# Testing the cosmic distance duality relation using Type Ia supernovae and
radio quasars through model-independent methods

Fan Yang  Xiangyun Fu 11footnotetext: Corresponding author<EMAIL_ADDRESS>Bing Xu 22footnotetext: Corresponding author<EMAIL_ADDRESS>Kaituo Zhang
Yang Huang  Ying Yang

###### Abstract

In this work, we perform a cosmological-model-independent test on the cosmic
distance duality relation (CDDR) by comparing the angular diameter distance
(ADD) obtained from the compact radio quasars (QSOs) with the luminosity
distance (LD) from the Pantheon Type Ia supernovae (SNIa) sample. The binning
method and Artificial Neural Network (ANN) are employed to match ADD data with
LD data at the same redshift, and three different parameterizations are
adopted to quantify the possible deviations from the CDDR. We initially
investigate the impacts of the specific prior values for the absolute
magnitude $M_{\rm B}$ from SNIa and the linear size scaling factor $l$ from
QSOs on the CDDR test, demonstrating that these prior values introduce
significant biases in the CDDR test. To avoid the biases, we propose a method
independent of $M_{\rm B}$ and $l$ to test CDDR, which treats the fiducial
value of a new variable $\kappa\equiv 10^{M_{\rm B}\over 5}\,l$ as a nuisance
parameter and then marginalize its impact with a flat prior in the statistical
analysis. The results show that the CDDR is consistent with the observational
data, and QSOs can serve as a powerful tool for testing the CDDR independent
of cosmological models.

##  1 Introduction

The cosmic distance duality relation (CDDR) is a fundamental relation in
modern cosmology [1], which relates the luminosity distance (LD) $D_{\rm
L}(z)$ and angular diameter distance (ADD) $D_{\rm A}(z)$ through the identity
equation ${D_{\rm L}}={D_{\rm A}}{(1+z)}^{2}$, where $z$ is the cosmological
redshift. This relation relies on three fundamental assumptions: the space-
time is described by the metric theory, the light travels along the null
geodesics between the source and the observer, and the photon number is
conserved. As a fundamental relation, the CDDR has undoubtedly been applied in
various research fields of astronomy, such as the large-scale distribution of
galaxies and the uniformity of the Cosmic Microwave Background (CMB)
temperature [2], as well as the gas mass density distribution and temperature
distribution of galaxy clusters [3, 4]. In astronomical observations, any
violation of the CDDR suggests the presence of new physics or unaccounted
errors in the observational data. Therefore, it is necessary to conduct
reliable testing on CDDR.

The CDDR test is usually conducted using a parametric method. Three different
forms can be used for the parametrization, namely $\eta(z)=1+\eta_{0}z$,
$\eta(z)=1+\eta_{0}z/(1+z)$, and $\eta(z)=1+\eta_{0}\ln(1+z)$, where the
$\eta_{0}$ indicates the possible violation from the CDDR. Considering the
advantages of the $\eta(z)$ expression, such as manageable one-dimensional
phase space and good sensitivity to observational data [5], the
parameterizations of $\eta(z)$ listed above were used to test the validity of
the CDDR. Some studies have been devoted to testing the validity of CDDR by
comparing LD data from observations of Type Ia supernovae (SNIa), HII
galaxies, or gamma-ray bursts with the various ADD data from the X-ray plus
Sunyaev-Zeldovich (SZ) effect and the gas mass fraction measurements in galaxy
clusters [6, 7, 8, 5, 9, 10, 11, 12, 13, 14, 15, 16, 17]. The results indicate
that within various redshift ranges, the CDDR is consistent with astronomical
observations [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. In addition, an
issue may be that it is difficult to obtain LD and ADD measurements from
astronomic observation at the same redshifts. To solve this problem, several
methods have been proposed in some literature. Using the galaxy cluster
samples [30, 31] and SNIa data, Holanda et al. [8] and Li et al. [10] selected
the closest one through a selection criterion ($\Delta z=|z_{\rm ADD}-z_{\rm
SNIa}|<0.005$) for CDDR test. To minimize statistical errors that could arise
from utilizing only a single SNIa data point from all those that meet the
selection criteria, Meng et al. [12], did not use the closest measurement, but
instead used a binning method to bin the available data that meets the
selection criterion to derive LD.

SNIa and compact radio quasars (QSO) measurements play important roles in
constraining cosmological parameters. It is worth noting that the LD derived
from SNIa observations is dependent upon its peak absolute magnitude $M_{\rm
B}$, which is assumed to be a constant value, unaffected by other variables.
Recently, efforts have been made to derive the value of $M_{\rm B}$ from a
cosmological observations [32, 33, 34]. Various values of $M_{\rm B}$ have
been determined by combining SNIa data, such as Pantheon, with other
observational datasets, including CMB observations, cosmic chronometer data
related to the Hubble parameter, and baryon acoustic oscillations (BAO). A
discrepancy in the absolute magnitude of SNIa, calibrated by Cepheids, was
observed between $z\leq 0.01$ and $z>0.01$ [35, 36]. Recent studies [37, 38]
have also indicated a potential weak evolution of $M_{\rm B}$. In addition,
due to the negligible dependence of the compact structure sizes of
intermediate-luminosity quasars on the source luminosity and redshift, these
quasars, with very-long-baseline interferometry (VLBI) observations, are
potentially promising standard rulers [39]. The ADD obtained from these QSO
observations depends on the linear size scaling factor $l$. The constraints on
$l$ have been made using both cosmological-model-dependent and cosmological-
model-independent methods [40, 41, 42], and the value of $l$ varies slightly
depending on the observational data and the cosmological model.

Obviously, if the exact values of $M_{\rm B}$ and $l$ are not determined by
astronomical observations, the specific prior values for $M_{\rm B}$ and $l$
may potentially introduce biases into the constraints on cosmological
parameters. Furthermore, the independence of the CDDR test, which relies on
the prior values of $M_{\rm B}$ and $l$, may be questionable, as the values of
$M_{\rm B}$ and $l$ are obtained from specific cosmological models. It is
worth noting that the measurement error of a single SNIa or QSO measurement is
not dependent on the parameter $M_{\rm B}$ or $l$, thus, theoretically, one
can use marginalization methods to eliminate $M_{\rm B}$ or $l$ parameters
during statistical analysis. In view of this, it is important to further study
the impacts of specific prior values for $M_{\rm B}$ or $l$ on CDDR test, and
to develop new testing methods that are not dependent on $M_{\rm B}$ and $l$,
which can improve the reliability of CDDR testing. This is also the main
motivation for us to carry out this research.

In this work, we perform the CDDR test by comparing the LD derived from
Pantheon SNIa data with the ADD from QSO data. The binning method and
Artificial Neural Network (ANN) are used to match the SNIa data with the QSO
data at the same redshifts. We first investigate the impacts on the CDDR test
by considering the specific value of $M_{\rm B}$ and $l$ to derive the LD and
ADD. The results indicate that the priors of $M_{\rm B}$ and $l$ may induce
significant biases in the CDDR test. To avoid these biases, we combine $M_{\rm
B}$ and $l$ into a new variable $\kappa$, defined as $\kappa\equiv 10^{M_{\rm
B}\over 5}\,l$, and consider it as a nuisance parameter with a flat prior in
statistical analysis, thereby marginalizing its impact on CDDR test.
Therefore, all of the quantities used in the CDDR test come directly from
observations, meaning that the absolute magnitudes from SNIa and the linear
size scaling factor from QSO measurements do not need to be calibrated. We
demonstrate that CDDR is consistent with the observed data, and the parametric
method of testing CDDR is independent of specific cosmological models.

##  2 Data and Methodology

###  2.1 Data

To verify the validity of CDDR, two kinds of cosmic distances are usually
required: LD ($D_{\rm L}$) and ADD ($D_{\rm A}$). The LD in this work are
obtained from the Pantheon SNIa observational data [43], which includes 1048
data points from the Pan-STARRS1 Deep Survey in the redshift range of
$0.01<z<2.26$. The distance modulus from the Pantheon compilation is
calibrated from the SALT2 light-curve fitter by applying the Bayesian
Estimation Applied to Multiple Species with Bias Corrections method to
determine the nuisance parameters and account for the distance bias
corrections, such as $\mu=m_{\rm B}-M_{\rm B}$, where $m_{\rm B}$ is the
observed peak apparent magnitude in the rest-frame B-band. Recently, some
research has been focused on the possible evolution of $M_{\rm B}$ with
redshift. The CMB constraint on the sound horizon forecasts that $M_{\rm
B}\sim-19.4\,{\rm mag}$ using an inverse distance ladder [34], while the
approximation from SH0ES gives that $M_{\rm B}\sim-19.2\,{\rm mag}$ [32].
Hence, we firstly investigate the impacts of different prior values of $M_{\rm
B}$ on the CDDR test. In this work, we consider two specific priors of $M_{\rm
B}$ derived from different observational data sets within various redshift
ranges: (a) $M_{\rm B}^{\rm D20}={-19.23\pm 0.0404\,{\rm mag}}$, obtained from
SNIa observation within the relatively low redshift range of $0.023<z<0.15$ by
Camarena and Marra [32] in $\Lambda$CDM, through a de-marginalization of the
SH0ES determination [44] (hereafter referred to as $M_{\rm B}^{\rm D20}$); and
(b) $M_{\rm B}^{\rm B23}={-19.396\pm 0.016\,{\rm mag}}$, obtained by combining
SNIa observations with BAO observations [34] (hereafter referred to as $M_{\rm
B}^{\rm B23}$). Taking into consideration the observational uncertainty of
$M_{\rm B}$, the error bar on $\mu$ can be represented as
$\sigma_{\mu}=\sqrt{\sigma_{M_{\rm B}}^{2}+\sigma_{m_{\rm B}}^{2}}$. The
relation between the LD $D_{\rm L}$ [45] and the distance modulus $\mu$ can be
expressed as

