross-inverse property (CIP) quasigroup. Then, the (12) parastrophe of G is also a CIP quasigroup.

Proof. 

Suppose $\left(xy\right)·x^{\rho }=y$, and let $xy=p$. Then, $p·x^{\rho }=y\stackrel{\left(12\right)}{\Rightarrow }$

$$x^{\rho }·p=y$$

(30)

If $xy=p\stackrel{\left(12\right)}{\Rightarrow }$

$$yx=p$$

(31)

Substituting (31) into (30), we obtain $x^{\rho }·\left(yx\right)=y$. Set $x⟷x^{\rho }$ and then $x·y{x}^{\rho }=y$. Thus, the (12) parastrophe of a CIP quasigroup is a CIP quasigroup. □

Theorem 16. 

Let G be a cross-inverse property quasigroup. Then, the (23) parastrophe of G is a symmetric quasigroup of order 2.

Proof. 

Suppose $\left(xy\right)·x^{\rho }=y$, and let $xy=p$

$P·x^{\rho }=y\stackrel{\left(23\right)}{\Rightarrow }$

$$p·y=x^p$$

(32)

Let $xy=p\stackrel{\left(23\right)}{\Rightarrow }$$xp=y$

$$p=x\setminus y$$

(33)

Substituting (33) into (32), we obtain $(x\setminus y)·y=x^p$. Set $q=x\setminus y\Rightarrow xq=y$,

$$q·\left(xq\right)=x^{\rho }$$

(34)

Set $q=e$ in (34) to obtain $x=x^p$. Also, set $x=e$ in (34) to obtain $q^2=e$. If we put $x=x^{\rho }$ in (34), then $q·xq=x$ □

Theorem 17. 

Let G be a cross-inverse property (CIP) quasigroup. Then, the (13) parastrophe of G is a symmetric quasigroup of order 2.

Proof. 

Suppose $\left(xy\right)·x^{\rho }=y$, and let $xy=p$. Then, $p·x^{\rho }=y\stackrel{\left(13\right)}{\Rightarrow }$

$$y·x^{\rho }=p$$

(35)

If $xy=p\stackrel{\left(13\right)}{\Rightarrow }py=x$

$$p=x/y$$

(36)

Substituting (36) into (35), we obtain $y·x^{\rho }=x/y$.

$$\left(y{x}^{\rho }\right)·y=x$$

(37)

Set $y=e$ in (37) to obtain $x^{\rho }=x$. Also, set $x=e$ in (37) to obtain $y^2=e$. If we put $x^{\rho }=x$ in (37), then $yx·y=x$. Thus, the (13) parastrophe of a CIP quasigroup is a symmetric quasigroup of order 2. □

Theorem 18. 

Let G be a cross-inverse property (CIP) quasigroup. Then, the (123) parastrophe of G is a symmetric quasigroup of order 2.

Proof. 

Suppose $\left(xy\right)·x^{\rho }=y$, and let $xy=p$. Then, $p·x^{\rho }=y\stackrel{\left(123\right)}{\Rightarrow }$

$$p·y=x^{\rho }$$

(38)

If $xy=p\stackrel{\left(123\right)}{\Rightarrow }px=y$, then

$$p=y/x$$

(39)

Substituting (39) into (38), we obtain $y·y/x=x^{\rho }$. Setting $q=y/x\Rightarrow qx=y$

$$\left(qx\right)·q=x^{\rho }$$

(40)

Set $q=e$ in (40) to obtain $x=x^{\rho }$. Also, set $x=e$ in (40) to obtain $q^2=e$. If we put $x=x^{\rho }$ in (40), then $qx·q=x$. Thus, the (123) parastrophe of a CIP quasigroup is a symmetric quasigroup of order 2. □

Theorem 19. 

Let G be a cross-inverse property (CIP) quasigroup. Then, the (132) parastrophe of G is a symmetric quasigroup of order 2.

Proof. 

Suppose $\left(xy\right)·x^{\rho }=y$, and let $xy=p$. Then, $p·x^{\rho }=y\stackrel{\left(132\right)}{\Rightarrow }$

$$x^p·y=p$$

(41)

If $xy=p\stackrel{\left(132\right)}{\Rightarrow }yp=x$, then

$$p=y\setminus x$$

(42)

Substituting (42) into (41), we obtain $x^{\rho }·y=y\setminus x$.

$$x=y·x^{\rho }y$$

(43)

Set $y=e$ in (43) to obtain $x=x^{\rho }$. Also, set $x=e$ in (43) to obtain $y^2=e$. If we put $x=x^{\rho }$ in (43), then $x=y·xy$. Thus, the (132) parastrophe of a CIP quasigroup is a symmetric quasigroup of order 2. □

Remark 4. 

