# Bright in the Black: Searching for Electromagnetic Counterparts to
Gravitational-Wave Candidates in LIGO-Virgo-KAGRA Observation Runs with
AstroSat-CZTI

Gaurav Waratkar Department of Physics, Indian Institute of Technology Bombay,
Powai, 400 076, India  Varun Bhalerao Department of Physics, Indian Institute
of Technology Bombay, Powai, 400 076, India  Dipankar Bhattacharya Ashoka
University, Department of Physics, Sonepat, Haryana 131029, India  Inter-
University Center for Astronomy and Astrophysics, Pune, Maharashtra-411007,
India G. Waratkar<EMAIL_ADDRESS>
###### Abstract

GW150914 marked the start of the gravitational wave (GW) era with the direct
detection of binary black hole (BBH) merger by the LIGO-Virgo GW detectors.
The event was temporally coincident with a weak signal detected by Fermi-GBM,
which hinted towards the possibility of electromagnetic emission associated
with the compact object coalescence. The detection of a short Gamma-Ray Burst
(GRB) associated with GW1708017, along with several multi-wavelength
detections, truly established that compact object mergers are indeed multi-
messenger events. The Cadmium Zinc Telluride Imager onboard AstroSat can
search for X-ray counterparts of the GW events and has detected over 600 GRBs
since launch. Here we present results from our searches for counterparts
coincident with GW triggers from the first three LIGO-Virgo-KAGRA (LVK) GW
Transient Catalogs. For 72 out of 90 GW events for which AstroSat-CZTI data
was available, we undertook a systematic search for temporally coincident
transients and we detected no X-ray counterparts. We evaluate the upper limits
on the maximum possible flux from the source in a 100 s window centered around
each trigger, consistent with the GW localization of the event. Thanks to the
high sensitivity of CZTI, these upper limits are highly competitive with those
from other spacecraft. We use these upper limits to constrain the theoretical
models that predict high-energy counterparts to the BBH mergers. We also
discuss the probability of non-detections of BBH mergers at different
luminosities and the implications of such non-detections from the ongoing
fourth observing run of the LVK detectors.

Gamma-ray bursts (629) — Gravitational waves (678) — Compact objects (288) —
Black holes (162) — Stellar mass black holes (1611)

##  1 Introduction

In the year 2015, LIGO-Virgo gravitational wave (GW) detectors reported the
first direct detection of gravitational waves from a binary black hole (BBH)
merger (Abbott et al., 2016). This event was accompanied by the detection of a
weak X-ray transient signal in Fermi-GBM (GRB150914-GBM; Connaughton et al.,
2016; Veres et al., 2019), which was not seen by any other mission. This
unproven association of the electromagnetic (EM) signal with the GW event
started the multi-messenger era. This era truly reached its peak when, in
2017, LIGO-Virgo GW detectors reported the first detection of GW from a binary
neutron star merger (BNS) in 2017. About 1.7 s after the detection of GW, a
short-duration gamma-ray burst (GRB) was detected by INTEGRAL, Fermi (Abbott
et al., 2017a). Soon, the counterpart to this GW event was detected across the
electromagnetic spectrum in the form of a kilonova (KN) emission (Abbott et
al., 2017b). To date, this remains the only known multi-messenger event where
gravitational waves from the merger, prompt emission from GRB, and a
corresponding KN emission were all detected (Petrov et al., 2022). It further
established that compact object mergers like BNS mergers are one of the
progenitors for GRBs that typically last for less than 2 s (Goldstein et al.,
2017; Berger, 2014; Metzger & Berger, 2012).

Other compact object mergers that the GW detectors can detect include the
merger of a neutron star and black hole (NSBH; Abbott et al., 2021b, 2020b,
2018). These NSBH mergers can also be accompanied by EM emission depending on
the mass ratios of the system, surrounding environment, and so on (Rosswog et
al., 2013; Singhal et al., 2022; Nakar, 2020). In addition, GW detectors have
been regularly detecting binary black hole (BBH) mergers (Abbott et al., 2016,
2019, 2021a; The LIGO Scientific Collaboration et al., 2021). While most
studies point towards the need for at least one neutron star in the binary
system to get an EM emission, some models suggest that even BBH mergers can
lead to EM emissions (McKernan et al., 2019; Perna et al., 2021; Bartos et
al., 2017).

In the past three observing runs of LIGO-Virgo GW detectors, there have been
several attempts to hunt for EM emission coming from these sources, none of
which led to any proven counterpart to the GW events (Petrov et al., 2022).
The non-detections from campaigns that hunted for the afterglow emission were
used to improve the models that predict the EM emission (see for instance
Kumar et al., 2022; Anand et al., 2021; Coughlin et al., 2019; Andreoni et
al., 2020). Kasliwal et al. (2020) used the optical non-detections to
constrain the KN luminosity function, and place constraints on bright-KN
rates. Recent studies have also discussed the association between Fast Radio
Bursts (FRBs) and GW events, although none conclusive yet (Moroianu et al.,
2023; Bhardwaj et al., 2023; The LIGO Scientific Collaboration et al., 2022).
Furthermore, the large distances at which most of the BBHs are detected and
the challenges associated with the formation of accretion disks during their
merger evolution make it difficult to detect any EM signatures from them. Yet,
there have been several claimed associations — for instance, there were
several candidate counterparts to BBH mergers during the third observing run
(O3) that were associated with active galactic nuclei (AGNs) with unusual
flaring activity (Graham et al., 2020, 2023).

In the hunt for coincident prompt high-energy emission from sources of GWs,
there were non-detections reported in the burst searches reported by various
space telescopes like Fermi-GBM, Swift-BAT, AGILE, Insight-HXMT, etc (Burns et
al., 2019; Hamburg et al., 2020; Fletcher et al., 2023; Ursi et al., 2022; Cai
et al., 2021). Offline searches undertaken to look for coincident events with
the GRB events detected by Fermi and Swift-BAT during O3 (Abbott et al.,
2021c, 2022) also yielded non-detections. The Fermi-GBM detected gamma-ray
signal coincident with the first BBH merger event GW150914 (Connaughton et
al., 2016) remains the only claimed association between the BBH merger event
and the detection of a GRB. Through this detection, Veres et al. (2019)
calculated the rates of such coincident signals given the sensitivity of LIGO-
Virgo GW detectors.