$\mu(z)=5\log_{10}(D_{\rm L}(z))+25\,,$ (2.1)

and the uncertainty in $D_{\rm L}$ can be obtained from the equation

$\sigma_{D_{\rm L}}={{\ln{10}}D_{\rm L}\sigma_{\mu}\over 5}\,.$ (2.2)

The angular size-distance relationship of QSO is utilized for cosmological
inference, originally proposed by Kellermann [46], who attempted to obtain the
deceleration parameter using VLBI observations of 79 compact radio sources at
5 GHz. Subsequently, Gurvits [47] extended this method and tried to study the
dependence of the observed characteristic sizes of 337 Active Galactic Nuclei
(AGNs) at 2.29 GHz on luminosity and redshift [48]. In the following analysis,
the angular size $\theta$ of the radio source is refined in Ref. [47] using
the visibility modulus $\Gamma=S_{\rm c}/S_{\rm t}$, which can be expressed as
$\theta={2\sqrt{-\ln{\Gamma}\ln 2}\over\pi{B}}$, where $B$ is the
interferometer baseline measured in multiple of wavelengths, and $S_{\rm c}$
and $S_{\rm t}$ are the correlated flux density and total flux density,
respectively. The linear size $l_{\rm m}$ of compact structures in radio
sources, the intrinsic luminosity $L$, and the redshift $z$ of the background
source supply the following relationship,

$l_{\rm m}=lL^{\beta}(1+z)^{n}\,,$ (2.3)

where $l$ represents the linear size scaling factor, describing the apparent
distribution of radio brightness within the core, $\beta$ and $n$ are used to
quantify the possible “angular size-luminosity” and “angular size-redshift”
relations, respectively. Moreover, for a cosmological rod with intrinsic
length, the relation of the angular size-redshift can be expressed as [49]

$\theta(z)={l_{\rm m}\over{D_{\rm A}(z)}}\,,$ (2.4)

where $\theta(z)$ is the observed angular size measured by VLBI techniques.
Combining Eq. 2.3 and Eq. 2.4, the angular diameter distance $D_{\rm A}(z)$
can be written as

$D_{\rm A}(z)={lL^{\beta}(1+z)^{n}\over{\theta(z)}}\,.$ (2.5)

Recently, Cao, et al. found that the linear size scaling factor is almost
independent on redshift and intrinsic luminosity ($|n|\simeq 10^{-3}$,
$|\beta|\simeq 10^{-4}$) [40, 41]. The sample of 120 intermediate-luminosity
radio quasars within redshift range of $0.4<z<2.8$ selected in [40] has been
widely used in various cosmological studies [39, 50, 51, 52]. The ADD obtained
from the QSO samples has already been used to test the CDDR along with the LD
obtained from HII galaxies and supernovae [53, 54], and to infer the value of
the Hubble constant $H_{0}$ together with the unanchored luminosity from
supernovae data [55].

The value of the linear size scaling factor $l$ can be constrained to
$l=11.19\pm 1.64{\,\rm pc}$ [41] (hereafter referred to as $l^{\rm C17}$) in
the flat $\Lambda$CDM model with WMAP9 observations. Then, in the manner of an
independent study on cosmological model, Cao et al. obtained $l={10.86\pm
1.58}{\,\rm pc}$ [41] by using 36 Hubble data points, some of which were
inferred from 30 cosmic chronometers [56, 57, 58], while the rest were derived
from 6 BAO measurements [59]. Furthermore, Cao et al. obtained the more
accurate value of $l$ [42] (hereafter referred to as $l^{\rm C19}$) by using
41 Hubble data points, some of which were inferred from 31 passively evolving
galaxies [57, 60, 61, 19, 62, 63], while the rest were derived from 10 BAO
measurements [64, 65, 66, 67, 68, 69, 70], which is $l^{\rm C19}={11.04\pm
0.40}\,{\rm pc}$. The values of $l$ provided by different observation data
exhibit slight variations. Consequently, the prior values of $l$ may
potentially induce bias in testing the CDDR. In this work, we consider the
values of the linear size scaling factors $l^{\rm C17}$ and $l^{\rm C19}$,
which are calibrated using methods that depend on and do not depend on the
cosmological model, respectively, to investigate their impacts on the CDDR
test.

###  2.2 Binning method

To test the validity of the CDDR, a straightforward method is to compare the
ADD and LD from different observations at the same redshift. Due to the lack
of observational ADD and LD data at the same redshift, we bin the SNIa data
points satisfying the selection criterion $\Delta z=|z_{\rm ADD}-z_{\rm
SNIa}|<0.005$, as proposed in the literature [8, 10, 27]. This method, known
as the binning method, can be used to avoid statistical errors caused by using
only one SNIa data point among those satisfying the selection criterion and
has been employed in discussing the CDDR test in the cited Ref. [12, 71].
Here, we take the inverse variance weighted average of all selected data. To
avoid correlations among the individual CDDR tests, we select the SNIa samples
following a procedure that the data points will not be used again if they have
been matched to a QSO sample. The weighted average LD $\bar{D_{\rm L}}$ and
its uncertainty $\sigma_{D_{\rm L}}$ can be obtained using conventional data
processing techniques in Chapter 4 of Ref. [72],

$\bar{D_{\rm L}}={\sum(D_{{\rm L}i}/\sigma_{D_{{\rm L}i}}^{2})\over\sum
1/\sigma_{D_{{\rm L}i}}^{2}},$ (2.6) $\sigma^{2}_{\bar{D_{\rm L}}}={1\over\sum
1/\sigma_{D_{{\rm L}i}}^{2}}\,.$ (2.7)

Here, $D_{{\rm L}i}$ denotes the $i$th appropriate luminosity distance data
points, and $\sigma_{D_{{\rm L}i}}$ corresponds the observational uncertainty.

Only 41 QSO data points satisfy the selection criterion. The distributions of
the QSO data and SNIa data derived from different priors of $l$ and $M_{\rm
B}$ are shown in Fig. 1.

Figure 1:  In binning method, the sample catalogs of the observed $D_{\rm
A}(1+z)^{2}$ distribution from the QSO data points and the corresponding LD
$D_{\rm L}$ from Pantheon data obtained with the priors of $M_{\rm B}^{\rm
D20}$ (upper panel), $M_{\rm B}^{\rm B23}$ (bottom panel), $l^{\rm C17}$ (left
panel) and $l^{\rm C19}$ (right panel), respectively.

###  2.3 Artificial Neural Network

It is important to note that when using selection criteria, one must be aware
of the errors caused by the mismatch between SNIa and QSO data points.
Additionally, most of the available QSO data points are excluded due to not
meeting the selection criterion, as the density distribution of SNIa data
differs from that of QSO data in certain redshift regions. To improve the
robustness of QSO data when testing CDDR, we employ the ANN to reconstruct the
smoothing $m_{\rm B}(z)$ function from the Pantheon SNIa observations.
Therefore, each ADD obtained from the QSO sample located within the redshift
range of Pantheon SNIa has a corresponding LD of SNIa at the same redshift.

An ANN is usually a Deep Learning algorithm consisting of three layers: an
input layer, a hidden layer, and an output layer. The input layer comprises
$n$ nodes, each of which corresponds to an independent variable, followed by
$m$ interconnected hidden layers and the output layer with activation function
nodes in the basic architecture [73]. ANN estimates the error gradient from
observations in the training dataset, and then updates the model weights and
bias estimates during back propagation process to iterate toward an optimal
solution through the Adam optimization [74]. The ANN process can be described
by vectorization representation, and more details can be found in Refs. [75,
76, 77].

We use the publicly available code, named Reconstructing Functions Using
Artificial Neural Networks (ReFANN)333https://github.com/Guo-Jian-Wang/refann
[75], to reconstruct the function of apparent magnitude $m_{\rm B}$ versus
redshift $z$, as shown in Fig. 2. It is easy to find that the uncertainty
obtained from the ANN-reconstructed function are close to the observational
uncertainty, and the reconstructed $1\sigma$ CL of the $m_{\rm B}$ can be
considered as the average level of observational error. The LD $D_{\rm L}$
corresponding to ADD $D_{\rm A}$ from QSO data points can be obtained through
the smoothing function $m_{\rm B}(z)$ reconstructed by ANN. For the QSO
samples, 116 QSO data points within the redshift range of SNIa observation
$0<z<2.26$ can be matched with those from SNIa observation at the same
redshift, and the remaining 4 QSO samples that are not within this redshift
range are discarded. The distributions of the QSO data and reconstructed SNIa
data derived from different priors of $M_{\rm B}$ and $l$ are shown in Fig. 3.