All the parastrophes of cross-inverse property quasigroups are symmetric quasigroups of order 2 except the (12) parastrophe, which is parastrophically invariant.

3.4. Parastrophes of Weak Inverse Property (WIP) Quasigroup

Theorem 20. 

Let G be a weak inverse property (WIP) quasigroup. Then, the (12) parastrophe of G is a WIP quasigroup.

Proof. 

Suppose $x·{\left(yx\right)}^{\rho }=y^{\rho }\stackrel{\left(12\right)}{\Rightarrow }$

$${\left(xy\right)}^{\rho }·x=y^{\rho }$$

(44)

Set $x=e$ in (44) to obtain $y^{\rho }·e=y^{\rho }$. Also, set $y=e$ in (44) to obtain $x^{\rho }·x=e$. Thus, the (12) parastrophe of WIP quasigroup is a WIP quasigroup. □

Theorem 21. 

Let G be a weak inverse property (WIP) quasigroup. Then, the (13) parastrophe of a WIPQ is a CIPQ.

Proof. 

$x·{\left(yx\right)}^{\rho }=y^{\rho }\stackrel{\left(13\right)}{\Rightarrow }$

$$y^{\rho }·{\left(yx\right)}^{\rho }=x$$

Let ${\left(yx\right)}^{\rho }=p$

$$y^{\rho }·p=x$$

(45)

Since ${\left(yx\right)}^{\rho }=p$, $yx=p^{\lambda }\stackrel{\left(13\right)}{\Rightarrow }$ $p^{\lambda }x=y\Rightarrow p^{\lambda }=y/x$

$$p={(y/x)}^{\rho }$$

(46)

Substituting (46) into (45), we obtain $y^{\rho }·{(y/x)}^{\rho }=x$. If $q=y/x\Rightarrow qx=y$

$${\left(qx\right)}^{\rho }·q^{\rho }=x$$

(47)

Set $q=e$ in (47); then, $x^{\rho }=x$. From ${\left(xy\right)}^{\lambda }·x=y^{\lambda }\stackrel{\left(13\right)}{\Rightarrow }{y}^{\lambda }·x={\left(xy\right)}^{\lambda }$, Let $xy=p$ and then $y^{\lambda }·x=p^{\lambda }$ From $xy=p\stackrel{\left(13\right)}{\Rightarrow }py=x$

$$p=x/y$$

$$y^{\lambda }·x={(x/y)}^{\lambda }$$

Set $q=x/y\Rightarrow qy=x$; thus, $y^{\lambda }·qy=q^{\lambda }$. Set $q^{\lambda }=q$ to obtain $y^{\lambda }·qy=q$. Also, set $y=y^{\rho }$ to obtain ${\left(y^{\rho }\right)}^{\lambda }·q{y}^{\rho }=q$. Therefore, $y·q{y}^{\rho }=q$. Thus, the (13) parastrophe of a WIPQ is a CIPQ. □

Theorem 22. 

Let G be a weak inverse property (WIP) quasigroup. Then, the (123) parastrophe of a WIPQ is a WIPQ.

Proof. 

Suppose $y·{\left(xy\right)}^{\rho }=x^{\rho }\stackrel{\left(123\right)}{\Rightarrow }$ $x^{\rho }·y={\left(xy\right)}^{\rho }$

If $xy=p$

$$x^{\rho }·y=p^{\rho }$$

(48)

$xy=p\stackrel{\left(123\right)}{\Rightarrow }$, $px=y$

$$p=y/x$$

(49)

Substituting (49) into (49), we obtain $x^{\rho }·y={(y/x)}^{\rho }$

$${\left(x^{\rho }y\right)}^{\lambda }·x=y$$

(50)

Set $x=e$ in (50) to obtain $y^{\lambda }=y$. Also, set $y=e$ in (50) to obtain $x^2=e$. Now, upon setting $x=x^{\lambda }$ in (50), we have ${\left({\left(x^{\lambda }\right)}^{\rho }y\right)}^{\lambda }·x^{\lambda }=y$. ${\left(xy\right)}^{\lambda }·x^{\lambda }=y\Rightarrow {\left(xy\right)}^{\lambda }·x=y^{\lambda }$. Thus, the (123) parastrophe of a WIPQ is a WIPQ. □

Theorem 23. 

Let G be a weak inverse property (WIP) quasigroup. Then, the (132) parastrophe of a WIPQ is a CIPQ.

Proof. 

$x·{\left(yx\right)}^{\rho }=y^{\rho }\stackrel{\left(132\right)}{\Rightarrow }{\left(yx\right)}^{\rho }·y^{\rho }=x$.

