# Quasi-Periodic Particle Acceleration in a Solar Flare

Brendan P. Clarke School of Cosmic Physics, Dublin Institute for Advanced
Studies, Dublin, D02 XF85, Ireland  School of Physics, Trinity College Dublin,
Dublin 2, Ireland  Laura A. Hayes Solar Physics Laboratory, Heliophysics
Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
Peter T. Gallagher School of Cosmic Physics, Dublin Institute for Advanced
Studies, Dublin, D02 XF85, Ireland  Shane A. Maloney School of Cosmic Physics,
Dublin Institute for Advanced Studies, Dublin, D02 XF85, Ireland  School of
Physics, Trinity College Dublin, Dublin 2, Ireland  Eoin P. Carley School of
Cosmic Physics, Dublin Institute for Advanced Studies, Dublin, D02 XF85,
Ireland  School of Physics, Trinity College Dublin, Dublin 2, Ireland

(Received 17 November, 2020; Revised 27 January, 2021; Accepted 7 February,
2021)

###### Abstract

A common feature of electromagnetic emission from solar flares is the presence
of intensity pulsations that vary as a function of time. Known as quasi-
periodic pulsations (QPPs), these variations in flux appear to include
periodic components and characteristic time-scales. Here, we analyse a GOES
M3.7 class flare exhibiting pronounced QPPs across a broad band of wavelengths
using imaging and time-series analysis. We identify QPPs in the timeseries of
X-ray, low frequency radio and EUV wavelengths using wavelet analysis, and
localise the region of the flare site from which the QPPs originate via X-ray
and EUV imaging. It was found that the pulsations within the 171 Å, 1600 Å,
soft X-ray (SXR), and hard X-ray (HXR) light curves yielded similar periods of
$122^{+26}_{-22}$ s, $131^{+36}_{-27}$ s, $123^{+11}_{-26}$ s, and
$137^{+49}_{-56}$ s, respectively, indicating a common progenitor. The low
frequency radio emission at 2.5 MHz contained a longer period of ${\sim}231$
s. Imaging analysis indicates that the location of the X-ray and EUV
pulsations originates from a HXR footpoint linked to a system of nearby open
magnetic field lines. Our results suggest that intermittent particle
acceleration, likely due to ‘bursty’ magnetic reconnection, is responsible for
the QPPs. The precipitating electrons accelerated towards the chromosphere
produce the X-ray and EUV pulsations, while the escaping electrons result in
low frequency radio pulses in the form of type iii@ radio bursts. The
modulation of the reconnection process, resulting in episodic particle
acceleration, explains the presence of these QPPs across the entire spatial
range of flaring emission.

Sun: flares – Sun: oscillations – Sun: EUV radiation Sun: X-rays Sun: radio

††journal: ApJ††facilities: SDO (AIA), RHESSI, WIND (WAVES), FERMI
(GBM),GOES/XRS††software: sunpy (The SunPy Community et al., 2020; Mumford et
al., 2020) pfsspy (Stansby et al., 2020) matplotlib (Hunter, 2007)

##  1 Introduction

Quasi-periodic pulsations (QPPs) are an important feature observed in solar
and stellar flare emission (Nakariakov & Melnikov, 2009; Van Doorsselaere et
al., 2016; Kupriyanova et al., 2020). This puzzling phenomenon lacks a
concrete definition, however, they are typically described by variations in
the flux from a flare as a function of time that appear to include periodic
components and time-scales that typically range from 1 s up to 1 min, and in
extreme cases from $\leq$1 s up to several minutes (Karlický et al., 2005; Tan
et al., 2010; Li et al., 2015; Hayes et al., 2019). QPPs are typically
observed during the impulsive phase of solar flares, however, in recent years
it has become clear that they can persist through to the decay phase, after
the impulsive energy release (Hayes et al., 2016; Dennis et al., 2017; Hayes
et al., 2019).

Figure 1: (a): SDO/AIA 171 Å passband image of the sun on 2015 Nov 4. The
active region associated with the event is visible in the dashed box at the
disk center. (b): The GOES SXR light curves showing the occurrence of the M3.7
class flare. The dashed grey line shows the time at which the image in panel a
was taken.

QPPs have been reported in a broad range of wavelengths from decametric radio
(Li et al., 2015; Carley et al., 2019), through to extreme ultra-violet (EUV)
and X-rays, (Dominique et al., 2018; Dolla et al., 2012), and even $\gamma$
rays (Nakariakov et al., 2009; Li et al., 2020). Statistical studies suggest
that QPPs are a common feature, especially in larger flaring events (Simões et
al., 2015; Inglis et al., 2016; Hayes et al., 2020). Within the decimetric
waveband, QPPs can manifest as a sequence of type iii@ radio bursts emanating
from the corona as a consequence of accelerated beams of electrons escaping
along open magnetic field lines away from the flare site (Aschwanden et al.,
1994; Ning et al., 2005; Kupriyanova et al., 2016). In contrast, QPPs in the
EUV are typically observed to originate from the hot plasma in the coronal
loops of a flaring region (Van Doorsselaere et al., 2016). In addition to
studies of QPPs analysed within specific spectral domains, some research has
been done focusing on events containing QPPs across a wide band of
wavelengths. For example, Aschwanden et al. (1993) investigated the timing of
HXR pulsations with respect to pulsations seen in radio wavelengths (100-300
MHz) and found evidence for a strong causal connection. Additionally, Tajima
et al. (1987) found that current loop coalescence can lead to quasi-periodic
amplitude oscillations in the microwave, X-ray, and $\gamma$-ray wavebands.
More recently, Kumar et al. (2016) presented a multi-wavelength analysis of
QPPs found to be occurring in HXR, radio (25–180, 245, 610 MHz), and EUV
wavelengths.

Several models have been proposed as explanations for the presence of QPPs in
solar and stellar flares (McLaughlin et al., 2018) which are typically
categorised as oscillatory or self-oscillatory processes. In the regime of
oscillatory processes, QPPs are interpreted as a signature of
magnetohydrodynamic (MHD) oscillations inducing periodic motions about an
equilibrium in the flaring region. This explanation has been promising for
some events, as some observed periodicities of QPPs are in good agreement with
that of the timescales of MHD waves in the corona (Nakariakov & Melnikov,
2009). There is widespread observational evidence for MHD waves existing in
the corona and it is possible that kink, toroidal, longitudinal, or sausage
modes could cause some of the thermal and non-thermal intensity variations
that we observe.

For example, kink mode oscillations have been reported that have an
overlapping timescale ($\sim$1.5-10 minutes) with observed QPP periodicities
(Anfinogentov et al., 2015). Such waves could periodically modulate emission
or influence particle dynamics (Nakariakov & Melnikov, 2009). It is also
possible that the presence of these waves could trigger magnetic reconnection
resulting in a periodicity related both to the type of wave mode involved and
the properties of the coronal loops (Nakariakov & Zimovets, 2011; Carley et
al., 2019).

In the other category of self-oscillatory processes, QPPs are interpreted as a
manifestation of time dependent, intermittent magnetic reconnection. Specific
mechanisms theorised to produce intermittently accelerated electron beams
resulting in QPPs that are relevant to this work include plasmoid magnetic
island reconnection and oscillatory reconnection (reconnection reversal). For
example, Kliem et al. (2000) demonstrated in a 2D numerical model of a long
current sheet that the formation of plasmoids and their coalescence could
result in quasi-periodic particle acceleration, which they used to explain the
presence of decimetric radio bursts with periods of 0.5-10s. In a more recent
study, Guidoni et al. (2016) built upon the work of Drake et al. (2006) to
show that the generation, contraction, and interaction of magnetic islands in
a multi-layered current sheet can efficiently accelerate charged particles. It
was demonstrated that this mechanism should occur in a manner that is sporadic
and intermittent, and hence would result in pulsating flare emission. The
period in this case would be related to the rate of island formation and their
interaction with the flaring arcade.

In addition to plasmoid-dominated reconnection models, numerical simulations
have demonstrated that the reconnection process itself can be oscillatory. In
this scenario of oscillatory reconnection, competition between the thermal-
pressure gradients and the Lorentz force provide a restoring force as each
aspect overshoots the other in search for equilibrium (e.g. McLaughlin et al.,
2009; Murray et al., 2009). Simulations have shown that this mechanism can
produce oscillatory reconnection which results in intermittent particle
acceleration in a self-consistent manner resulting in decaying QPPs with
periodicities of $\sim$105–212.5 s (McLaughlin et al., 2012). This work has
also been extended to a 3D null-point, and it has been shown that reconnection
can naturally proceed in a time-dependent oscillatory behaviour (Thurgood et
al., 2017).

Despite a plethora of studies and observations of QPPs in various contexts,
the underlying mechanism for their generation remains a topic of debate. The
challenge in identifying an underlying mechanism is that the QPPs are linked
to many aspects of flaring emission, and encompass electromagnetic emission
that originates from the very base HXR footpoints of a flare up to altitudes
of several solar radii in the corona. Hence detailed studies that identify
QPPs from multiple wavelengths are required to connect different aspects of
the flaring process in order to constrain the mechanism producing them. In
this paper we present a study that demonstrates prominent QPPs occurring over
an unusually wide range of frequencies. We observe non-thermal emission in the
form of HXRs produced via bremsstrahlung in the chromosphere, co-temporal
thermal emission via SXR and EUV emission in the flare loops/transition
region, as well as a sequence of low frequency type iii@ bursts emanating from
the high corona. We analyse the multiple types of emission mechanisms at play
including thermal emission, plasma emission, and non-thermal bremsstrahlung.
These mechanisms are all associated with the same intermittently accelerated
electron beams which result in the pulsations that we can localise to a
specific region of the flare site. This region is associated with a system of
open and closed field lines. Identifying the source region responsible for the
intermittent acceleration of the electron beams is novel from the perspective
of analysing QPPs. This work provides new observational evidence that QPPs can
originate from an identifiable specific regions of flares and manifest across
the entire electromagnetic spectrum via multiple emission mechanisms. The QPPs
observed in the radio regime are notably lower in frequency compared to what
is typically observed. This indicates the considerable distances over which
QPPs can manifest - from the solar chromosphere through to interplanetary
space (${\sim}16$ $R_{\odot}$)

Figure 2: (a): The normalised SXR light curve from GOES (1-8 Å) at the time of
the flare. (b): The time derivative of the SXR emission and the HXR light
curve from FERMI GBM (25-50 keV). The QPPs present are labelled one through
seven. Here, we see a clear illustration of the Neupert effect and indicate
the seven primary QPPs analysed in this work.

