# Parastrophes and Cosets of Soft Quasigroups 1112020 Mathematics Subject
Classification. Primary 20N05, 03E72; Secondary 03E75 ††thanks: Keywords and
Phrases : soft set, quasigroup, soft quasigroup, soft loop, left(right) coset,
quotient of soft quasigroup, parastrophes

Anthony Oyem
Department of Mathematics,
University of Lagos,
Akoka 100213 Nigeria.
<EMAIL_ADDRESS>All correspondence to be addressed to this author.
Tèmítọ́pẹ́ Gbọ́láhàn Jaiyéọlá
Department of Mathematics,
Obafemi Awolowo University,
Ile-Ife 220005, Nigeria.
<EMAIL_ADDRESS>
###### Abstract

This paper introduced the concept of soft quasigroup, its parastrophes, soft
nuclei, left (right) coset, distributive soft quasigroups and normal soft
quasigroups. Necessary and sufficient conditions for a soft set over a
quasigroup (loop) to be a soft quasigroup (loop) were established. It was
proved that a soft set over a group is a soft group if and only if it is a
soft loop or either of two of its parastrophes is a soft groupoid. For a
finite quasigroup, it was shown that the orders (arithmetic and geometric
means) of the soft quasigroup over it and its parastrophes are equal. It was
also proved that if a soft quasigroup is distributive, then all its
parastrophes are distributive, idempotent and flexible soft quasigroups. For a
distributive soft quasigroup, it was shown that its left and right cosets form
families of distributive soft quasigroups that are isomorphic. If in addition,
a soft quasigroup is normal, then its left and right cosets forms families of
normal soft quasigroups. On another hand, it was found that if a soft
quasigroup is a normal and distributive soft quasigroup, then its left (right)
quotient is a family of commutative distributive quasigroups which have a 1-1
correspondence with the left (right) coset of the soft quasigroup.

##  1 Introduction

####

A quasigroup is an algebraic structure which is not necessarily associative in
the sense of a group.

However, there exist some interesting properties of quasigroups that make them
different from group. A quasigroup may be homogeneous in nature in the sense
that its group of automorphisms may be transitive. The only group with
homogeneous property has just one element. Also a quasigroup has a rich outer
symmetry (or duality) because any quasigroup structure is associated with six
parastrophes. In general, only the transpose of a group is a group.

Effectiveness of the application of the theory of quasigroups is based on the
fact that quasigroups are “generalized permutations”. Namely, both left and
right translations in quasigroups are permutations and quasigroups are
characterized by this property.

The study of soft sets theory started with Molodtsov [1] as a better
generalization of set theory for the inquest and formal modeling of
mathematical problems represented by uncertainties and vagueness due to
partial and inadequate informations.

As a remedy for this defect in set theory, several advancements of set theory
have been proposed and developed like the vague set theory by Gau and Buehrer
[3], fuzzy set theory by Zadeh [4], rough set theory by Pawlak [5],
neutrosophic set by Smarandache [6]. However, all these theories have their
challenges, possibly mainly due to inadequacy of the parameterization tools of
the theories as pointed out in Molodtsov [1]. For instance, probability theory
can only deal with stochastically stable systems, where a limit of the sample
mean should exit in a long series of trials. The method of interval
mathematics is not adequate for problems with different type of uncertainties,
while rough set theory approach can only handle problems that involves
uncertainties caused by indiscernible elements with different values in
decision attributes. The fuzzy set theory approach is found most appropriate
for dealing with uncertainties. It provides a tool on how to set a membership
function, since the type of the membership function is defined on individual
attributes.

Compared to the above mentioned mathematical tools for computing
uncertainties, soft set has one important property that makes it different
from other tools; parametrization. For example, it is not necessarily like
membership grade in fuzzy set or approximation grade in rough set. Soft set
theory has a rich potential for applications in several directions, few of
which were shown by Molodtsov [1] in his pioneer work. Atkas and Cagman [2]
did a comparison of soft sets with fuzzy sets and rough sets and showed that
the both can be considered as a soft set.

Recently Oyem et al. [7, 8] extended the results of soft sets to quasigroup by
investigating the order of finite soft quasigroups and the algebraic
properties of soft quasigroups.

####

Inspired by the study of algebraic properties of soft sets, our aim in this
paper is to initiate research about soft quasigroup, its parastrophes and
their cosets. It was shown that every soft quasigroup is associated to five
soft quasigroups called its parastrophes. Necessary and sufficient conditions
for a soft set over a quasigroup (loop) to be a soft quasigroup (loop) were
established. It was proved that a soft set over a group is a soft group if and
only if it is a soft loop or either of two of its parastrophes is a soft
groupoid. For a finite quasigroup $Q$, it was shown that the orders
(arithmetic and geometric means) of the soft quasigroup $(F,A)_{Q}$ and it
parastrophes are equal.

It was established that if a soft quasigroup is distributive, then all its
parastrophes are distributive, idempotent and flexible soft quasigroups. For a
distributive soft quasigroup $(F,A)$, it was shown that its left and right
cosets form families of distributive soft quasigroups that are isomorphic. If
in addition, $(F,A)$ is normal, then its left and right cosets form families
of normal soft quasigroups. On another hand, it was found that if $(F,A)$ is a
normal and distributive soft quasigroup, then its quotient is a family of
commutative distributive quasigroups.

##  2 Preliminaries

####

In this section, we review some notions and results concerning quasigroups and
soft sets.

###  2.1 Groupoid and Quasigroups

######  Definition 2.1.

(Groupoid, Quasigroup)[9, 10, 11, 12]

Let $G$ be a non-empty set. Define a binary operation ($\cdot$) on $G$. If
$x\cdot y\in G$ for all $x,y\in G$, then the pair $(G,\cdot)$ is called a
groupoid or magma. If each of the equations:

$a\cdot x=b\qquad\textrm{and}\qquad y\cdot a=b$

has unique solutions in $G$ for $x$ and $y$ respectively for all $a,b\in G$,
then $(G,\cdot)$ is called a quasigroup. If there exists a unique element
$e\in G$ called the identity element such that for all $x\in G$, $x\cdot
e=e\cdot x=x$, $(G,\cdot)$ is called a loop.

We write $xy$ instead of $x\cdot y$, and stipulate that $\cdot$ has lower
priority than juxtaposition among factors to be multiplied. For instance,
$x\cdot yz$ stands for $x(yz)$. Let $x$ be a fixed element in a groupoid
$(G,\cdot)$. The left and right translation maps of $G$, $L_{x}$ and $R_{x}$
respectively are defined by

$yL_{x}=x\cdot y\qquad\textrm{and}\qquad yR_{x}=y\cdot x.$

It can now be said that a groupoid $(G,\cdot)$ is a quasigroup if its left and
right translation mappings are permutations. Since the left and right
translation mappings of a quasigroup are bijective, then the inverse mappings
$L_{x}^{-1}$ and $R_{x}^{-1}$ exist. Let

$x\backslash y=yL_{x}^{-1}\qquad\textrm{and}\qquad x/y=xR_{y}^{-1}$

and note that

$x\backslash y=z\Leftrightarrow x\cdot z=y\qquad\textrm{and}\qquad
x/y=z\Leftrightarrow z\cdot y=x.$

In the language of universal algebra, a quasigroup $(G,\cdot)$ can also be
represented as a quasigroup $(G,\cdot,/,\backslash)$.

For more on quasigroups, readers can check [13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23].

######  Definition 2.2.

(Subgroupoid, Subquasigroup)[9, 10, 11, 12]

Let $(Q,\cdot)$ be a groupoid quasigroup and $\emptyset\neq H\subseteq Q$.
Then, $H$ is called a subgroupoid (subquasigroup) of $Q$ if $(H,\cdot)$ is a
groupoid (quasigroup). This is often expressed as $H\leq Q$.

######  Definition 2.3.

(Normal Subquasigroup)[10, 11]

Let $(Q,\cdot)$ be a quasigroup. An equivalence relation $\theta$ on a
quasigroup $(Q,\cdot)$ is called a normal equivalence (or normal congruence)
relation if it satisfies the following conditions for all $a,b,c,d\in Q$:

  1. 1.

if $ca\theta cb\Rightarrow a\theta b$;

  2. 2.

if $ac\theta bc\Rightarrow a\theta b$;

  3. 3.

if $a\theta b$ and $c\theta d\Rightarrow~{}ac\theta bd$

Let $\emptyset\neq H\leq Q$. Then, $H$ is called a normal subquasigroup of
$Q$, written as $H\lhd Q$ if $H$ is an equivalence class with respect to some
normal equivalence relation $\theta$ on $(Q,\cdot)$.

######  Definition 2.4.

(Parastrophes of a Quasigroup [11, 20, 24])

Let $(Q,\star)$ be a quasigroup. The five parastrophes
$Q_{i}=(Q,\star_{i}),~{}i=1,2,3,4,5$ of $(Q,\star)$ are the quasigroups
$(Q,\circledast),(Q,/),(Q,\backslash),(Q,//),(Q,\backslash\backslash)$ whose
binary operations over $Q$ are defined in Table 1.

$\cdot$ | Parastrophic operation | Name
---|---|---
1 | $x\backslash y=z\Longleftrightarrow x\star z=y$ | left division
2 | $x/y=z\Longleftrightarrow z\star y=x$ | right division
3 | $x\star y=z\Longleftrightarrow y\circledast x=z$ | opposite multiplication
4 | $x//y=z\Longleftrightarrow y/x=z\Longleftrightarrow z\star x=y$ | opposite right division
5 | $x\backslash\backslash y=z\Longleftrightarrow y\backslash x=z\Longleftrightarrow y\star z=x$ | opposite left division
Table 1: Parastrophic operations on a quasigroup $(Q,\star)$

Every quasigroup belongs to a set of six quasigroups, called adjugates by
Fisher and Yates [17], conjugates by Stein [12], parastrophes by Sade [13].
Most other authors like Shchukin [14] and, Shchukin and Gushan [15] and Artzy
[16] adopted the last terminology. Ogunriade et al. [25] studied a class of
distributive quasigroup and their parastrophes as well as self-distributive
quasigroups with key laws in Ogunriade et al. [26].

######  Example 2.1.

Consider a quasigroup $(Q,\cdot)$. Its multiplication table and the
multiplication tables of its five parastrophes are as displayed in Table 2.

$\cdot$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$
---|---|---|---|---|---|---
1 | 1 | 2 | 3 | 4 | 6 | 5
2 | 2 | 1 | 5 | 6 | 3 | 4
3 | 3 | 5 | 4 | 1 | 2 | 6
4 | 4 | 6 | 1 | 3 | 5 | 2
5 | 5 | 4 | 6 | 2 | 1 | 3
6 | 6 | 3 | 2 | 5 | 4 | 1

$\star$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$
---|---|---|---|---|---|---
1 | 1 | 2 | 3 | 4 | 5 | 6
2 | 2 | 1 | 5 | 6 | 4 | 3
3 | 3 | 5 | 4 | 1 | 6 | 2
4 | 4 | 6 | 1 | 3 | 2 | 5
5 | 6 | 3 | 2 | 5 | 1 | 4
6 | 5 | 4 | 6 | 2 | 3 | 1

$/$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$
---|---|---|---|---|---|---
1 | 1 | 2 | 4 | 3 | 5 | 6
2 | 2 | 1 | 6 | 5 | 3 | 4
3 | 3 | 6 | 1 | 4 | 2 | 5
4 | 4 | 5 | 3 | 1 | 6 | 2
5 | 5 | 3 | 2 | 6 | 4 | 1
6 | 6 | 4 | 5 | 2 | 1 | 3

$\backslash$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$
---|---|---|---|---|---|---
1 | 1 | 2 | 3 | 4 | 6 | 5
2 | 2 | 1 | 5 | 6 | 3 | 4
3 | 4 | 5 | 1 | 3 | 2 | 6
4 | 3 | 6 | 4 | 1 | 5 | 2
5 | 5 | 4 | 6 | 2 | 1 | 3
6 | 6 | 3 | 2 | 5 | 4 | 1

$//$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$
---|---|---|---|---|---|---
1 | 1 | 2 | 3 | 4 | 5 | 6
2 | 2 | 1 | 5 | 6 | 4 | 3
3 | 3 | 5 | 1 | 4 | 6 | 2
4 | 4 | 6 | 3 | 1 | 2 | 5
5 | 6 | 3 | 2 | 5 | 1 | 4
6 | 5 | 4 | 6 | 2 | 3 | 1

$\backslash\backslash$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$
---|---|---|---|---|---|---
1 | 1 | 2 | 3 | 4 | 5 | 6
2 | 2 | 1 | 6 | 5 | 3 | 4
3 | 4 | 6 | 1 | 3 | 2 | 5
4 | 3 | 5 | 4 | 1 | 6 | 2
5 | 5 | 3 | 2 | 6 | 4 | 1
6 | 6 | 4 | 5 | 2 | 1 | 3

Table 2: Parastrophes
$(Q,\star),(Q,/),(Q,\backslash),(Q,//),(Q,\backslash\backslash)$ of
$(Q,\cdot)$

Parastrophes in general do not preserve the structure of a quasigroup but they
do preserve all subquasigroups; all individual generators as generators and
the number of generators is the same for each of the quasigroup’s
parastrophes. Belousov [27], Dudek [24], Duplak [28], Jaiyéọlá [22],
Samardziska [23] have studied the parastrophy of quasigroups and loops in
different fashions.