Figure 2:  The distributions of the reconstructed $m_{\rm B}(z)$ function with
the corresponding $1\sigma$ errors with the ANN (black line), and the
measurements of apparent magnitude from the Pantheon samples (red).

Figure 3:  The sample catalogs of the observed $D_{\rm A}(1+z)^{2}$
distribution from the QSO data points derived with the priors of $l^{\rm C17}$
(left panel) and $l^{\rm C19}$ (right panel), and the LD $D_{\rm L}$ curves
from Pantheon data derived with the priors of $M_{\rm B}^{\rm D20}$ (upper
panel) and $M_{\rm B}^{\rm B23}$ (bottom panel).

###  2.4 Methodology

We adopt the $\eta(z)$ function to verify any possible deviations from the
CDDR at any redshift by comparing the $D_{\rm L}$ from SNIa and the $D_{\rm
A}$ from QSO measurements. The $\eta(z)$ can be obtained through the following
expression:

$\eta(z)={D_{\rm L}\over D_{\rm A}}{(1+z)^{-2}}\,.$ (2.8)

At any redshift, $\eta(z)\neq 1$ indicates a deviation between the CDDR and
astronomical observations.

We adopt three types of parameterizations for $\eta(z)$: the linear form P1:
$\eta(z)=1+\eta_{0}z$, and two non-linear forms P2:
$\eta(z)=1+\eta_{0}z/(1+z)$, and P3: $\eta(z)=1+\eta_{0}\ln(1+z)$. The
observed $\eta_{\rm obs}(z)$ is obtained from Eq. 2.8, and the corresponding
error can be written as

$\sigma^{2}_{\eta_{\rm obs}}=\eta^{2}_{\rm obs}\left[\left({\sigma_{D_{\rm
A}(z)}\over{D_{\rm A}(z)}}\right)^{2}+\left(\sigma_{D_{\rm L}(z)}\over{D_{\rm
L}(z)}\right)^{2}\right]\,.$ (2.9)

Thus, we obtain

$\chi^{2}(\eta_{0})=\sum_{i}^{N}\frac{{\left[\eta(z)-\eta_{{\rm
obs},\,i}(z)\right]}^{2}}{\sigma^{2}_{\eta_{{\rm obs},i}}}\,.$ (2.10)

Here, $N$ represents the number of available QSO data points obtained the
binning method or ANN. The constraint results on $\eta_{0}$ are shown in Fig.
4, Fig. 5, and Tab. 1. It is evident that the results obtained from the
parametric method depend on the prior values of $M_{\rm B}$ or $l$. Thus,
specific prior values of $M_{\rm B}$ or $l$ cause biases in the CDDR test, if
their true values are not determined by astronomic observations.

Figure 4:  In binning method, the likelihood distribution functions obtained
with the priors of $M_{\rm B}^{\rm D20}$ (upper panel), $M_{\rm B}^{\rm B23}$
(bottom panel), $l^{\rm C17}$ (left panel) and $l^{\rm C19}$ (right panel).

Figure 5:  In ANN, the likelihood distribution functions obtained with the
priors of $M_{\rm B}^{\rm D20}$ (upper panel), $M_{\rm B}^{\rm B23}$ (bottom
panel), $l^{\rm C17}$ (left panel) and $l^{\rm C19}$ (right panel).

Recently, Liu et al. [78] used the fraction division $\eta(z_{i})/\eta(z_{j})$
to eliminate the impacts of $M_{\rm B}$ and $l$ on CDDR test, and the results
indicated agreement between the CDDR and observations. More recently, using
the latest five BAO measurements and the Pantheon SNIa sample, Xu et al. [28]
obtained CDDR test independently of the peak absolute magnitude $M_{\rm B}$
and the sound horizon scale $r_{\rm s}$ from transverse BAO measurements by
marginalizing analytically the likelihood function over the combination of
$M_{\rm B}$ and $r_{\rm s}$. Since the uncertainty in an individual SNIa or
QSO measurement is independent of $M_{\rm B}$ or $l$, these parameters can be
removed from the fits by analytically marginalizing over them in the analysis.
Following the process in Ref. [28], we treat the fiducial values of $M_{\rm
B}$ and $l$ as nuisance parameters to determine the LD $D_{\rm L}$ and ADD
$D_{\rm A}$, and then marginalize their effect by using a flat prior in the
statistic analysis. The likelihood distribution $\chi^{\prime\,2}$ can be
rewritten as

$\chi^{\prime\,2}(\eta_{0},\kappa)=\sum_{i}^{N}\frac{{{\alpha_{i}^{2}\over\beta_{i}^{2}}{\kappa}^{2}-2{\alpha_{i}\over\beta_{i}}{\kappa}+1}}{\sigma^{\prime\,2}_{{\eta_{{\rm
obs},i}}}}\,,$ (2.11)

Here, $\alpha_{i}=\eta(z_{i})$, $\beta_{i}=10^{({m_{{\rm B},i}\over
5}-5)}\theta_{{\rm QSO},i}(1+z_{i})^{-2}$, $\kappa=(10^{M_{\rm B}\over 5}l)$,
and

$\sigma_{\eta_{\rm obs},i}^{\prime\,2}=\left({\ln{10}\over{5}}{\sigma_{m_{{\rm
B},i}}}\right)^{2}+\left({\sigma_{\theta_{{\rm QSO},i}}\over{\theta_{{\rm
QSO},i}}}\right)^{2}\,.$ (2.12)

Thus, following the approach described in Refs. [28, 79, 80], the marginalized
$\chi^{\prime\,2}$ in Eq. 2.11 can be rewritten as:

$\chi_{\rm M}^{\prime\,2}(\eta_{0})=C-{B^{2}\over{A}}+\ln{A\over 2\pi}\,,$
(2.13)

where $A=\sum\alpha_{i}^{2}/(\beta_{i}^{2}{\sigma^{\prime\,2}_{{\eta_{{\rm
obs},i}}}})$, $B=\sum\alpha_{i}/(\beta_{i}{\sigma^{\prime\,2}_{{\eta_{{\rm
obs},i}}}})$, and $C=\sum 1/{\sigma^{\prime\,2}_{{\eta_{{\rm obs},i}}}}$.

It is evident that all of the quantities used in the CDDR test come directly
from observations, and $\chi_{\rm M}^{\prime\,2}$ in Eq. 2.13 is independent
of parameters such as $M_{\rm B}$ and $l$. In this way, we can remove $M_{\rm
B}$ and $l$ from the fit by analytically marginalizing them in Eq. 2.11. Since
this test is based on the observed data and does not require any assumptions
about cosmological models, the parametric method used to test CDDR is
independent of the cosmological model. The results are shown in Fig. 6 and
Tab. 1. To compare the capability of QSO data with that of other astronomic
observational data in testing CDDR, we list the results of the constraints on
$\eta_{0}$ obtained from different observational data sets in Tab. 2.