The flare presented in this paper is a Geostationary Operational Environmental
Satellite (GOES) M3.7 class flare that occurred on 04-Nov-2015. An overview of
the active region located at the center of the solar disk as observed in 171 Å
is shown in Figure 1a and the light curves of the flare from the GOES X-Ray
Sensor (XRS) in two channels (1-8 Å in red and 0.5 - 4 Å in blue) is shown in
Figure 1b. In Section 2, the instruments used for this study: The Atmospheric
Imaging Assembly (AIA) on board the Solar Dynamics Observatory (SDO) (Lemen et
al., 2012a), the Reuven Ramaty High Energy Solar Spectroscopic Imager (Lin et
al., 2002), the WAVES instrument on board the WIND satellite (Bougeret et al.,
2008), The Gamma-ray Space Telescope (GBM) onboard Fermi (Meegan et al.,
2009), and the GOES XRS are briefly introduced. Details of the analysed event
and data analysis techniques are also included in this section. In Section 3,
we present our observations alongside our analysis of the QPPs. In Section 4,
we present a discussion and interpretation of the work before concluding our
findings in Section 5.

##  2 Instrumentation, observations, and data analysis

###  2.1 Instrumentation and observations

The GOES XRS measures the solar SXR fluxes integrated over the entire solar
disk. It has a cadence of 2 s with two channels of 0.5-4 Å and 1-8 Å. In this
work, we primarily focus on the 1-8 Å channel as it exhibited the most
pronounced QPPs. Figure 2a shows this light curve. The pre-flare, impulsive,
and decay phases are also indicated. The event began at 13.31 UT and peaked at
13.52 UT. Figure 2b shows the time derivative of this light curve with the HXR
light curve observed by FERMI GBM (25-50 keV) overplotted. The Neupert effect,
which refers to the observed phenomenon that non-thermal HXR emission
coincides temporally with the rate of rise of the thermal SXR emission (i.e.
the derivative), is observed here as it is clear that the pulsations in the
SXR derivative are coincidental with those observed in the HXR emission
(Neupert, 1968). This relates the HXR flux from the flare ‘footpoints’ to the
thermal SXRs observed from the heated plasma. We identify seven distinct
pulsations throughout the event as shown in Figure 2b. One can see that these
pulsations all occur during the impulsive phase of the flare. This indicates
that the mechanism producing these QPPs must be able to modulate the
acceleration of electrons. However, it is clear that some pulsations do
persist into the decay phase within the SXR emission.

Figure 3: (a): Dynamic spectrum from WIND/WAVES RAD2 showing a series of
pulsed type iii@ radio bursts. (b): Multi-wavelength light curves observed
from a number of instruments. From top to bottom the lightcurves go from
longer to shorter wavelenghts. This is a proxy for altitude of the flaring
emission source with the radio data representing emission originating from
high in the corona down to the HXR emission originating from the footpoints.
The black, orange, red, and blue light curves have QPPs well correlated in
time. The green light curve shows the radio emission at 2.5 MHz. This
frequency was chosen as it captured each pulse most effectively. The radio
QPPs have a longer periodicity which we elaborate on in the discussion
section. Lines drawn from pulses 1 and 7 show the time lag needed for the
source electron beams of typical speeds to reach $16$ $R_{\odot}$ (approximate
height of the radio source) from the flare site.

The most pronounced pulsations we observed with AIA were from the 171 Å and
1600 Å passbands. Images from these passbands were used to analyse the
periodicity and spatial distribution of the QPPs in EUV. The cadences of these
images were 12 s and 24 s, respectively. The 171 Å passband is dominated by
the Fe xi@ line and most represents emission from the corona and upper
transition region while the 1600 Å passband is dominated by C iv@ and images
primarily the upper photosphere/transition region (Lemen et al., 2012b). These
images enable us to estimate the layer of the atmosphere from which the QPPs
we observe originate and how they relate to one another.

RHESSI observed the event up until 13:43 UT before entering spacecraft night.
This allowed us to image the location of the HXRs produced during the flare
for our analysis. Although RHESSI was unable to observe all the HXR emission
throughout the flare, Fermi GBM captured this information which we incorporate
into our analysis. The WIND/WAVES RAD2 instrument was used to gather radio
data. Dynamic spectra from 0.02-13.85 MHz (cadence: 16.188 s) were analysed to
investigate the low-frequency aspect of the QPPs in the event. The emission at
these wavelengths manifests in the form of type iii@ radio bursts which are a
result of plasma emission (Reid & Ratcliffe, 2014). Within this mechanism, the
frequency, $f_{p}$, of the radiation is proportional to the local electron
density $n_{e}$ via $f_{p}\sim 8980\sqrt{n_{e}}$. The electron density in the
corona decreases with height. Hence, by using an electron density model, the
height of a radio source produced through plasma emission may be estimated. At
the frequency of 2.5 MHz, most of these bursts are captured and so we use this
frequency to generate a light curve of the radio data. Using the electron
density model from Newkirk Jr. (1967), this corresponds to a height of
${\sim}16$ $R_{\odot}$.

Together, we can use these data to determine information about the periodicity
and location of the QPPs observed from the HXR footpoints through to the upper
transition region and corona. The altitude at which each different waveband
emits differs significantly. For example, the HXRs are produced through
bremsstrahlung by non-thermal electrons colliding with the chromosphere while
the type iii@ radio bursts are produced via plasma emission. Figure 3 is
summary of the QPPs analysed in this work. Figure 3a shows the dynamic
spectrum of the radio emission containing a sequence of type iii@ radio bursts
and Figure 3b shows the EUV, SXR and HXR light curves in which we identify 7
distinct QPPs. The EUV light curves were extracted from the QPP source region
we identified which is explained in Section 3.2.

###  2.2 Data analysis and imaging

Wavelet analysis using a Morlet wavelet was performed on the multiwavelength
light curves to determine their periodicities using the
software111http://atoc.colorado.edu/research/wavelets/ developed by Torrence &
Compo (1998). This technique is a powerful tool for searching within time-
series for periodic signatures as, unlike Fourier analysis, it provides a 2D
spectrum of both frequency and time allowing one to assess if a quasi-periodic
signature varies in time (De Moortel et al., 2002).

Figure 4: Example of the background subtraction technique used to isolate the
QPPs. (a): The time derivative of the SXR emission is shown in red with a
spline fit to the overall large-scale slowly varying emission overplotted in
black . The fit excludes the shorter time-scale variation of the QPPs. (b):
The subtraction of the fit from the SXR emission resulting in a time-series
containing the QPPs without the slowly varying background emission.

In order to more accurately determine the period of the QPPs via the wavelet
analysis, the slowly varying background trend of the flare emission was
removed from each time-series. This process is shown in Figure 4 for the case
of the SXR emission from GOES. To do this, a spline fit was constructed using
a 3rd order polynomial with 28 samples between nodes. The fit was then
subtracted from the original data resulting in a time-series containing only
the modulation of the emission resulting from the QPPs. This process was
repeated for each time-series we analysed. No subtraction was required for the
radio light-curve at 2.5 MHz as there was little background in this data. Care
was taken to ensure each fit accurately represented the slowly varying
background emission in order to avoid introducing any artefacts during the
subtractions.

In addition to carrying out the wavelet analysis on the multi-wavelength
detrended time-series, we also performed the same analysis on the relevant
data without detrending in order to cross-check our results. This analysis is
provided in the appendix of this paper. We also manually determined the period
of the QPPs. This was achieved by visually identifying the time of each pulse
and plotting these times against pulse number (1-7). The period can then be
simply estimated by fitting a straight line to this data and finding the
slope.

The PIXON algorithm was used to image the RHESSI HXR sources. It seeks a
superposition of circular sources of different sizes and parabolic profiles
that most replicate the modulations measured by the detectors, while
maintaining the fewest degrees of freedom possible. PIXON is thought to
provide accurate image photometry in comparison to the other faster algorithms
such as CLEAN (Hurford et al., 2002). Images taken by SDO/AIA were used to
analyse the most prominent pulsations in the EUV regime which were found in
the 171 Å and 1600 Å passbands. Time-series were constructed from these images
by integrating the emission over various regions of the flare in order to
localise the area producing the pulsations. This is discussed further in
Section 3.2. Additional data analysis was carried out to estimate the height
of the source producing the radio emission via the Newkirk Jr. (1967) electron
density model. This height was determined to be ${\sim}16$ $R_{\odot}$. Figure
3b shows the time lag required to reach this height from the flaring region
with beam speeds of 0.1-0.5 c. Type iii@ radio bursts typically have source
electron beam velocities of $\sim$0.3 c but have been found to vary from
0.1-0.5 c in some cases (Reid & Ratcliffe, 2014).

Figure 5: Wavelet analysis of the detrended (a): HXR and (b): SXR derivative
emission from the flare. The periods were found to be $137^{+49}_{-56}$ s and
$123^{+11}_{-26}$ s, respectively. The error is taken as the range over which
each global power spectrum is above 95% significance. Figure 6: (a): Wavelet
analysis of the detrended emission at 171 Å. (b): Wavelet analysis of the
emission at 1600 Å. The periods were found to be $122^{+26}_{-22}$ s and
$131^{+36}_{-27}$ s respectively.

##  3 Results

Across the electromagnetic spectrum, the impulsive nature of the event begins
at $\sim$13.37 UT and continues until $\sim$13.57 UT. Figure 3b shows the
normalised lightcurves under investigation that spans from the HXR to the low
frequency radio emission, all of which contain QPPs. The lightcurves were
normalised by dividing each time series by its peak value. They were then
scaled in order to plot them in a vertical sequence for ease of comparison,
going from higher frequency to lower frequency (bottom to top). The HXR light
curve (25-50 keV) from Fermi/GBM in blue is at the bottom of Figure 3b, with
the most prominent pulsations labelled one through seven.

§

Figure 7: Wavelet analysis of the radio emission at 2.5 MHz. The period was
found to be ${\sim}231$ s.

Above in red, is the derivative of the GOES light curve (1-8 Å), followed by
the 1600 Å curve in orange, and the 171 Å curve in black. The clear co-
temporal presence of the pulsations in each of these light curves obtained by
simultaneous observations from different instruments makes clear that these
QPPs are of solar origin and are not due to some instrumental effect.

Figure 8: Pulse number versus time of each pulse. The 7 HXR/SXR/EUV pulses
shown in Figure 3 are plotted using the circle symbols. The slope of the
straight line fitted to the data provides an estimate of the period: $\sim
109$ s. For the radio emission at 2.5 MHz, this analysis was done for the four
main peaks in the time-series, as well as for seven peaks in the case where
lower amplitude peaks are included. This resulted in periods of $\sim 230$ s
and $\sim 157$ s, respectively.