Aside the opposite parastrophe of a group, none of the other parastrophes is a
group. Jaiyéọlá [22] investigated the parastrophism of associativity in
quasigroups. However, the parastrophes of idempotent quasigroup is also
idempotent. In some instances, the parastrophes of a given quasigroup are
pairwise equal or pairwise distinct.

######  Theorem 2.1.

[11, 22]
Let $(Q,\star)$ be a quasigroup and $(Q,\star_{i})$ be its parastrophes,
$i=1,2,3,4,5$. If $(H,\star)$ is a subquasigroup of $(Q,\star)$, then
$(H,\star_{i})$ is a subquasigroup of $(Q,\star_{i})$ for $i=1,2,3,4,5$.

######  Theorem 2.2.

(Pflugfelder [11])

Let $(Q,\cdot)$ be a quasigroup (loop) and $\emptyset\neq H\subseteq Q$.
$(H,\cdot)\leq(Q,\cdot)$ if and only $(H,\cdot),(H,/),(H,\backslash)$ are
groupoids.

######  Theorem 2.3.

(Pflugfelder [11])

Let $(Q,\cdot)$ be a group. Then, for any $\emptyset\neq H\subseteq Q$, the
following statements are equivalent:

  1. 1.

$(H,\cdot)$ is a subloop of $(Q,\cdot)$.

  2. 2.

$(H,\cdot)$ is a subgroup of $(Q,\cdot)$.

  3. 3.

$(H,/)$ is a groupoid.

  4. 4.

$(H,\backslash)$ is a groupoid.

######  Definition 2.5.

(Distributive Quasigroup)[11]

A quasigroup $(Q,\cdot)$ is called a left distributive quasigroup if it
satisfies the left distributive law $x(yz)=xy\cdot xz$. A quasigroup
$(Q,\cdot)$ is called a right distributive quasigroup if it satisfies the
right distributive law $(yz)x=yx\cdot zx$. A quasigroup which is both a left
and right distributive quasigroup is called a distributive quasigroup.

######  Theorem 2.4.

(Pflugfelder [11])

If $(Q,\cdot,/,\backslash)$ is a distributive quasigroup, then the following
are true for all $x,y,z\in Q$.

  1. 1.

$Q$ is idempotent i.e. $x^{2}=x$.

  2. 2.

$L_{x}$ and $R_{x}$ are automorphisms of $(Q,\cdot)$.

  3. 3.

$Q$ is flexible i.e. $x(yx)=(xy)x$.

  4. 4.

$x(y\backslash
z)=(xy)\backslash(xz),~{}(y/z)x=(yx)/(zx),~{}x\backslash(yz)=(x\backslash
y)(x\backslash z),~{}(yz)/x=(y/x)(z/x)$.

######  Theorem 2.5.

(Pflugfelder [11])

If $(Q,\cdot,/,\backslash)$ is a distributive quasigroup, then
$(Q,\circledast),(Q,/),(Q,\backslash),(Q,//),(Q,\backslash\backslash)$ are
distributive quasigroups.

######  Theorem 2.6.

(Pflugfelder [11])

Let $Q$ be a distributive quasigroup such that $|Q|>1$, then
$N_{\rho}(G)=\emptyset=N_{\lambda}(G)$.

######  Theorem 2.7.

(Pflugfelder [11])

Let $H\leq Q$ where $(Q,\cdot)$ is a distributive quasigroup. Then, all left
cosets $x\cdot H$ and all right cosets $H\cdot x$ of $H$ are subquasigroups of
$Q$ for any $x\in Q$.

######  Theorem 2.8.

(Pflugfelder [11])

Let $H\leq Q$ where $Q$ is a distributive quasigroup. Then, every left and
right cosets of $H$ are isomorphic to $H$ and to each other. That is, $x\cdot
H=xH\cong H\cong Hx=H\cdot x$ for any $x\in Q$.

######  Theorem 2.9.

(Pflugfelder [11])

Let $H\lhd Q$ where $Q$ is a distributive quasigroup. Then, $xH,Hx\lhd Q$ for
any $x\in Q$.

######  Theorem 2.10.

(Pflugfelder [11])

Let $H\lhd Q$ where $Q$ is a distributive quasigroup. Then, $Q/H$ is a
commutative distributive quasigroup.

###  2.2 Soft Sets and Some Operations

####

We start with the notion of soft sets and operations defined on it. We refer
readers to [1, 2, 29, 30, 31] for earlier works on soft sets, soft groups and
their operations. Throughout this subsection, $Q$ denotes an initial universe,
$E$ is the set of parameters and $A\subseteq E$.

######  Definition 2.6.

(Soft Sets, Soft Subset, Equal Soft Sets)[1, 2, 31]

Let $Q$ be a set and $E$ be a set of parameters. For $A\subset E$, the pair
$(F,A)$ is called a soft set over $Q$ if $F(a)\subset Q$ for all $a\in A$,
where $F$ is a function mapping $A$ into the set of all non-empty subsets of
$Q$, i.e $F:A\longrightarrow 2^{Q}\backslash\\{\emptyset\\}$. A soft set
$(F,A)$ over a set $Q$ is identified or represented as a set of ordered pairs:
$(F,A)=\\{(a,F(a)):a\in A~{}\textrm{ and}~{}F(a)\in 2^{Q}\\}$. The set of all
soft sets, over $Q$ under a set of parameters $A$, is denoted by $SS(Q_{A})$.

#####

Let $(F,A)$ and $(H,B)$ be two soft sets over a common universe $U$, then
$(H,B)$ is called a soft subset of $(F,A)$ if

  1. 1.

$B\subseteq A$; and

  2. 2.

$H(x)\subseteq F(x)$ for all $x\in B$.

This is usually expressed as $(F,A)\supset(H,B)$ or $(H,B)\subset(F,A)$, and
$(F,A)$ is said to be a soft super set of $(H,B)$.

Two soft sets $(F,A)$ and $(H,B)$ over a common universe $U$ are said to be
soft equal if $(F,A)$ is a soft subset of $(H,B)$ and $(H,B)$ is a soft subset
of $(F,A)$.

######  Definition 2.7.

(Restricted Intersection)[29, 30, 31]

Let $(F,A)$ and $(G,B)$ be two soft sets over a common universe $U$ such that
$A\cap B\neq\emptyset$. Then their restricted intersection is
$(F,A)\cap_{R}(G,B)=(H,C)$ where $(H,C)$ is defined as $H(c)=F(c)\cap G(c)$
for all $c\in C$, where $C=A\cap B$.

######  Definition 2.8.

(Extended Intersection)[29, 30, 31]

The extended intersection of two soft sets $(F,A)$ and $(G,B)$ over a common
universe $U$ is the soft set $(H,C)$, where $C$ = $A\cup B$, and for all $x\in
C$, $H(x)$ is defined as

$H(x)=\left\\{\begin{array}[]{lll}F(x)&\mbox{if}\hskip 14.45377ptx\in A-B\\\
G(x)&\mbox{if}\hskip 14.45377ptx\in B-A\\\ F(x)\cap G(x)&\mbox{if}\hskip
14.45377ptx\in A\cap B.\end{array}\right.$

######  Definition 2.9.

(Extended Union)[29, 30, 31]

The union of two soft sets $(F,A)$ and $(G,B)$ over $U$ is denoted by
$(F,A)\bigcup(G,B)$ and is a soft set $(H,C)$ over $U$, such that $C$ = $A\cup
B$, $\forall x\in C$
and

$H(x)=\left\\{\begin{array}[]{lll}F(x)&\mbox{if}\hskip 14.45377ptx\in A-B\\\
G(x)&\mbox{if}\hskip 14.45377ptx\in B-A\\\ F(x)\cup G(x)&\mbox{if}\hskip
14.45377ptx\in A\cap B.\end{array}\right.$

##  3 Main Results

###  3.1 Soft Quasigroups and Soft Loops

######  Definition 3.1.

(Soft groupoid, Soft quasigroup and Soft loop)[7, 8]

Let $(Q,\cdot)$ be a groupoid (quasigroup, loop) and $E$ be a set of
parameters. For $A\subset E$, the pair $(F,A)_{(Q,\cdot)}=(F,A)_{Q}$ is called
a soft groupoid (quasigroup, loop) over $Q$ if $F(a)$ is a subgroupoid
(subquasigroup, subloop) of $Q$ for all $a\in A$, where $F:A\longrightarrow
2^{Q}\backslash\\{\emptyset\\}$. A soft groupoid (quasigroup, subloop)
$(F,A)_{(Q,\cdot)}$ over a groupoid (quasigroup, loop) $(Q,\cdot)$ is
identified or represented as a set of ordered pairs:
$(F,A)_{(Q,\cdot)}=\\{\left(a,(F(a),\cdot)\right):a\in A~{}\textrm{
and}~{}F(k)\in 2^{Q}\\}$.

######  Remark 3.1.

Based on Definition 3.1, a soft quasigroup is a soft groupoid, but the
converse is not necessarily true. A soft groupoid (quasigroup) is said to be
finite if its underlying groupoid (quasigroup) is finite.

$\cdot$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$
---|---|---|---|---|---|---|---|---
1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
2 | 2 | 1 | 4 | 3 | 6 | 5 | 8 | 7
3 | 3 | 4 | 1 | 2 | 7 | 8 | 6 | 5
4 | 4 | 3 | 2 | 1 | 8 | 7 | 5 | 6
5 | 6 | 5 | 8 | 7 | 2 | 1 | 4 | 3
6 | 5 | 6 | 7 | 8 | 4 | 2 | 3 | l
7 | 8 | 7 | 5 | 6 | 3 | 4 | 1 | 2
8 | 7 | 8 | 6 | 5 | 4 | 3 | 2 | 1
Table 3: Quasigroup $(Q,\cdot)$ of order $8$

######  Example 3.1.

Let Table 3 represents the Latin square of a finite quasigroup
$(Q,\cdot),\;Q=\\{1,2,3,4,5,6,7,8\\}$ and let
$A=\\{\gamma_{1},\gamma_{2},\gamma_{3}\\}$ be any set of parameters. Let
$F:A\longrightarrow 2^{Q}\backslash\\{\emptyset\\}$ be defined by

$F(\gamma_{1})=\\{1,2\\},\;F(\gamma_{2})=\\{1,2,3,4\\},\;F(\gamma_{3})=\\{1,2,7,8\\}.$

Then, the pair $(F,A)$ is called a soft quasigroup over quasigroup $Q$ because
each of $F(\gamma_{i})\leq Q,\;i=1,2,3.$

######  Theorem 3.1.

Let $(F,A)$ be a soft set over a group $(Q,\cdot)$. Then, $(F,A)_{(Q,\cdot)}$
is a soft group if and only if any of the following statements is true:

  1. 1.

$(F,A)_{(Q,\cdot)}$ is a soft loop.

  2. 2.

$(F,A)_{(Q,/)}$ is a soft groupoid.

  3. 3.

$(F,A)_{(Q,\backslash)}$ is a soft groupoid.

###### Proof.

Let $(F,A)$ be a soft set over a group $(Q,\cdot)$. $(F,A)_{(Q,\cdot)}$ is a
soft group if and only if $F(a)$ is a subgroup of $Q$ for all $a\in A$. Going
by Theorem 2.3, this possible if and only any of the following statements is
true:

  1. 1.

$F(a)$ is a subloop of $(Q,\cdot)$.

  2. 2.

$F(a)$ is a subgroupoid of $(Q,/)$.

  3. 3.

$F(a)$ is a subgroupoid of $(Q,\backslash)$.