Through these studies, it was seen that both coincident searches and the hunt
for the afterglows can vastly improve our understanding of these sources. Such
searches will be of even higher importance when the sensitivities of the GW
detectors improve in the future observing runs leading to a larger number of
GW events (Abbott et al., 2018). However, Petrov et al. (2022) showed that the
well-localised, high-profile events will make up a smaller proportion of the
overall population that LIGO-Virgo-KAGRA (LVK) GW detectors will be detecting
the future observing runs because of improvements in data analysis. Their
data-driven approach also pointed out that while the number of GW events for
which KN signatures are detectable increases, there is a significant reduction
in their fraction among total GW events. This combined with a large number of
events, most having poor localisation, underscores the need to have X-ray
coincident searches guiding these searches, similar to the case of GW170817
where the GRB detection set the tone for further follow-up. We see the need to
undertake searches for bursts that are coincident with the GW events, which
will help us understand the source properties, and also aid in the searches
for KNe by other facilities.

The Indian space telescope AstroSat (Singh et al., 2014) was launched in
September 2015, soon after the discovery of GW150914. One of the instruments
on board is the Cadmium Zinc Telluride Imager (CZTI; Bhalerao et al., 2017) —
a wide-field hard X-ray monitor operating in the 20-200 keV energy range. At
energies above $\sim 100$ keV, the collimators of CZTI become increasingly
transparent to radiation, and CZTI functions as an all-sky monitor with good
sensitivity to transients (Sharma et al., 2021). CZTI has detected over 600
Gamma Ray Bursts (GRBs) to date111CZTI GRB detections are reported at
http://astrosat.iucaa.in/czti/?q=grb., of which $\sim$ 50 have not been
detected by any other mission. This demonstrates the sensitivity of CZTI to
high-energy transients and makes it an important asset in the search for EM
counterparts to GW sources.

Since launch, AstroSat-CZTI is used to regularly search for these GW events
based on the low-latency information available immediately to the astronomers,
and has placed upper-limits on fluxes from the GW events (see for instance
Mate et al., 2017; Waratkar et al., 2023). Further, for the case of GW170817,
the primary localisation region was occulted by the earth. Based on the
reported source spectrum and flux (von Kienlin et al., 2017), it was inferred
that AstroSat-CZTI was sensitive enough to have detected the short GRB from
the event, but it did not. This non-detection was used in constraining the
combined localisation region of GW event and the Fermi-GBM GRB detection
localisation region (Kasliwal et al., 2017), highlighting how non-detections
can be of immense importance.

With an improved CZTI data processing pipeline (Ratheesh et al., 2021), and
availability of final confident events published by LVK through the
Gravitational Wave Transient Catalogs (GWTC), we report our comprehensive
search for bursts in AstroSat-CZTI coincident with the GW events reported in
these catalogs from the first three observing runs (O1, O2, and O3; hereafter
referred as O1-3) of LIGO-Virgo GW detectors. In Section 2, we describe the
procedure for CZTI transient analysis, focusing on the searches for GW
counterparts in CZTI data. We outline the GW event sample used, and the
results from our searches for coincident bursts in CZTI data in Section 3.
Finally, we discuss the implications of these results on the luminosity
function of different merger models and conclude with the prospects of
detecting prompt emission counterparts in the future observing runs of LVK GW
detectors in Section 4.

##  2 CZTI Data Processing

The procedure for CZTI transient analysis differs from that of usual
persistent sources, and more so for the analysis of GW coincident searches. In
this section, we outline the method used for the analysis to search for GW
counterparts in CZTI data.

###  2.1 Searches for coincident bursts

We broadly follow the data processing described in Sharma et al. (2021) and
Anumarlapudi et al. (2020). To reduce the raw data downlinked from the
AstroSat, we use the CZTI V3 pipeline222http://astrosat-ssc.iucaa.in/cztiData
which is based on the new generalised event selection algorithm (Ratheesh et
al., 2021). The unprocessed raw data files (Level 1 data products) are
processed through the standard workflow of the CZTI pipeline to get the Level
2 data products, which include the cleaned event data. We use the cztgtigen
module to make sure that the data are not discarded even when the on-axis
target of AstroSat is earth occulted. In the CZTI V3 pipeline, we employ the
new “generalized” event selection algorithm mode which is optimized for bright
GRB-like events. This generalized mode employs new modules cztnoisypixclean,
cztsuperbunchclean, cztheavybunchclean, cztflickpixclean, and czteventsep
(Ratheesh et al., 2021).

Using these cleaned event files, we generate lightcurves that are binned at
three timescales — 0.1 s, 1.0 s, and 10.0 s for all four quadrants. In these
lightcurves, we expect a slow variation in the background due to an increased
charged particle background when AstroSat is passing over the South Atlantic
Anomaly (SAA). We correct this variation by using a second-order Savitzky
Golay filter (time window width of 100 s) to estimate the broad background
trend, which is then subtracted from the time series data to obtain de-trended
light curves. The data are first inspected visually in the form of light
curves as well as spectrograms (2-dimensional histograms of photon energies
and arrival times), to search for transient candidates. This is followed by a
qualitative search for the EM counterpart in the data.

###  2.2 Calculating the minimum detectable flux

For each quadrant, we calculate the minimum counts needed from a source to get
a detection at a 90% confidence level. We then examine the data to see if the
light curve crosses these “cutoff count-rates” at any point in the search
window of 100 s. A valid transient should be detected above the minimum
detectable count-rate from any source for each quadrant (cutoff count-rates).
We check if anything crosses this threshold, and require that any
astrophysical event should be detected in at least three quadrants out of
four. If no such candidate is found, we use these count-rates to calculate the
upper limits on the source flux.