Figure 3a shows the dynamic spectrum from WIND/WAVES RAD2 (1.075–13.825 MHz)
that contains a series of pulsed type iii@ radio bursts during the same time
frame in which the QPPs were observed. The green light curve in Figure 3b
shows a slice from this dynamic spectrum at 2.5 MHz. This frequency was chosen
as it contained the majority of the bursts. Using the Newkirk Jr. (1967)
electron density model, we can estimate the height at which this radio
emission is emitted: ${\sim}16$ $R_{\odot}$. Lines drawn from the peaks of
pulses 1 and 7 from the 171 Å curve are shown that indicate the time delay
required for electron beams of velocities between 0.1 and 0.5 c to reach this
height. For both pulse 1 and 7, there appears to be radio pulsations that
occur at the expected times. This analysis suggests that the electron beam
speeds are close to the typical value of 0.3 c for type iii@ radio bursts. It
is clear that the radio QPPs at 2.5 MHz are less correlated with the higher
frequency radiation. There are a number of reasons for which one would not
expect a one-to-one relation between radio pulsations produced via plasma
emission in interplanetary space and the higher frequency emission produced
via bremsstrahlung/heating close to the flare site despite originating from
the same populations of accelerated electrons. These differences are
elaborated upon in the discussion section.

###  3.1 Periodicities

For each lightcurve, wavelet analysis was conducted over the same time period:
13:34-13:54 UT. The error for each result was taken as the range over which
the global power spectrum was above the 95% significance curve. The analysis
was carried out on the detrended light curves. However, the appendix includes
the same analysis for the data without detrending. The results agreed in both
cases.

Figure 9: Spatial analysis of QPPs. (a): Image of the flaring region in 1600 Å
taken at the time of the first QPP. The RHESSI image of the three HXR
footpoints is overlaid which are labelled within three kernels as K1, K2, and
K3. K1 and K2 lie along one flare ribbon while K3 is located on the other.
This image was constructed using the PIXON algorithm over the available time
period during which RHESSI captured the emission. This time period is shown in
magenta with the time-series data in (b). (b-d): Light curves of the HXRs
observed by GBM and RHESSI (25-50 keV) and the light curves extracted from the
1600 Å images taken by AIA at each HXR source location. The light curves were
constructed by integrating over the pixels contained in the boxes surrounding
the HXR sources in (a). The same analysis is done for the 171 Å images as
shown in (e-h). It was found that the emission in EUV from within K1 produced
light curves containing QPPs most correlated with those seen in the HXR
emission (b+f). This localises the source of the QPPs to this region of the
flare site which is close to a nearby system of open and closed magnetic field
lines. The animation related to this Figure, provided in the online version of
this article, shows the evolution of the flare at each time step from 13:34 -
13:53 UT.

Figure 5a shows the wavelet analysis that was carried out on the GBM 25-50 keV
lightcurve. In this plot, the HXR time-series, the wavelet power spectrum,
illustrating power at particular periodicities as a function of time, and the
global power spectrum are shown. A period of $137^{+49}_{-56}$ s was found in
this channel. Figures 5b, 6, and 7 show this same analysis for the SXR, EUV,
and radio wavelengths, respectively. The SXR emission contained a period of
$123^{+11}_{-26}$ s, while the pulsations within the 171 Å and 1600 Å light
curves yielded periodicities of $122^{+26}_{-22}$ s and $131^{+36}_{-27}$ s,
respectively. The 2.5 MHz light curve was found to have significant period at
a time-scale of ${\sim}231$ s. The time-scales of the QPPs in in the 171 Å,
1600 Å, SXR (1-8 Å), and HXR (25-50 keV) are therefore all in good agreement
within error. These results are summarised in Table 1.

In addition to calculating the periods of the light curves via wavelet
analysis, we also estimated them manually by visually identifying peaks.
Figure 8 shows the time of the HXR, EUV, and SXR pulsations versus pulse
number (see pulses 1-7 in Figure 3). The slope of this line provides an
estimate of the period. The result was found to be ${\sim}109$ s.

Table 1: Comparison of the periods found in the lightcurves for each analysed wavelength via wavelet analysis. The cadence of the data in each case, $\Delta t$, is also shown. $\lambda$ | 2.5 MHz | 171 Å | 1600 Å | 1-8 Å | 25-50 keV
---|---|---|---|---|---
Period |  $\sim$231 s |  $122^{+26}_{-22}$ s |  $131^{+36}_{-27}$ s |  $123^{+11}_{-26}$ s | $137^{+49}_{-56}$
$\Delta t$ | 16.2 s | 12 s | 24 s | 2 s | 1.6

This agrees with the results of the wavelet analysis within error. For the
radio emission at 2.5 MHz, this analysis was done for the four main peaks in
the time-series, shown in green in Figure 7 as well as seven peaks which
include lower amplitude pulsations, shown in blue in Figure 7. This resulted
in periods of $\sim 230$ s and $\sim 157$ s, respectively. Therefore, this
result matches well with the wavelet analysis when only the four main peaks
are accounted for. When the less significant peaks are included, we see that
the period draws closer to that of the higher frequency radiation. The
matching time-scales of the 171 Å, 1600 Å, SXR, and HXR light curves indicate
that the mechanism producing the QPPs in these wavebands must have the same
progenitor, that is also likely related to the radio emission observed. Our
interpretation of these results and the relationship between the emission in
each waveband is detailed in Section 4.

###  3.2 Spatial analysis

To investigate spatially the regions of the flare from which the QPPs
originate, we conducted imaging analysis using RHESSI and SDO/AIA. No radio
imaging instrument was available during the observation. Firstly, we used the
PIXON algorithm to determine where the non-thermal HXRs originated from. The
imaging was carried out over an energy band of 35-70 keV during the available
time period when RHESSI was observing the event before entering spacecraft
night: $\sim$13.35-13.42 UT. This includes the first two prominent pulsations
in the sequence of 7. The light curve showing the available RHESSI data is
shown in Figure 9 in magenta. It was found that there were three HXR sources
on the map which are labelled within three kernels as K1, K2, and K3. The
ribbons of the flare are clearly in visible in Figure 9a. K1 and K2 lie along
the higher ribbon while K3 is located on the lower ribbon. A system of flare
loops connects these ribbons. Figures 9a and 9e show these sources in red
overlaid on top of the 1600 Å and 171 Å backgrounds, respectively. The event
occurred close to the disk centre and had a loop footpoint separation of
$\sim$50 Mm and a loop height of $\sim$25 Mm. This was estimated by measuring
the separation of the HXR footpoints and assuming a semi-circular geometry of
the loops.

These HXRs are produced through non-thermal bremsstrahlung through interaction
of the flare accelerated electrons with the dense chromosphere which acts a
‘thick-target’ (Brown, 1971). The mechanism modulating the HXRs that produces
the observed QPPs must be causing a sequence of episodic or ‘bursty’ energy
releases that intermittently accelerates electrons resulting in a modulation
of the non-thermal bremsstrahlung emission. We discuss this further and its
relevance to the QPPs in the other wavebands in Section 4.

Figure 10: (a): SDO/AIA 171 Å passband image of the flare site on 2015 Nov 4.
Shown are the QPP source (within K1), the additional two HXR sources (within
K1 and K2), and two test kernels representing the arbitrary regions of the
map. (b): The time-series obtained from K1 (QPP source) compared against those
obtained from the test kernels. It is clear that the time-series constructed
using the test kernels are uncorrelated to the HXR QPPs while the light curve
obtained from K1 matches the HXR profile.

To determine the location within the flaring region producing the QPPs in the
171 Å and 1600 Å emission, we created time-series from the images taken from
SDO/AIA. To localise the QPP source, we integrated the emission from each
image over each region of the entire active region using various kernel sizes,
generated time-series for each of these kernels for the duration of the flare,
and compared the profiles of the time-series to that of the HXR emission. This
allowed us to obtain the flux from within each test kernel at each time-step
to compare to the HXR emission. It was found that the kernel that produced the
most prominent QPPs, as well as having the same characteristic periodicity as
the HXR QPPS, spatially coincided with the location and size of the HXR source
at K1. Figure 9b and 9f illustrate this in that there is a strong correlation
between the QPPs in the EUV emission extracted from K1 and the HXR emission.
The EUV emission from K2 and K3 is significantly less correlated to the HXR
emission as shown in Figures 9c, 9d, 9g, and 9f.

The animation associated with Figure 9, available in the online version of
this article, shows the evolution of the flare at each time step. It is clear
that there is an asymmetry between the light curves obtained for the EUV
emission at each HXR source location with K1 being the most correlated to the
HXR QPPs. This analysis suggests that K1 is the region of the flare site in
which the QPPs originate. Figure 10 shows an additional comparison of the EUV
emission from K1 (QPP source) and the emission obtained from two test regions
not associated with the HXR sources. Here, we can see again that integrating
each time step over K1 produces QPPs highly correlated with the HXR emission
while doing so for each test kernel does not. This trend continues no matter
which region of the flare is used to construct the EUV time-series.

K1 is associated with open magnetic field lines, identified in the potential
field source surface extrapolation (PFSS) shown in Figure 11. PFSS models
provide an approximation of the coronal magnetic field up to 2.5 $R_{\odot}$
based on the observed photospheric field (Schrijver & De Rosa, 2003). Here the
PFSS is calculated using pfsspy (Stansby et al., 2020). This magnetic field
geometry allows for a mechanism for the escape of the electrons responsible
for producing the radio emission. In the following section, we discuss the
interpretation of these data and what proposed models of QPP generation allow
for these observations.

Figure 11: Potential Field Source Surface (PFSS) extrapolation showing the
geometry of the magnetic field lines of the flaring region overlaid on the AIA
171 Å image. The open field lines are plotted in red and the closed lines in
white. It is clear that the K1 has an open field line source and we propose
that the interaction between the closed and open field lines at this footpoint
result in ‘bursty’ magnetic reconnection giving rise to the QPPs we observe.
The open field lines allow the flare-accelerated electrons to escape that
produce the Type iii@ radio emission.

##  4 Discussion and conclusions

Our results indicate that the EUV (171 Å and 1600 Å), SXR, and HXR QPPs
contain approximately the same periodicity. We also observe QPPs in the low
frequency radio domain in the form of a sequence of type iii@ radio bursts
that occur during the time of the flare that have a longer periodicity. Our
spatial analysis suggests that the EUV and HXR QPPs originate from the same
region of the flare - the HXR footpoint at K1. This points towards a scenario
in which intermittent particle acceleration is occurring due to a process
inducing time-dependent magnetic reconnection. This particle acceleration
occurs in a quasi-periodic fashion and results in bursty non-thermal
bremsstrahlung that modulates the HXR emission occurring at the footpoints.
The EUV emission would then be a consequence of this process as the ambient
plasma is heated as the precipitating accelerated particles lose their energy.