It can thus be concluded that $(F,A)_{(Q,\cdot)}$ is a soft group if and only
if $(F,A)_{(Q,\cdot)}$ is a soft loop if and only if $(F,A)_{(Q,/)}$ is a soft
groupoid if and only if $(F,A)_{(Q,\backslash)}$ is a soft groupoid. ∎

######  Definition 3.2.

(Soft Subquasigroup)[7, 8]

Let $(F,A)$ and $(G,B)$ be two soft quasigroups over a quasigroup $Q$. Then,

  1. 1.

$(F,A)$ is a soft subquasigroup of $(G,B)$, denoted $(F,A)\leq(G,B)$, if

    1. (a)

$A\subseteq B$, and

    2. (b)

$F(a)\leq G(a)$ for all $a\in A$.

  2. 2.

$(F,A)$ is soft equal to $(G,B)$, denoted $(F,A)=(G,B)$, whenever
$(F,A)\leq(G,B)$ and $(G,B)\leq(F,A)$.

######  Example 3.2.

Consider the Latin square Table 3 of a quasigroup
$(Q,\cdot),~{}Q=\\{1,2,3,4,5,6,7,8\\}$ in Example 3.1.
$E=\\{\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\gamma_{5},\gamma_{6}\\}.$
Then, let $A$ = $\\{\gamma_{1},\gamma_{2},\gamma_{3}\\}\subset E$ and $B$ =
$\\{\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4}\\}\subset E$ be two sets of
parameters, where
$F(\gamma_{1})=\\{1\\},F(\gamma_{2})=\\{1,2\\},F(\gamma_{3})=\\{1,2,7,8\\}$
and $G(\gamma_{1})=\\{1,2\\}$, $G(\gamma_{2})=\\{1,2,3,4\\}$,
$G(\gamma_{3})=Q$. Then, $(F,A)\leq(G,B)$; since $A\subseteq B$ and
$F(\gamma)\leq G(\gamma)$, for all $\gamma\in A$. But $(G,B)\not\leq(F,A)$.
Hence $(F,A)\neq(G,B)$.

This example shows that two soft quasigroups can not be equal if they have
different set parameters.

###  3.2 Parastrophes of Soft Quasigroups

######  Definition 3.3.

(Parastrophes of Soft Quasigroup)

Let $(F,A)$ be a soft quasigroup over a quasigroup $(Q,\star)$. The five
parastrophes of $(F,A)_{(Q,\star)}$ are the soft sets
$(F,A)_{(Q,\star_{i})}~{}i=1,2,3,4,5$.

######  Lemma 3.1.

Let $(F,A)$ be a soft set over a quasigroup $(Q,\cdot,\backslash,/)$, then the
following statements are equivalent.

  1. 1.

$(F,A)_{(Q,\cdot)}$ is a soft quasigroup.

  2. 2.

$(F,A)_{(Q,\circledast)}$ is a soft quasigroup.

  3. 3.

$(F,A)_{(Q,\backslash)}$ is a soft quasigroup .

  4. 4.

$(F,A)_{(Q,/)}$ is a soft quasigroup.

  5. 5.

$(F,A)_{(Q,\backslash\backslash)}$ is a soft quasigroup

  6. 6.

$(F,A)_{(Q,//)}$ is a soft quasigroup

###### Proof.

Let $(F,A)_{(Q,\cdot)}$ be a soft quasigroup. $(Q,\cdot)$ is a quasigroup if
and only if $(Q,\circledast)$ is a quasigroup if and only if $(Q,\backslash)$
is a quasigroup if and only if $(Q,/)$ is a quasigroup if and only if $(Q,//)$
is a quasigroup, if and only if $(Q,\backslash\backslash)$ is a quasigroup.
Thus, for any $F(a)\subset Q,~{}a\in A$ and based on Theorem 2.1, the
following are equivalent:

  * •

$(F(a),\cdot)$ is a subquasigroup of quasigroup $(Q,\cdot)$ for $a\in A$.

  * •

$(F(a),\circledast)$ is a subquasigroup of quasigroup $(Q,\circledast)$ for
$a\in A$.

  * •

$(F(a),\backslash)$ is a subquasigroup of quasigroup $(Q,\backslash)$ for
$a\in A$.

  * •

$(F(a),\backslash\backslash)$ is a subquasigroup of quasigroup
$(Q,\backslash\backslash)$ for $a\in A$.

  * •

$(F(a),/)$ is a subquasigroup of quasigroup $(Q,/)$ for $a\in A$.

  * •

$(F(a),//)$ is a subquasigroup of quasigroup $(Q,//)$ for $a\in A$.

Therefore, $(F,A)_{(Q,\star_{i})}$ is a soft quasigroup if and only if
$(F,A)_{(Q,\star_{j})}$ is a soft quasigroup for any $i,j=0,1,2,3,4,5$. ∎

######  Remark 3.2.

Based on Lemma 3.1, the soft quasigroup $(F,A)_{(Q,\cdot)}$ and its five
parastrophes have the same set of parameters but need not be equal.

######  Theorem 3.2.

Let $(Q,\cdot)$ be a quasigroup (loop). $(F,A)_{(Q,\cdot)}$ is a soft
quasigroup (loop) if and only the following are true:

  1. 1.

$(F,A)_{(Q,\cdot)}$ is a soft groupoid;

  2. 2.

$(F,A)_{(Q,\backslash)}$ is a soft groupoid;

  3. 3.

$(F,A)_{(Q,/)}$ is a soft groupoid;

###### Proof.

Given that $(F,A)_{(Q,\cdot)}$ is a soft quasigroup, then
$(F(a),\cdot)\leq(Q,\cdot)$ for all $a\in A$. By Theorem 2.2, this is possible
if and only $(F(a),\cdot),(F(a),/)$ and $(F(a),\backslash)$ are subgroupoids
of $(Q,\cdot),(Q,/)$ and $(Q,\backslash)$ respectively. This last statement is
true if and only if $(F,A)_{(Q,\cdot)},(F,A)_{(Q,/)}$ and
$(F,A)_{(Q,\backslash)}$ are soft groupoids. The proof for when
$(F,A)_{(Q,\cdot)}$ is a soft loop is similar. ∎

######  Example 3.3.

Consider the Latin squares in Table 2 of a quasigroup $(Q,\cdot)$ and its
parastrophes. Let $A=\\{\gamma_{1},\gamma_{2},\gamma_{3}\\}$ be any set of
parameters. Let $F:A\longrightarrow 2^{Q}\backslash\\{\emptyset\\}$ be defined
by

$F(\gamma_{1})=\\{1\\},\;F(\gamma_{2})=\\{1,2\\},\;F(\gamma_{3})=\\{1,3,4\\}.$

Then, by Lemma 3.1, the soft set $(F,A)_{(Q,\cdot)}$ and its parastrophes are
soft quasigroups with subquasigroups $F(\gamma_{i})\leq Q,\;i=1,2,3$ which are
represented by the Latin squares in Table 4.

$\cdot$ | 1
---|---
1 | 1

$\equiv\left(F(\gamma_{1}),\cdot\right)$ $\cdot$ 1 2 1 1 2 2 2 1
$\equiv\left(F(\gamma_{2}),\cdot\right)$ $\cdot$ 1 3 4 1 1 3 4 3 3 4 1 4 4 1 3
$\equiv\left(F(\gamma_{3}),\cdot\right)$

$\circledast$ | 1
---|---
1 | 1

$\equiv\left(F(\gamma_{1}),\circledast\right)$ $\circledast$ 1 2 1 1 2 2 2 1
$\equiv\left(F(\gamma_{2}),\circledast\right)$ $\circledast$ 1 3 4 1 1 3 4 3 3
4 1 4 4 1 3 $\equiv\left(F(\gamma_{3}),\circledast\right)$

$/$ | 1
---|---
1 | 1

$\equiv\left(F(\gamma_{1}),/\right)$ $/$ 1 2 1 1 2 2 2 1
$\equiv\left(F(\gamma_{2}),/\right)$ $/$ 1 3 4 1 1 4 3 3 3 1 4 4 4 3 1
$\equiv\left(F(\gamma_{3}),/\right)$

$\backslash$ | 1
---|---
1 | 1

$\equiv\left(F(\gamma_{1}),\backslash\right)$ $\backslash$ 1 2 1 1 2 2 2 1
$\equiv\left(F(\gamma_{2}),\backslash\right)$ $\backslash$ 1 3 4 1 1 3 4 3 4 1
3 4 3 4 1 $\equiv\left(F(\gamma_{3}),\backslash\right)$

$//$ | 1
---|---
1 | 1

$\equiv\left(F(\gamma_{1}),//\right)$ $//$ 1 2 1 1 2 2 2 1
$\equiv\left(F(\gamma_{2}),//\right)$ $//$ 1 3 4 1 1 3 4 3 3 1 4 4 4 3 1
$\equiv\left(F(\gamma_{3}),//\right)$

$\backslash\backslash$ | 1
---|---
1 | 1

$\equiv\left(F(\gamma_{1}),\backslash\backslash\right)$ $\backslash\backslash$
1 2 1 1 2 2 2 1 $\equiv\left(F(\gamma_{2}),\backslash\backslash\right)$
$\backslash\backslash$ 1 3 4 1 1 3 4 3 4 1 3 4 3 4 1
$\equiv\left(F(\gamma_{3}),\backslash\backslash\right)$

Table 4: Parastrophes of Soft quasigroup $(F,A)_{(Q,\cdot)}$

######  Definition 3.4.

(Distributive Soft Quasigroup)

A soft quasigroup $(F,A)_{Q}$ is called a distributive soft quasigroup if $Q$
is a distributive quasigroup.

######  Theorem 3.3.

Let $(F,A)_{(Q,\cdot)}$ be a distributive soft quasigroup. Then

  1. 1.

its five parastrophes
$(F,A)_{(Q,\circledast)},(F,A)_{(Q,\backslash)},(F,A)_{(Q,/)},(F,A)_{(Q,\backslash\backslash)},(F,A)_{(Q,//)}$
are distributive soft quasigroups.

  2. 2.

$(F,A)_{(Q,\cdot)},(F,A)_{(Q,\circledast)},(F,A)_{(Q,\backslash)},(F,A)_{(Q,/)},(F,A)_{(Q,\backslash\backslash)},(F,A)_{(Q,//)}$
are idempotent soft quasigroups.

  3. 3.

$(F,A)_{(Q,\cdot)},(F,A)_{(Q,\circledast)},(F,A)_{(Q,\backslash)},(F,A)_{(Q,/)},(F,A)_{(Q,\backslash\backslash)},(F,A)_{(Q,//)}$
are flexible soft quasigroups.

###### Proof.

Let $(F,A)_{(Q,\cdot)}$ be a soft quasigroup. Then, by Lemma 3.1,
$(F,A)_{(Q,\circledast)},(F,A)_{(Q,\backslash)},(F,A)_{(Q,/)},(F,A)_{(Q,\backslash\backslash)},(F,A)_{(Q,//)}$
are soft quasigroups.

$(F,A)_{(Q,\cdot)}$ is a distributive soft quasigroup means that $(Q,\cdot)$
is a distributive quasigroup. Thus, by Theorem 2.5,
$(Q,\circledast),(Q,/),(Q,\backslash),(Q,//),(Q,\backslash\backslash)$ are
distributive quasigroups. Consequently,
$(F,A)_{(Q,\circledast)},(F,A)_{(Q,\backslash)},(F,A)_{(Q,/)},(F,A)_{(Q,\backslash\backslash)},(F,A)_{(Q,//)}$
are distributive soft quasigroups.

Following Theorem 2.4(1,3),
$(F,A)_{(Q,\cdot)},(F,A)_{(Q,\circledast)},(F,A)_{(Q,\backslash)},(F,A)_{(Q,/)},(F,A)_{(Q,\backslash\backslash)},$
$(F,A)_{(Q,//)}$ are flexible soft quasigroups and idempotent soft quasigroups
∎

######  Lemma 3.2.

Let $(F,A)_{Q}$ be a distributive soft quasigroup, such that $|Q|>1$, then
$(F,A)$ is neither a left nor right nuclear.