parmetrization |  P1: $1+\eta_{0}{z}$ | P2: $1+\eta_{0}{z\over(1+z)}$ |  P3: $1+\eta_{0}{\ln(1+z)}$
---|---|---|---
$\eta_{0}^{\rm{A\,C}{\dagger}}$ | ${-0.088\pm 0.031\pm 0.062\pm 0.093}$ | ${-0.188\pm 0.059\pm 0.118\pm 0.177}$ | ${-0.132\pm 0.044\pm 0.087\pm 0.131}$
$\eta_{0}^{\rm{A\,C}{\ddagger}}$ | ${-0.100\pm 0.013\pm 0.025\pm 0.038}$ | ${-0.248\pm 0.031\pm 0.062\pm 0.093}$ | ${-0.165\pm 0.021\pm 0.041\pm 0.062}$
$\eta_{0}^{\rm{B\,C}{\dagger}}$ | ${-0.006\pm 0.033\pm 0.066\pm 0.100}$ | ${-0.030\pm 0.063\pm 0.126\pm 0.190}$ | ${-0.017\pm 0.047\pm 0.094\pm 0.141}$
$\eta_{0}^{\rm{B\,C}{\ddagger}}$ | ${-0.053\pm 0.014\pm 0.027\pm 0.041}$ | ${-0.127\pm 0.033\pm 0.067\pm 0.100}$ | ${-0.086\pm 0.022\pm 0.044\pm 0.067}$
$\eta_{0}^{\rm{A\,D}{\dagger}}$ | ${-0.065\pm 0.016\pm 0.032\pm 0.048}$ | ${-0.132\pm 0.030\pm 0.059\pm 0.089}$ | ${-0.095\pm 0.022\pm 0.045\pm 0.067}$
$\eta_{0}^{\rm{A\,D}{\ddagger}}$ | ${-0.050\pm 0.009\pm 0.017\pm 0.026}$ | ${-0.123\pm 0.021\pm 0.042\pm 0.063}$ | ${-0.082\pm 0.014\pm 0.028\pm 0.042}$
$\eta_{0}^{\rm{B\,D}{\dagger}}$ | ${0.024\pm 0.017\pm 0.034\pm 0.051}$ | ${0.035\pm 0.032\pm 0.063\pm 0.095}$ | ${0.029\pm 0.024\pm 0.048\pm 0.071}$
$\eta_{0}^{\rm{B\,D}{\ddagger}}$ | ${0.004\pm 0.009\pm 0.018\pm 0.028}$ | ${0.013\pm 0.022\pm 0.044\pm 0.067}$ | ${0.008\pm 0.015\pm 0.030\pm 0.045}$
${\eta_{0}}^{\star{\dagger}}$ | ${-0.044\pm^{0.049}_{0.046}\pm^{0.102}_{0.088}\pm^{0.160}_{0.128}}$ | ${-0.256\pm^{0.137}_{0.121}\pm^{0.292}_{0.230}\pm^{0.472}_{0.328}}$ | ${-0.114\pm^{0.084}_{0.076}\pm^{0.176}_{0.146}\pm^{0.280}_{0.211}}$
${\eta_{0}}^{\star{\ddagger}}$ | ${-0.014\pm^{0.024}_{0.023}\pm^{0.050}_{0.044}\pm^{0.077}_{0.065}}$ | ${-0.055\pm^{0.111}_{0.099}\pm^{0.237}_{0.189}\pm^{0.380}_{0.270}}$ | ${-0.029\pm^{0.053}_{0.049}\pm^{0.110}_{0.094}\pm^{0.173}_{0.136}}$
Table 1: The maximum likelihood estimation results for the parameterizations
with the binning method and ANN. The $\eta_{0}$ is represented by the best fit
value $\zeta_{0,{\rm best}}\pm 1\sigma\pm 2\sigma\pm 3\sigma$ for each
dataset. The superscripts A, B, C, and D represent the cases obtained from
$M_{\rm B}^{\rm D20}$, $M_{\rm B}^{\rm B23}$, $l^{\rm C17}$, and $l^{\rm
C19}$, respectively. The superscript $\star$ denotes the results obtained from
the flat marginalization for $M_{\rm B}$ and $l$, and ${\dagger}$ and
${\ddagger}$ denote the results obtained from the binning method and ANN,
respectively.

Figure 6:  The likelihood distribution obtained with flat priors on $\kappa$ using the binning method (left) and ANN (right).  Dataset used | P1: $1+\eta_{0}{z}$ | P2: $1+\eta_{0}{z\over(1+z)}$ |  P3: $1+\eta_{0}{\ln(1+z)}$
---|---|---|---
${\rm{Union2}}+{\rm{Galaxy(Prior)}}$ [10] | ${-0.22{\pm{0.11}}}$ | ${-0.33{\pm{0.16}}}$ |
${\rm{Union2.1}}+{\rm{91GMF(Prior)}}$ [81] | ${-0.08{\pm^{0.11}_{0.10}}}$ |  |
${\rm{Union2.1}}+{\rm{BAO(Prior)}}$ [71] | ${-0.086{\pm{0.064}}}$ | ${-0.131{\pm{0.098}}}$ |
${\rm{Union2.1}}+{\rm{BAO(Marg)}}$ [71] | ${-0.174{\pm^{0.253}_{0.199}}}$ | ${-0.409{\pm^{0.529}_{0.381}}}$ |
${\rm{Y_{SZ}}}-{\rm{Y_{X}}}\,{\rm{ratio}}+H(z)(\rm Prior)$ [17] | ${0.008{\pm{0.05}}}$ | ${0.019{\pm{0.11}}}$ | ${0.013{\pm{0.07}}}$
${\rm{GMF}}+{\rm{SNIa}}+{T_{\rm CMB}}(\rm Prior)$ [82] | ${-0.020{\pm{0.027}}}$ | ${-0.041{\pm{0.042}}}$ |
${\rm{SNIa}}+{\rm{BAO}(Marg)}$ [83] | ${-0.07{\pm{0.12}}}$ | ${-0.20{\pm{0.27}}}$ | ${-0.12{\pm{0.18}}}$
${\rm{SNIa}}+{\rm{BAO}(Marg)}$ [28] | ${-0.037{\pm^{0.110}_{0.097}}}$ | ${-0.101{\pm^{0.269}_{0.225}}}$ | ${-0.061{\pm^{0.173}_{0.149}}}$
Table 2: Summary of the constraints on parameter $\eta_{0}$ with different
data sets. “Prior” represents the results obtained using certain parameters
with specific priors, and “Marg” represents the results obtained by
marginalizing certain parameters with a flat prior.

##  3  Results and Analysis

From Fig. 4 and Tab. 1, it can be seen that through the binning method, CDDR
is consistent with the observed data at various confidence levels (CLs)
depending on the combination of $M_{\rm B}$ and $l$. Specifically, it is
consistent at $3\sigma$ CL with the combination $M_{\rm B}^{\rm D20}$ and
$l^{\rm C17}$; at $1\sigma$ CL with the combination of $M_{\rm B}^{\rm B23}$
and $l^{\rm C17}$; and at $2\sigma$ CL with the combination of $M_{\rm B}^{\rm
B23}$ and $l^{\rm C19}$. However, CDDR is not consistent with the observed
data at $3\sigma$ CL with the combination of $M_{\rm B}^{\rm D20}$ and $l^{\rm
C19}$. Similarly, from Fig. 5 and Tab. 1, through the ANN method, CDDR is only
consistent with the observed data at $1\sigma$ CL with the combination of
$M_{\rm B}^{\rm B23}$ and $l^{\rm C19}$. Nevertheless, when considering the
other combinations of $M_{\rm B}$ and $l$ values, specifically $M_{\rm B}^{\rm
B23}$ and $l^{\rm C17}$, $M_{\rm B}^{\rm D20}$ and $l^{\rm C17}$, as well as
$M_{\rm B}^{\rm D20}$ and $l^{\rm C19}$, the CDDR is not consistent with the
observed data. Therefore, the prior values of $M_{\rm B}$ and $l$ may induce
significant bias in the CDDR test.

From Fig. 4, Fig. 5, and Tab. 1, the parametrization P1 imposes the most
rigorous constraints on testing CDDR, although the result of the CDDR test is
nearly independent of the parametrization of $\eta(z)$. Now, we compare the
capability of QSO measurements to constrain parameter $\eta_{0}$ with that of
other astronomical observations obtained under specific prior conditions of
cosmological variables. With the binning method, the QSO measurements improve
the accuracy of $\eta_{0}$ by approximately 85% at $1\sigma$ CL when compared
to results obtained from the Union2+galaxy cluster observation (the elliptical
$\beta$ model) [10] or Union2.1+91gas mass fraction (GMF) observation [81];
and about 75% at $1\sigma$ CL when compared to results obtained from
Union2.1+BAO measurements [71], where the CDDR tests were conducted with
specific priors of $M_{\rm B}$ or $r_{\rm s}$ from the CMB observations. Our
results are also roughly 60% more stringent than those from the South Pole
Telescope-SZ clusters and X-ray measurements from Multi-mirror Mission-Newton
[17], where the prior of $M_{\rm B}$ and $H_{0}$ are utilized; and are 40%
more stringent than the result from the x-ray GMF of galaxy clusters jointly
with SNIa and CMB temperature [82], where $M_{\rm B}$ is fixed to derive the
LD. Using the ANN, while testing the CDDR with more available QSO
measurements, the uncertainties of $\eta_{0}$ at the $1\sigma$ CL are improved
by approximately 50% when compared to results obtained from the binning
method.

When testing the CDDR using a flat prior of $\kappa\equiv 10^{M_{\rm B}\over
5}l$, CDDR is consistent with the observed data at $2\sigma$ CL with the
binning method; and at $1\sigma$ CL with the ANN method. The constraints on
$\eta_{0}$ obtained from the flat prior of $\kappa$ are much weaker than those
obtained from the specific priors of $M_{\rm B}$ and $l$, due to marginalizing
$\kappa$ with a flat prior in our analysis. To assess the ability of testing
CDDR from QSO measurements, it is valuable to compare our results with
previous constraints on $\eta_{0}$ from other observational data by
marginalizing certain parameters with a flat prior. With the binning method,
the QSO measurements improve the accuracy of $\eta_{0}$ by approximately 75%
at $1\sigma$ CL when compared to the results obtained from the Union2.1+BAO
observations, where the dimensionless Hubble constant $h$ was marginalized
with a flat prior [71]. The constraints on $\eta_{0}$ are roughly 60% more
stringent than those from the most recent Pantheon samples and BOSS DR12 BAO
measurements within the redshift range $0.31\leq{z}\leq 0.72$ [83]; and they
are 50% more stringent than the result from 5 BAO measurements utilizing the
extended Baryon Oscillation Spectroscopic Survey data release 16 quasar
samples in conjunction with the Pantheon SNIa samples [28], where $M_{\rm B}$
and $r_{\rm s}$ were marginalized. As for the results obtained from the ANN
method, the uncertainties of $\eta_{0}$ at $1\sigma$ CL are reduced by
approximately 40% when compared to the results obtained from the binning
method. Therefore, the QSO measurements demonstrate a superior capability in
testing CDDR compared to BAO observations, which have been recognized as
powerful tools for testing CDDR [71, 83].