The asymmetry of the EUV pulsations present at each HXR source, as shown in
Figure 9, suggests that the electrons from the reconnection site must be
preferentially accelerated between the closed loops and open field lines close
to K1. Figure 11 shows these systems of open and closed field lines obtained
via a PFSS extrapolation. It is likely that the radio emission observed is a
consequence of the same intermittent particle acceleration that resulted in
the EUV and HXR pulsations. However, the electrons accelerated along the open
magnetic field lines from flare region result in the radio emission while it
is the precipitating electrons accelerated towards the chromosphere which
result in the HXR/SXR/EUV emission. Unfortunately no imaging observations at
these radio frequencies are available during this event, and so we could not
image the radio source to localise its origin. However, there are no nearby
active regions at the time of the flare that could have coincidentally
produced this radio emission.

To explain our observations we interpret the QPPs identified in this flare in
terms of pulsed electron acceleration caused by time-dependent intermittent
reconnection. In Figure 12 we show a cartoon scenario of the flare site to
illustrate how the QPP sources are related to the magnetic field
configuration. Following each burst of electron acceleration, those that
escape upwards along the open magnetic field lines result in the type iii@
QPPs, and those that travel along closed lines precipitate in the chromosphere
to cause the QPPs we observe in hard X-ray and EUV. But what causes the
reconnection and particle acceleration itself to be quasi-periodic? As
mentioned in the introduction, this could be due to either the process itself
being time-dependent (self-oscillatory) or indeed due quasi-periodic
triggering of magnetic reconnection due to external MHD waves. We can rule out
the latter, as it is unlikely as there are no active regions nearby.

Plasmoid magnetic island reconnection or oscillatory reconnection are both
good candidates. Given that the period of the QPPs analysed in this work match
well with the simulations in McLaughlin et al. (2012) (105-121.5 s), this
mechanism may be responsible. McLaughlin et al. (2012) outline how the
interaction of magnetic flux emerging from the tachocline with an existing
magnetic topology such as a flaring system can result in oscillatory
reconnection and pulsed particle acceleration. It is possible that this flux
emergence is localised to the region of the flare site we identified as the
QPP source. This could then give rise to the QPPs we observe across the
electromagnetic spectrum. However we are unable to rule out the possibility of
plasmoid magnetic island reconnection or other self-oscillatory processes.
There are a number of arguments that point towards a relation between the
HXR/SXR/EUV QPPs and the radio QPPs we observe despite them having different
periods according to our wavelet analysis. We outline below our argument that
they are indeed a consequence of the same progenitor.

1\. The emission mechanism involved that produces the radio (plasma emission)
versus the mechanism producing the EUV, SXR, and HXR (non-thermal
bremsstrahlung/heating) are very different in nature - i.e incoherent free-
free emission versus coherent collective emission. In the plasma emission
mechanism, accelerated electron beams travel to large heights (for the
frequencies we observe) along open magnetic field lines, induce the growth of
Langmuir waves, and then these Langmuir waves must interact to finally produce
radio emission (Melrose, 2017). Many factors, such as the electron energy
(which can vary from pulse to pulse), velocity dispersion, Coulomb collisions,
Langmuir wave growth and interaction, to name a few, play a role in generating
the emission. It is a multi-stage process, and variability in any of these
stages can change the characteristics of the radio pulses. In contrast, the
electrons producing the higher energy radiation, via non-thermal
bremsstrahlung and subsequent heating of the surrounding plasma, must only
travel from the acceleration site within the flaring region to the
chromosphere. Bremsstrahlung then occurs quickly followed by instantaneous
heating resulting in co-temporal pulsations in the EUV, SXR, and HXR wavebands
(White et al., 2011). Due to these factors, it is expected that not every
HXR/SXR/EUV pulsation would have a corresponding radio burst, as we observe,
despite being a consequence of the same intermittent particle acceleration.

Figure 12: Cartoon of the flaring region illustrating the likely mechanism
through which we observe the episodic particle acceleration resulting in QPPs
in EUV, radio, SXR, and HXR. The QPP source footpoint is related to the open
and closed field lines allowing for the escape of the electrons resulting in
the radio emission and the precipitation of the electrons giving rise to the
higher frequency emission.

2\. The region of the flare site we have identified as the QPP source is in
close proximity to open and closed magnetic field lines. This magnetic field
geometry allows for reconnection to occur between the open and closed field
lines, providing a natural route for the escaping electrons to produce the
radio emission and the precipitating electrons to produce the X-rays/EUV as
shown in Figure 11.

3\. The time delay between the HXR emission and the onset of the prominent
type iii@ radio bursts is consistent with the estimated distance over which
the radio electron beam sources must travel before they emit at 2.5 MHz, as
shown in Figure 3. This source height of ${\sim}16$ $R_{\odot}$ was obtained
by the Newkirk Jr. (1967) electron density model.

4\. The wavelet analysis of the 2.5 MHz light curve only picks up the four
main peaks in the time series. Smaller amplitude peaks fail to contribute
significantly to the result. In Figure 8 we manually find the period by
identifying the four most prominent peaks, which matches the result of the
wavelet analysis. However when the smaller amplitude peaks are accounted for,
amounting to a total of 7 pulses, the period of the radio emission comes
within error of the period of the HXR/SXR/EUV. An additional difficulty in
accurately calculating the period of the radio emission is that certain bursts
are more intense at different frequencies as is clear in the dynamic spectra.
However, from inspecting Figure 13 where the HXR emission is overplotted on
the dynamic spectrum, there is quite a clear relation between the radio bursts
and the HXR peaks when the entire frequency band is taken into account.

Figure 13: The HXR emission from FERMI GBM (25-50 KeV) overplotted against the
dynamic spectrum from WIND/WAVES showing the low frequency radio emission in
the form of a sequence of type iii@ radio bursts.

Cairns et al. (2020) point out that a type ii@ radio burst occurs at the time
of this flare and suggest that the associated shock may be responsible for
accelerating the electrons that result in the low frequency radio emission.
However, considering the arguments above (points 1-4), we conclude that it is
more likely that the type iii@ radio bursts are due to pulses of electron
beams accelerating along the open magnetic lines close to the QPP source
region. Additionally, the dynamic spectra of the radio emission from kHz to
GHz shows traces of type iii@ bursts that extend to high frequencies, above
the frequency of the type ii@ burst (see Cairns et al. (2020) Figure 15). This
suggests that they originate from a region closer to the flare site.

In summary, A multi-wavelength analysis of QPPs in an M-class flare has been
conducted. Several instruments were used to allow for the analysis of the HXR,
SXR, EUV, and radio emission detected during the event. The 171 Å, 1600 Å,
SXR, and HXR light curves yielded similar periods of $122^{+26}_{-22}$ s,
$131^{+36}_{-27}$ s, $123^{+11}_{-26}$ s, and $137^{+49}_{-56}$ s,
respectively, indicating a common underlying mechanism, while the radio
emission at 2.5 MHz contained a longer period of ${\sim}231$ s. X-ray and EUV
imaging enabled us to localise the QPP source to a region of the flare site
associated with open magnetic field lines. We found that the time delay
between the X-ray/EUV emission and the radio emission is consistent with the
estimated distance over which the electron beam sources must travel. We
discuss the differences between the emission mechanisms responsible for the
HXR/SXR/EUV emission versus the radio emission and determine that the QPPs in
each waveband are linked to the same populations of accelerated electrons. We
conclude that the QPPs in this event are due to some time-dependent self-
oscillatory reconnection mechanism. Magnetic reconnection occurring in this
bursty fashion injects populations of non-thermal electrons into the flare
site giving rise to the sequence of pulses we observe in the SXR, HXR, and EUV
as electrons collide with the chromosphere while the electrons accelerating
away from the flare site along open magnetic field lines produce the type iii@
radio bursts. This work provides new evidence that oscillatory reconnection
can naturally generate quasi-periodic periodic pulsations providing an
explanation for their presence across the entire spatial range of flaring
emission. This work also shines light onto the nature of energy release in
flares and provides new insight into how QPPs may be localised to specific
regions within flare sites. Future work that investigates the details and
conditions required for the triggering of magnetic reconnection in this bursty
fashion is needed.

This work has been supported by the European Space Agency PRODEX Programme
(BPC) and the Government of Ireland Studentship from the Irish Research
Council. L.A.H. is supported by an appointment to the NASA Postdoctoral
Program at Goddard Space Flight Center, administered by USRA through a
contract with NASA. We also thank the anonymous referee whose comments helped
to improve this paper.

##  Appendix A Wavelet analysis without detrending technique

Figure 14: Wavelet analysis of the (a): HXR and (b): SXR derivative emission
from the flare without detrending. The periods were found to be
$137^{+64}_{-61}$ s and $123^{+6}_{-34}$ s, respectively. The error is taken
as the range over which each global power spectrum is above 95% significance.
Figure 15: (a): Wavelet analysis without detrending of the emission at 171 Å.
(b): Wavelet analysis without detrending of the emission at 1600 Å. The
periods were found to be $122^{+17}_{-26}$ s and ${\sim}131$ s respectively.

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# Algebraic structures on the Cantor set

Evgenii Reznichenko<EMAIL_ADDRESS>Department of General Topology and
Geometry, Mechanics and Mathematics Faculty, M. V. Lomonosov Moscow State
University, Leninskie Gory 1, Moscow, 199991 Russia

###### Abstract

Below, by space we mean a separable metrizable zero-dimensional space. It is
studied when the space can be embedded in a Cantor set while maintaining the
algebraic structure. Main results of the work: every space is an open retract
of a Boolean precompact group; every strongly homogeneous space is
rectifiable. In this case, the space can be embedded in the Cantor set with
the preservation of the algebraic structure. An example of a strongly
homogeneous space is constructed which do not admit the structure of a right
topological group.

###### keywords:

Baire space , strongly homogeneous space , rectifiable space , retract of
group , separable metrizable zero-dimensional space , right topological group
,

###### MSC:

[2010] 54B10 , 54C30 , 54C05 , 54C20 , 22A05 , 54H11

##  1 Introduction

In this paper, we study what algebraic structures are possible on subspaces of
the Cantor set ${\mathbf{C}}$, that is, on separable metrizable zero-
dimensional (SMZD) spaces. Any SMZD space is Mal’tsev, moreover, it is a
retract of the topological group, that is, a retral space [1].