We follow the same treatment as described in Anumarlapudi et al. (2020) to
calculate the cutoff count-rates for each quadrant at three different search
timescales — at 0.1 s, 1.0 s, and 10.0 s. We choose a window of 100 s ($\pm$
50 s from the trigger time) and use de-trended data from ten neighboring
orbits to estimate the background count-rate histograms. We then estimate a
cutoff count-rate from these background count-rate distributions based on the
figure of merit of false alarm probability (FAP), which is set at 0.1 for each
quadrant. The combined four-quadrant FAP becomes $10^{-4}$, which states that
we expect an outlier in every $10^{4}$ bins. The choice of 10 neighbouring
orbits allows us to account for temporal background fluctuations of the CZT
detectors. It is observed that in some cases, one of the CZTI quadrants is
very noisy, and should be excluded from further analysis. We define such
“noisy” cases as ones where the cutoff count-rate for one quadrant is $>$
3$\sigma$ away from the median of the three quadrants with the lowest cutoff
count-rates and remove that quadrant from the further analysis.

The sensitivity of AstroSat-CZTI varies significantly over the entire sky, and
so does the conversion of detector counts to source flux. For every GW source,
we calculate direction-dependent flux limits using the AstroSat Mass Model
(Mate et al., 2021). The satellite response has been pre-calculated by the
CZTI team over a 2048-point “Level 4” Hierarchical Triangular Mesh (HTM) grid.
We downgrade the GW healpix skymap (Singer & Price, 2016) to NSIDE=16 to
roughly match this grid resolution, and assign nearest-neighbor sensitivity
for each point. We use a simple power-law spectrum $N(E)\sim E^{\Gamma}$, with
a $\Gamma=-1$, to convert the count-rates into a flux upper-limit for a given
grid point of the Mass Model. The final flux limits ($F_{\mathrm{UL}}$) are GW
probability-weighted averages over the sky visible to AstroSat at the time of
the GW event. We repeat this analysis for all three search timescales.

We estimate the isotropic equivalent luminosities ($L_{iso}$) in the
cosmological rest frame for the standard energy range of 1 keV – 10 MeV
through the following equation:

$L_{\mathrm{iso}}=4\pi\,D_{\mathrm{L}}^{2}\,F_{\mathrm{UL}}\,k$ (1)

where $k$ is the k-correction factor to correct for the source redshift as
well as the energy range of our detectors, defined as follows:

$k=\frac{\int_{1\,\mathrm{keV}/1+z}^{10\,\mathrm{MeV}/1+z}E\,\frac{\mathrm{d}N}{\mathrm{d}E}(E)\,\mathrm{d}E}{\int_{20\,\mathrm{keV}}^{200\,\mathrm{keV}}E\,\frac{\mathrm{d}N}{\mathrm{d}E}(E)\,\mathrm{d}E}$
(2)

where $F_{\mathrm{UL}}$ is the flux upper-limit calculated as described above,
$D_{\mathrm{L}}$ is the median luminosity distance, and z is the redshift for
the event.

##  3 Results

###  3.1 GW event sample

We used the GW transient sample within the period from the launch of AstroSat
(2015 October 6) till the end of the third observing run (O1-3) of LIGO-Virgo
GW detectors. This time window leads to the exclusion of GW150914, leaving us
with 89 events. All these events meet the thresholds of the event being
astrophysical as reported in Abbott et al. (2019, 2021a); The LIGO Scientific
Collaboration et al. (2021). For the analyses in Section 4, we used the
properties, skymaps, and posteriors of the GW events as reported in the
respective catalog paper data releases. Apart from these 90 events, more GW
events were found by other independent studies (Venumadhav et al., 2020; Olsen
et al., 2022), which are not included in our sample.

Out of these 89 GW events, AstroSat was passing through the South Atlantic
Anomaly (SAA) for 14 GW events, when CZTI detectors become non-functional.
Given the inclination and radius of orbit of AstroSat, we pass through the SAA
region for about 30% of it. We investigated whether the events for which
AstroSat was in SAA bring in any bias to the sample. We found that these
events are almost evenly distributed across the different mean luminosity
distances and hence are not expected to bring any distance bias to our sample.
Three GW events — GW151012, GW200128_022011, and GW200209_085452 occurred in a
data gap for AstroSat-CZTI and hence are excluded from our sample.

For each of the remaining 72 events, we estimated the instantaneous sky
coverage of the skymap, provided in the GWTC data releases, visible to
AstroSat given its position and attitude. We sum up the probabilities
associated with the pixels that are not occulted by Earth (71% of the entire
sky). These coverage statistics can be found in Tables 1 and 2. For six events
(GW190412, GW190413_134308, GW190512_180714, GW200129_065458, GW200202_154313,
GW200224_222234), AstroSat’s coverage was less than 10%. We do not include
these events in further analyses described in Section 4. For the special case
of GW170817 where the localization of the source was known with arcsecond
accuracy due to the discovery of the electromagnetic counterpart, the source
was occulted by Earth (Kasliwal et al., 2017) as seen from AstroSat. However,
we still include it in our analysis, using the final GW sky map.

For this work, we classify an event as a BNS if both components have masses
$m\leq 3~{}\mathrm{M}_{\odot}$, NSBH if only one component has mass $\leq
3~{}\mathrm{M}_{\odot}$, and BBH otherwise. GW literature often uses a “Mass
Gap” category for objects in the mass range from $3~{}\mathrm{M}_{\odot}$ to
$5~{}\mathrm{M}_{\odot}$, but we do not consider this as a separate category
for our analyses. This classification is on similar lines as in Fletcher et
al. (2023).

###  3.2 AstroSat-CZTI search results

Here, we summarise the searches for bursts in AstroSat-CZTI coincident with
the 72 GW events as described above. For each of the 72 events, we search for
a coincident GRB-like detection in CZTI data within the search window of 100 s
($\pm$ 50 s) around the merger time. We find no detections in the CZTI data
and hence we calculate flux upper limits as described in Section 2.

