# Strange hidden-charm tetraquarks in constituent quark models

Xin Jin1<EMAIL_ADDRESS>Xuejie Liu2<EMAIL_ADDRESS>Yaoyao Xue1
<EMAIL_ADDRESS>Hongxia Huang1<EMAIL_ADDRESS>(Corresponding
author) Jialun Ping1<EMAIL_ADDRESS>(Corresponding author) 1Department of
Physics, Nanjing Normal University, Nanjing 210023, P.R. China  2School of
Physics, Southeast University, Nanjing 210094, P. R. China

###### Abstract

Inspired by the newly reported $Z_{cs}(3985)^{-}$ by the BESIII Collaboration,
we systematically investigate the strange hidden-charm tetraquark systems
$cs\bar{c}\bar{u}$ with two structures: meson-meson and diquark-antidiquark.
Two quark models: the chiral quark model (ChQM) and the quark delocalization
color screening model (QDCSM) are used here. Similar results are obtained in
both two quark models. There is no any bound state in either ChQM or QDCSM,
which excludes the molecular state explanation
($D_{s}D^{*}/D_{s}^{*}D/D_{s}^{*}D^{*}$) of the reported $Z_{cs}(3985)^{-}$.
However, the effective potentials for the diquark-antidiquark
$cs\bar{c}\bar{u}$ systems shows the possibility of some resonance states with
mass range of $3916.5\sim 3964.6$ MeV for $IJ^{P}=\frac{1}{2}0^{+}$,
$4008.8\sim 4091.2$ MeV for $IJ^{P}=\frac{1}{2}1^{+}$, $4246.8\sim 4418.1$ MeV
for $IJ^{P}=\frac{1}{2}2^{+}$. So the observed $Z_{cs}(3985)^{-}$ state is
possible to be explained as a compact resonance state composed of
$cs\bar{c}\bar{u}$ with $IJ^{P}=\frac{1}{2}0^{+}$ or
$IJ^{P}=\frac{1}{2}1^{+}$. The study of the scattering process of
corresponding open channels is under way to check this conclusion.

###### pacs:

13.75.Cs, 12.39.Pn, 12.39.Jh

##  I Introduction

In 2013, the BESIII Collaboration studied the $e^{+}e^{-}\rightarrow
J/\psi\pi^{+}\pi^{-}$ process at a center-of-mass energy of 4.26 GeV and
reported a new charged charmonium-like structure in the $\pi^{\pm}J/\psi$
invariant spectrum, which is called $Z_{c}(3900)$ and has a mass of $3899.0\pm
3.6\pm 4.9$ MeV and a width of $46\pm 10\pm 20$ MeV Ablikim:2013mio . At the
same time, the Belle observed a $Z_{c}(3895)\pm$ state, with a mass of
$3894.5\pm 6.6\pm 4.5$ MeV and a width of $63\pm 24\pm 26$ MeV in the process
$Y(4260)\rightarrow J/\psi\pi^{+}\pi^{-}$ Liu:2013dau . The mass and width of
$Z_{c}(3900)$ and $Z_{c}(3895)\pm$ are very close within errors, so they are
the same state Liu:2013nqa . The state $Z_{c}(3900)$ has been further
confirmed by the CLEO-c Collaboration in the the decay $Y(4160)\rightarrow
J/\psi\pi^{+}\pi^{-}$ Xiao:2013iha . Subsequently, there is a lot of
theoretical work to study $Z_{c}(3900)$ and its associated states. Some
researches treated it as the tightly bound diquark-antidiquark state
Dias:2013xfa ; Wang:2013vex ; Deng:2014gqa ; Agaev:2016dev ; Liu:2019gmh ;
Chen:2019osl . However, many researches treated it as molecular states
Wilbring:2013cha ; Dong:2013iqa ; Gutsche:2014zda ; Chen:2015igx ;
Esposito:2014hsa ; Gong:2016hlt ; Ke:2016owt . Some indicated that the
$Z_{c}(3900)$ is not a usual resonance but a threshold cusp Swanson:2014tra ;
Ikeda:2016zwx . Lattice QCD method is also used to find $Z_{c}(3900)$, but it
has not been found Prelovsek:2013xba ; Prelovsek:2013sxa ; Chen:2014afa ;
Prelovsek:2014swa .

Very recently, the BESIII Collaboration reported their study of the processes
of $e^{+}e^{-}\rightarrow K^{+}(D_{s}^{-}D^{*0}+D_{s}^{*-}D^{0})$, and found a
new structure $Z_{cs}(3985)^{-}$ near the $D_{s}^{-}D^{*0}/D_{s}^{*-}D^{0}$
thresholds in the $K^{+}$ recoil-mass spectrum for events collected at
$\sqrt{s}=4.681$ GeV. The pole mass and width of this state are
($3982.5^{+1.8}_{-2.6}\pm 2.1$) Mev and ($12.8^{+5.3}_{-4.4}\pm 3.0$) Mev,
respectively Ablikim:2020hsk . From the production mode, it is easy to get
that the minimum quark component of $Z_{cs}(3985)^{-}$ is $c\bar{c}s\bar{q}$
($q=u/d$), and this state should be a partner structure of the well-known
$Z_{c}(3885)^{-}$ reported in $e^{+}e^{-}\rightarrow D^{*-}D^{0}\pi^{+}$
Ablikim:2014 . Besides, it is the first candidate of the charged hidden-charm
tetraquark state with strangeness, whose discovery can provide more hints to
the quest of charged exotic $Z$ structures. Therefore, the observation of
$Z_{cs}(3985)^{-}$ immediately stimulated a lot of theoretical discussions
Wang:2020kej ; Meng:2020ihj ; Liu:2020nge ; Yang:2020nrt ; Chen:2020yvq ;
Du:2020vwb ; Cao:2020cfx ; Sun:2020hjw ; Rossi:2020ezg ; Wang:2020rcx .

Actually, theoretical predictions of the charged hidden-charm tetraquark with
strangeness have been made in different models Lee:2009 ; Voloshin:2019 ;
Ferretti:2020 . D.Ebert used relativistic quark model based on the
quasipotential approach to calculate the mass spectra of tetraquarks
$[Qs][\bar{Q}\bar{q}]/[Qq][\bar{Q}\bar{s}](Q=c,b)$ Ebert:2005nc . They found
that all $S-$wave tetraquarks with hidden bottom lie considerably below open
bottom thresholds and they should be narrow states which can be observed
experimentally. However hidden-charm tetraquark states all above open charm
thresholds. Dianyong Chen indicated that there exist enhancement structures
with both hidden-charm and open-strange decays, which are near the
$D\bar{D^{*}_{s}}/D^{*}\bar{D_{s}}$ and
$D^{*}\bar{D^{*}_{s}}/\bar{D^{*}}D^{*}_{s}$ thresholds under the initial
single chiral particle emission (ISChE) mechanism Chen:2013wca . Chengrong
Deng studied the same charged tetraquark states using the variational method
GEM in the color flux-tube model with a four-body confinement potential. The
numerical results indicated that some compact resonance states can be formed
Deng:2014gqa .

Strong interaction is the strongest of the four interactions in nature, but
understanding its nature has always been a difficult problem in physics.
Quantum chromodynamics (QCD) is widely accepted as the basic theory of strong
interactions. QCD can deal with scattering problems by perturbation expansion
in high energy regions, but the spontaneous chiral symmetry breaking and color
confinement appear in low energy regions. To study hadron-hadron interactions
and multiquark states, many quark models based on QCD theory have been
developed. The chiral quark model (ChQM) is one of the accepted models
Salamanca1 . The interactions in this model include the colorful one-gluon-
exchange and confinement, colorless Goldstone boson exchange and the chiral
partner $\sigma$ meson-exchange. Another approach is the quark delocalization
color screening model (QDCSM), which was developed to study the similarities
between nuclear and molecular forces Wang:1992wi . Both of these two models
describe the properties of deuteron, nucleon-nucleon and hyperon-nucleon
interactions well ChenLZ ; ChenM . Very recently, both ChQM and QDCSM have
been used to explain the full-heavy tetraquark states Jin:2020jfc  and the
reported $X(6900)$ by LHCb collaboration Aaij:2020fnh  could be explained as a
compact resonance state in both two models. It is quite natural to extend the
study to the charged hidden-charm tetraquark systems with strangeness. So we
investigate the tetraquark systems composed of $c\bar{c}s\bar{q}$ ($q=u/d$) in
both ChQM and QDCSM in present work.

The structure of this paper is as follows. section II gives a brief
introduction of two quark models, and the construction of wave functions. The
numerical results and discussions are given in Section III. The summary is
presented in the last section.

##  II Models and wavefunctions

In this work, we investigate the charged charmonium-like tetraquarks with
hidden-charm and open-strange $c\bar{c}s\bar{q}$ ($q=u/d$) within two quark
models: ChQM and QDCSM. Two structures: meson-meson and diquark-antidiquark,
are considered. In this sector, we will introduce these two models and the
wave functions of the tetraquarks for two structures.

###  II.1 The chiral quark model (ChQM)

The ChQM has been successfully applied to describe the properties of hadrons
and hadron-hadron interactions Salamanca1 ; Salamanca2 . The model details can
be found in Ref. Salamanca1 ; Salamanca2 . We only show the Hamiltonian of the
model here.

$\displaystyle H={}$
$\displaystyle\sum_{i=1}^{4}(m_{i}+\frac{p_{i}^{2}}{2m_{i}})-T_{cm}+\sum_{i=1<j}^{4}(V^{CON}_{ij}+V^{OGE}_{ij})$
(1)

where $T_{cm}$ is the kinetic energy of the center of mass; $V^{CON}_{ij}$ and
$V^{OGE}_{ij}$ are the interactions of the confinement and the one-gluon-
exchange, respectively. For the $c\bar{c}s\bar{u}$ system, there is no
$\sigma$-exchange interactions, because the $\sigma$ meson cannot be exchanged
between $u/d$ quark and $s/c$ quark. The forms of $V^{CON}_{ij}$ and
$V^{OGE}_{ij}$ are shown below:

$\displaystyle V^{CON}_{ij}$
$\displaystyle=-\boldsymbol{\lambda}^{c}_{i}\cdot\boldsymbol{\lambda}^{c}_{j}(a_{c}r^{2}_{ij}+V_{0_{ij}})$
(2) $\displaystyle V^{OGE}_{ij}$
$\displaystyle=\frac{\alpha_{s_{ij}}}{4}\boldsymbol{\lambda}^{c}_{i}\cdot\boldsymbol{\lambda}^{c}_{j}[\frac{1}{r_{ij}}-\frac{\pi}{2}\delta(\mathbf{r}_{ij})(\frac{1}{m_{i}^{2}}+\frac{1}{m_{j}^{2}}+\frac{4\boldsymbol{\sigma}_{i}\cdot\boldsymbol{\sigma}_{j}}{3m_{i}m_{j}})$
$\displaystyle-\frac{3}{4m_{i}m_{j}r_{ij}^{3}}S_{ij}]$ (3) $\displaystyle
S_{ij}$
$\displaystyle=\\{3\frac{(\boldsymbol{\sigma}_{i}\cdot\boldsymbol{r}_{ij})(\boldsymbol{\sigma}_{j}\cdot\boldsymbol{r}_{ij})}{r_{ij}^{2}}-\boldsymbol{\sigma}_{i}\cdot\boldsymbol{\sigma}_{j}\\}$
(4)

where $S_{ij}$ is quark tensor operator; $\alpha_{s_{ij}}$ is the quark-gluon
coupling constant.

###  II.2 The quark delocalization color screening model (QDCSM)

Generally, the Hamiltonian of QDCSM is almost the same as that of ChQM, but
with two modifications Wang:1992wi . The one is that there is no
$\sigma$-meson exchange in QDCSM, and another one is that the screened color
confinement is used between quark pairs reside in different clusters, aiming
to take into account the QCD effect which has not yet been included in the
two-body confinement. The confining potential in QDCSM was modified as
follows:

$V^{CON}_{ij}=\left\\{\begin{array}[]{rcl}-\boldsymbol{\lambda}^{c}_{i}\cdot\boldsymbol{\lambda}^{c}_{j}(a_{c}r^{2}_{ij}+V_{0_{ij}})&&\text{i,j
in the same cluster}\\\
-\boldsymbol{\lambda}^{c}_{i}\cdot\boldsymbol{\lambda}^{c}_{j}a_{c}\frac{1-e^{-\mu_{ij}r^{2}_{ij}}}{\mu_{ij}}&&\text{,
otherwise}\\\ \end{array}\right.$

where $\mu_{ij}$ is the color screening parameter, which is determined by
fitting the deuteron properties, $NN$ scattering phase shifts, and $N\Lambda$
and $N\Sigma$ scattering phase shifts, respectively, with
$\mu_{uu}=0.45~{}$fm-2, $\mu_{us}=0.19~{}$fm-2 and $\mu_{ss}=0.08~{}$fm-2,
satisfying the relation, $\mu_{us}^{2}=\mu_{uu}\mu_{ss}=0.19~{}$fm-2 ChenM .
When extending to the heavy quark case, there is no experimental data
available, so we take it as a adjustable parameter $\mu_{cc}=0.01\sim
0.001~{}$fm-2 and we find the results are insensitive to the value of
$\mu_{cc}$. So in the present work, we take $\mu_{cc}=0.01~{}$fm-2,
$\mu_{uc}=0.067~{}$fm-2 and $\mu_{sc}=0.0283~{}$fm-2.