Let ${\left(yx\right)}^{\rho }=p$,

$$p·y^{\rho }=x$$

(51)

${\left(yx\right)}^{\rho }=p$. Then, $yx=p^{\lambda }\stackrel{\left(132\right)}{\Rightarrow }x{p}^{\lambda }=y$, $p^{\lambda }=x\setminus y$,

$$p={(x\setminus y)}^{\rho }.$$

(52)

Substituting (52) into (51), we obtain ${(x\setminus y)}^{\rho }·y^{\rho }=x$. If $q=x\setminus y\Rightarrow xq=y$,

$$q^{\rho }·{\left(xq\right)}^{\rho }=x.$$

(53)

Setting $q=e$ in (53) implies $x^{\rho }=x$. Also, setting $x=e$ in (53) gives $q^{\rho }·q^{\rho }=e=q^{2\rho }$.

From ${\left(xy\right)}^{\lambda }·x=y^{\lambda }\stackrel{\left(132\right)}{\Rightarrow }$

$$x·y^{\lambda }={\left(xy\right)}^{\lambda }.$$

(54)

Let $x·y^{\lambda }=p^{\lambda }$. Then, $xy=p\stackrel{\left(132\right)}{\Rightarrow }yp=x$,

$$p=y\setminus x.$$

(55)

Substituting (55) into (54), we obtain $x·y^{\lambda }=p^{\lambda }$.

$$x·y^{\lambda }={(y\setminus x)}^{\lambda }.$$

Set $q=y\setminus x\Rightarrow yq=x,$

$$yq·y^{\lambda }=q^{\lambda }.$$

(56)

From (56), $y^{\lambda }=y^{\rho }$, $yq·y^{\rho }=q^{\lambda }$. Also, $q=q^{\lambda }\Rightarrow yq·y^{\rho }=q$. Thus, the (132) parastrophe of a WIPQ is a CIPQ. □

Remark 5. 

The parastrophe of a WIPQ is either a WIPQ or a CIPQ, and, since every CIPQ is a WIPQ, the WIP is parastrophically invariant property of a quasigroup.

Example 2 

([23], p. 205). Consider the WIPQ $(G,·)$, $G=\{1,2,3,4,5,6,7\}$ (obtained from totally symmetric quasigroup), whose multiplication table is given in Table 3. The (132) parastrophe presented in Table 4 is a CIP quasigroup.

Table 3. A weak inverse property quasigroup.

∘	1	2	3	4	5	6	7
1	1	3	2	5	4	7	6
2	3	2	1	6	7	4	5
3	2	1	3	7	6	5	4
4	5	6	7	4	1	2	3
5	4	7	6	1	5	3	2
6	7	4	5	2	3	6	1
7	6	5	4	3	2	1	7

Table 4. The (132) parastrophe in Table 3.

∘	1	2	3	4	5	6	7
1	1	3	2	5	4	7	6
2	3	2	1	6	7	4	5
3	2	1	3	7	6	5	4
4	5	6	7	4	1	2	3
5	4	7	6	1	5	3	2
6	7	4	5	2	3	6	1
7	6	5	4	3	2	1	7

4. Discussion

This study examines the parastrophes of some notion of inverses in quasigroups. Our results showed that, of the five parastrophes in the LIP quasigroup, the (23) parastrophe is a LIP quasigroup, (12) and (132) parastrophes are RIP quasigroups, while (13) and (132) parastrophes are an anti-commutative quasigroup. Similarly, the (12) and (132) parastrophes of a RIP quasigroup are a LIP quasigroup, the (13) parastrophe of a RIP is a RIP quasigroup, while the (23) and (123) parastrophes are an anti-commutative quasigroup. As for the CIP quasigroup, only the (12) parastrophe is a CIP quasigroup;the other parastrophes are symmetric quasigroups of order two. Finally, the (12) parastrophe of a WIP quasigroup is a CIP quasigroup; the (13), (23), and (132) parastrophes of a WIP quasigroup are CIP quasigroups, while the (123) parastrophe of a WIP quasigroup is a WIP quasigroup.

5. Conclusions

As for the parastrophes of avLIP quasigroup, the (23) parastrophe is the only parastrophically invariant among the parastrophes of the LIP quasigroup. The (13) and (132) parastrophes are anti-commutative, while the (12) and (123) parastrophes are RIP. The (13) parastrophe is parastrophically invariant among the parastrophes of the RIP quasigroup. The (12) and (132) parastrophes are LIP, while the (23) and (123) parastrophes are anti-commutative. All the parastrophes of cross-inverse property quasigroups are symmetric quasigroups of order two, except the (12) parastrophe, which is invariant. The parastrophe of WIPQ is either a WIPQ or a CIPQ, and, since every CIPQ is a WIPQ, a WIP is a parastrophically invariant property of a quasigroup.