###### Proof.

If $(F,A)_{Q}$ is a distributive soft quasigroup, then $Q$ is a distributive
quasigroup. Going by Theorem 2.6, since $|Q|>1$, then,
$N_{\rho}(Q)=\emptyset=N_{\lambda}(Q)$. So, there does not exist $a\in A$ such
that $F(a)=N_{\rho}(Q)$ or $F(a)=N_{\lambda}(Q)$. Therefore, $(F,A)$ is
neither a left nor right nuclear. ∎

######  Definition 3.5.

Let $(F,A)_{Q}$ be a soft quasigroup. For a fixed $x\in Q$, let
$F_{x},{}_{x}F:A\longrightarrow 2^{Q}\backslash{\emptyset}$ such that
$F_{x}(a)=F(a)\cdot x$ and ${}_{x}F(a)=x\cdot F(a)$ for all $a\in A$.

  1. 1.

The soft set $(F_{x},A)$ will be called a $x$-right coset soft set of $(F,A)$
and at times denoted by $\left(F_{x}^{\rho},A\right)$. The family
$\left\\{\left(F_{x}^{\rho},A\right)_{Q}\right\\}_{x\in Q}$ of soft sets will
be represented by $\left(F_{Q}^{\rho},A\right)$ and called the right coset of
$(F,A)_{Q}$.

  2. 2.

The soft set $\left({}_{x}F,A\right)$ will be called a $x$-left coset soft set
of $(F,A)$ and at times denoted by $\left({}_{x}F^{\lambda},A\right)$. The
family $\left\\{\left({}_{x}F^{\lambda},A\right)_{Q}\right\\}_{x\in Q}$ of
soft sets will be represented by $\left(F_{Q}^{\lambda},A\right)$ and called
the left coset of $(F,A)_{Q}$.

######  Lemma 3.3.

Let $(F,A)_{Q}$ be a distributive soft quasigroup. Then,
$\left(F_{Q}^{\rho},A\right)=\left\\{\left(F_{x}^{\rho},A\right)_{Q}\right\\}_{x\in
Q}$ and
$\left(F_{Q}^{\lambda},A\right)=\left\\{\left({}_{x}F^{\lambda},A\right)_{Q}\right\\}_{x\in
Q}$ are both families of distributive soft quasigroups.

###### Proof.

If $(F,A)_{(Q,\cdot)}$ is a distributive soft quasigroup, then $(Q,\cdot)$ is
a distributive quasigroup and for all $a\in A,~{}F(a)\leq Q$. Going by Theorem
2.7, $x\cdot F(a)\leq Q$ for any fixed $x\in Q$. Thus, $(F,A)_{(Q,\cdot)}$ is
a distributive soft quasigroup for any fixed $x\in Q$ and consequently,
$\left(F_{Q}^{\lambda},A\right)=\left\\{\left({}_{x}F^{\lambda},A\right)_{Q}\right\\}_{x\in
Q}$ is a family of distributive soft quasigroups. A similar argument goes for
$\left(F_{Q}^{\rho},A\right)=\left\\{\left(F_{x}^{\rho},A\right)_{Q}\right\\}_{x\in
Q}$. ∎

######  Definition 3.6.

Let $(F,A)_{Q}$ and $(G,A)_{Q}$ be soft quasigroup over $Q$, then
$(F,A)_{Q}\cong(G,A)_{Q}$ if $F(a)\cong G(a)$ for all $a\in A$.

######  Definition 3.7.

Let $(F,A)_{Q}$ be a soft quasigroup over $Q$. $(F,A)_{Q}$ is called a normal
soft quasigroup if $F(a)\lhd Q$ for all $a\in A$.

######  Theorem 3.4.

Let $(F,A)_{Q}$ be a distributive soft quasigroup. Then:

  1. 1.

$(F,A)_{Q}\cong\left({}_{x}F^{\lambda},A\right)_{Q}\cong\left(F_{x}^{\rho},A\right)_{Q}$
for any fixed $x\in Q$. Furthermore,
$\left(F_{Q}^{\rho},A\right)\cong\left(F_{Q}^{\lambda},A\right)$.

  2. 2.

If $(F,A)_{Q}$ is a normal soft quasigroup, then
$\left(F_{Q}^{\rho},A\right)=\left\\{\left(F_{x}^{\rho},A\right)_{Q}\right\\}_{x\in
Q}$ and
$\left(F_{Q}^{\lambda},A\right)=\left\\{\left({}_{x}F^{\lambda},A\right)_{Q}\right\\}_{x\in
Q}$ are isomorphic families of normal soft quasigroups.

###### Proof.

  1. 1.

If $(F,A)_{(Q,\cdot)}$ is a distributive soft quasigroup, then, $(Q,\cdot)$ is
a distributive quasigroup.

Going by Lemma 3.3,
$\left(F_{Q}^{\rho},A\right)=\left\\{\left(F_{x}^{\rho},A\right)_{Q}\right\\}_{x\in
Q}$ and
$\left(F_{Q}^{\lambda},A\right)=\left\\{\left({}_{x}F^{\lambda},A\right)_{Q}\right\\}_{x\in
Q}$ are both families of distributive soft quasigroups. Using Theorem 2.8,
$F(a)\cong F_{x}(a)$ and $F(a)\cong{}_{x}F(a)$ for all $a\in A$ and for any
fixed $x\in Q$. So,
$(F,A)_{Q}\cong\left({}_{x}F^{\lambda},A\right)_{Q}\cong\left(F_{x}^{\rho},A\right)_{Q}$
for any fixed $x\in Q$. More so,
$\left(F_{Q}^{\rho},A\right)\cong\left(F_{Q}^{\lambda},A\right)$.

  2. 2.

If $(F,A)_{Q}$ is a normal soft quasigroup, then, $F(a)\lhd Q$ for all $a\in
A$.

Using Theorem 2.9, $F_{x}(a)\lhd Q$ and ${}_{x}F(a)\lhd Q$ for all $a\in A$
and for any fixed $x\in Q$. So, $\left({}_{x}F^{\lambda},A\right)_{Q}$ and
$\left(F_{x}^{\rho},A\right)_{Q}$ are normal soft quasigroups for any fixed
$x\in Q$.

Thus,
$\left(F_{Q}^{\rho},A\right)=\left\\{\left(F_{x}^{\rho},A\right)_{Q}\right\\}_{x\in
Q}$ and
$\left(F_{Q}^{\lambda},A\right)=\left\\{\left({}_{x}F^{\lambda},A\right)_{Q}\right\\}_{x\in
Q}$ are isomorphic families of normal soft quasigroups.

∎

######  Definition 3.8.

Let $(F,A)_{Q}$ be a soft quasigroup, then the family
$\left\\{Q/F(a)\right\\}_{a\in A}$ will be called the left quotient of
$(F,A)_{Q}$ in $Q$ while $\left\\{Q\backslash F(a)\right\\}_{a\in A}$ will be
called the right quotient of $(F,A)_{Q}$ in $Q$.

######  Theorem 3.5.

Let $(F,A)_{Q}$ be a normal and distributive soft quasigroup. Then:

  1. 1.

the left quotient of $(F,A)_{Q}$ in $Q$ is a family of commutative
distributive quasigroups and has a 1-1 correspondence with
$\left(F_{Q}^{\lambda},A\right)$.

  2. 2.

the right quotient of $(F,A)_{Q}$ in $Q$ is a family of commutative
distributive quasigroups and has a 1-1 correspondence with
$\left(F_{Q}^{\rho},A\right)$.

###### Proof.

  1. 1.

If $(F,A)_{Q}$ is a normal and distributive soft quasigroup, then, $F(a)\lhd
Q$ for all $a\in A$. Going by Theorem 2.10, $Q/F(a)$ is a commutative
distributive quasigroup for each $a\in A$. Thus, the left quotient of
$(F,A)_{Q}$ i.e. $\left\\{Q/F(a)\right\\}_{a\in A}$ is a family of commutative
distributive quasigroups.

  2. 2.

Similar argument.

∎

######  Definition 3.9.

(Order of Soft Quasigroup)([7])

Let $(F,A)$ be a soft quasigroup over a finite quasigroup $(Q,\cdot)$. The
order of the soft quasigroup $(F,A)$ or $(F,A)_{(Q,\cdot)}$ will be defined as

$|(F,A)|_{(Q,\cdot)}=|(F,A)|=\sum\limits_{a\in
A}|F(a)|,~{}~{}for~{}~{}F(a)\in(F,A)~{}~{}and~{}~{}a\in A.$

where the sum is over distinct proper subquasigroups $F(a)\in(F,A),~{}a\in A$.

######  Definition 3.10.

(Arithmetic and Geometric Means of Finite Soft Quasigroup)([7])

Let $(F,A)$ be a soft quasigroup over a finite quasigroup $(Q,\cdot)$. The
arithmetic mean and geometric mean of $(F,A)_{(Q,\cdot)}$ will be defined
respectively as

$\displaystyle\mathcal{AM}(F,A)_{(Q,\cdot)}=\frac{1}{|A|}\sum\limits_{a\in
A}|F(a)|\quad\textrm{and}\qquad\mathcal{GM}(F,A)_{(Q,\cdot)}=\sqrt[|A|]{\prod\limits_{a\in
A}|F(a)|}$

######  Theorem 3.6.

Let $(F,A)$ be a soft quasigroup over a quasigroup $(Q,\star)$. Then

  1. 1.

$|(F,A)|_{(Q,\star)}=|(F,A)|_{(Q,\star_{i})}$ for each $~{}i=1,2,3,4,5$.

  2. 2.

$\mathcal{AM}(F,A)_{(Q,\star)}=\mathcal{AM}(F,A)_{(Q,\star_{i})}$ for each
$~{}i=1,2,3,4,5$.

  3. 3.

$\mathcal{GM}(F,A)_{(Q,\star)}=\mathcal{GM}(F,A)_{(Q,\star_{i})}$ for each
$~{}i=1,2,3,4,5$.

###### Proof.

Let $(F,A)_{(Q,\star)}$ be a soft quasigroup, then by Lemma 3.1,
$(F,A)_{(Q,\star_{i})}$ is a soft quasigroup for each $~{}i=1,2,3,4,5$. Thus,

  1. 1.

$|(F,A)|_{(Q,\star)}=\sum\limits_{a\in A}|F(a)|=|(F,A)|_{(Q,\star_{i})}$ for
each $~{}i=1,2,3,4,5$.

  2. 2.

$\mathcal{AM}(F,A)_{(Q,\star)}=\frac{1}{|A|}\sum\limits_{a\in
A}|F(a)|=\mathcal{AM}(F,A)_{(Q,\star_{i})}$ for each $~{}i=1,2,3,4,5$.

  3. 3.

$\mathcal{GM}(F,A)_{(Q,\star)}=\sqrt[|A|]{\prod\limits_{a\in
A}|F(a)|}=\mathcal{GM}(F,A)_{(Q,\star_{i})}$ for each $~{}i=1,2,3,4,5$.

Thus, for any finite quasigroup, the orders (arithmetic and geometric means)
of the soft quasigroup over it and its parastrophes are equal. ∎

######  Remark 3.3.

Based on Theorem 3.6, the Maclaurin’s inequality and several other
inequalities for a finite soft quasigroup obtained in [7] are true in all the
five parastrophes of any finite soft quasigroup.

##  4 Conclusion and Further Studies

####

We considered soft set over non-associative algebraic structures groupoid,
quasigroup and loop, which was motivated by the study of algebraic structures
in the context soft sets, neutrosophic sets and rough sets. In this paper, we
introduced and studied parastrophes of soft quasigroup, soft nuclei, and
distributive soft quasigroups. It was cogently shown that any soft quasigroup
belongs to a particular family of six soft quasigroups (which are not
necessarily soft equal). A 1-1 correspondence was found between the left
(right) quotient of a soft quasigroup and the left (right) coset of the soft
quasigroup. Some examples were given for illustration. On the basis of these
results, soft quasigroup theory can be explored and their varieties studied.
Oyem et al. [32] just studied soft neutrosophic quasigroups.

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# Study of np-Scattering for S, P and D Waves using Deng-Fan Potential by
Phase Function Method

Ayushi Awasthi and O.S.K.S Sastri

###### Abstract

Background: Deng-Fan potential has been utilised to study neutron-proton and
neutron-Deuteron scattering phase shifts and in turn their corresponding
scattering cross-sections using Jost function method and phase function
method. It has been concluded that phase function method has certain
limitations in obtaining scattering phase shifts.

Purpose: In this paper, scattering phase shifts for various S, P and D states
of neutron-proton scattering have been obtained using Deng-Fan potential as
model of interaction.

Methods: The scattering phase shift for S, P and D channels are determined
using phase function method by incorporating Deng-Fan potential into
respective phase equations for $\ell=0,1,2$. The scattering phase shifts
obtained by phase function method are utilised to determine corresponding
scattering cross-section.