The single particle orbital wave functions in the ordinary quark cluster model
are the left and right centered single Gaussian functions:

$\displaystyle\phi_{\alpha}(\mathbf{S}_{i})=\left(\frac{1}{\pi
b^{2}}\right)^{\frac{3}{4}}e^{-\frac{(\mathbf{r}-\mathbf{S}_{i}/2)^{2}}{2b^{2}}},$
$\displaystyle\phi_{\beta}(-\mathbf{S}_{i})=\left(\frac{1}{\pi
b^{2}}\right)^{\frac{3}{4}}e^{-\frac{(\mathbf{r}+\mathbf{S}_{i}/2)^{2}}{2b^{2}}}.$
(5)

The quark delocalization in QDCSM is realized by writing the single particle
orbital wave function as a linear combination of the left and right Gaussians:

$\displaystyle{\psi}_{\alpha}(\mathbf{S}_{i},\epsilon)$ $\displaystyle=$
$\displaystyle\left({\phi}_{\alpha}(\mathbf{S}_{i})+\epsilon{\phi}_{\alpha}(-\mathbf{S}_{i})\right)/N(\epsilon),$
$\displaystyle{\psi}_{\beta}(-\mathbf{S}_{i},\epsilon)$ $\displaystyle=$
$\displaystyle\left({\phi}_{\beta}(-\mathbf{S}_{i})+\epsilon{\phi}_{\beta}(\mathbf{S}_{i})\right)/N(\epsilon),$
$\displaystyle N(\epsilon)$ $\displaystyle=$
$\displaystyle\sqrt{1+\epsilon^{2}+2\epsilon e^{{-S}_{i}^{2}/4b^{2}}}.$ (6)

where $\epsilon(\mathbf{S}_{i})$ is the delocalization parameter determined by
the dynamics of the quark system rather than adjusted parameters. In this way,
the system can choose its most favorable configuration through its own
dynamics in a larger Hilbert space.

The parameters used in our previous work are determined by fitting the mass
spectrum of mesons and baryons including light quarks ($u,~{}d,~{}s$), but the
mesons composed of heavy quarks like $\eta_{b}$($\eta_{c}$) or
$\Upsilon$($J/\psi$) do not fit well. To give the right mass of the mesons we
used in this work, we adjust the parameters by fitting the masses of mesons
that we need. Since there is no $\sigma$-exchange interactions in ChQM for the
$c\bar{c}s\bar{u}$ system and the delocalization and color screening in QDCSM
work in quarks of different clusters, there is no discrepancy of these two
models in describing mesons. So the parameters of two quark models are the
same, which are shown in Table 1. The calculated masses of the mesons are
shown in Table 2.

Table 1: Model parameters. State |  $b$ (fm) | 0.3
---|---|---
|  $m_{u}$ (MeV) | 313
|  $m_{s}$ (MeV) | 536
|  $m_{c}$ (MeV) | 1728
Confinement |  $a_{c}$ (MeV fm-2) | 101
|  $V_{0_{us}}$ (MeV) | -180.6
|  $V_{0_{uc}}$ (MeV) | -133.6
|  $V_{0_{sc}}$ (MeV) | -68.1
|  $V_{0_{cc}}$ (MeV) | 76
OGE | $\alpha_{s_{us}}$ | 0.33
| $\alpha_{s_{uc}}$ | 0.38
| $\alpha_{s_{uc}}$ | 0.66
| $\alpha_{s_{uc}}$ | 1.67
Table 2: The caculated masses (in MeV) of the mesons. Experimental values are taken from the Particle Data Group(PDG). | K | $K^{*}$ | $\eta_{c}$ | $J/\psi$ | $D$ | $D^{*}$ | $D_{s}$ | $D_{s}^{*}$
---|---|---|---|---|---|---|---|---
Exp. | 494 | 892 | 2984 | 3097 | 1865 | 2007 | 1968 | 2112
Model | 495 | 892 | 2984 | 3097 | 1865 | 2007 | 1968 | 2112

###  II.3 The wave function

In this work, we study a bound-state problem by a well-established method, the
resonating group method (RGM) RGM . In order to calculate the energy of the
$c\bar{c}s\bar{u}$ system, we first constructed the four-quark wave function
in the following form:

$\Psi=\mathcal{A}[[\psi^{L}\chi^{\sigma}]_{JM}\chi^{f}\chi^{c}].$ (7)

where $\psi^{L},\chi^{\sigma},\chi^{f}$ and $\chi^{c}$ are the orbital, spin,
flavor and color wave functions, respectively, which are given below. The
symbol $\mathcal{A}$ is the anti-symmetrization operator. For the
$c\bar{c}s\bar{u}$ system, $\mathcal{A}=1$ because the quarks are not
identical particles in the $SU(3)$ symmetry and no anti-symmetrization
requirement is needed here.

####  II.3.1 The orbital wave function

The total orbital wave function is composed of two internal cluster orbital
wave functions ($\psi_{1}(\mathbf{R}_{1})$ and $\psi_{2}(\mathbf{R}_{2})$),
and one relative motion wave function ($\chi_{L}(\mathbf{R})$) between two
clusters.

$\psi^{L}=\psi_{1}(\mathbf{R}_{1})\psi_{2}(\mathbf{R}_{2})\chi_{L}(\mathbf{R})$
(8)

where $\mathbf{R}_{1}$ and $\mathbf{R}_{2}$ are the internal coordinates for
the cluster 1 and cluster 2, respectively.
$\mathbf{R}=\mathbf{R}_{1}-\mathbf{R}_{2}$ is the relative coordinate between
the two clusters 1 and 2. $\chi_{L}(\mathbf{R})$ is expanded by gaussian bases

$\displaystyle\chi_{L}(\mathbf{R})={}$
$\displaystyle\frac{1}{\sqrt{4\pi}}(\frac{3}{2\pi b^{2}})\sum_{i=1}^{n}C_{i}$
$\displaystyle\times\int\exp[-\frac{3}{4b^{2}}(\mathbf{R}-\mathbf{s}_{i})^{2}]Y_{LM}(\hat{s}_{i})d\hat{s_{i}}$
(9)

where $\mathbf{s}_{i}$ is the generate coordinate, $n$ is the number of the
gaussian bases, which is determined by the stability of the results. By doing
this is expansion, we can simplify the integro-differential equation to an
algebraic equation, solve this generalized eigen-equation to get the energy of
the system more easily. The details of solving the RGM equation can be found
in Ref RGM .

####  II.3.2 The flavor wave function

The flavor wave function for the $c\bar{c}s\bar{u}$ system is very simple. For
the meson-meson structure,

$\displaystyle\chi_{m}^{f1}=[c\bar{c}][s\bar{u}]$ (10)
$\displaystyle\chi_{m}^{f2}=[c\bar{u}][s\bar{c}]$ (11)

where the superscript of the $\chi$ is the index of the flavor wave function
for meson-meson structure, and the subscript stands for the isospin $I$ and
the third component $I_{z}$.

For the diquark-antidiquark structure,

$\chi_{d}^{f1}=cs\bar{c}\bar{u}$ (12)

The upper and lower indices are similar to those of the meson-meson structure.

####  II.3.3 The spin wave function

For the spin part, the wave functions for two-body clusters are:

$\displaystyle\chi^{1}_{\sigma 11}=\alpha\alpha\qquad\chi^{2}_{\sigma
10}=\sqrt{\frac{1}{2}}(\alpha\beta+\beta\alpha)$
$\displaystyle\chi^{3}_{\sigma 1-1}=\beta\beta\qquad\chi^{4}_{\sigma
00}=\sqrt{\frac{1}{2}}(\alpha\beta-\beta\alpha)$ (13)

Then, the total spin wave functions for the four-quark system can be obtained
by coupling the wave functions of two clusters.

$\displaystyle\chi^{\sigma 1}_{00}$ $\displaystyle=\chi^{4}_{\sigma
00}\chi^{4}_{\sigma 00}$ $\displaystyle\chi^{\sigma 2}_{00}$
$\displaystyle=\sqrt{\frac{1}{3}}(\chi^{1}_{\sigma 11}\chi^{3}_{\sigma
1-1}-\chi^{2}_{\sigma 10}\chi^{2}_{\sigma 10}+\chi^{3}_{\sigma
1-1}\chi^{1}_{\sigma 10})$ $\displaystyle\chi^{\sigma 3}_{11}$
$\displaystyle=\chi^{4}_{\sigma 00}\chi^{1}_{\sigma 11}$
$\displaystyle\chi^{\sigma 4}_{11}$ $\displaystyle=\chi^{1}_{\sigma
11}\chi^{4}_{\sigma 00}$ $\displaystyle\chi^{\sigma 5}_{11}$
$\displaystyle=\sqrt{\frac{1}{2}}(\chi^{1}_{\sigma 11}\chi^{2}_{\sigma
10}-\chi^{2}_{\sigma 10}\chi^{1}_{\sigma 11})$ $\displaystyle\chi^{\sigma
6}_{22}$ $\displaystyle=\chi^{1}_{\sigma 11}\chi^{1}_{\sigma 11}$ (14)

The spin wave function of two structures is the same.

####  II.3.4 The color wave function

For the meson-meson structure, we give the wave functions for the two-body
clusters first, which are

$\displaystyle\chi^{1}_{c[111]}=\sqrt{\frac{1}{3}}(r\bar{r}+g\bar{g}+b\bar{b})$
(15) $\displaystyle\chi^{2}_{c[21]}=r\bar{b}\qquad\chi^{3}_{c[21]}=-r\bar{g}$
$\displaystyle\chi^{4}_{c[21]}=g\bar{b}\qquad\chi^{5}_{c[21]}=-b\bar{g}$
$\displaystyle\chi^{6}_{c[21]}=g\bar{r}\qquad\chi^{7}_{c[21]}=b\bar{r}$
$\displaystyle\chi^{8}_{c[21]}=\sqrt{\frac{1}{2}}(r\bar{r}-g\bar{g})$
$\displaystyle\chi^{9}_{c[21]}=\sqrt{\frac{1}{6}}(-r\bar{r}-g\bar{g}+2b\bar{b})$
(16)

where the subscript $[111]$ and $[21]$ stand for the color singlet and color
octet cluster respectively.

Then, the total color wave functions for the four-quark system with the meson-
meson structure can be obtained by coupling the wave functions of two
clusters.

$\displaystyle\chi^{c1}_{m}$
$\displaystyle=\chi^{1}_{c[111]}\chi^{1}_{c[111]}$ (17)
$\displaystyle\chi^{c2}_{m}$
$\displaystyle=\sqrt{\frac{1}{8}}(\chi^{2}_{c[21]}\chi^{7}_{c[21]}-\chi^{4}_{c[21]}\chi^{5}_{c[21]}-\chi^{3}_{c[21]}\chi^{6}_{c[21]}$
$\displaystyle+\chi^{8}_{c[21]}\chi^{8}_{c[21]}-\chi^{6}_{c[21]}\chi^{3}_{c[21]}+\chi^{9}_{c[21]}\chi^{9}_{c[21]}$
$\displaystyle-\chi^{5}_{c[21]}\chi^{4}_{c[21]}+\chi^{7}_{c[21]}\chi^{2}_{c[21]})$
(18)

where $\chi^{c1}_{m}$ and $\chi^{c2}_{m}$ represent the color wave function
for the color-singlet channel ($1\times 1$) and the hidden-color channel
($8\times 8$), respectively.

For the diquark-antidiquark structure, we firstly give the color wave
functions of the diquark clusters,

$\displaystyle\chi^{1}_{c[2]}=rr\qquad\chi^{2}_{c[2]}=\sqrt{\frac{1}{2}}(rg+gr)\qquad\chi^{3}_{c[2]}=gg$
$\displaystyle\chi^{4}_{c[2]}=\sqrt{\frac{1}{2}}(rb+br)\qquad\chi^{5}_{c[2]}=\sqrt{\frac{1}{2}}(gb+bg)$
$\displaystyle\chi^{6}_{c[2]}=bb\qquad\chi^{7}_{c[11]}=\sqrt{\frac{1}{2}}(rg-
gr)$ $\displaystyle\chi^{8}_{c[11]}=\sqrt{\frac{1}{2}}(rb-
br)\qquad\chi^{9}_{c[11]}=\sqrt{\frac{1}{2}}(gb-bg)$ (19)

and the color wave functions of the antidiquark clusters,

$\displaystyle\chi^{1}_{c[22]}=\bar{r}\bar{r}\qquad\chi^{2}_{c[22]}=\sqrt{\frac{1}{2}}(\bar{r}\bar{g}+\bar{g}\bar{r})\qquad\chi^{3}_{c[22]}=\bar{g}\bar{g}$
$\displaystyle\chi^{4}_{c[22]}=\sqrt{\frac{1}{2}}(\bar{r}\bar{b}+\bar{b}\bar{r})\qquad\chi^{5}_{c[22]}=\sqrt{\frac{1}{2}}(\bar{g}\bar{b}+\bar{b}\bar{g})$
$\displaystyle\chi^{6}_{c[22]}=\bar{b}\bar{b}\qquad\chi^{7}_{c[211]}=\sqrt{\frac{1}{2}}(\bar{r}\bar{g}-\bar{g}\bar{r})$
$\displaystyle\chi^{8}_{c[211]}=\sqrt{\frac{1}{2}}(\bar{r}\bar{b}-\bar{b}\bar{r})\qquad\chi^{9}_{c[211]}=\sqrt{\frac{1}{2}}(\bar{g}\bar{b}-\bar{b}\bar{g})$
(20)

After that, the total wave functions for the four-quark system with the
diquark-antidiquark structure are obtained as below,

$\displaystyle\chi^{c1}_{d}$
$\displaystyle=\sqrt{\frac{1}{6}}[\chi^{1}_{c[2]}\chi^{1}_{c[22]}-\chi^{2}_{c[2]}\chi^{2}_{c[22]}+\chi^{3}_{c[2]}\chi^{3}_{c[22]}$
$\displaystyle+\chi^{4}_{c[2]}\chi^{4}_{c[22]}-\chi^{5}_{c[2]}\chi^{5}_{c[22]}+\chi^{5}_{c[2]}\chi^{5}_{c[22]}]$
(21) $\displaystyle\chi^{c2}_{d}$
$\displaystyle=\sqrt{\frac{1}{3}}[\chi^{7}_{c[11]}\chi^{7}_{c[211]}-\chi^{8}_{c[11]}\chi^{8}_{c[211]}+\chi^{9}_{c[11]}\chi^{9}_{c[211]}]$
(22)

Finally, we can acquire the total wave functions by substituting the wave
functions of the orbital, the spin, the flavor and the color parts into the
Eq.(7) according to the given quantum number of the system.