In Tables 1 and 2, we summarize the 72 GW event sample, their classification
as per the mass cuts described in the previous section, skymap coverage for
AstroSat, and the flux upper-limits from our searches. For completeness, these
tables also include those events when AstroSat was in SAA and the events that
do not cross our coverage threshold as described in Section 3.1. Table 1
contains the flux upper-limits from all events that contain at least one
neutron star component ($<~{}3~{}\mathrm{M}_{\odot}$) and Table 2 contains all
potential BBH merger events, with both component masses
$>~{}3~{}\mathrm{M}_{\odot}$. Figure 1 shows a histogram of all flux upper-
limits from Table 1 and 2 at the three different search timescales. We find a
median flux upper-limit of 2.87 $\times$ $10^{-6}$
$\mathrm{ergs~{}cm^{-2}~{}s^{-1}}$ for 0.1 s, 6.37 $\times$ $10^{-7}$
$\mathrm{ergs~{}cm^{-2}~{}s^{-1}}$ for 1.0 s and 1.48 $\times$ $10^{-7}$
$\mathrm{ergs~{}cm^{-2}~{}s^{-1}}$ for 10.0 s.

We report that the flux upper-limits calculated through these methods
(described in Section 2) improve the limits reported previously through GCNs,
e.g. Shenoy et al. (2020a, b), that were based on the old CZTI V2
pipeline333http://astrosat-ssc.iucaa.in/cztiData. For comparison, Figure 1
also shows histograms the flux limits obtained using this older data pipeline,
which almost an order of magnitude shallower. We also report that our non-
detections are consistent with other analyses (Hamburg et al., 2020; Fletcher
et al., 2023; Ursi et al., 2022; Cai et al., 2021). We highlight that our
different coverage statistics for several of these sources draw different
implications on the emission models as discussed in Section 4.

Figure 1: Histogram of the flux upper-limits calculated for all 65 events as
described in Section 2.2, for three search time bins: 0.1 s (blue), 1.0 s
(orange), and 10.0 s (green). We also show improvement in our estimation of
flux upper-limits through the use of CZTI V2 pipeline (dotted) to CZTI V3
pipeline (solid) as described in Section 2. The median flux upper-limit for
each time bins are also indicated with a dotted line (V2), and a solid line
(V3). As you go from lower time bins to higher, the number of samples in the
witness window increases, and hence the cutoff count-rate has to increase to
achieve the same FAP. As a result, the corresponding flux upper-limits are
found to reduce with increasing time bins.

##  4 Discussion

We discuss the implications of these non-detections on different emission
models for the two classes of events — events with a neutron star component
(Table 1) and events that are possible BBH mergers (Table 2). We further
present the luminosity constraints that can be derived from our non-detections
and discuss the prospects from the ongoing fourth observing run (O4) of LVK GW
detectors.

###  4.1 Events with a neutron star component

In our sample of seven GW events (refer Table 1) that have a potential neutron
star component, for three events (GW190917_114630, GW190814, and GW190425)
AstroSat was in SAA. For GW170817, the EM counterpart was occulted by Earth
for AstroSat — but we still include the flux upper-limits in Table 1, with
corresponding probability coverage being 0. For the remaining three GW events
(GW200210_092254, GW200115_042309, and GW191219_163120), we have reported flux
upper-limits. From Figure 2, we can see that the flux upper-limits from these
three events are roughly consistent with our median sensitivity of CZTI at the
three search time bins — 0.1 s, 1.0 s, and 10.0 s. The median sensitivities,
derived from our median flux upper-limits of the complete GW event sample as
described in Section 3.1, also indicate that we are sensitive to fluxes from
events like GW170817at 10 s search timescales. We attribute the non-detections
of these three events to the fact that they are significantly further away
than the measured distance of GW170817 at 40 Mpc (Abbott et al., 2017b), with
median distances of about 960 Mpc, 300 Mpc, and 550 Mpc respectively.

Due to the relatively low number of detected events with a neutron star
component, we look ahead toward future observing runs of LVK GW detectors.
Petrov et al. (2022) have shown that even though the sensitivities of the GW
detectors will increase, most of the events will be distant, and hence their
EM signatures will be faint. They predicted that the combined rate of BNS and
NSBH detections would be $\sim 159$ in the 18 months of O4. We caution that
this rate would be lower because of Virgo not joining in the first half of O4.
In Figure 6, we estimate the coverage statistics of AstroSat-CZTI through the
O4 injections set done by Kiendrebeogo et al. (2023), which contained 8258
total events (BBH, NSBH, and BNS). We offset the injection times of these
events to the years 2018-2020 and used AstroSat’s known orbital positions to
estimate whether AstroSat was in SAA during the event, and subsequently
calculate the coverage for the skymaps. We note that there are no significant
orbital changes in AstroSat, hence this convenient time shift is an acceptable
proxy for the orbital behavior of the satellite in future observing runs. We
observe that among 1132 events that have a neutron star component in this
injection set, no inferences could be made for 29.5% events because AstroSat
is in SAA, there were data gaps, or the events occulted by Earth for AstroSat
(coverage less than 10%). For the remaining 70.5% events (corresponding to
$\sim 113$ events in the duration of O4), we would have coverage higher than
10%. Note that the median distance of these “observable” events is 435 Mpc,
much higher than the GW170817 distance. In Figure 2, we show these events
scaled from the luminosity of GW170817 and we show that they lie just above
our 10 s median sensitivity. The histogram (log-scaled) of the median
distances of these events is also shown, which shows that the majority of the
events would lie much farther away than 100 Mpc. Future missions like Daksha
would be able to detect GW170817 like events out to a distance of 76 Mpc
(Bhalerao et al., 2022a) and hence will be critical in the hunt for EM
counterparts from GW. Daksha, with its all-sky coverage, wide energy band, and
an order of magnitude higher sensitivity than CZTI, is expected to detect 12
joint counterparts with LVK every year (Bhalerao et al., 2022b).