Results: The obtained scattering phase shifts for ${}^{3}S_{1}$,
${}^{1}S_{0}$, ${}^{1}P_{1}$, ${}^{3}P_{0,1,2}$, ${}^{1}D_{2}$ and
${}^{3}D_{1,2,3}$ states are found to be closely matching with respect to
experimental data for lab energies up to 350 MeV. The total scattering cross-
sections are calculated for available energies and are in good match with
expected ones.

Conclusion: The phenomenological Deng-Fan potential has been shown to describe
the np-scattering results reasonably well.

Keywords: Deng-Fan potential, np-scattering, Phase function method, scattering
phase shifts, scattering cross-sections.

##  1 Introduction

One of the important goals of nuclear physics is to model the interaction, by
a phenomenological potential, responsible for explaining observed scattering
cross-sections (SCS) by phase shift analysis or phase wave analysis[1]. This
involves solving the non-relativistic radial time independent Schrodinger
equation (TISE) for the chosen potential for various $\ell$-channels, called
as partial waves, to obtain corresponding wavefunctions. One deduces
scattering phase shifts (SPS) by matching wavefunction within potential region
with asymptotic solution[2]. These SPS are utilised to calculate the partial
and total SCS. The partial SCS would give information about resonances. The
match between experimental resonances and total SCS and those obtained by
theoretical interaction potential validate the model. While most of the
techniques like R-matrix[3], Jost function method (JFM)[4], complex scaling
method (CSM)[5], J-matrix method[6], etc., are dependent on wavefunction,
variable phase approach (VPA) or phase function method (PFM)[7, 8, 9] directly
obtains SPS from interaction potential. Our group has successfully utilised
PFM for studying various two body nuclear scattering[10, 11].

###  1.1 Motivation for the Study:

Yukawa[12] was the first to successfully explain np-interaction based on meson
exchange theory using a phenomenological potential given by

$V_{Y}(r)=-V_{1}\frac{e^{-\alpha r}}{r}$ (1)

The numerical solution with Yukawa potential using PFM, was able to explain
successfully the expected SPS[13] for lab energies up to 50 MeV[14].
Hulthen potential[15], which is a modified form of Yukawa,

$V_{H}(r)=-V_{1}\frac{e^{-\alpha r}}{(1-e^{-\alpha r})}$ (2)

has advantage of having an analytical solution for TISE. One can observe that
for small r, denominator reduces to $\alpha r$ in first order, it goes as
$1/r$ as expected from Yukawa potential.
To explain expected SPS data up to 350 MeV, considered as threshold for pion
production[16], Malfliet-Tjon (MT)[17] added a repulsive core of Yukawa form
to Yukawa potential, as follows:

$V_{MT}(r)=V_{2}\frac{e^{-2\alpha r}}{r}-V_{1}\frac{e^{-\alpha r}}{r}$ (3)

This is one phenomenological potential with reasonable success in explaining
expected SPS data for both ${}^{3}S_{1}$ ground state and ${}^{1}S_{0}$
scattering state. But it’s disadvantage is, it has no analytical solution for
TISE and does not result in deuteron binding energy to good accuracy.
Recently, Deng-Fan potential, given by

$V_{DF}(r)=V_{2}\frac{e^{-2\alpha r}}{(1-e^{-\alpha
r})^{2}}-V_{1}\frac{e^{-\alpha r}}{(1-e^{-\alpha r})}$ (4)

has been utilised to perform a parallel study using both JFM and PFM to obtain
n-p and n-D SPS and in turn their corresponding SCS[18] for low energy data up
to 50 MeV. It is interesting to see that Deng-Fan potential is a variation of
Hulthen potential, just as MT is that of Yukawa. Notice that, repulsive term,
in Eq. 4, is of Hulthen form squared. This gives an $exp(-2\alpha r)$ as in MT
potential and goes as $\frac{1}{r^{2}}$ instead of $\frac{1}{r}$ for small r
values. The intent is to model the short range interaction to fall off
exponentially very quickly and retain the long range to have Yukawa form, as
required in successful one pion exchange potential (OPEP)[19]. Once again,
main advantage is that TISE for this potential has analytical solutions[18],
and ground state energy $E_{B}$ is given by

$E_{B}=\frac{\hbar^{2}}{2\mu}\left[\frac{V_{1}-V_{2}}{2\alpha\left(\frac{-\sqrt{\alpha^{2}+4V_{2}}}{2\alpha}-\frac{1}{2}\right)}\right]^{2}$
(5)

where $\mu$ is reduced mass. Keeping in mind that MT potential was successful
in explaining SPS data up to 350 MeV, similar performance is expected from DF
potential. Hence, we extend the range of energies for studying np interaction
using DF potential to 350 MeV, in current study. Saha et.al.[18], have worked
out SPS for ${}^{3}S_{1}$ and ${}^{1}S_{0}$ states of np-scattering by
choosing to fit different set of model parameters for each of the channels.
Considering this to be a valid procedure for phenomenological potentials, we
have included the P and D-states of np-interaction and determined respective
model parameters that result in interaction potentials responsible for
observed SPS.

##  2 Phase Wave Analysis:

The radial time independent Schr$\ddot{o}$dinger equation, is given by

$\frac{d^{2}u_{\ell}(k,r)}{dr^{2}}+\left(k^{2}-\frac{\ell(\ell+1)}{r^{2}}\right)u_{\ell}(k,r)=U(r)u_{\ell}(k,r)$
(6)

Where $k=\sqrt{\frac{2\mu E_{cm}}{\hbar^{2}}}$, $U(r)=\frac{2\mu
V(r)}{\hbar^{2}}$ and $\mu$ is reduced mass of the system.
The $E_{cm}$ is related to $E_{\ell ab}$ by the relation
$E_{cm}=\frac{m_{T}}{m_{T}+m_{P}}E_{\ell ab}.$ Here, $m_{T}$ and $m_{P}$ are
masses of target and projectile respectively.

###  2.1 Phase Function Method:

The second order TISE is transformed into Ricatti type equation[19] which is
given by

$\frac{d\delta_{\ell}(k,r)}{dr}=-\frac{U(r)}{k}\bigg{[}\cos(\delta_{\ell}(k,r))\hat{j}_{\ell}(kr)-\sin(\delta_{\ell}(k,r))\hat{\eta}_{\ell}(kr)\bigg{]}^{2}$
(7)

where $\hat{j}_{\ell}(kr)$ and $\hat{\eta}_{\ell}(kr)$ are the Ricatti-Bessel
and Ricatti-Neumann functions of order $\ell$. For $\ell$ = 0, 1 and 2 (S, P
and D waves) respectively, phase equations are:

$\frac{d\delta_{0}(k,r)}{dr}=-\frac{U(r)}{k}\sin^{2}\left[kr+\delta_{0}(k,r)\right]$
(8)
$\frac{d\delta_{1}(k,r)}{dr}=-\frac{U(r)}{k}\bigg{[}\frac{\sin\left[kr+\delta_{1}(k,r)\right]-k~{}cos\left[kr+\delta_{1}(k,r)\right]}{kr}\bigg{]}^{2}$
(9)
$\frac{d\delta_{2}(k,r)}{dr}=-\frac{U(r)}{k}\bigg{[}-\sin{\left[kr+\delta_{2}(k,r)\right]}-\frac{3\cos{\left[kr+\delta_{2}(k,r)\right]}}{kr}+\frac{3\sin{\left[kr+\delta_{2}(k,r)\right]}}{(kr)^{2}}\bigg{]}^{2}$
(10)

The SPS for S, P and D waves are obtained by numerically solving these
equations using 5th order Runge-Kutta method with initial condition chosen as
$\delta_{\ell}(r=0,k)=0$.

###  2.2 Scattering Cross-Section:

Once, SPS $\delta_{\ell}(E)$ are obtained for each orbital angular momentum
$\ell$, one can calculate the partial cross section $\sigma_{\ell}(E)$ using
the following formula [20] :

$\sigma_{\ell}(E)=\frac{4\pi(2\ell+1)}{k^{2}}\sin^{2}(\delta_{\ell}(E))$ (11)

Then, total cross section $\sigma_{T}$, is given as

$\sigma_{T}=\sigma_{S}+\sigma_{P}+\sigma_{D}$ (12)

where $\sigma_{S}$ , $\sigma_{P}$ and $\sigma_{D}$ are given as

$\sigma_{S}=\frac{1}{4}\sigma_{{}^{1}S_{0}}+\frac{3}{4}\sigma_{{}^{3}S_{1}}$
$\sigma_{P}=\frac{3}{12}\sigma_{{}^{1}P_{1}}+\frac{1}{12}\sigma_{{}^{3}P_{0}}+\frac{3}{12}\sigma_{{}^{3}P_{1}}+\frac{5}{12}\sigma_{{}^{3}P_{2}}$
$\sigma_{D}=\frac{5}{20}\sigma_{{}^{1}D_{2}}+\frac{3}{20}\sigma_{{}^{3}D_{1}}+\frac{5}{20}\sigma_{{}^{3}D_{2}}+\frac{7}{20}\sigma_{{}^{3}D_{3}}$

##  3 Results and Discussion:

Table 1: Model parameters of Deng-Fan Potential for various partial waves of S, P and D states of n-p scattering. States | $V_{1}(fm^{-2})$ | $V_{2}(fm^{-2})$ | $\alpha(fm^{-1})$
---|---|---|---
${}^{1}S_{0}$ | 10.8939 | 22.2299 | 1.9327
${}^{3}S_{1}$ | 8.0239 | 9.8525 | 1.4519
${}^{1}P_{1}$ | 1.0541 | 0.6111 | 0.5748
${}^{3}P_{0}$ | 6.8077 | 41.5311 | 1.5338
${}^{3}P_{1}$ | 0.0100 | 5.2844 | 0.9784
${}^{3}P_{2}$ | 9.8750 | 12.5749 | 2.4215
${}^{1}D_{2}$ | 4.5626 | 0.01 | 1.7670
${}^{3}D_{1}$ | 0.0100 | 1.4396 | 0.5073
${}^{3}D_{2}$ | 6.2594 | 6.1205 | 1.3185
${}^{3}D_{3}$ | 29.8481 | 450.0802 | 2.9657

The model parameters for various channels of S, P and D-states are given in
Table LABEL:table1. For ${}^{3}S_{1}$ ground state, we have utilised the
energy condition given by Eq. 5 and substituted $E_{B}=2.224549~{}MeV$ for
binding energy of deuteron[21]. So, choosing parameters $V_{2}$ and $\alpha$
in Deng- fan potential, one can obtain $V_{1}$. Only two parameters need to be
optimised for obtaining corresponding interaction potential. Hence, it is
possible to incorporate the energy condition while determining SPS using PFM.
This is contrary to one of the conclusions drawn by Saha et.al.,[18], where
they claim that Jost function method (JFM) has supremacy over PFM due to the
fact that the parameters of the potential are determined by ﬁtting proper
binding energies for the system. Further, they also claim that PFM gives
little discrepancies in SPS as compared to JFM. But, our phase-wave analysis
using PFM for both ${}^{3}S_{1}$ and ${}^{1}S_{0}$ states matches expected
data[13] for not only upto 50 MeV but all the way up to 350 MeV. The obtained
interaction potentials are shown in Fig. 1(a) and corresponding SPS for
S-states are plotted along with data taken from Perez et.al.,[13] in Fig.
1(b).

Figure 1: (a) ${}^{3}S_{1}$ and ${}^{1}S_{0}$ potentials and (b) Corresponding
scattering phase shifts

One can observe that potentials for triplet and singlet states look similar
except for their depth of interaction, which is expected, due to different
contributions from their spin-spin interactions. This gives us confidence that
the procedure to fit different parameters for different partial waves results
in appropriate potentials.