##  III Numerical results and discussions

In this work, we investigate the charged hidden-charm tetraquark systems with
strangeness in two structures, meson-meson and diquark-antidiquark. We take
into account all the possible quantum numbers for the $S-$wave
$cs\bar{c}\bar{u}$ systems, which are $IJ^{P}=\frac{1}{2}0^{+}$,
$\frac{1}{2}1^{+}$, and $\frac{1}{2}2^{+}$. For the meson-meson structure, we
take into account of two color configurations which are the color singlet-
singlet$(1\times 1)$ and color octet-octet ($8\times 8$) configurations in the
ChQM, while in the QDCSM, we only consider the color singlet-singlet$(1\times
1)$ configuration, since this model considers the effect of the hidden color
channel coupling to some extent PRC84064001 . For the diquark-antidiquark
structure, two color configurations, antitriplet-triplet ($\bar{3}\times 3$)
and sextet-antisextet ($6\times\bar{6}$), are considered in both models.

To find any bound states of the $cs\bar{c}\bar{u}$ systems, we carry out a
dynamical bound-state calculation. The energies of both the single channel and
the channel-coupling calculations are obtained. Table 3 shows the results of
the meson-meson structure, and Table 4 shows the results of the diquark-
antidiquark structure. In the tables, the column headed with
$[\chi^{\sigma_{i}}\chi^{f_{j}}\chi^{c_{k}}]$ denotes the combination in spin,
flavor and color degrees of freedom for each channel, respectively. The
columns headed with $E_{th}$ denotes the theoretical threshold of each channel
and $E_{sc}$ represents the lowest energies in the single channel calculation.
For meson-meson structure, the column ”Channel” represents the physical
contents of the channel. The tetraquarks composed of $c\bar{c}s\bar{u}$ can be
combined in two ways: $[c\bar{c}][s\bar{u}]$ and $[c\bar{u}][s\bar{c}]$. In
each combination, $E_{cc1}$ and $E_{cc2}$ denote the lowest energies of the
coupling of the color-singlet channels and the coupling of all channels
(including color-singlet channels and hidden-color channels). The coupling of
the two combinations is shown in $E_{cc3}$. For diquark-antidiquark structure,
$E_{cc1}$ and $E_{cc2}$ denote the lowest energies of the coupling of the
$6\times\bar{6}$ color configuration and the coupling of all channels
(including $6\times\bar{6}$ and $\bar{3}\times 3$ color configurations). All
the general features of the calculated results are as follows.

###  III.1 Meson-meson structure

For the meson-meson structure, $\eta_{c}K$ and $\eta_{c8}K_{8}$ in Table 3
represent the color singlet-singlet$(1\times 1)$ and color octet-octet
($8\times 8$) configurations, respectively. From Table 3 we can see that the
energies of every single channel are above the corresponding theoretical
threshold in both two models. The channel-coupling calculation cannot help too
much, and the energies by the channel-coupling are still above the theoretical
threshold, which indicates that the effect of the channel-coupling is very
small here. This is mainly due to the large energy difference between each
single channel. As a result, in the meson-meson structure, there is no any
bound states in either ChQM or QDCSM. So the reported $Z_{cs}(3985)^{-}$
cannot be explained as $D_{s}D^{*}/D_{s}^{*}D/D_{s}^{*}D^{*}$ molecular state
in present calculation.

Table 3: The energies (in MeV) of $c\bar{c}s\bar{u}$ systems with the meson-meson structure in the ChQM and QDCSM. |  |  |  | ChQM | QDCSM
---|---|---|---|---|---
$IJ^{P}$ | $[\chi^{\sigma_{i}}\chi^{f_{j}}\chi^{c_{k}}]$ | Channel | $E_{th}$ | $E_{sc}$ | $E_{cc1}$ | $E_{cc2}$ | $E_{cc3}$ | $E_{sc}$ | $E_{cc1}$ | $E_{cc3}$
$\frac{1}{2}0^{+}$ | $\chi^{\sigma 1}_{00}\chi_{m}^{f1}\chi^{c1}_{m}$ | $\eta_{c}K$ | 3478.2 | 3484.9 | 3484.9 | 3484.9 | 3484.9 | 3484.9 | 3484.9 | 3484.9
| $\chi^{\sigma 2}_{00}\chi_{m}^{f1}\chi^{c1}_{m}$ | $J/\psi K^{*}$ | 3988.1 | 3994.8 |  |  |  | 3994.1 |  |
| $\chi^{\sigma 1}_{00}\chi_{m}^{f1}\chi^{c2}_{m}$ | $\eta_{c8}K_{8}$ |  | 4542.6 |  |  |  |  |  |
| $\chi^{\sigma 2}_{00}\chi_{m}^{f1}\chi^{c2}_{m}$ | $J/\psi_{8}K^{*}_{8}$ |  | 4363.8 |  |  |  |  |  |
| $\chi^{\sigma 1}_{00}\chi_{m}^{f2}\chi^{c1}_{m}$ | $DD_{s}$ | 3833.4 | 3840.6 | 3840.6 | 3838.3 |  | 3840.1 | 3840.1 |
| $\chi^{\sigma 2}_{00}\chi_{m}^{f2}\chi^{c1}_{m}$ | $D^{*}D_{s}^{*}$ | 4119.5 | 4126.2 |  |  |  | 4121.2 |  |
| $\chi^{\sigma 1}_{00}\chi_{m}^{f2}\chi^{c2}_{m}$ | $D_{8}D_{s8}$ |  | 4395.5 |  |  |  |  |  |
| $\chi^{\sigma 2}_{00}\chi_{m}^{f2}\chi^{c2}_{m}$ | $D^{*}_{8}D_{s8}^{*}$ |  | 4125.3 |  |  |  |  |  |
$\frac{1}{2}1^{+}$ | $\chi^{\sigma 3}_{11}\chi_{m}^{f1}\chi^{c1}_{m}$ | $\eta_{c}K^{*}$ | 3875.2 | 3881.9 | 3597.8 | 3597.8 | 3597.8 | 3881.6 | 3597.8 | 3597.7
| $\chi^{\sigma 4}_{11}\chi_{m}^{f1}\chi^{c1}_{m}$ | $J/\psi K$ | 3591.1 | 3597.8 |  |  |  | 3597.8 |  |
| $\chi^{\sigma 5}_{11}\chi_{m}^{f1}\chi^{c1}_{m}$ | $J/\psi K^{*}$ | 3988.1 | 3994.8 |  |  |  | 3994.1 |  |
| $\chi^{\sigma 3}_{11}\chi_{m}^{f1}\chi^{c2}_{m}$ | $\eta_{c8}K^{*}_{8}$ |  | 4493.0 |  |  |  |  |  |
| $\chi^{\sigma 4}_{11}\chi_{m}^{f1}\chi^{c2}_{m}$ | $J/\psi_{8}K_{8}$ |  | 4528.5 |  |  |  |  |  |
| $\chi^{\sigma 5}_{11}\chi_{m}^{f1}\chi^{c2}_{m}$ | $J/\psi_{8}K^{*}_{8}$ |  | 4423.6 |  |  |  |  |  |
| $\chi^{\sigma 3}_{11}\chi_{m}^{f2}\chi^{c1}_{m}$ | $DD_{s}^{*}$ | 3977.7 | 3984.4 | 3982.4 | 3980.2 |  | 3983.1 | 3981.1 |
| $\chi^{\sigma 4}_{11}\chi_{m}^{f2}\chi^{c1}_{m}$ | $D^{*}D_{s}$ | 3975.7 | 3982.4 |  |  |  | 3981.1 |  |
| $\chi^{\sigma 5}_{11}\chi_{m}^{f2}\chi^{c1}_{m}$ | $D^{*}D_{s}^{*}$ | 4119.5 | 4126.2 |  |  |  | 4121.2 |  |
| $\chi^{\sigma 3}_{11}\chi_{m}^{f2}\chi^{c2}_{m}$ | $D_{8}D_{s8}^{*}$ |  | 4377.5 |  |  |  |  |  |
| $\chi^{\sigma 4}_{11}\chi_{m}^{f2}\chi^{c2}_{m}$ | $D^{*}_{8}D_{s8}$ |  | 4377.7 |  |  |  |  |  |
| $\chi^{\sigma 5}_{11}\chi_{m}^{f2}\chi^{c2}_{m}$ | $D^{*}_{8}D_{s8}^{*}$ |  | 4247.2 |  |  |  |  |  |
$\frac{1}{2}2^{+}$ | $\chi^{\sigma 6}_{11}\chi_{m}^{f1}\chi^{c1}_{m}$ | $J/\psi K^{*}$ | 3988.1 | 3994.8 |  | 3994.8 | 3994.8 | 3994.1 |  | 3994.1
| $\chi^{\sigma 6}_{11}\chi_{m}^{f1}\chi^{c2}_{m}$ | $J/\psi_{8}K^{*}_{8}$ |  | 4529.6 |  |  |  |  |  |
| $\chi^{\sigma 6}_{11}\chi_{m}^{f2}\chi^{c1}_{m}$ | $D^{*}D_{s}^{*}$ | 4119.5 | 4126.2 |  | 4124.0 |  | 4121.2 |  |
| $\chi^{\sigma 6}_{11}\chi_{m}^{f2}\chi^{c2}_{m}$ | $D^{*}_{8}D_{s8}^{*}$ |  | 4462.0 |  |  |  |  |  |

To study the interaction between two mesons, we carry out the adiabatic
calculation of the effective potentials for the $c\bar{c}s\bar{u}$ system. For
the color singlet channels, the effective potentials in both two quark models
are shown in Fig. 1 and Fig. 2, respectively. In ChQM, all effective
potentials are repulsive, and that’s why we cannot get any bound state. In
QDCSM, although the effective potential of every channel is attractive, it is
not strong enough to form any bound state. For the hidden color channel, we
only gives the potential of ChQM in Fig. 3, because the effect of the hidden
color channel coupling is considered to some extent in QDCSM. From Fig. 3 we
can see that the minimum potential of each channel appears at the separation
of $0.3$ or $0.4$ fm, which indicates that two colorful subclusters are not
willing to huddle together or fall apart, so it is possible to form some
resonance states here. However, the energy of the hidden color channel listed
in Table 3 is among $4.1\sim 4.6$ GeV, which indicates that these resonances
may not be suitable to explain the observed $Z_{cs}(3985)^{-}$. The scattering
process of the corresponding open channels should be studied to confirm if
there is any resonance state or not.

Figure 1: The effective potentials of the color singlet channels for the
meson-meson $c\bar{c}s\bar{u}$ systems in ChQM. Figure 2: The effective
potentials of the color singlet channels for the meson-meson
$c\bar{c}s\bar{u}$ systems in QDCSM. Figure 3: The effective potentials of the
hidden-color channels for the meson-meson $c\bar{c}s\bar{u}$ systems in ChQM.

We also try to investigate why the effective potentials of the two models are
different in the color singlet channels. The Hamiltonian of the ChQM and the
QDCSM consists of mass, the kinetic energy ($V_{VK}$), the confinement
($V_{CON}$), the Coulomb interaction ($V_{Coul}$) and the color-magnetic
interaction ($V_{CMI}$). We take the results of the $IJ^{P}=\frac{1}{2}0^{+}$
$\eta_{c}K$ channel in meson-meson structure as an example in Fig. 4. We can
see that the kinetic energy term in the ChQM provides repulsion, while the one
in QDCSM provides attractive interactions. In the ChQM, the confinement
($V_{CON}$), the Coulomb interaction ($V_{Coul}$) and the color-magnetic
interaction ($V_{CMI}$) do not contribute to the effective potential between
two color singlet clusters $c\bar{c}$ and $s\bar{u}$, so the interaction
between the two mesons is only affected by the kinetic energy term($V_{VK}$).
There is no exchange terms between $c\bar{c}$ and $s\bar{u}$ in ChQM, so the
confinement interaction, the Coulomb interaction and the color-magnetic
interaction don’t work between $c\bar{c}$ and $s\bar{u}$. Therefore, there is
no any term which provides attractive interaction between two color singlet
clusters in ChQM, which leads to absence of bound states in this system. In
QDCSM, although the quark delocalization is largely weakened because of the
heavy quarks in the system, it still affects the kinetic energy, which
provides attractive interactions. The other interaction terms are also
affected by the quark delocalization and provide repulsive interaction to this
system. That’s why the total effective potentials are possible to be
attractive in QDCSM, although it is still difficult to form any bound state.

Figure 4: The contributions to the effective potential of the $IJ^{P}=00^{+}$
$\eta_{c}K$ channel from various terms of interactions in the ChQM and QDCSM.