##  4 conclusion

The CDDR plays an important role in astronomical observations and modern
cosmology, and any deviation from the CDDR may indicate new physical signals.
SNIa and QSO measurements can be regarded as effective observational data for
testing the CDDR. However, due to the uncertainty in the absolute magnitude
$M_{\rm B}$ and the linear size scaling factor $l$ , which are constrained by
different astronomical observations and cosmological models, it is necessary
to investigate the impact of the prior values of $M_{\rm B}$ and $l$ on the
CDDR test, and to verify the validity of the CDDR using new methods.

In this work, we test the CDDR by comparing the LD derived from the Pantheon
SNIa compilation with ADD from QSO measurements, using parametric methods. We
employ the binning method and ANN to match the SNIa data with the QSO
measurements at the same redshift, and adopt the function $\eta(z)=D_{\rm
L}(z)/D_{\rm A}(z)(1+z)^{-2}$ to probe the possible deviations from the CDDR
at any redshift. Two specific prior values of $M_{\rm B}$ and $l$ are used to
obtain the LDs from the SNIa observations and the ADDs from QSO measurements,
respectively. The results show that the specific prior values of $M_{\rm B}$
and $l$ cause significant biases in the CDDR test, if the astronomical
observations do not provide accurate values for $M_{\rm B}$ and $l$.

To avoid the bias in the CDDR test caused by the prior values of $M_{\rm B}$
and $l$, we treat the fiducial values of $M_{\rm B}$ and $l$ as nuisance
parameters to determine the LD ${D_{\rm L}}$ and ADD ${D_{\rm A}}$. We then
marginalize their impacts on the CDDR test by applying a flat prior on the new
variable $\kappa$ $\equiv$ $10^{M_{\rm B}\over 5}l$ in the statistical
analysis. Our results indicate no violation of the CDDR. However, the
capability of the QSO measurements to test CDDR is reduced compared to the
results obtained from specific values of $M_{\rm B}$ and $l$, due to
marginalizing $\kappa$ with a flat prior in our analysis. In comparison to the
previous results, the capability of QSO measurements to test the CDDR is much
stronger than that of other previous astronomic observations, regardless of
whether the method used is dependent on $M_{\rm B}$ and $l$ or not. It is
noteworthy that the method for testing the CDDR in this work is not only
independent of the cosmological model but also independent of the prior values
of the absolute magnitude $M_{\rm B}$ and the linear size scaling factor $l$.
Therefore, QSO measurement can serve as a powerful tool for testing CDDR
independently of cosmological model.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China
under Grants No. 12375045, No. 12305056, No. 12105097 and No. 12205093, the
Hunan Provincial Natural Science Foundation of China under Grants No. 12JJA001
and No. 2020JJ4284, the Natural Science Research Project of Education
Department of Anhui Province No. 2022AH051634, and the Science Research Fund
of Hunan Provincial Education Department No. 21A0297.

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#  Circular orderability and quandles

Idrissa Ba  Department of Mathematics
Marianopolis College
Westmount
QC H3Y 1X9<EMAIL_ADDRESS>and  Mohamed Elhamdadi  Department of
Mathematics, University of South Florida, Tampa, FL 33620, U.S.A.
<EMAIL_ADDRESS>http://shell.cas.usf.edu/ emohamed/

###### Abstract.

In this paper, we introduce the notion of circular orderability for quandles.
We show that the set all right (respectively left) circular orderings of a
quandle is a compact topological space. We also show that the space of right
(respectively left) orderings of a quandle embeds in its space of right
(respectively left) circular orderings. Examples of quandles that are not left
circularly orderable and examples of quandles that are neither left nor right
circularly orderable are given.

###### Key words and phrases:

circular orderability, quandles, left-orderability

###### 2010 Mathematics Subject Classification:

Primary: 57M07, 06F15, 20F99.

##  1\. Introduction

Quandles are algebraic structures whose axioms are motivated by the three
Reidemeister moves in classical knot theory. They were introduced
independently by Joyce [Joyce] and Matveev [Matveev] in the early 1980’s.
Since then they have been used to construct invariants of knots in the
$3$-space and knotted surfaces in $4$-space (see for example [EN] for more on
quandles). However, the notion of a quandle can be traced back to the 1940’s
in the work of Mituhisa Takasaki [Takasaki]. Joyce and Matveev introduced the
notion the fundamental quandle of a knot leading to their main theorem which
states that two knots $K$ and $K^{\prime}$ are equivalent (up to reverse and
mirror image) if and only if the fundamental quandles $Q(K)$ and
$Q(K^{\prime})$ are isomorphic. This theorem turns the topological problem of
classification of knots into the algebraic problem of classification of
quandles. Recently, there has been many works on classifying some families of
quandles, such as connected quandles, Alexander quandles, medial quandles etc.
Quandles also have been investigated recently from other point of views such
as relations to Lie algebras [CCES1, CCES2], Hopf algebras [AG, CCES2],
quasigroups and Moufang loops [Elhamdadi], ring theory [EFT], Yang-Baxter
equation and its homology [CES] and topology[EM]. Orderability of quandles
appeared in [BPS2, DDHPV] where it was shown for example that conjugation
quandles (respectively core quandles) of bi-orderable groups are right
orderable (respectively left orderable) quandles. The set of left ordering on
a group $G$ can be given a topology, and this space is called the space of
left orderings. In [Sikora] it was shown that the space of left orderings of a
group is compact. The analogous of this theorem was proved for magmas in
general (and quandles in particular) in [DDHPV].

Recently, the notions of circular orderability and left-orderability have been
intensively studied in low dimensional topology, particularly for the
fundamental group of 3-manifolds (see [[BaC1], [[BaC2], [BS], [BRW] and
[Cal]). In [[BaC1] the authors have shown that a compact, connected,
$P^{2}$-irreducible $3$-manifold has a left circularly orderable fundamental
group if and only if there exists a finite cyclic cover with left-orderable
fundamental group. A consequence of this result is that, if a $3$-manifold
admits a cooriented taut foliation then it has a finite cyclic cover with
left-orderable fundamental group, and this give a characterization of a
topological property of a $3$-manifold (taut foliation) by an algebraic
property to its fundamental group (left-orderability and circular
orderability). This may also be a motivation to study some algebraic
properties of quandles (circular orderability and orderability).

In this article we introduce the notion of circular orderability for quandles.
We show that the set all right (respectively left) circular orderings of a
quandle is a compact topological spaces.

Let $Q$ be a quandle, we denote by $RCO(Q)$ (respectively $LCO(Q)$) the set of
right circular orderings on $Q$ (respectively the set of left circular
orderings on $Q$)

######  Theorem 1.1.

The space $RCO(Q)$ (respectively $LCO(Q)$) is compact.

We also show that the space of right (respectively left) orderings [DDHPV] of
a quandle embeds in its space of right (respectively left) circular orderings.

######  Theorem 1.2.

If $Q$ is a quandle, then the space of right (respectively left) orderings of
$Q$ is a subspace of its space of right (respectively left) circular
orderings.

Given a group with a circular ordering which is both right and left invariant,
we prove that the conjugation quandle is right circularly orderable and more
strongly we shown that the conjugation quandle is right orderable. We also
give examples of quandles that are not left circularly orderable and examples
of quandles that are neither left nor right circularly orderable.

The following is the organization of this article. In section 2, we recall the
basics of quandles and also orderability. Section 3 introduces the notion of
circular orderability of quandles, and show that a right (respectively left)
orderable quandle is right (respectively left) circularly orderable. In this
section, it is also shown that the conjugation quandle of a bi-orderable group
is a right orderable quandle and we give examples of non-left circularly
orderable quandles, and explicit examples of quandles of small cardinalities
that are neither left nor right circularly orderable. Section 4 deals with the
topology of the set of right (respectively left) circular orderings on
quandles. Precisely, we introduce a topology on the set of right (respectively
left) circular orderings on quandles and prove that it is compact.

##  2\. Review of quandles and orderability

A quandle is a non-empty set $Q$ with a binary operation $*$ which satisfies
the following conditions:

  1. (1)

For any $s\in Q$, $s*s=s$;

  2. (2)

For any $s_{1},s_{2}\in Q$, there exists a unique $s_{3}\in Q$ such that
$s_{3}*s_{2}=s_{1}$; and

  3. (3)

For any $s_{1},s_{2},s_{3}\in Q$,
$(s_{1}*s_{2})*s_{3}=(s_{1}*s_{3})*(s_{2}*s_{3})$.

Let consider right multiplications $R_{s}:Q\rightarrow Q$ given by $t\mapsto
t*s$. Then axioms (2) and (3) state that $R_{s}$ are automorphisms of the
quandle $Q$ and axiom (1) states that $s$ is a fixed point of $R_{s}$. An
element $e$ of a quandle $Q$ is called stabilizer element if $s*e=s$ for any
$s\in Q$.