Figure 2: (a) P-wave interaction potentials and (b) Corresponding scattering
phase shifts Figure 3: (a) D-wave interaction potentials (b) Corresponding
scattering phase shifts

The methodology is then extended to study SPS for P and D waves wherein one
expects spin-orbit interaction to play an important role. The centrifugal
terms for P and D states are taken care of in respective phase equations for
$\ell=1$ and $\ell=2$ in PFM. Typically, spin-orbit term is modeled as
derivative of central potential. Since, DF potential is basically combination
of exponential terms, its derivative would also result in a more complicated
combination of exponential terms and with larger powers for expressions in
their denominators. Certainly, one way of determining model parameters would
be to simultaneously optimise them to fit expected SPS data for all channels.
That would increase the total number of parameters to be simultaneously
optimised and hence the computational cost. Actually, one is interested only
in interaction potential, responsible for observed SPS, for each of the
channels. This would basically be obtained by substituting overall model
parameters in a potential consisting of various contributions from central,
spin, spin-orbit, etc. If the same can be achieved by refitting model
parameters of the phenomenological potential, there is no loss in information.
This procedure of obtaining model parameters has been undertaken using PFM
while studying np, pp, n$\alpha$ and p$\alpha$ systems already[23, 24, 25,
26]. The overall potentials, that include all contributions of underlying
interactions, for various P and D states are shown in Figs. 2(a) and 3(a)
respectively. The obtained SPS, for data up to 350 MeV, are shown in Figs.
2(b) and 3(b) respectively. The observed match between SPS obtained using PFM
and expected SPS[13], very much confirm the points raised in the above
discussion.

Table 2: The differential and total elastic scattering cross-section(SCS):
The $\%$-contribution of each channel to experimental SCS is given in brackets next to differential SCS E |  $\sigma_{exp}$(b) | $\sigma_{{}^{1}S_{0}}$ | $\sigma_{{}^{3}S_{1}}$ | $\sigma_{P}$ | $\sigma_{D}$ |  $\sigma_{sim}$(b)
---|---|---|---|---|---|---
(MeV) | [22] |  |  |  |  |
0.1 | - | 8.727 | 3.661 | 3.75$\times 10^{-7}$ | 2.50$\times 10^{-14}$ | 12.388
0.5 | 6.135 | 3.561(57%) | 2.716(43%) | 9.23$\times 10^{-6}$ | 1.58$\times 10^{-11}$ | 6.276
1 | 4.253 | 2.029 (48%) | 2.247(52%) | 3.62$\times 10^{-5}$ | 2.45$\times 10^{-10}$ | 4.276
10 | 0.9455 | 0.1978(21%) | 0.7395(78%) | 0.0023 | 1.72$\times 10^{-6}$ | 0.9396
50 | 0.1684 | 0.0200 (12%) | 0.1180(70%) | 0.0118(7%) | 0.0004 | 0.1503
100 | 0.07553 | 0.00439(6%) | 0.0334(44%) | 0.01523(20%) | 0.00151(2%) | 0.05453
150 | 0.05224 | 0.00118(2%) | 0.01238(24%) | 0.01597(31%) | 0.00213(4%) | 0.03166
200 | 0.04304 | 0.00027(1%) | 0.00496(12%) | 0.01576(37%) | 0.00236(5%) | 0.02335
250 | 0.03835 | 0.00003 | 0.00193(5%) | 0.01522(40%) | 0.00238(6%) | 0.01955
300 | 0.03561 | 7.02$\times 10^{-6}$ | 0.00064(2%) | 0.0146(41%) | 0.00227(6%) | 0.01751
350 | 0.03411 | 6.88$\times 10^{-5}$ | 0.00013 | 0.01397(41%) | 0.00211(6%) | 0.01629

The partial and total scattering cross-sections are obtained using Eq.11 and
Eq. 12 respectively. The individual contributions due to ${}^{1}S_{0}$ and
${}^{3}S_{1}$ and overall contribution due to P and D waves are given in table
LABEL:table2. One can observe that the contributions from P and D states
become comparable for higher energies. It is seen that the discrepancies
between experimental and observed SCS increases with increasing energy. The
differential SCS for both states of S wave are plotted in Fig. 4(a), with
those of P and D states as inset. The total SCS plot with logarithmic energy
scale is shown in Fig. 4(b), with an inset of contributions from ${}^{3}S_{1}$
and ${}^{1}S_{0}$.
In Fig. 4(a), it is observed that, contribution from ${}^{1}S_{0}$ state is
much larger than the ${}^{3}S_{1}$ state. The contribution from P and D waves
are very less as compared to those from S waves at low energies. The obtained
total cross sections are very well matched with experimental cross sections as
shown in Fig. 4(b). In its inset, one can observe that beyond 1 MeV,
${}^{3}S_{1}$ has greater contribution to total scattering cross-section than
${}^{1}S_{0}$. This is because, while the scattering state has energy close to
zero, about 77 keV, the ground state has energy of 2.2245 MeV.
It would be interesting to see the performance of Deng-Fan potential by
considering all other higher $\ell$-channels of np-interaction. It is also
important to cross-check it’s effectiveness in explaining the experimentally
observed Deuteron properties. In this paper, we have limited the scope of
study to only understand np-scattering through interaction modeled using DF
potential and obtained total cross-sections to validate its effectiveness, in
explaining experimentally observed SCS.

Figure 4: (a) Partial scattering cross-sections for Singlet and Triplet
S-state and those of P and D-states are shown in inset (b) Total elastic
scattering cross-section for n-p interaction.

##  4 Conclusion

The Deng-Fan potential, which is a combination of attractive Hulthen potential
and a repulsive part which is square of the Hulthen term, has the advantage of
having analytical solutions for time independent Schrodinger equation. Being a
combination of Hulthen terms, it should have been utilised as model of
interaction for understanding np scattering. This has been achieved for
S-waves for lab energies up to 50 MeV using Jost function method and parallely
using phase function method[18]. In this work, we have extended the phase wave
analysis, for lab energies up to 350 MeV, by obtaining scattering phase shifts
for not only S-waves but also P and D waves. The total scattering cross-
sections have been obtained by determining partial cross-sections for each of
the S, P and D states and are shown to match very closely with experimental
ones over the entire range of energies. Hence, one can conclude that Deng-Fan
potential is a suitable phenomenological potential to study np-interaction. It
would be interesting to see its performance in studying other scattering
systems such as n-D, p-D, n-$\alpha$, p-$\alpha$, $\alpha-^{3}He$,
$\alpha-^{3}H$ etc.

Acknowledgments
A. Awasthi acknowledges financial support provided by Department of Science
and Technology (DST), Government of India vide Grant No. DST/INSPIRE
Fellowship/2020/IF200538. The authors dedicate this effort to memory of Late
Prof. H.S. Hans, during his birth centenary celebrations.

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# Arc-distinguishing of orientations of graphs

Aleksandra Gorzkowska, Jakub Kwaśny

(AGH University of Krakow)

###### Abstract

A distinguishing index of a (di)graph is the minimum number of colours in an
edge (or arc) colouring such that the identity is the only automorphism that
preserves that colouring. We investigate the minimum and maximum value of the
distinguishing index over all orientations of a given graph $G$. We present
sharp results for these parameters in terms of the distinguishing index of $G$
for trees, unbalanced bipartite graphs, traceable graphs and claw-free graphs.
With this, we answer the question of Meslem and Sopena [8].

##  1 Introduction

We follow the terminology and notation of [10]. We consider edge colourings of
graphs, which are not necessarily proper. We say that a colouring $c\colon
E(G)\to[k]$ _breaks an automorphism_ $\varphi\in\operatorname{Aut}(G)$ if
there exists an edge $xy\in E(G)$ such that $c(\varphi(x)\varphi(y))\neq
c(xy)$. An edge colouring is _distinguishing_ if it breaks all non-trivial
automorphisms of $G$. The _distinguishing index_ of a graph $G$ is the least
number of colours in a distinguishing edge colouring, and it is denoted by
$D^{\prime}(G)$. Clearly, it is not well-defined for $K_{2}$. We consider only
connected graphs other than $K_{2}$.

The study of the distinguishing index was started by Kalinowski and Pilśniak
[4] in 2015 and since then, there have been a number of results on the
subject. In particular, the optimal bounds for the distinguishing index have
been determined, among others, for the classes of traceable graphs [9], claw-
free graphs [2], or regular graphs [7]. A general upper bound of $\Delta(G)$
is known, as well as the classification of graphs satisfying
$D^{\prime}(G)=\Delta(G)$ [9].

Recently, a variant of this problem for digraphs has attracted some interest.
With a notion of an automorphism of a digraph, which preserves the arcs as
well as their direction, we can similarly as above define arc distinguishing
colourings of a digraph, and subsequently the distinguishing index of a
digraph. In particular, the study of symmetric digraphs has been started,
which are constructed from graphs by substituting each edge by a pair of
opposite arcs, see [5, 6].

In 2020, Meslem and Sopena [8] started a study of determining the minimum and
maximum value of distinguishing index among all possible orientations of a
given graph $G$ (we recall that an orientation of a graph $G$ is a digraph
$\overrightarrow{G}$ obtained from $G$ by chosing an orientation,
$\overrightarrow{xy}$ or $\overrightarrow{yx}$, for each edge $xy\in E(G)$).
The corresponding parameters are $OD^{\prime-}(G)$ and $OD^{\prime+}(G)$. They
computed the values of these parameters for paths, cycles, complete graphs and
balanced complete bipartite graphs. We extend their results to some wider
classes of graphs. However, we use a different approach – rather than
computing the specific values of these parameters, we establish a relationship
with the distinguishing index of the underlying graph.

The relationship between the distinguishing index of a graph and of its
orientation is often based on an underlying relationship between their
automorphism groups. Therefore, the following simple observation will be
helpful in our work.

######  Observation 1.

Let $\overrightarrow{G}$ be an orientation of a graph $G$. Then:

  1. (i)

$\operatorname{Aut}(\overrightarrow{G})\subseteq\operatorname{Aut}(G)$,

  2. (ii)

if $\operatorname{Aut}(\overrightarrow{G})=\operatorname{Aut}(G)$, then
$D^{\prime}(\overrightarrow{G})=D^{\prime}(G)$,

  3. (iii)

if $\operatorname{Aut}(\overrightarrow{G})=\\{\operatorname{id}\\}$, then
$D^{\prime}(\overrightarrow{G})=1$.

We say that a set of vertices $S$ of a graph $G$ (or a digraph $D$) is setwise
fixed, if for every vertex $v\in S$ and every automorphism $\varphi$ of $G$
(or $D$) we have $\varphi(v)\in S$. We say that $S$ is pointwise fixed, if for
every vertex $v\in S$ and every automorphism $\varphi$ of $G$ (or $D$) we have
$\varphi(v)=v$. Whenever we say that a vertex $v$ is fixed, we mean $\\{v\\}$
is pointwise fixed.

The paper is organised as follows. In Section 2 we study orientations of
bipartite graphs. We determine the values of $OD^{\prime-}$ and $OD^{\prime+}$
for bipartite graphs with no automorphism that interchanges the partition
classes. In particular, our result answers the question of Meslem and Sopena.
Then, we show that there are only two possible values of $OD^{\prime-}$ and
$OD^{\prime+}$ in the case of trees, and we give an equivalent condition for
determining these values. In Section 3 we study some classes of graphs with
$D^{\prime}(G)=2$ for the existence of a rigid orientation, i.e., whether
there exists an orientation of $G$ that has no non-trivial automorphisms. In
particular, we confirm this for traceable and claw-free graphs.

##  2 Bipartite graphs

In this section, we consider the bipartite graphs. We begin by citing the
result of Meslem and Sopena [8]. We do it only partially, including the
parameters which are of interest to us in this paper.

######  Theorem 2.

[8] For every two integers $m$ and $n$, $2\leq m<n$, the following hold:

  1. 1.

$OD^{\prime+}(K_{m,n})=D^{\prime}(K_{m,n})$.

  2. 2.

If $K_{m,n}$ admits a rigid orientation, then $OD^{\prime-}(K_{m,n})=1$.

  3. 3.

If $K_{m,n}$ does not admit any rigid orientation, then
$OD^{\prime-}(K_{m,n})\leq D^{\prime}(K_{m,\lceil\frac{n}{m-1}\rceil})$.

We expand on these results by considering bipartite graphs in a general
setting, not necessarily the complete graphs. We begin with the following
Lemma, which applies to multipartite graphs with a special condition imposed
on the partition sets. We then draw conclusions for the bipartite graphs. In
particular, the Lemma is applied to unbalanced bipartite graphs, which allows
us to answer the question left by Meslem and Sopena in their paper.

######  Lemma 3.

Let $G=(V,E)$ be a graph. If there exists a partition $V=V_{1}\cup\dots\cup
V_{k}$ into $k\geq 1$ independent sets which are setwise fixed by any
automorphism, then $OD^{\prime+}(G)=D^{\prime}(G)$ and $OD^{\prime-}(G)=\lceil
D^{\prime}(G)/2\rceil$.