###  III.2 Diquark-antidiquark structure

Table 4: The energies (in MeV) of $cs\bar{c}\bar{u}$ systems with the diquark-antidiquark structure in the ChQM and QDCSM. |  |  | ChQM | QDCSM
---|---|---|---|---
$IJ^{P}$ | $[\chi^{\sigma_{i}}\chi^{f_{j}}\chi^{c_{k}}]$ | $E_{th}$ | $E_{sc}$ | $E_{cc1}$ | $E_{cc2}$ | $E_{sc}$ | $E_{cc1}$ | $E_{cc2}$
$\frac{1}{2}0^{+}$ | $\chi^{\sigma 1}_{00}\chi_{d}^{f1}\chi^{c1}_{d}$ | 3478.2 | 4462.3 | 4161.1 | 3964.6 | 4224.1 | 4034.3 | 3916.5
| $\chi^{\sigma 2}_{00}\chi_{d}^{f1}\chi^{c1}_{d}$ |  | 4171.6 |  |  | 4041.3 |  |
| $\chi^{\sigma 1}_{00}\chi_{d}^{f1}\chi^{c2}_{d}$ |  | 4263.3 |  |  | 4146.6 |  |
| $\chi^{\sigma 2}_{00}\chi_{d}^{f1}\chi^{c2}_{d}$ |  | 4350.1 |  |  | 4178.5 |  |
$\frac{1}{2}1^{+}$ | $\chi^{\sigma 3}_{11}\chi_{d}^{f1}\chi^{c1}_{d}$ | 3591.1 | 4426.8 | 4245.0 | 4091.2 | 4205.3 | 4089.2 | 4008.8
| $\chi^{\sigma 4}_{11}\chi_{d}^{f1}\chi^{c1}_{d}$ |  | 4426.3 |  |  | 4205.0 |  |
| $\chi^{\sigma 5}_{11}\chi_{d}^{f1}\chi^{c1}_{d}$ |  | 4285.7 |  |  | 4116.3 |  |
| $\chi^{\sigma 3}_{11}\chi_{d}^{f1}\chi^{c2}_{d}$ |  | 4334.2 |  |  | 4188.3 |  |
| $\chi^{\sigma 4}_{11}\chi_{d}^{f1}\chi^{c2}_{d}$ |  | 4335.2 |  |  | 4188.8 |  |
| $\chi^{\sigma 5}_{11}\chi_{d}^{f1}\chi^{c2}_{d}$ |  | 4379.1 |  |  | 4203.6 |  |
$\frac{1}{2}2^{+}$ | $\chi^{\sigma 6}_{11}\chi_{d}^{f1}\chi^{c1}_{d}$ | 3988.1 | 4486.5 |  | 4418.1 | 4251.7 |  | 4246.8
| $\chi^{\sigma 6}_{11}\chi_{d}^{f1}\chi^{c2}_{d}$ |  | 4431.1 |  |  | 4251.8 |  |

With regards to the diquark-antidiquark structure, the results of both ChQM
and QDCSM are listed in Tables 4. As shown in the table, the diquark-
antidiquark structure has higher energy than the meson-meson structure so
there is no any bound state. Besides, the energies in QDCSM are generally
lower than that in ChQM. Since the color symmetry of the diquark and
antidiquark are color octet, the color screening will make the quark
delocalization work in QDCSM, which leads to lower energy in this model. We
also find that the channel coupling of the diquark-antidiquark structure makes
the energy a little lower, but the energy of each state is still higher than
the corresponding threshold. However, since the confinement potential requires
that the colorful subclusters diquark and antidiquark cannot fall apart
directly, resonance states are possible in this configuration. We perform an
adiabatic calculation to determine the possibility of the existence of any
resonance state, the results of which are shown in Fig. 5.

Obviously, in both ChQM and QDCSM, the energy of each single channel will rise
when the two subclusters are too close, so there is a hinderance for the state
changing structure to meson-meson even if the energy of the diquark-
antidiquark state is higher than the meson-meson state. So it is possible to
form a resonance state. The minimum energy of each channel appears at the
separation of $0.3\sim 0.4$fm in ChQM, the one appears at the separation of
$0.6\sim 0.7$fm in QDCSM, which indicates that the diquark and antidiquark
subclusters are close to each other in both two models. Therefore, the
resonance state may be the compact resonance state. According to the Table 4,
after the channel coupling calculation, the lowest resonance energies in ChQM
are $3964.6$ MeV for $IJ^{P}=\frac{1}{2}0^{+}$, $4091.2$ MeV for
$IJ^{P}=\frac{1}{2}1^{+}$, $4418.1$ MeV for $IJ^{P}=\frac{1}{2}2^{+}$; and
those in QDCSM are $3916.5$ MeV for $IJ^{P}=\frac{1}{2}0^{+}$, $4008.8$ MeV
for $IJ^{P}=\frac{1}{2}1^{+}$, and $4246.8$ MeV for $IJ^{P}=\frac{1}{2}2^{+}$.
By comparing the mass of the observed $Z_{cs}(3985)^{-}$, this $Z_{cs}$ state
is possible to be explained as a compact resonance state $c\bar{c}s\bar{u}$
with $IJ^{P}=\frac{1}{2}0^{+}$ or $IJ^{P}=\frac{1}{2}1^{+}$. However, the
scattering process of the corresponding open channels should be studied to
confirm if there is any resonance state or not.

Figure 5: The effective potentials for the diquark-antidiquark
$cs\bar{c}\bar{u}$ systems in two quark models.

##  IV Summary

In this work, we systematically investigate the low-lying charged hidden-charm
tetraquark systems with strangeness in two quark models ChQM and QDCSM. Two
configurations, meson-meson and diquark-antidiquark, are considered. The
dynamical bound-state calculation is carried out to search for any bound state
in the $cs\bar{c}\bar{u}$ systems. To investigate the effect of the channel
coupling, both the single channel and the channel coupling calculation are
performed. Meanwhile, an adiabatic calculation of the effective potentials is
added to study the interactions of the systems and to find any resonance
state.

The results are similar in both two quark models. The bound-state calculation
shows that there is no any bound state in either ChQM or QDCSM, which excludes
the reported $Z_{cs}(3985)^{-}$ as a molecular state
$D_{s}D^{*}/D_{s}^{*}D/D_{s}^{*}D^{*}$. The study of the interaction between
two mesons shows that the confinement interaction, the Coulomb interaction and
the color-magnetic interaction don’t work between two mesons of the
$c\bar{c}s\bar{u}$ systems, because there is no exchange terms between them.
So it is difficult to obtain a molecular state in this $cs\bar{c}\bar{u}$
system in present work. However, the effective potentials for the diquark-
antidiquark $cs\bar{c}\bar{u}$ systems shows the possibility of some resonance
states with mass of $3916.5\sim 3964.6$ MeV for $IJ^{P}=\frac{1}{2}0^{+}$,
$4008.8\sim 4091.2$ MeV for $IJ^{P}=\frac{1}{2}1^{+}$, $4246.8\sim 4418.1$ MeV
for $IJ^{P}=\frac{1}{2}2^{+}$. The newly reported $Z_{cs}$ state is possible
to be explained as a compact resonance state $cs\bar{c}\bar{u}$ with
$IJ^{P}=\frac{1}{2}0^{+}$ or $IJ^{P}=\frac{1}{2}1^{+}$. To confirm this
conclusion, the scattering process of the corresponding open channels is
needed, and the decay width can also be obtained by the scattering cross
section, which is our further work.

###### Acknowledgements.

This work is supported partly by the National Science Foundation of China
under Contract Nos. 11675080, 11775118 and 11535005.

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# On correct structures and similarity measures of soft sets along with
historic comments of Prof. D. A. Molodtsov

Santanu Acharjee1 and Amlanjyoti Oza2
1,2Department of Mathematics
Gauhati University
Guwahati-781014, Assam, India.
e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract

After the paper of Molodtsov [ D. Molodtsov, Soft set theory—first results.
Computers and Mathematics with Applications, 37(4-5) (1999), 19-31], soft set
theory grew at a breakneck pace. Several authors have introduced various
operations, relations, results, as well as other aspects in soft set theory
and hybrid structures incorrectly, despite their widespread use in mathematics
and allied areas. In his paper [D. A. Molodtsov, Equivalence and correct
operations for soft sets. International Robotics and Automation Journal, 4(1)
(2018), 18-21], Molodtsov, the father of soft set theory, pointed out the
wrong results and notions. Molodtsov [D. A. Molodtsov, Structure of soft sets.
Nechetkie Sistemy i Myagkie, 12(1) (2017), 5-18] also stated that the concept
of soft set had not been fully understood and used everywhere. As a result, it
seemed reasonable to revisit the quirks of those conceptions and provide a
formal account of the notion of soft set equivalency. Molodtsov already
explored the correct operations on soft sets. We use some notions and results
of Molodtsov [Molodtsov D.A. (2017) Structure of soft sets. Nechetkie Sistemy
i Myagkie, 12(1) (2017), 5-18] to create matrix representations as well as
certain associated operations of soft sets, and to quantify the similarity
between two soft sets.

2020 AMS Classifications: 03C99, 03B52,03B70, 68T27, 68T09.
Keywords: Soft set, operations of soft sets, matrix representation, similarity
measure.

##  1 Introduction

Molodtsov [1] introduced soft set theory in 1999. It is important to note that
soft set theory has contributed to create a new area of analysis named soft
rational analysis [16]. But, the idea [4] of soft set theory was developed in
1980. Later, various authors e.g. [5, 6, 7, 8, 9, 10, 11, 12, 13] changed
several outcomes that contradicted the accuracy of the ideas and operations
established in [1, 2, 3, 4] and others. In [2], Molodtsov mentioned, “… many
authors have introduced new operations and relations for soft sets and used
these structures in various areas of mathematics and in applied science.
Unfortunately in some works the introduction of operations and relationships
for soft sets were carried out without due regard to the specific of soft set
definition”. In section 2 of [3], the notion of correctness of soft operations
is explicitly stated. Almost all the authors of the previously stated papers
have used the incorrect notion of soft subset of a soft set [2]. Molodtsov
also stated in [2] that the authors of [6] established definitions of
complement, union, and intersection of soft sets incorrectly. Moreover, many
researchers have utilised these incorrect notions in their works. Çağman and
Enginoğlu [9] also defined a new notion of soft set and presented the matrix
form of it, which takes into account a subset of the collection of parameters.
This representation is erroneous because even if we are given the soft set’s
matrix, we cannot determine the soft set along with original set of
attributes. Moreover, this notion has difference with the original notion of
soft sets defined by Molodtsov [1]. In this regard, we want to attract the
attention of the community of soft set researchers to a comment on
ResearchGate [21]. On 15/11/2018, Molodtsov commented on a paper of Al-Qudah
and Hassan [20], which is available in ResearchGate till writing of this
paper. He commented as follows [21]:

“Dear Colleagues, I did not write such a definition of a soft set. You have
added one extra condition that the set of parameters is a subset of a fixed
set.”

We provide screenshot ( figure 1) of this very important comment in the
history of soft set theory for our readers. It was taken by the first author
of this paper through his ResearchGate account.

Figure 1: Comment of Prof. D. Molodtsov on ResearchGate

In [8], the authors considered the sets of attributes of the two soft sets for
measuring similarity between them. But, it is not necessary that the sets of
attributes should always contain some common elements. In [12], Kamacı defined
a formula for measuring similarity, which results, if there is no any common
attribute between the two soft sets then the similarity between them is always
0. During early research career in soft set theory and hybrid structures [14,
15], the first author of this paper also followed wrong notions of soft set
and associated structures, which were developed by others other than
Molodtsov. But, it is not a matter of shame for the first author as he learnt
correct concepts of soft set theory and related structures while working with
Molodtsov in [16] and in other two papers. Since then, he has been trying to
follow the correct paths in soft set theory as set by Molodtsov.

In [3], Molodtsov mentioned, “The principle of constructing correct operations
for soft sets is very simple. Operations should be defined through families of
sets $\tau(S,A)$”. Hence, the results of soft sets must be dependent on the
sets $\tau(F,A)$ and $\tau(G,B)$ of two soft sets $(F,A)$ and $(G,B)$
respectively, but not on the sets of attributes. In this paper, we try to
propose various correct operations of soft sets accurately in matrix forms, as
well as we find a similarity measure between two soft sets that is independent
of the set of parameters. We also consider the condition where the set of
attributes can have varying cardinality and no shared parameters, which
follows philosophy of Molodtsov behind introducing soft set theory [1, 2, 3,
4].

##  2 Preliminaries

In this section, we discuss some preliminary notions of soft set theory and
related results. Let $A$ be a set of parameters and $X$ be a universal set,
over which a soft set is defined. The formal definition of soft set is as
follows:

######  Definition 2.1.

[1] A pair $(S,A)$ will be called a soft set over $X$, if $S$ is a mapping
from the set $A$ to the set of subsets of the set $X$, i.e., $S:A\rightarrow
2^{X}$. In fact, a soft set is a parametrized family of subsets.

If the soft set $(S,A)$ is given, then the family $\tau(S,A)$ can be defined
as $\tau(S,A)=\\{S(a),a\in A\\}$ [3].

######  Definition 2.2.

[3] Two soft sets $(S,A)$ and $(S^{\prime},A^{\prime})$ given over the
universal set $X$ will be called equal and we write
$(S,A)=(S^{\prime},A^{\prime})$ if and only if $S=S^{\prime}$ and
$A=A^{\prime}$.

######  Definition 2.3.

[3] Two soft set $(S,A)$ and $(S^{\prime},A^{\prime})$ given over the
universal set $X$ will be called equivalent and written as
$(S,A)\cong(S^{\prime},A^{\prime})$ if and only if
$\tau(S,A)=\tau(S^{\prime},A^{\prime})$.

It is easy to note that equivalence of soft set is an equivalence relation.