We are primarily interested in Mal’tsev, rectifiable, retral, homogeneous SMZD
spaces, (right) topological groups. It is also studied when a SMZD space with
algebraic structure can be embedded in SMZD a compact space with the same
structure. In particular, when the SMZD space can be embedded in
${\mathbf{C}}$ with the algebraic structure preserved.

Main results of the work: each SMZD space is an open retract of a Boolean
precompact SMZD group (Theorem 1); every SMZD strongly homogeneous space is
rectifiable (Theorem 4).

Rectifiable spaces are also characterized as spaces on which there is a
Mal’tsev operation with some additional identity ($\Phi_{3}$), the so-called
homogeneous Mal’tsev operations (Definition 1). A strongly Mal’tsev operation
is introduced and studied, which defines strongly rectifiable spaces. This
class lies between rectifiable spaces and topological groups.

In the 6 section, an example of a strongly homogeneous SMZD space is
constructed which do not admit the structure of a right topological group.

##  2 Definitions and notation

By a space we understand a regular topological space.

The weight and character of the space $X$ will be denoted by $w(X)$ and
$\chi(X)$, respectively.

Denote by $\operatorname{Aut}(X)$ the set of all homeomorphisms of the space
$X$ onto itself,
$\operatorname{Aut}(X|M)=\\{f\in\operatorname{Aut}(X)\,:\,f(M)=M\\}$ for
$M\subset X$. Denote by $\operatorname{id}_{X}$ the identity mapping of $X$
onto itself.

A space $X$ is called strongly homogeneous if every non-empty clopen subset of
$X$ is homeomorphic to $X$. Every strongly homogeneous SMZD space is
homogeneous.

If $f:X\to Y$ is a mapping and $A\subset X$, then $\left.f\right|_{A}$ denotes
the restriction of $f$ to $A$.

###  2.1 Non-Archimedean spaces

A subset $M$ of a linearly ordered set $X$ is called convex if $a\leq b\leq c$
holds for $a,c\in M$ and $b\in X$, then $b\in M$.

A family of sets ${\mathcal{B}}$ is called non-Archimedean if for
$U,V\in{\mathcal{B}}$ the following holds: if $U\cap V\neq\varnothing$, then
either $U\subset V$ or $V\subset U$. Non-Archimedean base is the base
${\mathcal{B}}$ of the space $X$, which is a non-Archimedean family. Non-
Archimedean bases called bases of rank 1 were introduced and studied in [2,
3].

Every non-Archimedean space has a base which is a tree by reverse inclusion
(Theorem 2.9 [4]). Each a base which is a tree by reverse inclusion is a non-
Archimedean base.

A linear order $\leq$ on a space $X$ is said to be consistent with a non-
Archimedean base ${\mathcal{B}}$ if any element of $U\in{\mathcal{B}}$ is
convex with respect to the order $\leq$.

###  2.2 Cantor set

Denote ${\mathbf{D}}=\\{0,1\\}$,
${\mathbf{D}}^{<\omega}=\bigcup_{n\in\omega}{\mathbf{D}}^{n}$. Here
${\mathbf{D}}^{0}=\\{\varnothing\\}$.

Denote by ${\mathbf{C}}$ the Cantor set, ${\mathbf{C}}={\mathbf{D}}^{\omega}$.
Let $x=(x_{n})_{n\in\omega}\in{\mathbf{C}}$, $k\in\omega$ and
$c=(c_{n})_{n<k}\in{\mathbf{D}}^{k}\subset{\mathbf{D}}^{<\omega}$. Denote

$\displaystyle x|k$ $\displaystyle=(x_{0},...,x_{k-1}),$ $\displaystyle x(k)$
$\displaystyle=x_{k},$ $\displaystyle B(c)$
$\displaystyle=\\{x\in{\mathbf{C}}\,:\,x|k=c\\}.$

The family

$\mathcal{C}=\\{B(c)\,:\,c\in{\mathbf{D}}^{<\omega}\\}$

is the standard non-Archimedean base ${\mathbf{C}}$. The base $\mathcal{C}$ is
a tree by reverse inclusion. On ${\mathbf{C}}$ we will consider the
lexigraphic order: for different $x,y\in{\mathbf{C}}$, $x<y$ if $x|k=y|k$,
$x(k)=0$ and $y(k)=1$ for some $k\in\omega$. The lexigraphic order is
consistent with the non-Archimedean base $\mathcal{C}$.

Denote $\operatorname{supp}x=\\{{n\in\omega}\,:\,x(n)=1\\}$ for
$x\in{\mathbf{C}}$,
${\mathbf{Q}}=\\{x\in{\mathbf{C}}\,:\,|\operatorname{supp}x|<\omega\\}$ and
${\mathbf{P}}={\mathbf{C}}\setminus{\mathbf{Q}}$. The space ${\mathbf{Q}}$ is
homeomorphic to rational numbers, and the space ${\mathbf{P}}$ is homeomorphic
to irrational numbers.

###  2.3 Rectifiable, Maltsev, retral spaces

Let $X$ be a space, $e\in X$, $\Psi:X^{2}\to X^{2}$. A mapping is called
rectiﬁcation if it is a homeomorphism and there exists a mapping $p:X^{2}\to
X$ such that $\Psi(x,y)=(x,p(x,y))$ and $p(x,x)=e$ for $x,y\in X$. A space $X$
is called rectiﬁable if there exists a rectiﬁcation on $X$. We define the
operation $q:X^{2}\to X$ by the identity $q(x,p(x,y))=y$. The operation $q$ is
continuous and the identities $q(x,e)=x$ and $p(x,q(x,y))=y$ hold for it.
Also, $\Psi^{-1}(x,y)=(x,q(x,y))$. The operations $p$ and $q$ are called
homogeneous algebra [5, 6].

A biternary algebra on a space $X$ is a pair of ternary continuous operations
$\alpha,\beta:X^{3}\to X$ such that

$\alpha(y,y,x)=\alpha(\beta(x,y,z),y,z)=\beta(\alpha(x,y,z),y,z)=x,$

for all $x,y,z\in X$ (see [7]).

For a space $X$ the following conditions are equivalent: $X$ is a rectiﬁable
space; $X$ is homeomorphic to a homogeneous algebra; there exists a structure
of a biternary algebra on $X$ [5].

A mapping $\Phi:X^{3}\to X$ is called a Maltsev operation if
$\Phi(x,y,y)=\Phi(y,y,x)=x$ for $x,y\in X$. A space $X$ is called Mal’tsev if
there exists a continuous Mal’tsev operation on $X$. Rectifiable spaces are
Maltsev: $\Phi(x,y,z)=q(x,p(y,z))$.

A space $X$ is called retral if there exists a topological group $G$ into
which $X$ is embedded in such a way that $X$ is a retract of $G$ [8]. The
retral space is Mal’tsev, the reverse is not true [1].

##  3 Non-Archimedean spaces

In this section, we prove a more detailed and specialized version of Theorem 7
(version 2) from [1] and apply it to ${\mathbf{C}}$.

Let $X$ be a zero-dimensional space. Denote by $B(X)$ the set of all finite
subsets of $X$. We consider $B(X)$ as an adiative Boolean group with zero
$\varnothing$ and the group operation of addition is a symmetric difference.
Denote by $\Gamma(X)$ the family of all open partitions of $X$. For
$\gamma\in\Gamma(X)$ we set

$H(\gamma)=\\{g\in B(X)\,:\,|g\cap U|\text{ is even for }U\in\gamma\\}.$

The set $H(\gamma)$ is a subgroup of $B(X)$. Let
$\mathcal{G}\subset\Gamma(X)$. Denote by $B_{z}(X,\mathcal{G})$ the group
$B(X)$ with the group topology whose prebase at unity is formed by sets of the
form $H(\gamma)$ for $\gamma\in\mathcal{G}$. The set $X$ naturally embeds in
$B(X)$: $i:X\to B(X),\ x\mapsto\\{x\\}$. This embedding is topological and
$i(X)$ is closed in $B(X)$ if and only if $\bigcup\mathcal{G}$ is a subbase of
$X$. In what follows we will identify $X$ and $i(X)$.

Denote by $B_{z}(X)=B_{z}(X,\Gamma(X))$. The space $X$ is close embedded in
$B_{z}(X)$. Let $Y\subset X$. Denote

$\displaystyle B_{z}^{0}(X)$ $\displaystyle=\\{g\in B_{z}(X)\,:\,|g|\text{ is
even}\\},$ $\displaystyle B_{z}^{1}(X)$ $\displaystyle=\\{g\in
B_{z}(X)\,:\,|g|\text{ is odd}\\},$ $\displaystyle B_{z}(X|Y)$
$\displaystyle=\\{g\in B_{z}(X)\,:\,g\subset Y\\},$ $\displaystyle
B_{z}^{0}(X|Y)$ $\displaystyle=B_{z}(X|Y)\cap B_{z}^{0}(X),$ $\displaystyle
B_{z}^{1}(X|Y)$ $\displaystyle=B_{z}(X|Y)\cap B_{z}^{1}(X).$

The set $B_{z}^{0}(X)=H(\\{X\\})$ is a clopen subgroup of index two. The set
$B_{z}^{1}(X)=B_{z}(X)\setminus B_{z}^{0}(X)$ is openly closed in $B_{z}(X)$
and is a coset of the group $B_{z}^{0}(X)$ [1].

Let $X$ be a space with non-Archimedean base ${\mathcal{B}}$ is a tree by
reverse inclusion and some linear order $<$ consistent with base
${\mathcal{B}}$.

For ${\mathcal{B}}^{\prime}\subset{\mathcal{B}}$ denote by
$\Lambda_{*}({\mathcal{B}}^{\prime})$ the set of maximal by inclusion elements
${\mathcal{B}}^{\prime}$. Then
$\Lambda_{*}({\mathcal{B}}^{\prime})\subset{\mathcal{B}}^{\prime}$; the family
${\mathcal{B}}^{\prime}$ is inscribed in
$\Lambda_{*}({\mathcal{B}}^{\prime})$; $\Lambda_{*}({\mathcal{B}}^{\prime})$
is an open partition of $\bigcup{\mathcal{B}}^{\prime}$.