###### Proof.

We start with $OD^{\prime+}(G)$. Let $\overrightarrow{G}=(V,A)$ be an
orientation of $G$ such that any arc $\overrightarrow{uv}$, $u\in V_{i}$,
$v\in V_{j}$ is directed such that $i<j$ (note that there are no edges in $G$
with both ends in the same $V_{i}$). We show that
$\operatorname{Aut}(\overrightarrow{G})=\operatorname{Aut}(G)$, which, by
Observation 1, gives us the claim. Assume this is not the case, i.e., that
there is an automorphism $\varphi$ of $G$ which is not an automorphism of
$\overrightarrow{G}$. Then there must exist an arc $\overrightarrow{uv}\in A$,
$u\in V_{i}$, $v\in V_{j}$, $i<j$, such that
$\overrightarrow{\varphi(v)\varphi(u)}\in A$. However, $V_{i}$ and $V_{j}$ are
setwise fixed by $\varphi$, therefore $\varphi(u)\in V_{i}$ and $\varphi(v)\in
V_{j}$, which is a contradiction with the definition of $\overrightarrow{G}$.

We now turn to $OD^{\prime-}(G)$. We shall construct a bijection between the
set of colourings of $G$ and the pairs of the colourings of
$\overrightarrow{G}$ and the directions of the arcs of $\overrightarrow{G}$.
More formally, let $C_{r}=\\{0,1\\}\times\\{1,2,\dots,r\\}$ and
$c:E\rightarrow C_{r}$, $c=(c_{1},c_{2})$ be a colouring of $G$. We associate
with $c_{1}$ an orientation $\overrightarrow{G}$ of $G$ such that any edge
$uv$, $u\in V_{i}$, $v\in V_{j}$, $i<j$, is directed from $u$ to $v$ if
$c_{1}(uv)=0$ and from $v$ to $u$ otherwise. We show that $c$ is a
distinguishing colouring of $G$ if and only if $c_{2}$ is a distinguishing
colouring of $\overrightarrow{G}$.

Assume that $c$ is a distinguishing colouring of $G$ and $c_{2}$ is not a
distinguishing colouring of $\overrightarrow{G}$. Then there is an
automorphism $\varphi$ of $\overrightarrow{G}$ which preserves $c_{2}$.
However, the same automorphism $\varphi$ acting on $G$ would preserve both
$c_{2}$ (by the assumption on $\varphi$) and $c_{1}$ (since $V_{i}$ are
setwise fixed), hence also $c$, which is a contradiction. Conversely, let
$c_{2}$ be a distinguishing colouring of $\overrightarrow{G}$ and take any
$\varphi\in\operatorname{Aut}(G)$. If
$\varphi\in\operatorname{Aut}(\overrightarrow{G})$, then there is an edge $xy$
such that $c_{2}(xy)\neq c_{2}(\varphi(x)\varphi(y))$. If
$\varphi\not\in\operatorname{Aut}(\overrightarrow{G})$, then for some edge
$xy$ the orientation of $xy$ is different from the orientation of
$\varphi(x)\varphi(y)$, hence $c_{1}(xy)\neq c_{1}(\varphi(x)\varphi(y))$. In
both cases, $c$ is a distinguishing colouring of $G$.

For $r=\lceil D^{\prime}(G)/2\rceil$ there exists a distinguishing colouring
$c:E\rightarrow C_{r}$ of $G$, and therefore there exists an orientation
$\overrightarrow{G}$ of $G$ (constructed above) such that
$D^{\prime}(\overrightarrow{G})=r$ which gives us $OD^{\prime-}(G)\leq\lceil
D^{\prime}(G)/2\rceil$. If there was an orientation $\overrightarrow{G}$ such
that $D^{\prime}(\overrightarrow{G})=r<\lceil D^{\prime}(G)/2\rceil$, then the
above construction would yield a distinguishing colouring of $G$ with
$2r<D^{\prime}(G)$ colours, therefore $OD^{\prime-}(G)\geq\lceil
D^{\prime}(G)/2\rceil$.

∎

This lemma gives us an immediate result for the bipartite graphs with
bipartition sets setwise fixed.

######  Corollary 4.

Let $G=(X\cup Y,E)$ be a bipartite graph such that there is no automorphism
that interchanges $X$ and $Y$. Then $OD^{\prime-}(G)=\lceil
D^{\prime}(G)/2\rceil$ and $OD^{\prime+}(G)=D^{\prime}(G)$.

###### Proof.

Take $V_{1}=X$ and $V_{2}=Y$ and apply Lemma 3. ∎

This answers the question of Meslem and Sopena [8] about determining the value
of $OD^{\prime-}(K_{m,n})$ where $n$ is substantially larger than $m$. To give
the full answer, we use the result of Fisher and Isaak [1], and Imrich,
Jerebic and Klavžar [3].

######  Theorem 5.

[1, 3] Let $m,n$ and $r$ be integers such that $r\geq 2$ and $(r-1)^{m}<n\leq
r^{m}$. Then

$D^{\prime}(K_{m,n})=\left\\{\begin{array}[]{ll}r,&\textrm{if }n\leq
r^{m}-\lceil\log_{r}m\rceil-1;\\\ r+1,&\textrm{if }n\geq
r^{m}-\lceil\log_{r}m\rceil+1.\end{array}\right.$

Moreover, if $n=r^{m}-\lceil\log_{r}m\rceil$, then $D^{\prime}(K_{m,n})$ is
either $r$ or $r+1$ and can be computed recursively in time $O(\log^{*}(n))$.

We use this theorem to determine the value of $OD^{\prime-}(K_{m,n})$ in
relation to the sizes of the partition sets.

######  Corollary 6.

Let $m,n$ and $r$ be integers such that $r\geq 2$ and $(r-1)^{m}<n\leq r^{m}$.
Then

$OD^{\prime-}(K_{m,n})=\left\\{\begin{array}[]{ll}\lceil\frac{r}{2}\rceil,&\textrm{if
}n\leq r^{m}-\lceil\log_{r}m\rceil-1;\\\ \lceil\frac{r+1}{2}\rceil,&\textrm{if
}n\geq r^{m}-\lceil\log_{r}m\rceil+1.\end{array}\right.$

Moreover, if $n=r^{m}-\lceil\log_{r}m\rceil$, then $OD^{\prime-}(K_{m,n})$ is
either $\lceil r/2\rceil$ or $\lceil(r+1)/2\rceil$ and can be computed
recursively in time $O(\log^{*}(n))$.

In particular, an unbalanced complete bipartite graph admits a rigid
orientation if and only if $D^{\prime}(K_{m,n})=2$.

We will now devote some attention to a particular family of bipartite graphs,
namely trees. In the context of the distinguishing colourings, one of the
important concepts is the _center_ of a graph, which in the case of trees
consists of a single vertex, or two vertices joined by an edge. It is easy to
see that the center of any graph is setwise fixed by any automorphism.

Since trees are bipartite graphs, Corollary 4 applies to them. In this
particular case, the assumptions of Corollary 4 can be reformulated using the
notion of a center of a graph.

######  Corollary 7.

Let $T$ be a tree with either a central vertex, or a central edge, which is
fixed pointwise by any automorphism. Then $OD^{\prime-}(T)=\lceil
D^{\prime}(T)/2\rceil$ and $OD^{\prime+}(T)=D^{\prime}(T)$.

The remainder of this section will be devoted to cases that are not covered by
Corollary 7. It will require some additional concepts, which we will now
introduce.

Let $T$ be a tree which does not satisfy the assumptions of Corollary 7.
Therefore, $T$ has a central edge $e$, and there exists automorphism which
interchange the end-vertices of $e$. Therefore, $T-e$ consists of two
isomorphic connected components, which are subtrees of $T$. Denote by
$(T^{\prime},r)$ a rooted tree isomorphic with these subtrees, with an end of
the central edge $e$ as a root.

The automorphism group of a rooted tree $(T^{\prime},r)$ consists of these
automorphisms of $T^{\prime}$ which fix $r$. The distinguishing index
$D^{\prime}((T^{\prime},r))$ of a rooted tree is the least number of colours
in an edge colouring, which breaks all non-trivial automorphisms of
$(T^{\prime},r)$. We call any such colouring which uses
$D^{\prime}((T^{\prime},r))$ colours an _optimal distinguishing colouring_.

We call two edge colourings $c_{1},c_{2}$ of a rooted tree $(T^{\prime},r)$
_isomorphic_ if there exists an automorphism $\varphi$ of $(T^{\prime},r)$
such that for every edge $xy$ of $G$ we have
$c_{2}(xy)=c_{1}(\varphi(x)\varphi(y))$. If no such automorphism exists, we
call the colourings _non-isomorphic_. We will be interested in the number of
non-isomorphic optimal distinguishing colourings of rooted trees.

######  Theorem 8.

Let $T$ be a tree of order $n\geq 3$ which does not satisfy the assumptions of
Corollary 7. Then $OD^{\prime+}(T)=D^{\prime}(T)$ and $OD^{\prime-}(T)=\lceil
D^{\prime}(T)/2\rceil$, if $(T^{\prime},r)$ has two non-isomorphic optimal
distinguishing colourings, and $OD^{\prime+}(T)=D^{\prime}(T)-1$ and
$OD^{\prime-}(T)=\lceil(D^{\prime}(T)-1)/2\rceil$, otherwise.

###### Proof.

Let $e$ be the central edge of $T$ and let $(T^{\prime},r)$ be a rooted tree
isomorphic with the components of $T-e$. In any orientation of $T$, the fact
that the central edge $e$ is directed makes both connected components of $T-e$
fixed setwise and the ends of $e$ fixed pointwise.

If $(T^{\prime},r)$ has two non-isomorphic optimal distinguishing colourings,
then $D^{\prime}(T)=D^{\prime}((T^{\prime},r))$. Then, the natural bipartition
of $T^{\prime}$ gives us a partition of $V(T^{\prime})$ into two independent
sets, and since $r$ is fixed, these sets are also setwise fixed by any
automorphism. Therefore, we can apply Lemma 3 and claim that there exists an
orientation $\overrightarrow{T^{\prime}}$ of $(T^{\prime},r)$ such that
$D^{\prime}(\overrightarrow{T^{\prime}})=D^{\prime}((T^{\prime},r))$. We use
that orientation on both components of $T-e$ and direct $e$ arbitrarily to
construct an orientation of $T$ with the distinguishing index of
$D^{\prime}(T)$. The same reasoning using Lemma 3 gives us the claim about
$OD^{\prime-}(T)$.

In the other case, note that $D^{\prime}(T)=D^{\prime}((T^{\prime},r))+1$,
since if both copies of $(T^{\prime},r)$ receive isomorphic distinguishing
colourings, there is an automorphism which interchanges the copies and
preserves the colouring. However, any such automorphism is not an automorphism
of any orientation of $T$. The remainder of the proof follows again from Lemma
3. ∎

Note that the class of rooted trees $(T^{\prime},r)$ which have a unique (up
to an automorphism) distinguishing colouring with $D^{\prime}((T^{\prime},r))$
colours is large. For example, start with any rooted tree $(T_{0},r_{0})$ and
any number $k\geq D^{\prime}((T_{0},r_{0}))$, then take $k$ times as many
copies of $(T_{0},r_{0})$ as there are non-isomorphic distinguishing
colourings of $(T_{0},r_{0})$ with $k$ colours and connect the root of each
copy by an edge to a new vertex $r$. The constructed tree rooted at $r$
belongs to the discussed class. Since the problem of finding all such trees is
not related to digraphs, we leave the following question for further
consideration.

Question. Characterise all rooted trees $(T^{\prime},r)$, which have a unique
(up to an automorphism) distinguishing colouring with
$D^{\prime}((T^{\prime},r))$ colours.

##  3 Graphs with $D^{\prime}(G)=2$

In this section, we investigate a few classes of graphs which are known to
have a distinguishing index equal two. A naive approach would suggest that if
two colours are enough to break all non-trivial automorphisms, then two
directions on the edges would also suffice and such graphs have a rigid
orientation. Surprisingly, this is indeed true for the classes of graphs we
consider.

We first study traceable graphs. Pilśniak [9] proved that any traceable graph
$G$ of order at least seven has $D^{\prime}(G)\leq 2$. As shown in the
following theorem, these graphs have a rigid orientation. Moreover, traceable
graphs with smaller order than seven are also included in our reasoning.

######  Theorem 9.

For any traceable graph $G$, $OD^{\prime-}(G)=1$.