######  Definition 2.4.

[3] A soft set $(S,A)$ internally approximates a soft set $(F,D)$ denoted by
$(S,A)\subseteq(F,D)$ if for any $d\in D$ such that $F(d)\neq\phi$, there
exists $a\in A$ for which $\phi\neq S(a)\subseteq F(D)$.

######  Definition 2.5.

[3] A soft set $(S,A)$ externally approximates a soft set $(F,D)$ denoted by
$(S,A)\supseteq(F,D)$ if for any $d\in D$ such that $F(d)\neq X$, there exists
$a\in A$ for which $X\neq S(a)\supseteq F(D)$.

######  Definition 2.6.

[3] If $(S,A)\subseteq(F,D)$ but the relation $(F,D)\subseteq(S,A)$ has no
place, then we say that the soft set $(S,A)$ internally strictly approximate
the soft set $(F,D)$, denoted by $(S,A)\subset(F,D)$.

######  Definition 2.7.

[3] If $(S,A)\supseteq(F,D)$ but the relation $(F,D)\supseteq(S,A)$ has no
place, then we say that the soft set $(S,A)$ outwardly strictly approximate
the soft set $(F,D)$, denoted by $(S,A)\supset(F,D)$.

######  Definition 2.8.

[3] The soft set $(S,A)$ is internally equivalent to the soft set $(F,D)$,
denoted by $(S,A)\begin{matrix}\subset\\\ \approx\end{matrix}(F,D)$ if
$(S,A)\subseteq(F,D)$ and $(F,D)\subseteq(S,A)$.

######  Definition 2.9.

[3] The soft set $(S,A)$ is externally equivalent to the soft set $(F,D)$,
denoted by $(S,A)\begin{matrix}\supset\\\ \approx\end{matrix}(F,D)$ if
$(S,A)\supseteq(F,D)$ and $(F,D)\supseteq(S,A)$.

######  Definition 2.10.

[3] The soft set $(S,A)$ is weakly equivalent to the soft set $(F,D)$, denoted
by $(S,A)\approx(F,D)$ if $(S,A)\begin{matrix}\subset\\\
\approx\end{matrix}(F,D)$ and $(S,A)\begin{matrix}\supset\\\
\approx\end{matrix}(F,D)$.

######  Definition 2.11.

[3] The minimal and maximal on inclusion of sets of the family $\tau(S,A)$ is
defined as follows.

$MIN(\tau(S,A))=\\{B\in\tau(S,A)|B\neq\phi,\nexists
B^{\prime}\in\tau(S,A):B^{\prime}\subset B\neq B^{\prime}\neq\phi\\}$

$MAX(\tau(S,A))=\\{B\in\tau(S,A)|B\neq X,\nexists
B^{\prime}\in\tau(S,A):B^{\prime}\supset B\neq B^{\prime}\neq X\\}$

######  Definition 2.12.

[3] A relation $\Omega$ is called correct if for any quadruple of pairwise
equivalent soft sets $(S,A)\cong(S^{\prime},A^{\prime})$,
$(F,D)\cong(F^{\prime},D^{\prime})$, given over the universal set $X$, the
following equality is satisfied.

$\Omega((S,A),(F,D))=\Omega((S^{\prime},A^{\prime}),(F^{\prime},D^{\prime}))$

######  Definition 2.13.

[3] The unary operation complement of $(S,A)$, $C(S,A)=(W,A)$ is defined as
follows. The set of parameters remains the same and the mapping is given by
$W(a)=X\setminus S(a)$ for any $a\in A$.

######  Definition 2.14.

[3] The binary operation union $(S,A)\cup(F,D)=(H,A\times D)$ for a pair of
soft sets $(S,A)$ and $(F,D)$ given over a universal set $X$ is defined as
follows: The parameter set is chosen equal to the direct product of the
parameter sets, i.e., equal to $A\times D$, and the corresponding mappings are
given by the formula $H(a,d)=S(a)\cup F(d)$, $(a,d)\in A\times D$.

######  Definition 2.15.

[3] The binary operation intersection $(S,A)\cap(F,D)=(W,A\times D)$ for a
pair of soft sets $(S,A)$ and $(F,D)$ given over a universal set $X$ is
defined as follows: The parameter set is chosen equal to the direct product of
the parameter sets, i.e., equal to $A\times D$, and the corresponding mappings
are given by the formula $W(a,d)=S(a)\cap F(d)$, $(a,d)\in A\times D$.

######  Definition 2.16.

The binary operation product $(S,A)\times(F,D)=(X,A\times D)$ for a pair of
soft sets $(S,A)$ and $(F,D)$ given over a universal set $X$ is defined as
follows. The parameter set is chosen equal to the direct product of the
parameter sets, i.e., equal to $A\times D$ and the corresponding mappings are
given by the formula $X(a,d)=S(a)\times F(d),(a,d)\in A\times D$.

Now, we consider an illustrative example to discuss the above operations.

######  Example 2.1.

Let us consider two soft sets $(F,A)$ and $(G,B)$ over a universal set $X$,
where $A$ and $B$ be two sets of attributes. We consider $X=\\{a,b,c\\}$,
$A$=$\\{x,y,z\\}$, and $B$=$\\{m,n,o\\}$. We define the soft set $(F,A)$ as
follows:

$F(x)=\\{b,c\\}$, $F(y)=\\{c\\}$, $F(z)=\\{a\\}$.

Hence, $\tau(F,A)=\\{\\{b,c\\},\\{c\\},\\{a\\}\\}$. Similarly, we define the
soft set $(G,B)$ as follows:

$G(m)=\\{a\\}$, $G(n)=\\{c\\}$, $G(o)=\\{c\\}$.

Hence, $\tau(G,B)=\\{\\{a\\},\\{c\\}\\}$.
Now. we consider the unary operation complement ‘$C$’ of the soft set $(F,A)$
which is given by $C(F,A)=(CF,A)$, where $CF(a)=X\setminus S(a),\forall a\in
A$. So, $CF(x)=\\{a\\}$, $CF(y)=\\{a,b\\}$, $CF(z)=\\{b,c\\}$. Therefore,
$\tau(CF,A)=\\{\\{a\\},\\{a,b\\},\\{b,c\\}\\}$.
Again, we consider the binary operation union ‘$\bigcup$’ between the soft
sets $(F,A)$ and $(G,B)$ denoted by $(F,A)\bigcup(G,B)=(H,A\times B)$, where
$A\times B$ is the Cartesian product of $A$ and $B$, and the mapping is given
by $H(a,b)=F(a)\cup G(a)$, where $(a,b)\in A\times B$. Thus, $A\times
B=\\{\\{x,m\\},\\{x,n\\},\\{x,o\\},\\{y,m\\},\\{y,n\\},\\{y,o\\},\\{z,m\\},\\{z,n\\},\\{z,o\\}\\}$.
So, we obtain $H(x,m)=\\{a,b,c\\},\\\
H(x,n)=\\{b,c\\},H(x,o)=\\{b,c\\},H(y,m)=\\{a,c\\},H(y,n)=\\{c\\},H(y,o)=\\{c\\},H(z,m)=\\{a\\},H(z,n)=\\{a,c\\}$,
and $H(z,o)=\\{a,c\\}$. Hence, $\tau(H,A\times
B)=\\{\\{a,b,c\\},\\{b,c\\},\\{a,c\\},\\{c\\},\\{a\\}\\}$.
Now, we also consider the binary operation intersection ‘$\bigcap$’ between
$(F,A)$ and $(G,B)$ denoted by $(F,A)\bigcap(G,B)=(W,A\times B)$, and the
mapping is given by $W(a,b)=F(a)\cap G(b)$, where $(a,b)\in A\times B$. Thus,
we get
$W(x,m)=\phi,W(x,n)=\\{c\\},W(x,o)=\\{c\\},W(y,m)=\phi,W(y,n)=\\{c\\},W(y,o)=\\{c\\},W(z,m)=\\{a\\},$
$W(z,n)$ =$\phi$, and $W(z,o)=\phi$. Therefore, $\tau(W,A\times
B)=\\{\\{a\\},\\{c\\}\\}$.
Also, we consider the binary operation product ‘$\times$’ between $(F,A)$ and
$(G,B)$, denoted by $(F,A)\times(G,B)=(X,A\times B)$, and the mapping is given
by $X(a,b)=F(a)\times G(b),$ where $(a,b)\in A\times B$.
$W(x,m)=\\{b,c\\}\times\\{a\\}=\\{(b,a),(c,a)\\}$,
$W(x,n)=\\{b,c\\}\times\\{c\\}=\\{(b,c),(c,c)\\}$,
$W(x,o)=\\{b,c\\}\times\\{c\\}=\\{(b,c),(c,c)\\}$,
$W(y,m)=\\{c\\}\times\\{a\\}=\\{(c,a)\\}$,
$W(y,n)=\\{c\\}\times\\{c\\}=\\{(c,c)\\}$,
$W(y,o)=\\{c\\}\times\\{c\\}=\\{(c,c)\\}$,
$W(z,m)=\\{a\\}\times\\{a\\}=\\{(a,a)\\}$,
$W(z,n)=\\{a\\}\times\\{c\\}=\\{(a,c)\\}$, and
$W(z,o)=\\{a\\}\times\\{c\\}=\\{(a,c)\\}$. Hence, $\tau(X,A\times
B)=\\{\\{(b,a),(c,a)\\},\\{(b,c),(c,c)\\},\\{(c,a)\\},\\{(c,c)\\},\\{(a,a)\\},\\{(a,c)\\}\\}$.

##  3 Matrix representation of soft sets

In this section, we shall discuss the matrix representation of a soft set. A
soft set $(F,A)$ defined over a universal set $X$ can be represented by a
matrix $M$ such that the number of rows of $M$ is equal to the cardinality of
the universal set $X$, and the number of columns of $M$ is equal to the
cardinality of the set of attribute $A$.

We consider the cardinalities of $X$ and $A$ to be finite for the practical
feasibility of computation and other real life purposes. Let us consider
$|X|=n$ and $|A|=m$, where $X=\\{x_{1},x_{2},x_{3},\dots,x_{n}\\}$ and
$A=\\{a_{1},a_{2},a_{3},\dots,a_{m}\\}$. Then, the binary representation table
of $(F,A)$ is given below.

| $F(a_{1})$ | $F(a_{2})$ | $F(a_{3})$ | $\dots$ | $F(a_{m})$
---|---|---|---|---|---
$x_{1}$ | $m_{11}$ | $m_{12}$ | $m_{13}$ | $\dots$ | $m_{1m}$
$x_{2}$ | $m_{21}$ | $m_{22}$ | $m_{23}$ | $\dots$ | $m_{2m}$
$x_{3}$ | $m_{31}$ | $m_{32}$ | $m_{33}$ | $\dots$ | $m_{3m}$
⋮ | ⋮ | ⋮ | ⋮ | $\ddots$ | ⋮
$x_{n}$ | $m_{n1}$ | $m_{n2}$ | $m_{n3}$ | … | $m_{nm}$
Table 1: Binary representation table of soft set $(F,A)$

Here,

$m_{ij}=\begin{cases}1;\>if\>x_{i}\in F(a_{j}),\\\ 0;\>if\>x_{i}\notin
F(a_{j}).\end{cases}$

Moreover, $i=1,2,3,\dots,n$ and $j=1,2,3,\dots,m$. Thus, we get the following
matrix $M$ as a matrix representation of $(F,A)$ from table 1.

$M=\begin{pmatrix}m_{11}&m_{12}&m_{13}&\dots&m_{1m}\\\
m_{21}&m_{22}&m_{23}&\dots&m_{2m}\\\ m_{31}&m_{32}&m_{33}&\dots&m_{3m}\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
m_{n1}&m_{n2}&m_{n3}&...&m_{nm}\end{pmatrix}$

So, every soft set can be transformed into it’s matrix form and if the matrix
of a soft set is given, then, we can uniquely determine the soft set. Here, we
should concern about the order of the elements of $X$ in case of matrix
representation, i.e., two different orderings of the elements of the set $X$
may led to two different representations of the same soft set, which may cause
difficulties to define various operations related to the matrix
representation. Hence, the order of $x_{i}$’s must be mentioned in case of
matrix representation, where $x_{i}\in X$. Let us consider example 2.1 to
illustrate the matrix representation of soft set $(F,A)$.

######  Example 3.1.

Given, $X=\\{a,b,c\\}$, $A=\\{x,y,z\\}$, $F(x)=\\{b,c\\}$, $F(y)=\\{c\\}$, and
$F(z)=\\{a\\}$. Then, the binary representation of $(F,A)$ is given below.

$M=$ $F(x)$ $F(y)$ $F(z)$ $a$ $0$ $0$ $1$ $b$ $1$ $0$ $0$ $c$ $1$ $1$ $0$

Hence, we obtain the following matrix $M$ of the soft set $(F,A)$.

$M=\begin{pmatrix}0&0&1\\\ 1&0&0\\\ 1&1&0\end{pmatrix}$

##  4 Operations on soft sets in matrix form

In this section, we formulate four operations viz. complement, union,
intersection and product of soft sets in matrix form.

###  4.1 Soft complement

Let $M=(x_{ij})_{m\times n}$ be the matrix representation of the soft set
$(F,A)$ defined over a universal set $X$ and order of $M$ is $m\times n$.
Then, $M^{\prime}=(x^{\prime}_{ij})_{m\times n}$ is the matrix representation
of $C(F,A)$, the complement of the soft set $(F,A)$ and it is defined as
below.