For a family $\gamma$ of open subsets $X$, denote
$\Lambda(\gamma)=\Lambda_{*}({\mathcal{B}}^{\prime})$, where
${\mathcal{B}}^{\prime}=\\{U\in{\mathcal{B}}\,:\,U\subset V\text{ for some
}V\in\gamma\\}$. Then $\Lambda(\gamma)$ is inscribed in $\gamma$;
$\Lambda(\gamma)$ is an open partition of $\bigcup\gamma$; if
$V\in{\mathcal{B}}$ and $V\subset W$ for some $W\in\gamma$, then $V\subset U$
for some $U\in\Lambda(\gamma)$. If $\gamma$ is an open cover of $X$, then
$\Lambda(\gamma)\in\Gamma(X)$ and $\Lambda(\gamma)$ are inscribed in $\gamma$.
Therefore, $B_{z}(X)=B_{z}(X,{\mathcal{P}})$, where

${\mathcal{P}}=\\{\gamma\in\Gamma(X)\,:\,\gamma\subset{\mathcal{B}}\\}.$

For an open $W\subset X$, denote $\Lambda(W)=\Lambda(\\{W\\})$. Then
$\Lambda(W)$ is an open partition of $W$; if $V\in{\mathcal{B}}$ and $V\subset
W$, then $V\subset U$ for some $U\in\Lambda(W)$.

Recall the construction of the retraction $r:B_{z}^{1}(X)\to X$ from [1]. For
$g\in B_{z}^{1}(X)$ we set $\Upsilon(g)=\Lambda_{*}(\gamma)$, where
$\gamma=\\{U\in{\mathcal{B}}\,:\,g\cap U\neq\varnothing\text{ and }|g\cap
U|\text{ is even}\\}$. Let $R(g)=g\setminus\bigcup\Upsilon(g)$ and $r(g)=\min
R(g)$. Theorem 7 (version 2) [1] proves that $r$ is a continuous retraction.

######  Lemma 1.

The mapping $r$ is open. If $Y\subset X$ and $S=B_{z}^{1}(X|Y)$, then the
mapping $\left.r\right|_{S}:S\to Y$ is a continuous open retraction of $S$ to
$Y$.

###### Proof.

Let $g\in B_{z}^{1}(X)$ and $W\subset B_{z}(X)$ be a neighborhood of the point
$g$. Then $g+H(\gamma)\subset W$ for some $\gamma\in\Gamma(X)$. Let $x=r(g)$.
Let $x\in V\in\gamma$. The family $\Upsilon(g)$ is finite and
$x\notin\bigcup\Upsilon(g)$. Take $U\in{\mathcal{B}}$ such that $x\in U\subset
V\setminus\bigcup\Upsilon(g)$.

Let $y\in U$ and $h=g+\\{x,y\\}$. Then $\Upsilon(g)=\Upsilon(h)$,
$R(h)=R(g)+\\{x,y\\}$ and $r(h)=y$. Let $M=\\{g+\\{x,y\\}\,:\,y\in U\\}$. Then
$M\subset g+H(\gamma)$ and $r(M)=U$. Hence $r(g)\in U\subset r(W)$.

Let $g\in Y$, $Q=U\cap Y$ and $L=\\{g+\\{x,y\\}\,:\,y\in Q\\}$. Then $L=M\cap
S$ and $r(L)=Q$. Hence $r(g)\in Q\subset r(W\cap S)$. ∎

Lemma 1 and Theorem 7 (version 2) [1] imply the following assertion.

######  Theorem 1.

Let $X$ be a non-Archimedean space. Then there exists a continuous open
retraction $r:B_{z}^{1}(X)\to X$ such that for any $Y\subset X$ and
$S=B_{z}^{1}(X|Y)$ the map $\left.r\right|_{S}:S\to Y$ is a continuous open
retraction.

If $X$ is a compact space, then $B_{z}(X)$ is a precompact group. The set
$\Gamma(X)$ is countable. Hence $B_{z}({\mathbf{C}})$ is a precompact SMZD
group. Let’s apply the theorem 1 to ${\mathbf{C}}$.

######  Theorem 2.

There exists a continuous open retraction
$r:B_{z}^{1}({\mathbf{C}})\to{\mathbf{C}}$ such that for any
$X\subset{\mathbf{C}}$ and $S=B_{z}^{1}({\mathbf{C}}|X)$ the map
$\left.r\right|_{S}:S\to X$ is a continuous open retraction.

The set $B_{z}^{1}({\mathbf{C}})$ is a clopen subset of the precompact SMZD
group $B_{z}({\mathbf{C}})$. Let $y\in Y$ and

$h_{y}:B_{z}^{1}({\mathbf{C}})\to B_{z}^{0}({\mathbf{C}}),\ g\mapsto g+y.$

The mapping $h_{y}$ is a homeomorphism and
$h_{y}(B_{z}^{1}({\mathbf{C}}|X))=B_{z}^{0}({\mathbf{C}}|X)$, i.e. $X$ is an
open continuous retract of the precompact Boolean SMZD group
$B_{z}^{0}({\mathbf{C}}|X)$.

######  Corollary 1.

Any SMZD space is a continuous open retract of a precompact Boolean group.

##  4 Mal’tsev and rectifiable spaces

Let $X$ be a space with the operation $\Phi:X^{3}\to X$. Consider the
following conditions for the operation $\Phi$:

  * ($\Phi_{1}$)

$\Phi(x,y,y)=\Phi(y,y,x)=x$;

  * ($\Phi_{2}$)

$\Phi(x,y,\Phi(y,z,u))=\Phi(x,z,u)$;

  * ($\Phi_{3}$)

$\Phi(x,y,\Phi(y,x,u))=u$.

for $x,y,z,u\in X$. The $\Phi$ operation is a Mal’tsev operation if
($\Phi_{1}$) is satisfied.

######  Definition 1.

Let $X$ be a space and $\Phi:X^{3}\to X$ be a continuous map. We call a
mapping $\Phi$ homogeneous Mal’tsev operation if $\Phi$ satisfies the
conditions ($\Phi_{1}$) and ($\Phi_{3}$). We call a mapping $\Phi$ a strong
Mal’tsev operation if the conditions ($\Phi_{1}$) and ($\Phi_{2}$) are
satisfied for $\Phi$. A space $X$ is said to be strongly rectifiable if there
exists a continuous strong Mal’tsev operation on $X$.

######  Proposition 1.

The space $X$ is rectifiable if and only if there exists a homogeneous
Mal’tsev operation $\Phi$ on $X$.

###### Proof.

Assume that $X$ is a rectifiable space. Let the operations $p$ and $q$ define
the structure of a homogeneous algebra, $\Psi$ is a rectification,
$\Psi(x,y)=(x,p(x,y))$. We put $\Phi(x,y,z)=q(x,p(y,z))$. $q(x,e)=x$ and
$p(x,x)=e$ imply $\Phi(x,y,y)=x$. $q(x,p(x,y))=y$ implies $\Phi(y,y,x)=x$.
Let’s check ($\Phi_{3}$). Since $p(y,q(y,p(x,u)))=p(x,u)$ and $q(x,p(x,u))=u$,
then

$\Phi(x,y,\Phi(y,x,u))=q(x,p(y,q(y,p(x,u))))=u.$

Suppose that $\Phi$ satisfies the conditions ($\Phi_{1}$) and ($\Phi_{3}$).
Take $e\in X$ arbitrarily. Let $p(x,y)=\Phi(e,x,y)$ and $q(x,y)=\Phi(x,e,y)$.
From ($\Phi_{1}$) follows $p(x,x)=e$ and $q(x,e)=x$. From ($\Phi_{3}$) it
follows

$\displaystyle q(x,p(x,y))$ $\displaystyle=\Phi(x,e,\Phi(e,x,y))=y,$
$\displaystyle p(x,q(x,y))$ $\displaystyle=\Phi(e,x,\Phi(x,e,y))=y.$

∎

######  Proposition 2.

If $X$ is strongly rectifiable, then $X$ is rectifiable.

###### Proof.

The condition ($\Phi_{3}$) follows from ($\Phi_{1}$) and ($\Phi_{2}$), it
suffices to substitute $z=x$ into ($\Phi_{3}$). It remains to apply the
Proposition 1. ∎

######  Proposition 3.

Let $Y$ be a Hausdorff space, $X\subset Y=\overline{X}$, mapping
$\widehat{\Phi}:Y^{3}\to Y$ is continuous, and $\Phi(X^{3})=X$. Denote
$\Phi=\left.\widehat{\Phi}\right|_{X^{3}}$.

  1. 1.

If $\Phi$ is a Mal’tsev operation, then $\widehat{\Phi}$ is a Mal’tsev
operation.

  2. 2.

If $\Phi$ is a homogeneous Mal’tsev operation, then $\widehat{\Phi}$ is a
homogeneous Mal’tsev operation.

  3. 3.

If $\Phi$ is a strong Mal’tsev operation, then $\widehat{\Phi}$ is a strong
Mal’tsev operation.

###### Proof.

If the identities ($\Phi_{1}$), ($\Phi_{2}$), and ($\Phi_{3}$) hold on a dense
subspace, then they hold everywhere. ∎

######  Proposition 4.

Let $X$ be a space with continuous Mal’tsev operation $\Phi$, $Y_{1}$ and
$Y_{2}$ be compact Hausdorff extensions of $X$, the operation $\Phi$ extends
to $Y_{1}$ and $Y_{2}$ up to continuous operations $\Phi_{1}$ and $\Phi_{2}$,
respectively. Then $Y_{1}$ and $Y_{2}$ coincide, that is, there exists a
homeomorphism $f:Y_{1}\to Y_{2}$ for which
$\left.f\right|_{X}=\operatorname{id}_{X}$.

###### Proof.

We identify $X$ with $\\{(x,x)\in Y_{1}\times Y_{2}\,:\,x\in X\\}$. Let
$Y_{3}$ be the closure of $X$ in $Y_{1}\times Y_{2}$, $\pi_{i}$ be the
projection of $Y_{1}\times Y_{2}$ onto $Y_{i}$,
$f_{i}=\left.\pi_{i}\right|_{Y_{3}}$ for $i=1.2$,
$\Phi_{3}=\left.\Phi_{1}\times\Phi_{2}\right|_{X^{3}}$. Then
$\Phi_{3}(X^{3})=X$ and, by the proposition 3, the operations $\Phi_{i}$ for
$i=1,2,3$ are a Mal’tsev operation and $f_{i}$ are morphisms with respect to
them. Since the spaces $Y_{i}$ are compact, the mappings $f_{i}$ are
factorial. Therefore, mappings $f_{i}$ are open (Mal’tsev’s Theorem 4.11 [9],
[7]). An open continuous map that is a homeomorphism on a dense set is itself
a homeomorphism. Hence the mappings $f_{i}$ are homeomorphisms. Therefore, the
mapping $f=f_{1}\circ f_{2}^{-1}:Y_{1}\to Y_{2}$ is a homeomorphism. ∎

######  Remark 1.

Propositions 3, 4 and their proofs remain valid if we consider separately
continuous Mal’tsev operations.