Figure 2: Comparison of limits on derived isotropic luminosities for the three
events in our GW event sample that contain a neutron star as discussed in
Section 4.1. All values are scaled to the 1 keV – 10 MeV band. The downward
pointing triangles mark the limits on source luminosity at the median distance
of these events as given in The LIGO Scientific Collaboration et al. (2021),
using AstroSat-CZTI flux upper-limits calculated in Section 2.2 for the three
time bins: 0.1 s (green), 1.0 s (red), and 10.0 s (purple). The shaded
inclined bands show the limits for the 1-$\sigma$ distance ranges for these
events. For comparison, the green, red, and purple lines show the median
sensitivity of CZTI as a function of distance for searches at these three
timescales. Note that due to variations in data quality, the limits for
individual events may be deeper or shallower than the median values. For
comparison, we also show the distance and luminosity of GW170817 reported by
Abbott et al. (2017a, b) though it was not detected by CZTI. The orange
histogram at the bottom shows the distribution of median distances to the
detected NS events from the Kiendrebeogo et al. (2023) simulations, which
gives us an idea of the relevant range of luminosities that CZTI can probe.

###  4.2 Binary Black Hole events

We have summarized the non-detections of 83 BBH events in Table 2. In this
section, we discuss the implications of these non-detections on the luminosity
function of these sources, and some models that predict the emission of EM
radiation from them.

####  4.2.1 Emission models

While BBH mergers are not expected to emit gamma-ray emission due to the lack
of matter in the accretion disk, Connaughton et al. (2016) reported a 2.9
sigma detection of a gamma-ray signal GRB150914-GBM by Fermi-GBM which was
temporally coincident with GW150914. Later, Veres et al. (2019) put forth
strong constraints on some emission models that provided mechanisms that
predict EM emission from BBHs. We refer to the same models for our analysis in
this work: a neutrino–antineutrino annihilation powered jet mechanism (Ruffert
& Janka, 1998), the Blandford–Znajek mechanism (Blandford & Znajek, 1977;
Komissarov & Barkov, 2010), and a charged black hole model. These models have
been critiqued by others (see for instance Li et al., 2016) but are still
widely discussed especially in the context of the association of GRB150914-GBM
with GW150914 (Connaughton et al., 2016; Fletcher et al., 2023; Veres et al.,
2019).

In this section, we explain the methodology to study the implications of our
AstroSat-CZTI flux upper-limits for the BBH non-detections reported in Table
2, which is broadly based upon Veres et al. (2019) and (Hamburg et al., 2020).
Each model used below uses several physical properties of the events. We use
one free parameter whose value is derived from GRB150914-GBM observed
properties, and the remaining parameters for the model are taken from the
physical parameters of the event posteriors. For all these posteriors of an
event, we estimate the expected flux for a given model and compare it with the
flux upper-limits calculated from our searches. We further consider three
cases of GRB jet geometries, through the inclination of the posteriors: an
isotropic emitter (no inclination constrain), a uniform emitter (opening angle
within the range of 10-40 $\deg$), and a fixed GRB-like jet (opening angle of
less than 20 $\deg$). In Figure 3, we show how our upper limits compare
against the calculated fluxes for all posteriors, grouped by the inclination
angles, for a single case of GW151226 as explained below. An important
assumption behind the implications stated below is that the source lies within
the skymap covered by AstroSat (for instance, the coverage for GW151226 is
95%). While this assumption needs to be kept in mind while interpreting
single-event analyses, in Section 4.2.2 we undertake a joint evaluation of all
non-detections including the coverage of individual events. For comparison
with our limits, we highlight the joint–marginal 1 s 5-$\sigma$ sensitivity of
Swift-BAT and Fermi-GBM (15-350 keV) for this event (Fletcher et al., 2023)
along with the 1 s 5-$\sigma$ sensitivity of the proposed Daksha mission
(Bhalerao et al., 2022b) in the same Figure 3.

In the neutrino-antineutrino annihilation powered jet model, relativistic jets
are launched by the electron-positron pairs formed because of collisions from
the neutrino winds from the thick accretion disk around the BBH merger system
(Ruffert & Janka, 1998; Zalamea & Beloborodov, 2011; Page & Thorne, 1974). For
this model, the accretion rate is the free parameter which is derived to match
the luminosity of the GRB150914-GBM detection. Here we note that there is a
typographical error in a term of Equation 4 of Veres et al. (2019), and we
have used the correct Equation 15 of Page & Thorne (1974). We find that for
GW151226 our 10 s flux upper-limits are deep enough to rule out more than 98%
of the posteriors of the event (top panel in Figure 3).

In the Blandford-Znajek mechanism model, we expect that the rotational energy
of the progenitor would be extracted with the help of a magnetic field of the
BH (Blandford & Znajek, 1977; Komissarov & Barkov, 2010; Reynolds et al.,
2006). Here, the free parameter is the value of the magnetic field, which is
scaled to match the GRB150914-GBM luminosity. We find that, unlike the other
two models, our flux upper-limits are not deep enough to rule out any
particular model, after considering the posterior distributions for each
event. Our flux upper-limits are an order of magnitude higher than the median
posterior distributions as shown in the middle panel of Figure 3 for the case
of GW151226. Even future higher-sensitivity missions like Daksha will be able
to explore these models only to a limited extent unless the BBH event occurs
nearby.

Some models predict that Poynting flux can be extracted to generate gamma-ray
emissions from BBHs when one of the black holes is charged (Zhang, 2016; Liu
et al., 2016). Here, we scale the charge of the BH to match the luminosity of
GRB150914-GBM. For GW151226, our flux upper-limits are deep enough to rule out
more than 99% of the posteriors of this event. We demonstrate an example of
the same in the bottom panel of Figure 3 for the case of GW151226.