$M^{\prime}=\left\\{\begin{array}[]{ll}x^{\prime}_{ij}=1&,\mbox{if
}x_{ij}=0,\\\ x^{\prime}_{ij}=0&,\mbox{if }x_{ij}=1.\end{array}\right.$

Here, $i=1,2,\dots,m$, and $j=1,2,\dots,n$. It’s easy to see that order of the
matrix $M^{\prime}$ is identical to that of order of matrix $M$.

###  4.2 Soft union

Let, $K=(x_{ij})_{m\times n}$ and $L=(y_{ij})_{m\times p}$ be matrix
representations of soft sets $(F,A)$ and $(G,B)$ respectively, defined over a
universal set $X$. Since both the soft sets are defined over $X$, hence the
matrix $K$ and $L$ may have orders $m\times n$ and $m\times p$ respectively,
where $m$, $n$ and $p$ are positive integers and $|n-p|\geq 0$. Now, we
represent union of the two soft sets $(F,A)$ and $(G,B)$ by matrix
representation and we consider $M$ to be the matrix of
$(F,A)\cup(G,B)=(H,A\times B)$. Then, the matrix $M$ is of order
$m\times(n.p)$. Let us define the matrix $M=(m_{ij})_{m\times(n.p)}$ as
follows.

$M=\begin{cases}m_{ij}=max\\{x_{i1},y_{ij}\\},where\>i=1,2,\dots,m\>and\>j=1,2,\dots,p\\\
m_{ij}=max\\{x_{i2},y_{i(j-p)}\\},where\>i=1,2,\dots,m\>and\>j=p+1,p+2,\dots,2p\\\
m_{ij}=max\\{x_{i3},y_{i(j-2p)}\\},where\>i=1,2,\dots,m\>and\>j=2p+1,2p+2,\dots,3p\\\
...................................................................................................................\\\
m_{ij}=max\\{x_{in},y_{i(j-(n-1)p)}\\},where\>i=1,2,\dots,m\>and\>j=(n-1)p+1,(n-1)p+2,\dots,n.p\par\end{cases}$

The matrix $M$ follows the definition of union of two soft sets $(F,A)$ and
$(G,B)$ given in [3].

###  4.3 Soft intersection

Let, $K=(x_{ij})_{m\times n}$ and $L=(y_{ij})_{m\times p}$ be matrix
representations of two soft sets $(F,A)$ and $(G,B)$ respectively, defined
over a universal set $X$. Since both the soft sets are defined over $X$, hence
the matrix $K$ and $L$ may have the orders $m\times n$ and $m\times p$, where
$m$, $n$ and $p$ are positive integers and $|n-p|\geq 0$. Now, we represent
intersection of the two soft sets $(F,A)$ and $(G,B)$ by matrix representation
and we consider $N$ to be the matrix of $(F,A)\cap(G,B)=(H,A\times B)$. Then,
the matrix $N$ is of order $m\times(n.p)$. Let us define the matrix
$N=(n_{ij})_{m\times(n.p)}$ as follows:

$N=\begin{cases}n_{ij}=min\\{x_{i1},y_{ij}\\},where\>i=1,2,\dots,m\>and\>j=1,2,\dots,p\\\
n_{ij}=min\\{x_{i2},y_{i(j-p)}\\},where\>i=1,2,\dots,m\>and\>j=p+1,p+2,\dots,2p\\\
n_{ij}=min\\{x_{i3},y_{i(j-2p)}\\},where\>i=1,2,\dots,m\>and\>j=2p+1,2p+2,\dots,3p\\\
...................................................................................................................\\\
n_{ij}=min\\{x_{in},y_{i(j-(n-1)p)}\\},where\>i=1,2,\dots,m\>and\>j=(n-1)p+1,(n-1)p+2,\dots,n.p\par\end{cases}$
The matrix $N$ also follows the definition of intersection of two soft sets
$(F,A)$ and $(G,B)$ given in [3].

###  4.4 Soft product

Let $(F,A)$ and $(G,B)$ be two soft sets defined over a universal set $X$,
where $X=\\{x_{1},x_{2},\dots x_{m}\\}$, $A=\\{e_{1},e_{2},\dots e_{n}\\}$,
and $B=\\{g_{1},g_{2},\dots g_{p}\\}$. Then, we define the mapping
$prod:(X\times X)\times(A\times B)\rightarrow\\{0,1\\}$ such that,

$prod\\{(x_{k},x_{j}),(e_{i},g_{j})\\}=\begin{cases}1,\>if\>x_{k}\in
F(e_{i})\>and\>x_{j}\in G(g_{j})\\\ 0,\>otherwise\\\ \end{cases}$

Here, $k=1,2,\dots m$; $i=1,2,\dots n$, and $j=1,2,\dots p$ Then, the matrix
representation of $(F,A)\times(G,B)$ is given by
$P=(p_{ij})_{m^{2}\times(n.p)}$ such that,

$P=\begin{cases}p_{kj}=1,\>if\>prod\\{(x_{k},x_{j}),(e_{i},g_{j})\\}=1\\\
p_{kj}=0,\>if\>prod\\{(x_{k},x_{j}),(e_{i},g_{j})\\}=0\\\ \end{cases}$

Now, we consider some examples to illustrate the matrix representations with
the operators complement, union, intersection and product of soft sets.

######  Example 4.1.

From example 3.1, the matrix $M^{\prime}$ of $C(F,A)$ can be found below.

$M^{\prime}=\begin{pmatrix}1&1&0\\\ 0&1&1\\\ 0&0&1\end{pmatrix}$

######  Example 4.2.

Let us consider union of two soft sets $(F,A)$ and $(G,B)$ defined over a
universal set $X$. Let $X=\\{a,b,c\\}$, $A=\\{x,y,z\\}$, and
$B=\\{m,n,o,p\\}$. We define the soft set $(F,A)$ as follows:
$F(x)=\\{b,c\\}$, $F(y)=\\{c\\}$, $F(z)=\\{a\\}$. Similarly, we define the
soft set $(G,B)$ as follows: $G(m)=\\{a\\}$, $G(n)=\\{c\\}$, $G(o)=\\{c\\}$,
$G(p)=\\{a,c\\}$. Then, the set of parameters for $(F,A)\bigcup(G,B)$ is
$A\times
B=\\{\\{x,m\\},\\{x,n\\},\\{x,o\\},\\{x,p\\},\\{y,m\\},\\{y,n\\},\\{y,o\\},\\{y,p\\},\\{z,m\\},\\{z,n\\},\\{z,o\\},\\{z,p\\}\\}$.
Moreover, we have the matrices $M=(x_{ij})_{3\times 3}$ and
$N=(y_{ij})_{3\times 4}$ for $(F,A)$ and $(G,B)$ respectively as shown below.

$M=\begin{pmatrix}0&0&1\\\ 1&0&0\\\ 1&1&0\end{pmatrix}$,
$N=\begin{pmatrix}1&0&0&1\\\ 0&0&0&0\\\ 0&1&1&1\end{pmatrix}$

Now, the matrix representation of $(F,A)\bigcup(G,B)$ is obtained as a matrix
$U=(m_{ij})_{3\times 12}$, which is shown below. We can easily calculate
$m_{ij}$ for $i=1,2,3$ and $j=1,2,3,4$. Thus, we obtain the matrix $U$ as
shown below.

$U=\setcounter{MaxMatrixCols}{12}\begin{pmatrix}1&0&0&1&1&0&0&1&1&1&1&1\\\
1&1&1&1&0&0&0&0&0&0&0&0\\\ 1&1&1&1&1&1&1&1&0&1&1&1\par\end{pmatrix}$

The matrix $U$ is identical to the matrix obtained by calculating the union of
the soft sets $(F,A)$ and $(G,B)$ as in definition 1.5 and then constructing
the matrix from the obtained soft set.
Similarly, the intersection between $(F,A)$ and $(G,B)$ can also be calculated
using 3.3.

######  Example 4.3.

Let $(F,A)$ and $(G,B)$ be two soft sets defined over a universal set $X$.
Also, we consider $X=\\{a,b,c\\}$, $A=\\{m,n\\}$, and $B=\\{x,y\\}$. Let the
binary representations of $(F,A)$ and $(G,B)$ be $M$ and $N$ respectively.

$M=$ $F(m)$ $F(n)$ $a$ 1 0 $b$ 1 1 $c$ 0 0 , $N=$ $G(x)$ $G(y)$ $a$ 0 0 $b$ 1
0 $c$ 1 1

Then, the binary representation of $(F,A)\times(G,B)=(H,A\times B)$ is given
below.

| $H(m,x)$ | $H(m,y)$ | $H(n,x)$ | $H(n,y)$
---|---|---|---|---
$(a,a)$ | 0 | 0 | 0 | 0
$(a,b)$ | 1 | 0 | 0 | 0
$(a,c)$ | 1 | 1 | 0 | 0
$(b,a)$ | 0 | 0 | 0 | 0
$(b,b)$ | 1 | 0 | 1 | 0
$(b,c)$ | 1 | 1 | 1 | 1
$(c,a)$ | 0 | 0 | 0 | 0
$(c,b)$ | 0 | 0 | 0 | 0
$(c,c)$ | 0 | 0 | 0 | 0

Hence, the matrix $P$ of $(F,A)\times(G,B)$ is given by
$P=\begin{pmatrix}0&0&0&0\\\ 1&0&0&0\\\ 1&1&0&0\\\ 0&0&0&0\\\ 1&0&1&0\\\
1&1&1&1\\\ 0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix}$ .

Now, we want to mention an important idea related to soft set theory.
Molodtsov in [2] wrote, “Of course, when you specify a soft set, you have some
semantic interpretation of this soft set. However, the mathematical formalism
of soft sets does not imply any semantic sense on family of subsets or on the
parameters. The parameters serve only the purpose to indicate a specific
subset ………To determine topology we have to define only the family of
vicinities of a point. No comparison of vicinities and no other properties of
these subsets are needed. The situation is quite similar for soft sets, as the
soft set is a family of vicinities of a point except that the initial point
(as in topology) may not exist. Thus, the role of parameters in definition of
soft sets is only auxiliary. Parameters are used only as names of subsets.
Therefore, the introduction of the notion of equivalence of soft sets $(S,A)$
and $(S^{\prime},A^{\prime})$ should be based on equality of families of sets
$\tau(S,A)$ and $\tau(S^{\prime},A^{\prime})$, but not on equality of point-
to-set mappings $S$ and $S^{\prime}$.” Similar assumption can also be found in
[4]. Hence, it may be noted that in case of union, intersection and product of
soft sets $(F,A)$ and $(G,B)$, the set of parameters $A\times B$ and $B\times
A$ are indistinguishable. Hence, we propose the following theorem.

######  Theorem 4.1.

Let $(F,A)$, $(G,B)$ and $(H,C)$ be three soft sets defined over a universal
set $X$, then the following properties hold.
(i) $C(C(F,A))=(F,A)$,
(ii) $(F,A)\cup(G,B)=(G,B)\cup(F,A)$,
(iii) $\\{(F,A)\cup(G,B)\\}\cup(H,C)=(F,A)\cup\\{(G,B)\cup(H,C)\\}$,
(iv) $(F,A)\cap(G,B)=(G,B)\cap(F,A)$,
(v) $\\{(F,A)\cap(G,B)\\}\cap(H,C)=(F,A)\cap\\{(G,B)\cap(H,C)\\}$.

###### Proof.

We only prove $(i)$ and $(ii)$.
(i) For the complement $C(F,A)=(W,A)$ of $(F,A)$, the set of parameters
remains the same and the mapping is given by $W(a)=X\setminus F(a)$, for $a\in
A$. Again, taking complement of the set $(W,A)$, we get, $C(W,A)=(U,A)$ and
the mapping is given by $U(a)=X\setminus W(a)=X\setminus\\{X\setminus
F(a)\\}=F(a),$ for $a\in A$. Thus, $C(C(F,A))=(F,A)$.

(ii) Let $(F,A)\cup(G,B)=(H,A\times B)$. Then, the set of parameters for
$(F,A)\cup(G,B)$ is $A\times B$, and the corresponding mapping is given by
$H(a,b)=F(a)\cup G(b)$, where $(a,b)\in A\times B$. Again, we consider
$(G,B)\cup(F,A)=(I,B\times A)$. In this case, the set of parameter is $B\times
A$, and the corresponding mapping is given by $I(b,a)=G(b)\cup F(a)$. Since,
according to Molodtsov [2], the sets of parameters are auxiliaries in case of
soft sets, thus $A\times B$ and $B\times A$ are indistinguishable in the sense
that $(a,b)\in A\times B$ and $(b,a)\in B\times A$ are indistinguishable. So,
$H(a,b)=I(b,a)$ because $F(a)\bigcap G(b)=G(b)\bigcap F(a).$ Hence,
$(F,A)\cup(G,B)=(G,B)\cup(F,A)$.

∎

##  5 Similarity between two soft sets

In this section, we define similarity measure between two soft sets $(F,A)$
and $(G,B)$ defined over a universal set $X$. We consider the concept of
matrix representation of a soft set while measuring the similarity between two
soft sets. For real world phenomena, we consider both the universal set and
the set of parameters for a soft set to be finite.

######  Definition 5.1.

Let $X$ be a universal set, $A$ and $B$ are two sets of attributes such that
$|X|=m$, $|A|=n$ and $|B|=p$. Here, $|K|$ denotes the cardinality of a set
$K$. Without loss of generality, we consider $n\geq p$. Also, we assume
$Y=(y_{ij})_{m\times n}$ and $Z=(z_{ij})_{m\times p}$ to be the matrices of
$(F,A)$ and $(G,B)$ respectively. Now, we consider following two cases.