######  Definition 2.

Let $X$ be a space and $\Phi:X^{3}\to X$ be a continuous Mal’tsev operation.
An operation $\Phi$ is called precompact if there exists a Hausdoff compact
extension $bX$ of the space $X$ onto which the mapping $\Phi$ extends to the
mapping $\widehat{\Phi}:bX^{3}\to bX$.

A compact extension $bX$ is uniquely defined up to isomorphism (Proposition
4), we denote it by $\widehat{X}$ or $\widehat{(X,\Phi)}$.

  1. 1.

A space $X$ is called precompact Mal’tsev if there exists a continuous
precompact Mal’tsev operation on $X$.

  2. 2.

A space $X$ is called precompact rectifiable if there exists a continuous
precompact homogeneous Mal’tsev operation on $X$.

  3. 3.

We call a space $X$ precompact strongly rectifiable if there exists a
continuous precompact strong Mal’tsev operation on $X$.

######  Proposition 5.

If $X$ is a topological group, then $X$ is strongly rectifiable.

If $X$ is a precompact topological group, then $X$ is precompact and strongly
rectifiable.

###### Proof.

We put $\Phi(x,y,z)=xy^{-1}z$. The operation $\Phi$ is the standard Mal’tsev
operation on a group. Let’s check ($\Phi_{2}$).

$\Phi(x,y,\Phi(y,z,u))=xy^{-1}(yz^{-1}u)=xz^{-1}u=\Phi(x,z,u).$

If $G$ is precompact, then $\Phi$ extends to the completion of $G$. ∎

######  Proposition 6.

Let $X$ be a Mal’tsev precompact space. Then $w(\widehat{X})=w(X)=\chi(X)$.

###### Proof.

Let $\tau=\chi(X)$. The compact Mal’tsev space $\widehat{X}$ is a Dugundzhi
compact (Theorem 1 [10]). The Dugunji compactum is a dyadic compactum. If a
compact dyadic compact contains a dense subspace whose character does not
exceed $\tau$, then its weight does not exceed $\tau$ ([11]). ∎

######  Lemma 2.

Let $\Phi$ be a precompact homogeneous Mal’tsev operation on the space $X$ and
$e\in X$. For any non-empty open $U\subset X$ there exists a finite set
$M\subset X$ so that $X=\bigcup_{y\in M}\Phi(e,y,U)$.

###### Proof.

Let $W\subset\widehat{X}$ be open and $W\cap X=U$. Take a non-empty open
$V\subset\overline{V}\subset W$. The family
$\\{\Phi(e,y,V)\,:\,y\in\widehat{X}\\}$ forms an open cover of $\widehat{X}$.
Then $\bigcup_{i=1}^{n}\Phi(e,x_{i},V)$ for some
$x_{1},x_{2},...,x_{n}\in\widehat{X}$. Then
$\Phi(e,x_{i},V)\subset\Phi(e,y_{i},U)$ for some $y_{i}\in X$. Let
$M=\\{y_{1},y_{2},...,y_{n}\\}$. ∎

##  5 Mal’tsev operations on the Cantor set ${\mathbf{C}}$

######  Theorem 3.

Let $X$ be the SMZD space. Then $X$ is a pre-compact Mal’tsev space.

Moreover, there is a precompact Mal’tsev operation $\Phi$ on ${\mathbf{C}}$,
so that $\Phi(X^{3})=X$ for any $X\subset{\mathbf{C}}$.

###### Proof.

Let $r:B_{z}^{1}({\mathbf{C}})\to{\mathbf{C}}$ as in Theorem 2. Let
$\Phi(x,y,z)=r(\\{x,y,z\\})$. Then $\Phi(x,y,z)\in\\{x,y,z\\}$ and hence
$\Phi$ is a Mal’tsev operation. ∎

######  Theorem 4.

Let $X$ be a strongly homogeneous SMZD space. Then $X$ is a precompact
strongly rectifiable space.

Moreover, there exists a precompact strongly Malcev operation $\Phi$ on $X$,
so that $\widehat{(X,\Phi)}$ is homeomorphic to ${\mathbf{C}}$.

###### Proof.

There is a compact zero-dimensional extension $bX$ of the space $X$
homeomorphic to ${\mathbf{C}}$, so that for any non-empty open-closed
$U,V\subset bX$, $U,V\neq Y$ there exists $f\in\operatorname{Aut}(Y|X)$ so
$f(U)=V$, Lemma 2.2 [12]. We will assume that $bX={\mathbf{C}}$.

######  Lemma 3.

For every non-empty open-closed $U\subset{\mathbf{C}}$ there exists a
homeorphism $f_{U}:{\mathbf{C}}\to U$, so that $f_{U}(X)=U\cap X$.

###### Proof.

Let $V_{1}$ and $V_{2}$ be a partition of ${\mathbf{C}}$ into two non-empty
open-closed subsets and $U_{1}$ and $U_{2}$ be a partition of $U$ into two
non-empty open-closed subsets. For $i=1,2$, let $f_{i}:V_{i}\to U_{i}$ be a
homeorphism such that $f_{i}(V_{i}\cap X)=U_{i}\cap X$. Then $f_{U}$ is the
union of $f_{1}$ and $f_{2}$. ∎

Denote $\mathcal{C}^{*}=\mathcal{C}\setminus\\{{\mathbf{C}}\\}$. For
$U,V\in\mathcal{C}^{*}$ we set

$\Theta\left[U,V\right]=f_{U}\circ f_{V}^{-1}:V\to U.$

It follows from the construction

  * ($\Theta_{1}$)

$\Theta\left[U,V\right]:V\to U$ is a homeomorphism and
$\Theta\left[U,V\right](V\cap X)=U\cap X$ for $U,V\in\mathcal{C}^{*}$;

  * ($\Theta_{2}$)

$\Theta\left[U,U\right]=\operatorname{id}_{U}$ for $U\in\mathcal{C}^{*}$;

  * ($\Theta_{3}$)

$\Theta\left[U,V\right]\circ\Theta\left[V,W\right]=\Theta\left[U,W\right]$ for
$U,V,W\in\mathcal{C}^{*}$.

Let $x=(x_{n})_{n\in\omega}\in{\mathbf{C}}$, $k\in\omega$. Let’s put

$\displaystyle U_{k}(x)$
$\displaystyle=\\{y\in{\mathbf{C}}\,:\,x|k+1=y|k+1\\}=B(x|k+1),$
$\displaystyle V_{k}(x)$ $\displaystyle=\\{y\in{\mathbf{C}}\,:\,x|k=y|k\text{
and }x(k)\neq y(k)\\}=U_{k-1}(x)\setminus U_{k}(x)=B(\tilde{x}),$

where $\tilde{x}=(x_{0},...,x_{k-1},1-x_{k})$. Note that

  * ($U_{1}$)

$U_{k}(x),V_{k}(x)\in\mathcal{C}^{*}$;

  * ($U_{2}$)

the family $\\{V_{n}(x)\,:\,n\leq k\\}$ is an open partition of the set
${\mathbf{C}}\setminus U_{k}(x)$;

  * ($U_{3}$)

the family $\\{V_{n}(x)\,:\,{n\in\omega}\\}$ is an open partition of the set
${\mathbf{C}}\setminus\\{x\\}$;

  * ($U_{4}$)

the family $\\{U_{n}(x)\,:\,{n\in\omega}\\}$ is the base at the point $x$;

  * ($U_{5}$)

$U_{k}(x)=U_{k}(y)$, $V_{k}(x)=V_{k}(y)$ and $x\in U_{k}(y)$ if $y\in
U_{k}(x)$;

  * ($U_{6}$)

$V_{k}(x)=U_{k}(y)$, $V_{k}(y)=U_{k}(x)$ and $x\in V_{k}(y)$ if $y\in
V_{k}(x)$.

For $x,y\in{\mathbf{C}}$ we define the mapping
$h_{x,y}\in\operatorname{Aut}({\mathbf{C}})$. Let $z\in{\mathbf{C}}$. Let’s
put

$h_{x,y}(z)=\begin{cases}x,&z=y\\\
\Theta\left[V_{k}(x),V_{k}(y)\right](z),&\text{if $z\in V_{k}(x)$ for some
$k\in\omega$}\end{cases}$

It follows from the construction

  * ($h_{1}$)

$h_{x,y}\in\operatorname{Aut}({\mathbf{C}})$ for $x,y\in{\mathbf{C}}$ and
$h_{x,y}\in\operatorname{Aut}({\mathbf{C}}|X)$ for $x,y\in X$;

  * ($h_{2}$)

$h_{x,x}=\operatorname{id}_{\mathbf{C}}$;

  * ($h_{3}$)

$h_{x,y}\circ h_{y,z}=h_{x,z}$;

  * ($h_{4}$)

$h_{x,y}(y)=x$;

  * ($h_{5}$)

$h_{x,y}(U_{k}(y))=U_{k}(x)$ and $h_{x,y}(V_{k}(y))=V_{k}(x)$.

  * ($h_{6}$)

if $x^{\prime}\in U_{k}(x)$ and $y^{\prime}\in U_{k}(y)$ then
$\left.h_{x,y}\right|_{V}=\left.h_{x^{\prime},y^{\prime}}\right|_{V}$, where
$V=V_{k}(y)$.

Let’s put

$\Phi:{\mathbf{C}}^{3}\to{\mathbf{C}},\quad(x,y,z)\mapsto h_{x,y}(z).$

The condition ($\Phi_{1}$) follows from ($h_{2}$) and ($h_{4}$). The condition
($\Phi_{2}$) follows from ($h_{3}$). Hence the operation $\Phi$ is a strong
Mal’tsev operation.

Let us prove that $\Phi$ is continuous. Let $x,y,z,u\in{\mathbf{C}}$,
$\Phi(x,y,z)=u$, $U$ be a neighborhood of the point $u$. We need to find a
neighborhood $W\subset{\mathbf{C}}^{3}$ of the point $(z,y,z)$, so that
$\Phi(W)\subset U$.

Consider the case $y=z$. Then $x=u$. Then $S=U_{k}(x)\subset U$ for some
$k\in\omega$. Let $V=U_{k}(y)$ and $W=S\times V\times V$. Let’s check
$\Phi(W)\subset U$. Let $(x^{\prime},y^{\prime},z^{\prime})\in W$. From
($U_{5}$) and ($h_{5}$) it follows that $h_{x^{\prime},y^{\prime}}(V)=S$.
Hence $\Phi(x^{\prime},y^{\prime},z^{\prime})\in U$.