###### Proof.

Take a Hamiltonian path in $G$ and orient all the edges of $G$ from the vertex
with a smaller index on the path to the vertex with a larger index on that
path. In that orientation, each vertex has a unique number of vertices
achievable by a path, which is an isomorphism invariant. Therefore,
constructed orientation has no non-trivial automorphism. ∎

Now, we devote some attention to the properties of the automorphisms of a
graph. Let $G$ be a graph and $\varphi\in\operatorname{Aut}(G)$. We call
$\varphi$ _twisted_ if there is a positive integer $n$ such that $\varphi^{n}$
has a transposition which interchanges two end-vertices of an edge, and _non-
twisted_ , otherwise. We shall see that no such automorphism is present in the
automorphism group of any orientation of $G$.

######  Theorem 10.

Let $G$ be a graph such that $D^{\prime}(G)=2$. Then $OD^{\prime+}(G)=2$ if
$G$ has a non-trivial, non-twisted automorphism. Otherwise,
$OD^{\prime-}(G)=OD^{\prime+}(G)=1$.

###### Proof.

We first claim that a twisted automorphism $\varphi$ of $G$ cannot be an
automorphism of $\overrightarrow{G}$ for any orientation $\overrightarrow{G}$
of $G$. Otherwise, there would exist some power
$\varphi^{n}\in\operatorname{Aut}(\overrightarrow{G})$ of that automorphism
that interchanges two neighbouring vertices and cannot preserve the
orientation of the arc between these vertices. Therefore, if there is no non-
trivial, non-twisted automorphism in $\operatorname{Aut}(G)$, then
$\operatorname{Aut}(\overrightarrow{G})=\\{\operatorname{id}\\}$ for any
orientation $\overrightarrow{G}$ of $G$, and consequently,
$OD^{\prime-}(G)=OD^{\prime+}(G)=1$.

Now assume that $G$ has a non-trivial, non-twisted automorphism $\varphi$. It
suffices to show that there exists an orientation $\overrightarrow{G}$ of $G$
such that $\varphi\in\operatorname{Aut}(\overrightarrow{G})$. Note that
$\varphi$ induces a permutation $\varphi^{\prime}$ on the set
$A(G)=\\{(u,v):u,v\in V(G),uv\in E(G)\\}$. We note that for every edge $uv$
there are two pairs $(u,v)$ and $(v,u)$ in the set $A(G)$. Since $\varphi$ is
non-twisted, these pairs are in different cycles of the permutation
$\varphi^{\prime}$. Moreoever, for each pair $(u^{\prime},v^{\prime})$ which
belongs to the same cycle of $\varphi^{\prime}$ as $(u,v)$, the pair
$(v^{\prime},u^{\prime})$ belongs to the cycle with $(v,u)$. We call the
cycles that contain $(u,v)$ and $(v,u)$ mirror cycles. We take a cycle
decomposition of $\varphi^{\prime}$ and consider its cycles one by one,
assigning an orientation for all the edges in the cycle which is compatible
with $\varphi^{\prime}$ (i.e. if we already assigned for $(u,v)\in A(G)$ an
orientation $\overrightarrow{uv}$, then for
$(u^{\prime},v^{\prime})=\varphi^{\prime}((u,v))$ we assign an orientation
$\overrightarrow{u^{\prime}v^{\prime}}$). This can be done with no conflict
for each of the cycles. For otherwise, the number of steps in the cycle
leading to the conflict would define the integer $n$ such that $\varphi^{n}$
interchanges the end-vertices of some edge. If we encounter a cycle with an
edge that is already directed, then it is a mirror cycle of some other cycle
that was already considered and all the edges in this cycle are already
oriented correctly. This way, we construct an orientation $\overrightarrow{G}$
of $G$ such that $\varphi\in\operatorname{Aut}(\overrightarrow{G})$ and
therefore $OD^{\prime+}(G)=2$. ∎

Another known result about the distinguishing index is by Gorzkowska et al.
[2] who proved that any connected claw-free graph $G$ of order at least six
has $D^{\prime}(G)\leq 2$. They proposed a greedy algorithm that constructs a
desired colouring. We adapt this algorithm to show that each such graph has a
rigid orientation.

We define a path cover of a graph $G$ to be a set of paths
$\mathcal{P}=\\{P_{i}\colon i\in I\\}$ such that every vertex of $G$ belongs
to exactly one path from the chosen set. For each of the paths, we choose one
of its end-vertices and call it a first vertex of this path. A minimal path
cover of the graph $G$ is a path cover whose number of paths is the smallest.

We shall use the following lemmas from [2] which are provided there as Lemma 5
and Claim 13.

######  Lemma 11 ([2]).

Let $G$ be a connected claw-free graph and let $xy$ be an edge of $G$. If
$A\subset N(x)$ and $B\subset N(x)\setminus N[y]$, then:

  1. 1.

There exists a path cover of $G[A]$ with at most two paths.

  2. 2.

There exists a path cover of $G[B]$ with one path.

######  Lemma 12 ([2]).

Let $G$ be a connected claw-free graph of order at least six and let $C$ be
the longest cycle of $G$. Then there is a vertex $s\in V(C)$ such that
$N(s)\subseteq V(C)$.

We will show that for every claw-free graph $G$ of sufficiently large order,
there exists an orientation $\overrightarrow{G}$ such that
$\operatorname{Aut}(\overrightarrow{G})=\\{\operatorname{id}\\}$. This proves
the following theorem.

######  Theorem 13.

If $G$ is a connected, claw-free graph of order at least six, then
$OD^{\prime-}(G)=1$.

###### Proof.

First, assume that $G$ is 2-connected. Therefore, $G$ has a cycle of length at
least four. Let $C$ be the longest cycle in $G$. If all vertices of $G$ lie on
$C$, then $G$ is traceable and the claim follows from Theorem 9. Otherwise,
there exists a vertex $u$ outside $C$ which has a neighbour $v$ on $C$. Since
$G$ is claw-free and $C$ is the longest cycle, the two neighbours of $v$ on
$C$ must be adjacent. Therefore, $C$ has at least one chord.

From Lemma 12 there exists a vertex in $V(C)$ such that its neighbourhood is
contained in $C$. We denote this vertex $v_{1}$ and let
$V(C)=\\{v_{1},v_{2},v_{3},\ldots v_{n}\\}$. We orient the edges of $C$ to
obtain an oriented cycle. Then, we orient the remaining edges between the
vertices of $C$ from the smaller to the larger number. This breaks all the
symmetries of $C$. We will ensure that $C$ remains the only directed cycle of
length $||C||$ in the resulting orientation of $G$.

We define two sets of vertices: the ones that we have reached ($R$) and the
ones which we have processed ($P$). At the beginning, let $R=V(C)$ and
$P=\emptyset$. We note that in the process, all vertices in $R$ will be
adjacent to already oriented edges and all the vertices in $V\setminus R$ will
not be adjacent to any oriented edges. We orient the edges of $G$ recursively.
In the first step, we take $v_{1}$, and we add $v_{1}$ to $P$. Note that all
the neighbours of $v_{1}$ are already in $R$. The step of the recursion starts
with taking the vertex $v$ from $R\setminus P$ with the smallest label. Each
time we choose a vertex from $R\setminus P$ with no neighbours outside $R$, we
add it to $P$ and proceed with the next vertex. Otherwise, by Lemma 11, the
subgraph induced by $N[v]\setminus R$ is traceable. It is true for $v_{2}$,
since $v_{1}$ and its entire neighbourhood is in $R$. We will make sure it is
true in further steps as well. We orient all the edges from the preceding to
the following vertex on the Hamiltonian path. Moreover, we orient the edges
from $v$ towards its neighbours in $N(v)\setminus R$. The step concludes with
adding $v$ to $P$ and adding all the vertices of $N(v)\setminus R$ to $R$,
labelling them with consecutive integers from the first vertex of the
Hamiltonian path to the last one. This way we ensure that at each point in our
procedure the subgraph of $G$ induced by the first $k$ vertices is connected
for every $k\leq|G|$. Therefore, at each step of the recursion the vertex $v$
has a neighbour $v^{\prime}$ which has already been processed. We repeat the
step until there are no vertices in $R\setminus P$. Since the graph is
connected, the process terminates when $P=V$. In each step, we orient the
edges adjacent to vertices that did not have any edges oriented before the
step in a way that does not create any oriented cycle.

After the process has terminated, there may still remain some edges without a
given orientation. We orient them one by one, so as not to create any oriented
cycle. Note that it is possible, assuming that the only oriented cycle before
this part of the algorithm consisted only of the vertices of $C$. Indeed, if
for some edge $xy$ any orientation would create a cycle, that would mean there
were two oriented paths from $x$ to $y$ and from $y$ to $x$, which together
would form a previously existing oriented cycle. The only such cycle could
consist only of the vertices of $C$, but $xy$ is not a chord of $C$ (since all
chords of $C$ were given an orientation at the beginning), a contradiction.

We show that the orientation of $G$ we have created has no non-trivial
automorphisms. Since $C$ is the only oriented cycle with length $||C||$ and we
have broken all the symmetries of this cycle, then every vertex of $C$ is
fixed. Moreover, we claim that if $v$ chosen in any step is fixed, then after
this step, all the vertices from $N(v)\setminus R$ are also fixed in any
orientation of $G$ that agrees on the already oriented edges. Indeed, let
$\varphi$ be an automorphism of any such orientation of $G$ and $u\in
N(v)\setminus R$. Then $\varphi(u)$ cannot be any other vertex from
$N(v)\setminus R$, as each such vertex has a different length of the longest
path from $v$. Therefore, $\varphi(u)\not\in N(v)\setminus R$ which means that
it must lie in $R$ and therefore has been reached before through some other
vertex $v^{\prime}\in P$. However, $v^{\prime}$ is fixed by $\varphi$.
Therefore, $v^{\prime}u\in E(G)$ which is a contradiction, since $u$ must have
been reached before $v$ was processed.

Now consider the case when $G$ is not 2-connected. Consider a 2-connected
component $B$ of $G$ that contains only one cut-vertex $v$ (there must be one,
since the block and cut-vertex graph of $G$ is a tree which has a leaf). Let
$u$ be a neighbour of $v$ in $B$. Then $G-u$ is a connected graph, claw-free
graph. Consider the neighbourhood of $v$ in that graph. It either can be
covered by two paths, first in $B$ and second in the other 2-connected
component containing $v$; or by one path in the other 2-connected component
containing $v$. We orient the edges between the vertices of $N[v]$ from $v$ to
its neighbours and then along these paths. Then we set $R=N[v]$, $P=\\{v\\}$
and repeat the step of the algorithm as described in the previous case. At the
end we orient the edges incident to $u$ so that $u$ is a source.

We shall now verify that $u$ is the only source in the resulting oriented
graph. The vertex $v$ has an incoming arc from $u$. The neighbours of $v$ have
incoming arcs from $v$. Every other vertex in $G$ was at some point added to
$R$, and in that step it received an incoming arc from the currently processed
vertex. So $u$ is the only source, and it is therefore fixed by all
automorphisms. Then $v$ is also fixed as the only cut-vertex adjacent to $u$.
Even if $v$ has two paths covering its neighbours, one of these paths is in
the 2-connected component containing $u$ so they cannot be interchanged by an
automorphism. The rest of the reasoning is the same as in the case of $G$
being 2-connected. Which concludes the proof.

∎

##  4 Conclusions

We have determined the exact value of the parameters $OD^{\prime-}(G)$ and
$OD^{\prime+}(G)$ in terms of the distinguishing index of $G$ for unbalanced
bipartite graphs, trees, traceable graphs and claw-free graphs. It looks like
a well-chosen orientation may reduce the number of required colours by half,
especially in the situation where it is possible to objectively decide which
direction of a given edge is called ,,left”, and which is ,,right”. However,
we postulate that this reduction in the number of colours cannot be greater.

######  Conjecture 14.

If $G$ is a connected graph, then $OD^{\prime-}(G)\geq\lfloor
D^{\prime}(G)/2\rfloor$.

In particular, this would imply that any graph with a rigid orientation has
the distinguishing index at most three.

The section about the graphs with distinguishing index equal two leads us to
another conjecture.

######  Conjecture 15.

If $G$ is a connected graph with $D^{\prime}(G)=2$, then $OD^{\prime-}(G)=1$.

Both conjectures are supported with our results for trees, traceable graphs
and claw-free graphs.

Another open question is about the values of $OD^{\prime+}$ and $OD^{\prime-}$
for balanced bipartite graphs, which have an automorphism that interchanges
the bipartition sets. The results in this paper only cover the case if such a
graph is a tree, or if it is traceable, or claw-free.

## References

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