### case 1 : when $n=p$

If $n=p$, then we construct a new matrix $D=(d_{ij})$ which depends on
matrices $Y$ and $Z$ in the following way:

$D=\begin{cases}d_{ij}=1,\ if\ y_{ij}=z_{ij}\\\ d_{ij}=0,\ if\ y_{ij}\neq
z_{ij}\end{cases}$

Then, similarity measure between the two soft sets $(F,A)$ and $(G,B)$ is
denoted by $Sim\\{(F,A),(G,B)\\}$ and defined as:

$Sim\\{(F,A),(G,B)\\}=\frac{{\sum_{i,j}}\\{d_{ij}\ :\
d_{ij}=1\\}}{|X|\times|A|}$.

### case 1 : when $n>p$

Suppose $n-p=t$, i.e., the matrix $Z=(z_{ij})_{m\times p}$ has exactly $t$
columns more than the matrix $Y=(y_{ij})_{m\times n}$. Now, we first consider
the submatrix $W=(w_{ij})_{m\times n}$ of $Z$ with the first $n$ columns of
$Z$ and with all the rows of $Z$ i.e., with $m$ rows. Now, the matrices $Y$
and $W$ are of the same order. Again, let us define a new matrix
$D=(d_{ij})_{m\times n}$ as follows.

$D=\begin{cases}d_{ij}=1,\ if\ y_{ij}=w_{ij}\\\ d_{ij}=0,\ if\ y_{ij}\neq
w_{ij}\end{cases}$

Next, we consider the remaining submatrix of $Z$ with $m$ rows and $(p-t)$
columns beginning from the $(t+1)^{th}$ column up to the $p^{th}$ column. We
denote it by $M=(m_{ij})_{m\times(p-t)}$, which is of order $m\times(p-t)$.
Now, we construct a zero matrix $N=(n_{ij})_{m\times(p-t)}$, in which all the
entries are 0 and is of order $m\times(p-t)$. Again, we define the matrix
$C=(c_{ij})_{m\times(p-t)}$ as follows.

$C=\begin{cases}c_{ij}=1,\ if\ m_{ij}=n_{ij}=0,\\\ c_{ij}=0,\
otherwise.\end{cases}$

Then, similarity measure between the two soft sets $(F,A)$ and $(G,B)$ is
defined as below.

$Sim\\{(F,A),(G,B)\\}=\frac{\sum_{i,j}\\{d_{ij}\ :\ d_{ij}=1\\}+\sum_{i,j}\
\\{c_{ij}\ :\ c_{ij}=1\\}}{|X|\times max\\{|A|,|B|\\}}$

These two similarity measures are independent of the set of attributes chosen
and they only depend on the families $\tau(F,A)$ and $\tau(G,B)$. Two soft
sets is said to be completely similar if their similarity measure is 1 or
completely dissimilar if their similarity measure is 0. Thus, we obtain
similarity measures using Molodtsov’s ideas [2] as stated above. Let us
consider an illustrative example.

######  Example 5.1.

Let $M$ and $N$ be two matrices of the soft sets $(F,A)$ and $(G,B)$.

$M=\begin{pmatrix}1&0&1\\\ 1&0&0\\\ 1&0&1\end{pmatrix}$
$N=\begin{pmatrix}0&1&1&0\\\ 1&0&0&1\\\ 1&1&1&0\end{pmatrix}$

Thus, $d_{11}=0$, $d_{12}=0$, $d_{13}=1$, $d_{21}=1$, $d_{22}=1$, $d_{23}=1$
$d_{31}=1$, $d_{32}=0$, and $d_{33}=1$. Now, we construct the matrix
$D=(d_{ij})_{3\times 3}$, $(i=1,2,3)$, $(j=1,2,3)$ of order $3\times 3$ as
follows.

$D=\begin{pmatrix}0&0&1\\\ 1&1&1\\\ 1&0&1\end{pmatrix}$

Next, we construct a zero matrix $R=(r_{ij})_{3\times 1}$, and a column matrix
$S=(s_{ij})_{3\times 1}$, each of order $3\times 1$, where $i=1$, and
$j=1,2,3$. It is easy to observe that $S$ is the submatrix of $N$ consisting
of $4^{th}$ column of $N$.

$R=\begin{pmatrix}0\\\ 0\\\ 0\end{pmatrix}$, $S=\begin{pmatrix}0\\\ 1\\\
0\end{pmatrix}$

Thus, $c_{11}=1$, $c_{21}=0$, and $c_{31}=1$. Now, construct the matrix $C$ as
follows.

$C=\begin{pmatrix}1\\\ 0\\\ 1\end{pmatrix}$

Hence,

$Sim\\{(F,A),(G,B)\\}=\frac{\sum_{i,j}\\{d_{ij}\ :\ d_{ij}=1\\}+\sum_{i,j}\
\\{c_{ij}\ :\ c_{ij}=1\\}}{|X|\times max\\{|A|,|B|\\}}=\frac{2}{3}$.

######  Theorem 5.1.

Let $(F,A)$ and $(G,B)$ be two soft sets over a universal set $X$. Then, the
following results hold:
(i) $0\leq Sim\\{(F,A),(G,B)\\}\leq 1$
(ii) $Sim\\{(F,A),(G,B)\\}=Sim\\{(G,B),(F,A)\\}$
(iii) $Sim\\{(F,A),(F,A)\\}=1$

###### Proof.

(i) Let, $|X|=m$, $|A|=n$ and $|B|=p$. Without loss of generality, we consider
$n\geq p$. Let, $Y=(y_{ij})_{m\times n}$ and $Z=(z_{ij})_{m\times p}$ be the
matrices of the soft sets $(F,A)$ and $(G,B)$ respectively. We consider
following two cases.

case 1($n=p$)

$\sum_{i,j}\\{d_{ij}\ :\ d_{ij}=1\\}\leq|X|\times|A|$

$\implies\frac{\sum_{i,j}\\{d_{ij}\ :\ d_{ij}=1\\}}{|X|\times|A|}\leq 1$

$\implies Sim\\{(F,A),(G,B)\\}\leq 1$

Also,

$\sum_{i,j}\\{d_{ij}\ :\ d_{ij}=1\\}\geq 0$

$\implies\frac{\sum_{i,j}\\{d_{ij}\ :\ d_{ij}=1\\}}{|X|\times|A|}\geq 0$

$\implies Sim\\{(F,A),(G,B)\\}\geq 0$

case 2($n>p$)

$\sum_{i,j}\\{d_{ij}\ :\ d_{ij}=1\\}+\sum_{i,j}\ \\{c_{ij}\ :\
c_{ij}=1\\}\leq|X|\times max\\{|A|,|B|\\}$

$\implies\frac{\sum_{i,j}\\{d_{ij}\ :\ d_{ij}=1\\}+\sum_{i,j}\ \\{c_{ij}\ :\
c_{ij}=1\\}}{|X|\times max\\{|A|,|B|\\}}\leq 1$

$\implies Sim\\{(F,A),(G,B)\\}\leq 1$

Also,

$\sum_{i,j}\\{d_{ij}\ :\ d_{ij}=1\\}+\sum_{i,j}\ \\{c_{ij}\ :\ c_{ij}=1\\}\geq
0$

$\implies\frac{\sum_{i,j}\\{d_{ij}\ :\ d_{ij}=1\\}+\sum_{i,j}\ \\{c_{ij}\ :\
c_{ij}=1\\}}{|X|\times max\\{|A|,|B|\\}}\geq 0$

$\implies Sim\\{(F,A),(G,B)\\}\geq 0$

Thus, combining the above results we get $0\leq Sim\\{(F,A),(G,B)\\}\leq 1$.

(ii) Let $M$ and $N$ be the matrices of order $m\times n$ and $m\times p$ of
the soft sets $(F,A)$ and $(G,B)$ respectively. If $n=p$, then the result is
obvious.
If $n>p$ and $n-p=t$, then we construct a zero matrix $N$ of order
$m\times(n-t)$ to obtain the matrix $C$ as defined in definition 5.1. If $n<p$
and $p-n=t$, then similarly, we construct the zero matrix $N^{\prime}$ of
order $m\times(p-t)$ to obtain the matrix $C$. In both the cases, we obtain
the same matrix $C$. Hence, it does not affect if we interchange the order of
the soft sets for which we have to calculate the similarity measure.

(iii) Directly follows from definition 5.1. ∎

######  Theorem 5.2.

If $(F,A)$ be a soft set defined over a universal set $X$ and $(CF,A)$ denotes
the complement of the soft set $(F,A)$, then $Sim\\{(F,A),(CF,A)\\}=0$.

###### Proof.

The proof is obvious. ∎

######  Theorem 5.3.

Let $(F,A)$ and $(G,B)$ be two soft sets defined over a universal set $X$. If
the mappings $F$ and $G$ are one-one and $(F,A)\cong(G,B)$, then,
$Sim\\{(F,A),(G,B)\\}=1$, provided the ordering of the attributes is
independent of choice.

###### Proof.

For two soft sets $(F,A)$ and $(G,B)$ defined over a universal set $X$,
$(F,A)\cong(G,B)$ implies $\tau(F,A)$ = $\tau(G,B)$. Since
$\tau(F,A)=\tau(G,B)$, then the mappings $F$ and $G$ are one-one imply that
the $|A|=|B|$. It implies that the matrices of both the soft sets will be of
same order. Now, an element of $\tau(F,A)$ represents one column in the
respective matrix of $(F,A)$ and we obtain a matrix similarly for $(G,B)$. It
is also given that the ordering of the attributes is independent of choice.
Hence, $\tau(F,A)$ = $\tau(G,B)$ implies that all the columns in the matrices
of the two soft sets are identical. Thus, $Sim\\{(F,A),(G,B)\\}=1$. ∎

Now, we consider a case where $\tau(F,A)\cap\tau(G,B)=\phi$, for two soft sets
$(F,A)$ and $(G,B)$. For the practical feasibility, let us discuss a situation
by a real life example. Granular computing is an emerging computing paradigm
of information processing that concerns the processing of complex information
entities called “information granules”, which arise in the process of data
abstraction and derivation of knowledge from information or data [18].

Figure 2: Representations of two granular structures

In this process, there are granules and several granular layers in a granular
structure. A granule may be a subset, class, object or cluster of a universe
[17]. In granular computing [18, 19], we may construct a granular layer by
using a soft set say $(F,A)$ and the granules of the layer may be represented
by the elements of $\tau(F,A)$ of the soft set $(F,A)$. Then, we can have
similarity measures between the different layers formed by different soft
sets. It is often feasible to find that the information in a specific layer of
a granular structure does not match with an another layer of a different
granular structure. In spite of it, we cannot conclude that there is no
similarity between the two layers of different granules. For example, we
consider two granular layers as shown in figure 2 represented by two soft sets
$(F,A)$ and $(G,B)$ defined over the same universe $X=\\{a,b,c\\}$. The
elements in the families $\tau(F,A)$ and $\tau(G,B)$ will represent the
granules in two granular layers. Suppose $\tau(F,A)=\\{\\{a,b\\},\\{c\\}\\}$
and $\tau(G,B)=\\{\\{a,c\\},\\{b\\},\\{d,e\\}\\}$ are representing the sets of
two and three granules in granular structure 1 and granular structure 2
respectively as shown in the figure 2. Here, $\tau(F,A)\cap\tau(G,B)=\phi$,
but we can not say that there is no similarity between the granular layers
because the elements $a$, $b$ and $c$ are in some granules in both the layers.
From this discussions, we conclude the following theorem which may be useful
for granular computing.

######  Theorem 5.4.

If $(F,A)$ and $(G,B)$ be two soft sets defined over a universal set $X$, then
$\tau(F,A)\cap\tau(G,B)=\phi$ does not imply that $Sim\\{(F,A),(G,B)\\}=0$ in
general.

Let us consider an example to illustrate the above theorem.

######  Example 5.2.

Let $(F,A)$ and $(G,B)$ be two soft sets representing two granular layers in
two different granular structures defined over a universal set $X$. Let,
$X=\\{a,b,c,d,e\\}$, $A=\\{m,n\\}$, and $B=\\{x,y\\}$. The mappings $F$ and
$G$ are given by $F(m)=\\{a,b\\},F(n)=\\{e\\},$ $G(x)=\\{b,c\\}$, and
$G(y)=\\{c,d,e\\}$. Hence, $\tau(F,A)=\\{\\{a,b\\},\\{e\\}\\}$ and
$\tau(G,B)=\\{\\{b,c\\},\\{c,d,e\\}\\}$. In these two families, each element
will represent a granule in the granular layer of the respective granular
system. We find that $\tau(F,A)\cap\tau(G,B)=\phi$. It is now easy to find
that $Sim\\{(F,A),(G,B)\\}=\frac{3}{5}$.

Now, let us now consider some properties of soft sets with respect to the
internal and external approximations of Molodtsov [3].

######  Definition 5.2.

Let $M$ be a matrix of a soft set $(F,A)$ defined over a universal set $X$. We
consider $A=\\{e_{1},e_{2},\dots,e_{n}\\}$, i.e., $|A|=n$. Then, gravity of an
attribute $e_{i}\in A$ is denoted by $\chi(e_{1})$ and it is defined to be the
total number of 1’s in the ordered $i^{th}$ column of the matrix $M$ of
$(F,A)$, where, $i=1,2,\dots,n$.

######  Theorem 5.5.

Consider two soft sets $(F,A)$ and $(G,B)$ defined over a universal set $X$
such that $(F,A)\subseteq(G,B)$. Let $|A|=m$ and $|B|=n$. Then,
$\chi(e_{i})\leq\chi(g_{j})$ in some order, where $i=1,2,\dots m$ and
$j=1,2,\dots n$.

######  Corollary 5.1.

If $(F,A)\subseteq(G,B)$ and $|A|=|B|=n$, then $\chi(e_{i})\leq\chi(g_{i})$,
where $i=1,2,\dots n$, if the ordering of attributes is independent of choice.