We further studied the expected isotropic equivalent luminosities
($\mathrm{L_{iso}}$) (1 keV - 10 MeV) from the posteriors of each of these
events to probe the luminosity parameter space that can be ruled out for each
event. We show the same in Figure 4 for the case of GW151226, for the neutrino
model described above, where our flux upper-limits allow us to place
constraints on the luminosity space of $10^{49}$ to $2\times 10^{50}$
$\mathrm{ergs~{}s^{-1}}$, using our weighted 10 s sensitivities for that
event.

We note that GW151226, with a median distance of 440 Mpc, is among the
relatively nearby BBHs. The majority of the remaining BBHs in our GW sample
are much fainter than the current sensitivities of the gamma-ray detectors.
While our sensitivity is not high enough to place meaningful constraints on
individual event properties, in the next section we explore the implications
of a joint analyses of all our non-detections.

Figure 3: Expected fluxes for GW151226 predicted from the three models —
neutrino–antineutrino annihilation powered jet mechanism, the Blandford Znajek
mechanism, and a charged black hole model; that predict EM radiation from BBH
mergers, as discussed in Section 4. The vertical solid lines show our flux
upper-limits for this event for the three search time bins, the yellow dashed
vertical line shows the 1 s 5-$\sigma$ flux sensitivity for the 20–200 keV
band of Daksha (Bhalerao et al., 2022b), and the blue dashed vertical line
marks the 1 s 5-$\sigma$ sensitivity for the 15–350 keV band search with
Swift-BAT and Fermi-GBM (Fletcher et al., 2023). The three histograms within
each panel show the three jet models discussed in Section 4.2.1. For this
particular event, our flux upper limits are deep enough to rule out the
Neutrino Wind and Charged BH models. Figure 4: We show the isotropic
equivalent luminosity $\mathrm{L_{iso}}$ $\mathrm{ergs~{}s^{-1}}$ (1 keV - 10
MeV) upper-limits from AstroSat-CZTI searches as a function of calculated
luminosities derived from GRB150914-GBM for the Neutrino Model, for the case
of GW151226. As described in Section 4, we can rule out the luminosities below
the diagonal, mainly using our 10 s weighted sensitivity.

####  4.2.2 GRB150914-GBM Constraints

We now estimate the joint probability of AstroSat detecting zero out of 58 BBH
mergers. For each of these 58 events, we assume a certain source luminosity.
Then for each pixel in the Multi-Order-Coverage (MOC) skymap (Singer et al.,
2016a), we calculate the distance to which AstroSat-CZTI could have detected
this source using the flux upper-limit calculated as described in Section 2.2.
Next, we compute the cumulative probability of detection based on the
posterior distributions of each pixel given in the event skymap using the
conditional_cdf function from the ligo.skymap.distance module (Singer et al.,
2016b), and multiply this cumulative probability with the GW probability of
the pixel from the skymap. We sum these individual probabilities for all
pixels in the skymap that are not occulted by Earth for AstroSat to get the
combined event detection probability at an assumed luminosity of the source.
We repeat this for several luminosities from $10^{45}$ to $10^{55}$
$\mathrm{ergs~{}s^{-1}}$ as shown in Figure 5. We expect that the probability
of CZTI detecting an event to saturate to the coverage probability at higher
luminosities as the source would be bright enough for CZTI to detect it, and
observe the same in Figure 5. Note that out of 62 BBH events in our GW sample,
we exclude 4 events — GW190527_092055, GW190929_012149, GW200308_173609, and
GW200322_091133 — from this analysis since the location parameters of majority
pixels in their skymaps (based on the waveform IMRPhenomXPHM) are defined as
(inf). The probability in such pixels is extremely small to have a conditional
distance distribution represented by ansatz (Singer et al., 2016b).

Further, we combine all of these non-detection probabilities to get the
probability of detection of at least one of the 58 equal-luminosity BBHs, as a
function of luminosity (black–dashed line in Figure 5). We note that this
assumption of all BBHs having the same luminosity is simplistic, but it is the
most basic model-independent case that can be considered. We show that if all
58 BBH mergers were at the luminosity of $3.6\times 10^{49}$
$\mathrm{ergs~{}s^{-1}}$ or higher, we can say with a 99.99% confidence that
AstroSat-CZTI would have detected at least one counterpart among them. Through
this, we can also rule out the luminosity of GRB150914-GBM (Connaughton et
al., 2016) with a 97% confidence, if we were to assume that all 58 BBH events
had the same luminosity ($1.8\times 10^{49}$ $\mathrm{ergs~{}s^{-1}}$) as that
of the claimed counterpart of GW150914.

We further observe that the combined non-detection curve in Figure 5 is
dominated by a select few events that are relatively nearby, and
well–localised. In our sample, the three closest events are not the most
impactful for various reasons. For the closest event GW170608, with a median
distance of 320 Mpc, AstroSat’s coverage was relatively low (24%), and our
flux sensitivity at that time was only half the median value. For the event
GW191216_213338, which had a median distance of 340 Mpc, our flux upper-limits
were $\sim 4$ times worse than our median flux upper-limits. The third closest
event, GW200202_154313, was fully occulted by the Earth. As a result, the
combined non-detection curve closely follows the non-detection plot of
GW151226 which, at a distance of 440 Mpc — much closer than the median value
D${}_{\mathrm{L}}\sim 1730$ Mpc for BBH events in our GW sample. This is
likely to remain true in the future: the constraints from nearby events will
continue to dominate the statistics as compared to the rest of the sample
combined. We explore this further in Section 4.3.

Figure 5: Probability of detection of all 58 BBH events as a function of
isotropic equivalent luminosity. The black dashed curve shows the joint
detection probability across these 58 BBH events as described in Section 4.2.2
which shows the detection probability of at least one event out of 58 BBH
events. The vertical red line shows the isotropic luminosity of GRB150914-GBM
(Connaughton et al., 2016), which is the claimed counterpart to GW150914.
Through the non-detections of 58 BBHs, all assumed at the luminosity of
GRB150914-GBM, we can rule out this luminosity of the claimed counterpart with
a confidence of 97%.