A quandle $Q$ is called trivial if all its elements are stabilizer elements. A
trivial quandle can have an arbitrary number of element, this is one
difference between a group and a quandle.

By the conditions $(2)$ and $(3)$ of the definition of quandle for any $b\in
Q$, there exists an automorphism $R_{b}:Q\longrightarrow Q$ defined by
$R_{b}(s)=s*b$, for any $s\in Q$. We call $id_{Q}$ the identity automorphism
of $Q$. A quandle is called involutary if $R_{s}\circ R_{s}=id_{Q}$ for any
$s\in Q$.

By the conditions $(2)$ and $(3)$ of the definition of quandle, there exists a
dual binary operation $*^{-1}$ defined by $s*^{-1}r=t$ if $s=t*r$ for any
$r,s$ and $t\in Q$. Hence, we have also that $(s*r)*^{-1}r=s$ and
$(s*^{-1}r)*r=s$ for any $r,s\in Q$, this is called right cancellation.
Therefore, by the condition $(1)$ of the definition of quandle and right
cancellation we have that $s*^{-1}s=s$ for any $s\in Q$.

A quandle is called latin if for any $s\in Q$ the map $L_{s}:Q\longrightarrow
Q$ defined by $L_{s}(t)=s*t$ is a bijection for any $t\in Q$. A quandle is
called semi-latin if the map $L_{s}$ is injective for any $s\in Q$.

A subquandle of a quandle, $Q$, is a subset $X\subset Q$ which is closed under
the quandle operation.

The following are some examples of quandle.

  * •

Any set with the binary operation $x*y=x$ is a quandle called _trivial_
quandle.

  * •

Let $G$ be a group. The set $S=G$ with the binary operation $*$ defined by
$g*h=h^{-1}gh$ is a quandle, denoted Conj$(G)$, and called _conjugation_
quandle of $G$.

  * •

The set $T=G$, where $G$ is a group, with the binary operation $*$ defined by
$g*h=hg^{-1}h$ is a quandle, denoted by Core$(G)$, and called _core_ quandle
of $G$. If furthermore $G$ is an abelian group then this quandle is called
_Takasaki_ quandle [Takasaki].

  * •

If $\phi$ is an automorphism of a group $G$, then the set $R=G$ with the
binary operation $*$ defined by $g*h=\phi(gh^{-1})h$ is a quandle, denoted by
Aut$(G,\phi)$, and called _generalized Alexander_ quandle of $G$ with respect
to $\phi$.

Recall that a group $G$ is called _left-orderable_ (respectively right-
orderable) if there exists a strict total ordering $<$ on $G$ such that $g<h$
implies $fg<fh$ (respectively $gf<hf$) for all $f,g,h\in G$, the relation $<$
is called a _left-ordering_ of $G$. A left-orderable (respectively right-
orderable) quandle was defined in the same fashion in [BPS]. A quandle $(Q,*)$
is called left-orderable (respectively right-orderable) if there exists a
strict total ordering $\prec$ on $Q$ such that $s\prec t$ implies $r*s\prec
r*t$ (respectively $s*r\prec t*r$) for all $r,s,t\in Q$, the relation $\prec$
is called a _left-ordering_ (respectively right-ordering) on $Q$. The notion
of a homomorphism between two ordered quandles is natural. Let
$(X,*,\prec_{X})$ and $(Y,\diamond,\prec_{Y})$ be two right ordered quandles.
A map $f:X\rightarrow Y$ is called an order preserving homomorphism of right
ordered quandles if $f$ is a quandle homomorphism (that
is,$f(x*y)=f(x)\diamond f(y),\forall x,y\in X$) and $x\prec_{X}y$ implies
$f(x)\prec_{Y}f(y)$.

######  Example 2.1.

We use quandle structures defined in [CES] over the real line $\mathbb{R}$ to
give explicit structures of right and left orderable quandles. Consider the
real line $\mathbb{R}$ with its natural ordering and with quandle operation
$x*y=\alpha x+(1-\alpha)y$, where $\alpha\neq 0,1$

  1. (1)

If $\alpha>0$, then the quandle $\mathbb{R}$ is right orderable.

  2. (2)

If $\alpha<1$, then the quandle $\mathbb{R}$ is left orderable.

  3. (3)

Thus, if $0<\alpha<1$, then the quandle $\mathbb{R}$ is bi-orderable.

##  3\. circular orderability of quandles

A circular ordering on a set $T$ is a map $c:T\times T\times
T\rightarrow\\{-1,0,1\\}$ satisfying:

  1. (1)

If $(t_{1},t_{2},t_{3})\in T^{3}$ then $c(t_{1},t_{2},t_{3})=0$ if and only if
$\\{t_{1},t_{2},t_{3}\\}$ are not all distinct;

  2. (2)

For all $t_{1},t_{2},t_{3},t_{4}\in T$ we have

$c(t_{1},t_{2},t_{3})-c(t_{1},t_{2},t_{4})+c(t_{1},t_{3},t_{4})-c(t_{2},t_{3},t_{4})=0.$

A set $T$ with a circular ordering is called circularly orderable. A group $G$
is called left-circularly orderable (respectively right-circularly orderable)
if it admits a circular ordering $c:G\times G\times G\rightarrow\\{-1,0,1\\}$
such that $c(g_{1},g_{2},g_{3})=c(gg_{1},gg_{2},gg_{3})$ (respectively
$c(g_{1},g_{2},g_{3})=c(g_{1}g,g_{2}g,g_{3}g)$) for any $g,g_{1},g_{2}$ and
$g_{3}\in G$. In this case we say that the circular ordering $c$ is left-
invariant (respectively right-invariant).

We say that a quandle $Q$ is left-circularly orderable (respectively right-
circularly orderable) if it admits a circular ordering $c:Q\times Q\times
Q\rightarrow\\{-1,0,1\\}$ such that
$c(s_{1},s_{2},s_{3})=c(s*s_{1},s*s_{2},s*s_{3})$ (respectively
$c(s_{1},s_{2},s_{3})=c(s_{1}*s,s_{2}*s,s_{3}*s)$) for any $s,s_{1},s_{2}$ and
$s_{3}\in Q$.

######  Lemma 3.1.

A left (respectively right) orderable quandle is also left (respectively
right) circularly orderable.

###### Proof.

Let $\prec$ be a left ordering on a quandle $Q$ then define the map
$c_{\prec}:Q^{3}\rightarrow\\{-1,0,1\\}$ by

$(1)\;\;\;\;c_{\prec}(s_{1},s_{2},s_{3})=\left\\{\begin{array}[]{cl}1&\text{if
$s_{1}\prec s_{2}\prec s_{3}$ or $s_{2}\prec s_{3}\prec s_{1}$ or $s_{3}\prec
s_{1}\prec s_{2}$}\\\ -1&\text{if $s_{1}\prec s_{3}\prec s_{2}$ or $s_{3}\prec
s_{2}\prec s_{1}$ or $s_{2}\prec s_{1}\prec s_{3}$ }\\\
0&\text{otherwise.}\end{array}\right.$

By definition, $c_{\prec}$ is a circular ordering. Now, it is left to check
that $c_{\prec}$ is invariant under left-multiplication. Since $Q$ is left
orderable, multiplying on the left by any element $s\in Q$ will not change the
inequalities of $(1).$ Similarly, if a quandle $Q$ is right orderable then the
same proof show that $Q$ is right circularly orderable. ∎

######  Proposition 3.2.

Let $G$ be a circularly orderable group with a circular ordering $c$ which is
left and right-invariant. Then, the quandle $\mathrm{Conj}(G)$ is a right
circularly orderable quandle;

###### Proof.

Let
$d:\mathrm{Conj}(G)\times\mathrm{Conj}(G)\times\mathrm{Conj}(G)\rightarrow\\{-1,0,1\\}$
defined by $d(g_{1},g_{2},g_{3})=c(g_{1},g_{2},g_{3})$ for any $g_{1},g_{2}$
and $g_{3}\in G$. Let $g_{1},g_{2}$ and $g_{3}\in G$, since
$d(g_{1},g_{2},g_{3})=c(g_{1},g_{2},g_{3})$, $d$ is a circular ordering on
$\mathrm{Conj}(G)$ as a set. Now it is left to show the right-invariant
condition. Let $g,g_{1},g_{2}$ and $g_{3}\in G$, we have that
$d(g_{1}*g,g_{2}*g,g_{3}*g)=d(g^{-1}g_{1}g,g^{-1}g_{2}g,g^{-1}g_{3}g)=c(g^{-1}g_{1}g,g^{-1}g_{2}g,g^{-1}g_{3}g)=c(g_{1},g_{2},g_{3})$.
Hence, $\mathrm{Conj}(G)$ is a right circularly orderable quandle.