Let us consider an illustrative example.

######  Example 5.3.

Let $(F,A)$ and $(G,B)$ be two soft set defined over a universal set $X$. Let
$A=\\{e_{1},e_{2},e_{3}\\}$ and $B=\\{g_{1},g_{2},g_{3},g_{4}\\}$. If $M$ and
$N$ are the matrices of $(F,A)$ and $(G,B)$ respectively, where,

$M=\begin{pmatrix}1&1&1\\\ 1&1&0\\\ 0&1&0\end{pmatrix}$,
$N=\begin{pmatrix}1&1&1&1\\\ 1&0&0&1\\\ 0&0&1&1\end{pmatrix}$

then obviously, $(F,A)\subseteq(G,B)$. Hence, $\chi(e_{1})=\chi(g_{1})$,
$\chi(e_{2})=\chi(g_{4})$, $\chi(e_{3})\leq\chi(g_{3})$.

######  Corollary 5.2.

Let $(F,A)$ and $(G,B)$ be two soft set defined over a universal set $X$ and
$(F,A)\subseteq(G,B)$. Let $A=\\{e_{1},e_{2},\dots,e_{m}\\}$ and
$B=\\{g_{1},g_{2},\dots,g_{n}\\}$. Then, it is not always true that
$\sum_{i=1}^{m}\chi(e_{i})\leq\sum_{j=1}^{n}\chi(g_{j})$.

The following example is used to support the preceding corollary.

######  Example 5.4.

Let $(F,A)$ and $(G,B)$ be two soft set defined over a universal set $X$. Let
$A=\\{e_{1},e_{2},e_{3}\\}$ and $B=\\{g_{1},g_{2},g_{3},g_{4},g_{5}\\}$. If
$M$ and $N$ are the matrices of $(F,A)$ and $(G,B)$ respectively, where,

$M=\begin{pmatrix}1&1&1&1&1\\\ 0&0&0&0&0\\\ 0&0&0&0&0\end{pmatrix}$,
$N=\begin{pmatrix}1&1&1\\\ 0&0&0\\\ 0&1&0\end{pmatrix}$

then obviously, $(F,A)\subseteq(G,B)$. But,
$\sum_{i=1}^{3}\chi(e_{i})=5\nleq\sum_{j=1}^{5}\chi(g_{j})=4$.

######  Theorem 5.6.

Let $(F,A)$ and $(G,B)$ be two soft sets defined over a universal set $X$ such
that $(F,A)\supseteq(G,B)$. Let $|A|=m$ and $|B|=n$. Then,
$\chi(e_{i})\geq\chi(g_{j})$ in some order, where $i=1,2,\dots m$ and
$j=1,2,\dots n$.

######  Corollary 5.3.

If $(F,A)\supseteq(G,B)$ and $|A|=|B|=n$, then $\chi(e_{i})\geq\chi(g_{i})$,
where $i=1,2,\dots n$, if the ordering of attributes is independent of choice.

######  Corollary 5.4.

Let $(F,A)$ and $(G,B)$ be two soft sets defined over a universal set $X$ and
$(F,A)\supseteq(G,B)$. Let $A=\\{e_{1},e_{2},\dots e_{m}\\}$ and
$B=\\{g_{1},g_{2},\dots g_{n}\\}$. Then, it is not always true that
$\sum_{i=1}^{m}\chi(e_{i})\geq\sum_{j=1}^{n}\chi(g_{j})$.

######  Theorem 5.7.

Let $(F,A)$ and $(G,A)$ be two soft sets defined over a universal set $X$. If
$(F,A)\begin{matrix}\subset\\\ \approx\end{matrix}(G,A)$, then
$Sim\\{(F,A),(G,A)\\}=1$, for some specific ordering of the attributes in the
matrix representation of the soft sets.

###### Proof.

Since the sets of attributes are same for both of the soft sets $(F,A)$ and
$(G,A)$, hence the matrices are of same order. Also, the condition
$(F,A)\begin{matrix}\subset\\\ \approx\end{matrix}(G,A)$ implies that
$(F,A)\subseteq(G,A)$ and $(G,A)\subseteq(F,A)$. Now, using Theorem 5.5, we
get, $\chi(e)_{(F,A)}$ = $\chi(e)_{(G,A)}$, for all $e\in A$. Here, we
indicate $\chi(e)_{(F,A)}$ to represent gravity of $e\in A$ in $(F,A)$ and
$\chi(e)_{(G,A)}$ to indicate gravity of $e\in A$ in $(G,A)$. Again, since $X$
is ordered while representing the matrices and the ordering of the attributes
is independent of choice, hence the 1’s in each of the matrix will be placed
in the same ordered place, which results the same matrices for both the soft
sets. Thus, we get $Sim\\{(F,A),(G,B)\\}=1$. ∎

######  Theorem 5.8.

Let $(F,A)$ and $(G,A)$ be two soft sets defined over a universal set $X$. If
$(F,A)\begin{matrix}\supset\\\ \approx\end{matrix}(G,A)$, then
$Sim((F,A),(G,A))=1$, in some specific ordering of the attributes in the
matrix representation of the soft sets.

###### Proof.

Similar to the above proof. ∎

######  Theorem 5.9.

Let $(F,A)$ and $(G,A)$ be two soft sets defined over a universal set $X$. We
consider $M$ and $N$ to be the matrices of $(F,A)$ and $(G,A)$ respectively,
where $|A|=n$. If the columns of $M$ and $N$ are linearly independent and span
$R^{n}$, then $Sim\\{(F,A),(G,A)\\}=1$, in some specific ordering of the
attributes in the matrix representation of the soft sets.

###### Proof.

Since the columns of $M$ and $N$ are linearly independent and span $R^{n}$,
hence they form a basis for $R^{n}$. Also, since the entries of $M$ and $N$
are $0$ and $1$ only, they will form the identical basis i.e., the set
$\\{(1,0,0,\dots,0),(0,1,0,\dots,0),\dots,(0,0,0,\dots,1)\\}$. Again, the set
of attributes are same for both the matrices, hence the matrices will be
identical to each other if we arrange the columns in a specific order. Thus,
obviously for two identical matrices, we get $Sim\\{(F,A),(G,A)\\}=1$. ∎

######  Corollary 5.5.

For two soft sets $(F,A)$ and $(G,A)$ defined over a universal set $X$, if $F$
and $G$ are one-one and every element of the families $\tau(F,A)$ and
$\tau(G,A)$ belongs to $MIN(\tau(F,A))$ and $MIN(\tau(G,A))$ respectively,
then $Sim\\{(F,A),(G,A)\\}=1$, in some specific ordering of the attributes in
the matrix representation of the soft sets.

###### Proof.

If every element of the families $\tau(F,A)$ and $\tau(G,A)$ belongs to
$MIN(\tau(F,A))$ and $MIN(\tau(G,A))$ respectively, then in the matrix
representation of $(F,A)$ and $(G,B)$; every column in each matrix will
linearly independent, since all the elements of $\tau(F,A)$ and $\tau(G,B)$
are minimal and hence does not contain in any other element of $\tau(F,A)$ and
$\tau(G,B)$ respectively. Again, the mappings $F$ and $G$ are one-one, hence,
let $|\tau(F,A)|=|A|=|\tau(G,A)|=n$. Thus, the matrices of $(F,A)$ and $(G,B)$
will be of same order and the linearly independent columns will span $R^{n}$.
Hence, by Theorem 5.9, $Sim\\{(F,A),(G,A)\\}=1$. ∎

Let us consider an example to illustrate the above result.

######  Example 5.5.

For two soft sets $(F,A)$ and $(G,A)$ defined over a universal set $X$, let
$X=\\{a,b,c\\}$, $A$=$\\{x,y,z\\}$. We define $(F,A)$, where $F(x)=\\{b,c\\}$,
$F(y)=\\{c,a\\}$, $F(z)=\\{a,b\\}$. Hence,
$\tau(F,A)=\\{\\{a,b\\},\\{b,c\\},\\{c,a\\}\\}$ Similarly, we define $(G,A)$,
where $G(x)=\\{c,a\\}$, $G(y)=\\{a,b\\}$, $G(z)=\\{b,c\\}$. Hence,
$\tau(G,A)=\\{\\{a,b\\},\\{b,c\\},\\{c,a\\}\\}$ Here, the mappings $F$ and $G$
are one-one and all the elements of the families $\tau(F,A)$ and $\tau(G,A)$
belong to $MIN(\tau(F,A))$ and $MIN(\tau(G,A))$ respectively. The binary
representations of $(F,A)$ and $(G,A)$ are given by $M^{\prime}$ and
$N^{\prime}$.

$M=$ F(x) F(y) F(z) $a$ 0 1 1 $b$ 1 0 1 $c$ 1 1 0 , $N=$ G(z) G(x) G(y) $a$ 0
1 1 $b$ 1 0 1 $c$ 1 1 0

Thus, we get following two matrices $M$ and $N$ for $(F,A)$ and $(G,A)$
respectively.

$M=\begin{pmatrix}0&1&1\\\ 1&0&1\\\ 1&1&0\end{pmatrix}$,
$N=\begin{pmatrix}0&1&1\\\ 1&0&1\\\ 1&1&0\end{pmatrix}$

Thus, $Sim\\{(F,A),(G,B)\\}=1$.

######  Corollary 5.6.

For two soft set $(F,A)$ and $(G,A)$ defined over a universal set $X$, if $F$
and $G$ are one-one and every element of the families $\tau(F,A)$ and
$\tau(G,A)$ belongs to $MAX(\tau(F,A))$ and $MAX(\tau(G,A))$ respectively,
then $Sim\\{(F,A),(G,A)\\}=1$, in some specific ordering of the attributes in
the matrix representation of the soft sets.

###### Proof.

Similar to the above proof. ∎

##  6 Discussion

The theory of soft set has been applied wrongly to many areas of mathematics
and allied areas by almost all the researchers of soft set theory except
Molodtsov [2, 3].As he has stated multiple times [2, 3], correctness of soft
set theory is required to maintain the genuine philosophy of soft set theory’s
importance. Thus, we have tried to establish some results and notions based on
the correct structures introduced by only Molodtsov [1, 2, 3, 4]. Similarity
measure discussed here may be applied in granular computing as well as in
other computing paradigm. Based on our present study, we propose the following
conjecture.

Conjecture 1: If a relation $\Omega$ is correct for any quadruple of pairwise
equivalent soft sets $(F,A)\cong(F^{\prime},A^{\prime})$ and
$(G,B)\cong(G^{\prime},B^{\prime})$, defined over a universe $X$, then it is
not necessary that $Sim\\{(F,A),(G,B)\\}$ =
$Sim\\{(F^{\prime},A^{\prime}),(G^{\prime},B^{\prime})\\}$.

Not only we are trying to follow correct notions and philosophy of soft set
theory, but also several pioneers of fuzzy set theory corrected several
misconceptions related to fuzzy set theory. Fuzzy pioneer George J. Klir and
his co-authors [22] pointed out several misconceptions available in literature
of philosophy of concepts and fuzzy set theory. But in our case, not only a
part of soft set theory is wrong but almost all the fundamental operations of
soft sets are completely wrong along with the available notion of soft set
theory by Çağman and Enginoğlu [9]. For example, we can raise question on
definition of soft topology [23, 24, 25, 26] because the definition of empty
soft set provided by Molodtsov [3] is completely different than that of the
notion of empty set available literature of soft set theory. In case of
Molodtsov’s definition of empty soft set, an empty set of parameters plays a
crucial role, but this idea is absent in the definition of empty set provided
by Maji et al. [6]. Since Molodtsov, the father of soft set theory, raised
concern about incorrect notions, operations and thus related results of soft
set theory, hence it is now prime duty of the community of soft set
researchers to look back to the beginning of soft set theory and hybrid
structures by following the correct path of Molodtsov. Although there are
thousands of published papers available on soft set theory and related areas ,
but it is not our intention to encourage wrong ideas related to soft set
theory and hybrid structures. Thus in this paper, we do not focus to apply our
results in different areas but our main intention is to develop correct
theories of soft sets by following Molodtsov [1, 2, 3, 4].

##  7 Conclusion

In this paper, we develop some results based on correct notions of soft sets
introduced by Molodtsov [1, 2, 3, 4]. Some unary and binary operations on soft
sets are defined correctly in matrix forms. Also, similarity measure between
two soft sets $(F,A)$ and $(G,B)$ are defined based on the families
$\tau(F,A)$ and $\tau(G,B)$, but not on the set of parameters [11, 12]. We
hope to continue the process of establishing various ideas related to correct
notions of Molodtsov in forth coming papers. We also hope that this paper will
attract the attention of scientific community related to soft set and hybrid
structures; and it will considered one of important paper in the history of
soft set theory because this paper reports for the first time the comment of
Molodtsov on social platform regarding incorrectness of available definition
of soft set .

CRediT authorship contribution statement
Santanu Acharjee: Conceptualization, Formal analysis, Investigation,
Methodology, Writing - original draft, Writing- review $\&$ editing.

Amlanjyoti Oza: Conceptualization, Formal analysis, Investigation,
Methodology, Writing - original draft, Writing- review $\&$ editing.

Declaration of Competing Interest The authors declare that they have no known
competing financial interests or personal relationships that could have
appeared to influence the work reported in this paper.

## References

  * [1] Molodtsov, D. (1999). Soft set theory—first results. Computers and Mathematics with Applications, 37(4-5), 19-31.
  * [2] Molodtsov, D. A. (2018). Equivalence and correct operations for soft sets. International Robotics and Automation Journal, 4(1), 18-21.
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