###  4.3 Prospects for O4, and beyond

With higher expected BBH merger detection rates in the subsequent observing
runs of LVK GW detectors, we expect more non-detections given the improved
horizon distances of the GW detectors, and lack of comparable increase in the
sensitivities of EM detectors (Petrov et al., 2022; Kiendrebeogo et al.,
2023). The number of BBH detections is expected to increase to $\sim$ 159 in
O4 (Petrov et al., 2022; Abbott et al., 2020a). We use the same methodology as
described in Section 4.1 and use the 7126 O4 BBH injections by Kiendrebeogo et
al. (2023) to simulate AstroSat’s coverage statistics. We again offset the
injection times of these events to the years 2018-2020. As shown in Figure 6,
we have a coverage higher than 10% for about 74% or $\sim~{}118$ events
according to the projected rates given in Petrov et al. (2022). No inferences
could be made for the remaining 25% due to AstroSat being in SAA, coverage
being less than 10% coverage, or data gaps in CZTI.

We repeat a similar study as shown in Section 4.2.2 to estimate the
luminosities that can be ruled out through these non-detections in O4. For all
simulated events, we assume the sensitivity of CZTI as the 10.0 s median flux
upper-limit that was obtained in Section 3.2. We consider a random subset of
118 BBH non-detections from these 7126 O4 BBH injections and repeat our
calculations from this section to create a combined probability curve similar
to the dashed black line in Figure 5. To avoid any sampling bias, we repeat
the procedure a total of 500 times. All these realisations are shown as solid
blue lines in the top panel in Figure 7. The bottom panel of the same figure
shows the histogram of isotropic luminosities that we estimate that we can
rule out with confidence of 50% (brown dashed) and 90% (black dashed).

Similar to the O1-3 combined non-detection curve (Figure 5), the O1-4 combined
non-detection curve (Figure 7) is also heavily dominated by nearby well-
localised sources. We draw attention to two such BBH mergers – at a distance
of 106 Mpc and 113 Mpc. The individual non-detection curves for these are
shown by dashed orange and dashed green lines respectively. We find that
multiple realisations overlap with these lines. The explanation is similar to
what we observed with GW151226 in Section 4.2.2: whenever our 118-source
subset had any of these events present, their luminosity constraints were far
stronger than the rest of the sample, causing the combined curve to nearly
overlap with the individual source curve. In these cases, we can rule out a
BBH merger luminosity of $10^{48}$ $\mathrm{ergs~{}s^{-1}}$ with 90%
confidence.

Figure 6: AstroSat coverage statistics for the injection set of the fourth
observing run (O4) of LVK GW detectors (Kiendrebeogo et al., 2023) as
described in Section 4.3. The histogram (solid) shows the recovery of 7126
binary black holes, where we have 74.33% events are recovered with a coverage
higher than 10% of the skymap (black), while 7.35% events have a coverage
lower than that (green). The other histogram (dashed) shows the recovery of
1132 injections with at least one neutron star component, where we have 70.49%
recovered injections with at least 10% coverage of the skymap (black), while
5.21% had a coverage lower than that (green). The blue and orange markers show
the events in SAA and Data Gaps respectively for BBH and NS component
injections (blue and orange respectively). Figure 7: Top: Detection
probability curves of 500 realisations (blue) of 118 BBH non-detections from
the ongoing fourth observing run of LVK GW detectors, combined with the non-
detection data from previous runs of LVK GW detectors computed in Section
4.2.2. Each of the blue lines shows the joint detection probability of
AstroSat detecting at least one out of 118 BBH events predicted in O4, along
with the 58 BBH non-detections from previous runs, as described in Section
4.3. Bottom: Histogram of the isotropic luminosities we can rule out with 50%
(brown dashed) and 90% (black dashed) confidence given these 500 realisations.
We show that typically these realisations cluster around a $\mathrm{L_{iso}}$
of $10^{49}$ $\mathrm{ergs~{}s^{-1}}$, apart from cases where nearby BBH
events (at distances of 106, 113 Mpc) become a part of the realisation, and
hence completely dominate the joint-probability curve. In such cases, we would
be able to probe even fainter luminosities of $10^{49}$
$\mathrm{ergs~{}s^{-1}}$.

##  5 Summary

Over the three past observing runs, LVK GW detectors have detected 90 CBC GW
triggers, out of which 83 were potential BBHs, and the remaining 7 had a
neutron star component (either a BNS or an NSBH). We undertook a systematic
search for hard X-ray bursts in AstroSat-CZTI data coincident with these
triggers. For the events during which CZTI detectors were functioning, we
found no coincident detections. We calculated GW-skymap probability weighted
X-ray flux upper-limits in the operating range of CZTI (20-200 keV) for all
these events.

For the three mergers in our sample that potentially contained a neutron star
component, the luminosity distance of the mergers was too large to have
produced any detectable flux given the median sensitivity of AstroSat-CZTI.
For these three events, the partial Earth occultation of the coverage skymap
also meant that we could have missed the real event.

The majority of our GW sample comprised of BBHs, accounting for 58 events. We
use their non-detections to constrain the emission models. We saw that for a
relatively nearby BBH event GW151226, at a distance of 440 Mpc, we were able
to rule out the neutrino–antineutrino annihilation powered jet mechanism (95%)
and the emission due to a charged BH model (99%), assuming that the source was
not occulted by earth for AstroSat. We further estimated the detection
probability of 58 of these events that had good distance posteriors, and a
combined probability of detecting at least one out of these 58 BBHs. We show
that we can rule out an isotropic luminosity of $3.6\times 10^{49}$
$\mathrm{ergs~{}s^{-1}}$ with a confidence of 99.99%. Through the same
estimate, with a confidence of 97%, we can rule out the possibility that all
BBH mergers had the luminosity of GRB150914-GBM, the putative counterpart to
GW150914.