∎

Recall that a group $G$ is called bi-circularly orderable (respectively bi-
orderable) if it admits a circular ordering (respectively an ordering) which
is left and right-invariant. Similarly, we say that a quandle $Q$ is bi-
circularly orderable if it admits a circular ordering (respectively an
ordering) which is left and right-invariant. The notion of bi-circular
orderability for groups have been studied since the $1950$’s, and in $1959$
Swierczkowski had given in [Sw] a complete classification of these type of
groups.

######  Theorem 3.1.

[Sw] If a group $G$ is bi-circularly orderable, then there exists a bi-
orderable group $\Gamma$ such that $G$ is a subgroup of
$\Gamma\times\mathbb{S}^{1}$. Moreover, the bi-circular order on $G$ is
determined by the natural bi-circular order on $\Gamma\times\mathbb{S}^{1}$.

By the natural bi-circular order it is meant that if you equip
$\Gamma\times\mathbb{S}^{1}$ with the lexicographic bi-circular ordering, then
the bi-circular ordering on $G$ is just the restriction of this lexicographic
circular ordering on $\Gamma\times\mathbb{S}^{1}$ to the subgroup $G$. The
Goursat’s Lemma Give a classification of all the subgroups of
$\Gamma\times\mathbb{S}^{1}$. Hence, Theorem 3.1 gives infinitely many example
of bi-circularly orderable groups that are not bi-orderable.

Let $\\{Q_{i},*_{i}\\}_{\i\in I}$ be a family of quandles. Then the Cartesian
product $Q=\prod_{i\in I}Q_{i}$ with the operation
$(x_{i})\star(y_{i})=(x_{i}*_{i}y_{i})$ is a quandle called the quandle
product.

######  Proposition 3.3.

The conjugation quandle of any bi-circularly orderable group $G$ is right
orderable.

###### Proof.

Let $H$ be any bi-orderable group. Since
$\mathrm{Conj}(H\times\mathbb{S}^{1})=\mathrm{Conj}(H)\times\mathrm{Conj}(\mathbb{S}^{1})$,
and $\mathbb{S}^{1}$ is commutative, so $\mathrm{Conj}(\mathbb{S}^{1})$ is
right orderable as trivial quandle. Since $H$ is bi-orderable,
$\mathrm{Conj}(H)$ is right orderable. Hence, the quandle
$\mathrm{Conj}(H\times\mathbb{S}^{1})=\mathrm{Conj}(H)\times\mathrm{Conj}(\mathbb{S}^{1})$
is right orderable by [RSS, Proposition 4.4 ]. Therefore, the result follows
from Theorem 3.1. ∎

######  Lemma 3.4.

If $G$ is a nontrivial group with at least three elements, then the quandle
$\mathrm{Conj}(G)$ is not left circularly orderable.

###### Proof.

Let $G$ be any nontrivial group with at least three elements. Consider the
conjugation quandle $\mathrm{Conj}(G)$ with operation $x*y=y^{-1}xy$. By
contradiction, let $c:\mathrm{Conj}(G)^{3}\rightarrow\\{\pm 1,0\\}$ be a left
circular ordering. Then
$c(s_{1},s_{2},s_{3})=c(e*s_{1},e*s_{2},e*s_{3})=c(e,e,e)=0$, where $e$ is the
identity element of $G$. Thus the function $c$ is the constant zero map
implying that $\mathrm{Conj}(G)$ cannot be have a left circular ordering. ∎

######  Example 3.5.

The trivial quandle with _two_ elements is bi-circularly orderable since the
trivial circular ordering which is always equal to zero is both left and right
invariant. Since this quandle is not left orderable, this show that the notion
of circular orderability for quandles is different to the notion orderability.

######  Example 3.6.

Consider the three element quandle $X=\\{1,2,3\\}$ with orbit decomposition
$\\{1,2\\}\sqcup\\{3\\}$. It’s Cayley table is given by
$\begin{bmatrix}1&1&2\\\ 2&2&1\\\ 3&3&3\end{bmatrix}.$ The $(i,j)$-entry of
this matrix correspond to the element $i*j$ in the quandle. We show that this
quandle is neither left nor right circularly orderable.
(I) Now assume that $c$ is left-invariant circular ordering on $X$, then for
any pairwise distinct elements $x,y,z\in X$, we have
$c(x,y,z)=c(3*x,3*y,3*z)=c(3,3,3)=0,$ and thus the quandle $X$ can’t have a
left-invariant circular ordering.
(II) Now assume that $c$ is right-invariant circular ordering on $X$, then we
have $c(1,2,3)=c(1*3,2*3,3*3)=c(2,1,3)$. Since $c$ is a circular ordering, it
satisfies condition $(2)$ of the definition, so

$c(1,2,1)-c(1,2,3)+c(1,1,3)-c(2,1,3)=0.$

Hence, $c(1,2,3)=-c(2,1,3)$ which is a contradiction to the fact that

$c(1,2,3)=c(1*3,2*3,3*3)=c(2,1,3).$

######  Example 3.7.

Consider the dihedral quandle $X=\mathbb{Z}_{3}$, where $x*y=2y-x=2y+2x$. We
show that this quandle is neither left nor right circularly orderable. The
quandle operation satisfies $x*y=y*x$ for all $x,y\in X$, and then left
circular orderings and right circular orderings are the same. By contradiction
assume that there exists a right circular ordering $c$ on $X$, and let
$x,y,z\in X$. If at least two of $x,y,z$ are equal then by definition
$c(x,y,z)=0$. Now assume that $x,y,z$ are pairwise distinct. Thus the product
of any two of them gives the third element. Then
$c(x,y,z)=c(x*x,y*x,z*x)=c(x,z,y)$, $c(x,y,z)=c(x*y,y*y,z*y)=c(z,y,x)$ and
$c(x,y,z)=c(x*z,y*z,z*z)=c(y,x,z)$. We thus obtain
$c(x,y,z)=c(x,z,y)=c(z,y,x)=c(y,x,z)$ and consequently any of the 6
permutations of the set $\\{x,y,z\\}$ leaves the value of $c(x,y,z)$
unchanged. Since $c$ is a circular ordering, it satisfies condition $(2)$ of
the definition, so

$c(x,y,x)-c(x,y,z)+c(x,x,z)-c(y,x,z)=0.$

Hence, $c(x,y,z)=-c(y,x,z)$ which is a contradiction to the fact that

$c(x,y,z)=c(y,x,z).$

Therefore, the dihedral quandle $X=\mathbb{Z}_{3}$ with quandle operation
$x*y=2y-x=2y+2x$ is neither left nor right circular orderable.

##  4\. The space of circular orderings of quandles

In this section we study the set of circular orderings of a quandle. Let $Q$
be a quandle, we denote by $RCO(Q)$ (respectively $LCO(Q)$) the set of right
circular orderings on $Q$ (respectively the set of left circular orderings on
$Q$). Since a right ordering (respectively left ordering) on a quandle gives a
right circular ordering (respectively left circular ordering) by Lemma 3.1,
the set of right orderings $RO(Q)$ (respectively left orderings $LO(Q)$) on a
quandle $Q$ can be seen as a subset of the set of left circular orderings
$LCO(Q)$. The set of left-orderings (respectively right orderings) on a
quandle was already studied in [DDHPV] and they showed that for any magma
$\mathcal{M}$, the set of left orderings $LO(M)$ (respectively right orderings
$RO(M)$) is a compact topological space. A magma is a set with a binary
operation $\cdot:\mathcal{M}\times\mathcal{M}\longrightarrow\mathcal{M}.$ So,
a quandle is in fact a magma.

We topologize the set $RCO(Q)$ (respectively $LCO(Q)$) as a subspace of the
space $\\{-1,0,1\\}^{Q^{3}}$ of all maps from $Q^{3}$ to $\\{-1,0,1\\}$ with
the Tychonoff topology. We define

$\Gamma(Q):=\\{S=(x,y,z)\in Q^{3}\;|\;x=y\;{\rm or}\;y=z\;{\rm or}\;x=z\\},$
$R_{S}=\\{c\in RCO(Q)\;|\;c(S)=1\\}$

and

$L_{S}=\\{c\in LCO(Q)\;|\;c(S)=1\\}$

for any $S\in Q^{3}\setminus\Gamma(Q)$.

######  Lemma 4.1.

The set $\\{R_{S}\\}_{S\in Q^{3}\setminus\Gamma(Q)}$ (respectively
$\\{L_{S}\\}_{S\in Q^{3}\setminus\Gamma(Q)}$) is a subbasis for the topology
on $RCO(Q)$ (respectively $LCO(Q)$).

###### Proof.

Since the $\\{-1,0,1\\}^{Q^{3}}$ is a Cantor set with a subbasis
$\\{V_{s,-1},V_{S,0}\;{\rm and}\;V_{S,1}\\}_{S\in Q^{3}}$ Where

$V_{S,-1}=\\{f:Q^{3}\rightarrow\\{-1,0,1\\}\;|\;f(S)=-1\\},$
$V_{S,0}=\\{f:Q^{3}\rightarrow\\{-1,0,1\\}\;|\;f(S)=0\\}$

and

$V_{S,1}=\\{f:Q^{3}\rightarrow\\{-1,0,1\\}\;|\;f(S)=1\\}.$

Since $RCO(Q)\cap V_{S,0}=RCO(Q)$ (respectively $LCO(Q)\cap V_{S,0}=LCO(Q)$)
for any $S\in\Gamma(Q)$ and $RCO(Q)\cap V_{S,0}=\emptyset$ (respectively
$LCO(Q)\cap V_{S,0}=\emptyset$) for any $S\in Q^{3}\setminus\Gamma(Q)$, we can
throw out all sets of the form $RCO(Q)\cap V_{S,0}$.