Consider the case $y\neq z$. Then $z\in V_{m}(y)$ for some $m\in\omega$. Let
$S=U_{m}(x)$ and $V=U_{m}(y)$. Since the mapping $h_{x,y}$ is continuous, then
$h_{x,y}(Q^{\prime})\subset U$ for some neighborhood $Q^{\prime}$ of the point
$z$. Let $Q=Q^{\prime}\cap V_{m}(y)$ and $W=S\times V\times Q$. Let’s check
$\Phi(W)\subset U$. Let $(x^{\prime},y^{\prime},z^{\prime})\in W$. From
($U_{6}$) it follows that
$\left.h_{x^{\prime},y^{\prime}}\right|_{Q}=\left.h_{x,y}\right|_{Q}$. Hence
$\Phi(x^{\prime},y^{\prime},z^{\prime})\in U$.

The continuity of $\Phi$ is proved. From ($h_{1}$) it follows that
$\Phi(X^{3})=X$. ∎

######  Proposition 7.

Let $X$ be a zero-dimensional first countable space. Then
$X^{\omega}\times{\mathbf{C}}$ is strongly homogeneous.

###### Proof.

Let $Y=X\times{\mathbf{D}}$. Then the spaces $X^{\omega}\times{\mathbf{C}}$,
$Y^{\omega}$ and $Y^{\omega}\times{\mathbf{D}}$ are homeomorphic. Therefore,
$Y^{\omega}$ has a proper open-closed subspace homeomorphic to $Y^{\omega}$.
Proposition 24 (5) [13] implies that $Y^{\omega}$ is strongly homogeneous. ∎

######  Proposition 8.

Let $X$ be the SMZD space. There is a space SMZD space $Y$ such that $X\times
Y$ is strongly homogeneous and strongly rectifiable.

###### Proof.

Let $Y=X^{\omega}\times{\mathbf{C}}$. Then, by the Proposition 7, $Y$ is
strongly homogeneous. The space $X\times Y$ is homeomorphic to $Y$. It remains
to apply the Theorem 4. ∎

##  6 Homogeneous spaces that are not right topological groups

Van Mill in [14] constructed a homogeneous space $A$, a subspace of the real
numbers, which is not a topological group. In fact, the space $A$ is not a
right topological group either.

We call a set $A\subset X$ be a clumsy if $h(A)\cap A$ is non empty for any
homeomorphism $h:X\to X$.

######  Theorem 5.

Let $X$ be an space and $A\subset X$ be a clumsy subset. If $|A|<|X|$, then
$X$ is not a right topological group.

###### Proof.

From the contrary. Take $g\in X\setminus A^{-1}A$. Then $Ag\cap
A=\varnothing$. Contradiction. ∎

Theorem 5 generalizes Lemma 3.4. from [14] and the proof of the theorem is
much simpler.

There is a countable dense set $E\subset A$ such that $E$ is clumsy in $A$
(Theorem 3.3 [14]). Therefore, by the Theorem 5, the space $A$ from [14] is
not a right topological group.

In [15] Taras Banakh noticed that the zero-dimensional homogeneous first-
countable van Douwen compactum [16] contains a countable clumsy set and,
therefore, is not a right topological group.

Let
${\mathbf{T}}={\mathbf{P}}\times{\mathbf{P}}\cup{\mathbf{Q}}\times{\mathbf{Q}}$
is van Douwen space and ${\mathbf{S}}={\mathbf{C}}\times{\mathbf{T}}$ is van
Mill space. Spaces ${\mathbf{T}}$ and ${\mathbf{S}}$ are SMZD strongly
homogeneous spaces [17, 18].

######  Statement 1.

${\mathbf{Q}}\times{\mathbf{Q}}$ is clumsy in ${\mathbf{T}}$.

###### Proof.

Assume the contrary, that is,
$h({\mathbf{Q}}\times{\mathbf{Q}})\cap{\mathbf{Q}}\times{\mathbf{Q}}=\varnothing$
for some homeomorphism $h:{\mathbf{T}}\to{\mathbf{T}}$. Let
$F={\mathbf{Q}}\times\\{0\\}$. Then $F$ is closed in ${\mathbf{T}}$ and
homeomorphic to ${\mathbf{Q}}$. Hence $h(F)$ is closed in
${\mathbf{P}}\times{\mathbf{P}}$ and homeomorphic to ${\mathbf{Q}}$.
Contradiction. ∎

The Theorem 5 and Statement 1 imply

######  Proposition 9.

${\mathbf{T}}$ is not a right topological group.

It is not clear about the van Mill space ${\mathbf{S}}$ whether the structure
a right topological group is allowed on it.

######  Proposition 10 (Corollary 5.4 [19]).

If a topological group $G$ contains a dense Čech complete subspace, then $G$
is Čech complete.

Recall that Polish spaces are exactly Čech complete separable metrizable
spaces.

The space ${\mathbf{S}}$ contains a dense subspace homeomorphic to
${\mathbf{P}}$ and is not Polish. Then ${\mathbf{S}}^{\omega}$ also contains a
dense subspace homeomorphic to ${\mathbf{P}}$ and is not Polish. The following
sentence follows from the Proposition 10.

######  Proposition 11.

${\mathbf{S}}^{\omega}$ is not a topological group.

##  7 Examples and questions

######  Question 1.

(A.V. Arhangelskii) Let $G$ be a right topological group.

Is it true that if $G$ contains a dense Čech complete subspace, then $G$ is
Čech complete?

Is it true that if $G$ is a separable metrizable and contains a dense Polish
subspace, then $G$ is Polish?

######  Example 1.

The van Mill space ${\mathbf{S}}$ and the van Dawn space ${\mathbf{T}}$ have
the following properties:

  * 1.

are strongly homogeneous [18];

  * 2.

absolutely Borel [18, 17];

  * 3.

coset sets of some absolutely Borel groups [12];

  * 4.

have a dense Polish subspace and are not Polish [18];

  * 5.

precompact strongly rectifiable (Theorem 4);

  * 6.

${\mathbf{T}}$ is not a right topological group (Proposition 9);

  * 7.

${\mathbf{S}}$ is not a topological group (Proposition 10);

######  Question 2.

Is the van Mill space ${\mathbf{S}}$ a right topological group?

######  Question 3.

Let $X\in\\{{\mathbf{S}},{\mathbf{T}}\\}$.

  1. 1.

Is the space $X\times{\mathbf{Q}}$ homeomorphic to a (boolean) (precompact)
(right) topological group?

  2. 2.

Is the space $X\times{\mathbf{P}}$ homeomorphic to a (boolean) (precompact)
(right) topological group?

  3. 3.

Is there a SMZD space $Y$ such that $X\times Y$ is homeomorphic to a (boolean)
(precompact) (right) topological group?

A topological group $G$ is called locally precompact if there exists a
neighborhood $U$ of the neutral element $e$ of $G$ such that $U$ can be
covered by ﬁnitely many left and right translates of each neighborhood of $e$
in $G$. A group is locally precompact iff its completion is locally compact.

######  Proposition 12 (Theorem 8 [20]).

Let $G$ be a SMZD topological group which is not locally precompact. Then the
space $G$ is strongly homogeneous.

There exist a dense subgroup $G_{M}$ of the real line [21, Theorem 5.1] and
dense subgroup $G_{D}$ of the circle [16], so $G\in\\{G_{M},G_{D}\\}$
satisfies the condition:

  * if $A,B\subset\widehat{G}$ are open sets and $A\cap G$ is homeomorphic to $B\cap G$, then $\mu(A)=\mu(B)$,

where $\widehat{G}$ is the completion of $G$ (for $G_{M}$ the completion is
the real line, and for $G_{D}$ the completion is the circle) and $\mu$ is the
standard invariant Lebesgue measure on $\widehat{G}$. The group $G_{D}$ is
precompact, the group $G_{M}$ is locally precompact but not precompact.

######  Example 2.

SMZD the groups $G_{D}$ and $G_{M}$ are not strongly homogeneous, it suffices
to take two segments $A,B$ of different lengths, the ends of which do not lie
in the group [16]. The group $G_{M}$ is not homeomorphic to a precompact
rectifiable space, a finite number of shifts of the set $(-1,1)\cap G_{M}$
cannot cover $G_{M}$ (see Lemma 2).

######  Proposition 13 (Corollary 5 [20]).

If $X$ is a homogeneous SMZD space, then $X\times{\mathbf{Q}}$ is a strongly
homogeneous space.

######  Question 4.

Let $G\in\\{G_{M},G_{D}\\}$. Which of the spaces listed below are strongly
homogeneous: $G\times{\mathbf{P}}$, $G\times{\mathbf{C}}$, $G^{\omega}$?

######  Question 5.

Which of the following spaces are homeomorphic to (1) precompact group; (2)
precompact (strongly) rectifiable space: $G_{M}\times{\mathbf{Q}}$,
$G_{M}\times{\mathbf{P}}$, $G_{M}\times{\mathbf{C}}$, $G_{M}^{\omega}$?

######  Question 6.

Let $G\in\\{G_{M},G_{D}\\}$. Which of the spaces listed below are homeomorphic
to a Boolean group: $G$, $G\times{\mathbf{Q}}$, $G\times P$,
$G\times{\mathbf{C}}$, $G^{\omega}$?

######  Proposition 14 ([22, 23]).

Let $X$ be a SMZD space. Then $X^{\omega}$ is homogeneous.

######  Question 7 (Terada, [24]).

Let $X$ be a SMZD space. Is it true that $X^{\omega}$ is a strongly
homogeneous space?

######  Question 8.

Let $X$ be the SMZD space.

  1. 1.

Which of the following spaces are (1) (strongly) (precompactly) rectifiable
spaces; (2) is homeomorphic to a right topological group: $X^{\omega}$,
$X^{\omega}\times{\mathbf{Q}}$, $X^{\omega}\times{\mathbf{P}}$,
$X^{\omega}\times{\mathbf{C}}$?

  2. 2.

Is the space $X^{\omega}\times{\mathbf{Q}}$ homeomorphic to a topological
(boolean, precompact) group?

  3. 3.

Is there a SMZD space $Y$ such that $X\times Y$ is homeomorphic to a (right)
topological group?

######  Question 9.

Let $X$ be a homogeneous SMZD space.

  1. 1.

Will the space $X$ be a (strongly) (precompact) rectifiable space?

  2. 2.

Is the space $X\times{\mathbf{Q}}$ homeomorphic to a (precompact) (right)
topological group?

######  Question 10.

In the class SMZD spaces, distinguish the following classes: rectifiable;
strongly straightened; pre-compact straightening; strongly precompactly
rectifiable.

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