Since $RCO(Q)\cap V_{S,-1}=RCO(Q)\cap V_{\tau.S,1}$ (respectively $LCO(Q)\cap
V_{s,-1}=LCO(Q)\cap V_{\tau.S,1}$) for any transposition $\tau$, we can also
throw out all sets of the form $RCO(Q)\cap V_{S,-1}$ (respectively $LCO(Q)\cap
V_{\tau.S,1}$).

∎

######  Theorem 4.1.

The space $RCO(Q)$ (respectively $LCO(Q)$) is compact.

###### Proof.

We have that

$RCO(Q)=\\{c:Q^{3}\rightarrow\\{-1,0,1\\}\;|\;{\rm if}\;(t_{1},t_{2},t_{3})\in
Q^{3}\;{\rm then}\;c(t_{1},t_{2},t_{3})=0\\\ {\rm
if\;and\;only\;if}\;(t_{1},t_{2},t_{3})\in\Gamma(Q);\\\ {\rm
for\;all}\;(t_{1},t_{2},t_{3},t_{4})\in Q^{4}\;\\\ {\rm
we\;have}\;c(t_{1},t_{2},t_{3})-c(t_{1},t_{2},t_{4})+c(t_{1},t_{3},t_{4})-c(t_{2},t_{3},t_{4})=0;\\\
{\rm and}\;c(t_{1}*t,t_{2}*t,t_{3}*t)=c(t_{1},t_{2},t_{3})\;{\rm
for\;any}\;(t,t_{1},t_{2},t_{3})\in Q^{4}\\}$

Let

$\displaystyle A:$ $\displaystyle=\\{c:Q^{3}\rightarrow\\{-1,0,1\\}\;|\;{\rm
if}\;S=(t_{1},t_{2},t_{3})\in Q^{3}\;{\rm then}\;c(t_{1},t_{2},t_{3})=0\;{\rm
if\;and\;only\;if}\;$ $\displaystyle S\in\Gamma(Q)\\}$
$\displaystyle=(\bigcap_{S\in\Gamma(Q)}V_{S,0})\cap(\bigcap_{S\in
Q^{3}\setminus\Gamma(Q)}(V_{S,-1}\cup V_{S,1})),$

$\displaystyle B:$ $\displaystyle=\\{c:Q^{3}\rightarrow\\{-1,0,1\\}\;|\;{\rm
for\;all}\;W=(t_{1},t_{2},t_{3},t_{4})\in Q^{4}\;{\rm we\;have}$
$\displaystyle
d_{W}(c)=c(t_{1},t_{2},t_{3})-c(t_{1},t_{2},t_{4})+c(t_{1},t_{3},t_{4})-c(t_{2},t_{3},t_{4})=0\\}$
$\displaystyle=\bigcap_{W\in
Q^{4}}\\{c\in\\{-1,0,1\\}^{Q^{3}}\;|\;d_{W}(c)=0\\}$

where for any $W=(t_{1},t_{2},t_{3},t_{4})\in Q^{4}$ we define
$d_{W}:\\{-1,0,1\\}^{Q^{3}}\longrightarrow\mathbb{Z}$ by

$d_{W}(c)=c(t_{1},t_{2},t_{3})-c(t_{1},t_{2},t_{4})+c(t_{1},t_{3},t_{4})-c(t_{2},t_{3},t_{4})$

and

$\displaystyle C:$
$\displaystyle=\\{c:Q^{3}\rightarrow\\{-1,0,1\\}\;|\;c(t_{1}*t,t_{2}*t,t_{3}.t)=c(t_{1},t_{2},t_{3})\;{\rm
for\;any}\;(t,t_{1},t_{2},t_{3})\in Q^{4}\\}$ $\displaystyle=\bigcap_{t\in
Q,(t_{1},t_{2},t_{3})\in
Q^{3}}\\{c\in\\{-1,0,1\\}^{Q^{3}}\;|\;c(t_{1}*t,t_{2}*t,t_{3}*t)=c(t_{1},t_{2},t_{3})\\}.$

Since $RCO(Q)=A\cap B\cap C$ and the subsets $V_{S,0}$, $V_{S,-1}$, $V_{S,1}$,
$\\{c\in\\{-1,0,1\\}^{Q^{3}}\;|\;d_{W}(c)=0\\}$ and
$\\{c\in\\{-1,0,1\\}^{Q^{3}}\;|\;c(t_{1}*t,t_{2}*t,t_{3}*t)=c(t_{1},t_{2},t_{3})\\}$
are closed, $RCO(Q)$ is closed in $\\{-1,0,1\\}^{Q^{3}}$. Therefore, $RCO(Q)$
is compact.

A similar proof show that $LCO(Q)$ is compact. ∎

Given a quandle $Q$, recall that a topology on $LO(Q)$ is constructed in
[DDHPV] by choosing as a subbasis the collection $\\{V_{(a,b)}\\}_{(a,b)\in
Q\times Q\setminus\Delta}$, where $\Delta=\\{(a,a)\in Q\;|\;a\in Q\\}$, and
$V_{(a,b)}=\\{<\in LO(Q)\;|\;a<b\\}$. Similarly, a topology on $RO(Q)$ is
constructed by choosing as a subbasis the collection
$\\{V_{(a,b)}\\}_{(a,b)\in Q\times Q\setminus\Delta}$, where
$\Delta=\\{(a,a)\in Q\;|\;a\in Q\\}$ and $V_{(a,b)}=\\{<\in RO(Q)\;|\;a<b\\}$.

######  Proposition 4.2.

The inclusion map $i:LO(Q)\rightarrow LCO(Q)$ (respectively
$j:RO(Q)\rightarrow RCO(Q)$) given by

$<\mapsto c_{<},$

is an embedding.

###### Proof.

Recall that the collection $\\{L_{S}\\}_{S\in Q^{3}\setminus\Gamma(Q)}$
(respectively $\\{R_{S}\\}_{S\in Q^{3}\setminus\Gamma(Q)}$) is a subbasis for
the topology on $LCO(Q)$ (respectively $RCO(Q)$), where

$R_{S}=\\{c\in RCO(Q)\;|\;c(S)=1\\}$

and

$L_{S}=\\{c\in LCO(Q)\;|\;c(S)=1\\}$

for any $S=(x,y,z)\in Q^{3}\setminus\Gamma(Q)$.

For any $S=(x,y,z)\in Q^{3}\setminus\Gamma(Q)$, we have that

$\displaystyle i^{-1}(L_{S})$ $\displaystyle=\\{<\in LO(Q)\;|\;c_{<}(S)=1\\}$
$\displaystyle=\\{<\in LO(Q)\;|\;x<y<z\;{\rm or}\;y<z<x\;{\rm or}\;z<x<y\\}$
$\displaystyle=(V_{(x,y)}\cap V_{(y,z)})\bigcup(V_{(y,z)}\cap
V_{(z,x)})\bigcup(V_{(z,x)}\cap V_{(x,y)})$

and

$\displaystyle i^{-1}(R_{S})$ $\displaystyle=\\{<\in RO(Q)\;|\;c_{<}(S)=1\\}$
$\displaystyle=\\{<\in RO(Q)\;|\;x<y<z\;{\rm or}\;y<z<x\;{\rm or}\;z<x<y\\}$
$\displaystyle=(V_{(x,y)}\cap V_{(y,z)})\bigcup(V_{(y,z)}\cap
V_{(z,x)})\bigcup(V_{(z,x)}\cap V_{(x,y)}).$

Therefore, both of the injective maps $i$ and $j$ are continuous. Since both
$LO(Q)$ and $RO(Q)$ are compact and $\\{-1,0,1\\}^{Q^{3}}$ is Hausdorff, the
maps $i$ and $j$ are embeddings. ∎

######  Corollary 4.2.

The inclusion map $k:BO(Q)=LO(Q)\cap RO(Q)\longrightarrow LCO(Q)$
(respectively $k^{\prime}:BO(Q)=LO(Q)\cap RO(Q)\longrightarrow RCO(Q)$) is an
embedding.

We end this section by asking few questions about the notion of circular
orderability for quandles.

######  Question 4.3.

For exactly which quandles $Q$, the space $RCO(Q)$ (respectively $LCO(Q)$) is
finite?

######  Question 4.4.

Is there a characterization of the notion of right circular orderability
(respectively left circular orderability) of a quandle in term of quandles
actions?

######  Question 4.5.

For exactly which quandles $Q$, the embeddings $i:LO(Q)\rightarrow LCO(Q)$ and
$j:RO(Q)\rightarrow RCO(Q)$ are homeomorphisms?

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