We expect a higher number of BBH mergers in the ongoing fourth observing run
(O4) of LVK GW detectors. By calculating the combined probability of non-
detections of all BBH events in O4, we estimate to rule out a luminosity of
$10^{49}$ $\mathrm{ergs~{}s^{-1}}$ with a 90% confidence. If LVK GW detectors
can detect a few relatively nearby BBH mergers (around 100 Mpc), then we would
be able to probe a luminosity as low as $10^{48}$ $\mathrm{ergs~{}s^{-1}}$ as
well. Thus, we highlight the importance of the detection of nearby BBH
mergers.

To improve our estimates of AstroSat-CZTI’s flux upper-limits, which currently
are quite conservative, we are currently developing a search pipeline that
would further improve AstroSat-CZTI’s flux upper-limit estimates. AstroSat-
CZTI continues to carry out automated searches for EM counterparts for GW in
the currently ongoing observing run O4 which are reported regularly in the GCN
circulars and on our webpage444http://astrosat.iucaa.in/czti/?q=emgw. Our
results from this observing run will be summarised after the end of O4.

To better constrain the BBH emission models and to probe a deeper isotropic
luminosity, we emphasise the need for a more sensitive X-ray all-sky monitor
suitable for GW counterpart searches, to make up for the increased sensitivity
of the GW detectors. Future missions like Daksha would be ideal to further
constrain the BBH emission models discussed in this work, potentially rule
them out consistently, and probe even fainter luminosities.

##  6 Software and third party data repository citations

This work utilised various software including Python (Van Rossum & Drake,
2009), AstroPy (Astropy Collaboration et al., 2022), NumPy (Harris et al.,
2020), and Matplotlib (Hunter, 2007). We used the public data release of the
three GWTCs for skymaps, posterior samples, and properties of all individual
events, hosted on Zenodo (European Organization For Nuclear Research &
OpenAIRE, 2013). This research has made use of NASA’s Astrophysics Data
System.

##  7 Acknowledgments

We thank Matthew Graham, Sanjit Mitra, A. R. Rao, and Gulab Dewangan for
fruitful discussions regarding the analyses. We also thank Vedant Shenoy for
his contributions to the initial setting up of the codes that were then
further developed by the current team. Lastly, we thank the CZTI Interface for
Fast Transients (CIFT) Team at IIT Bombay, and the Payloads Operations Center
(POC) team at IUCAA, Pune, for their valuable contributions to the regular
transient searches in the AstroSat-CZTI data. We also thank Vishwajeet Swain,
Utkarsh Pathak, Anirudh Salgundi, and Anuraag Arya for their valuable comments
on the paper text.

CZT–Imager is built by a consortium of Institutes across India. The Tata
Institute of Fundamental Research, Mumbai, led the effort with instrument
design and development. Vikram Sarabhai Space Centre (VSSC) at
Thiruvananthapuram provided the electronic design, assembly, and testing. ISRO
Satellite Centre (ISAC) at Bengaluru provided the mechanical design, quality
consultation, and project management. The Inter-University Centre for
Astronomy and Astrophysics (IUCAA) at Pune did the Coded Mask design,
instrument calibration, and the Payload Operation Centre. Space Application
Centre (SAC) at Ahmedabad provided the analysis software. Physical Research
Laboratory (PRL) Ahmedabad, provided the polarisation detection algorithm and
ground calibration. A vast number of industries participated in the
fabrication and the University sector pitched in by participating in the test
and evaluation of the payload.

The Indian Space Research Organisation (ISRO) funded, managed, and facilitated
the project.

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Table 1: AstroSat-CZTI results for GW events, with at least one neutron star component (M${}_{2}<3\mathrm{M}_{\odot}$), from the first three observing runs of the LIGO-Virgo GW detectors. The column labeled Dl, z, M1, and M2 correspond to the luminosity distance, redshift, mass of the more massive component of the binary and the mass of the less massive component in the binary, respectively. The Merger column corresponds to whether both the components of the binary are neutron stars (BNS) or just a neutron star (NSBH), based on the 3 $\mathrm{M}_{\odot}$ mass threshold. The columns — FUL0.1, FUL1.0, FUL10.0 are the flux upper-limits for the three search time bins 0.1 s, 1.0 s, and 10.0 s respectively. The column labeled SAA shows the status of AstroSat being in the South Atlantic Anomaly (SAA), and the last column labeled Coverage corresponds to the probability covered by AstroSat for each GW event.  Name | DL | z | M1 | M2 | Merger |  | FUL ($10^{-7}~{}\mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$) | SAA | Coverage
---|---|---|---|---|---|---|---|---|---
| (Mpc) |  | ($\mathrm{M}_{\odot}$) | ($\mathrm{M}_{\odot}$) |  |  | 0.1 s | 1.0 s | 10.0s |  |
GW170817 | 40${}^{+7}_{-15}$ | 0.01${}^{+0.00}_{-0.00}$ | 1.46 | 1.27 | BNS |  | 62.04 | 11.97 | 3.77 | No | 0.00
GW190425 | 150${}^{+80}_{-60}$ | 0.03${}^{+0.02}_{-0.01}$ | 2.1 | 1.3 | BNS |  |  $\cdots$ |  $\cdots$ |  $\cdots$ | Yes |  $\cdots$
GW190814 | 230${}^{+40}_{-50}$ | 0.05${}^{+0.01}_{-0.01}$ | 23.3 | 2.6 | NSBH |  |  $\cdots$ |  $\cdots$ |  $\cdots$ | Yes |  $\cdots$
GW190917_114630 | 720${}^{+300}_{-310}$ | 0.15${}^{+0.05}_{-0.06}$ | 9.7 | 2.1 | NSBH |  |  $\cdots$ |  $\cdots$ |  $\cdots$ | Yes |  $\cdots$
GW191219_163120 | 550${}^{+250}_{-160}$ | 0.11${}^{+0.05}_{-0.03}$ | 31.1 | 1.17 | NSBH |  | 14.87 | 2.74 | 0.61 | No | 0.60
GW200115_042309 | 290${}^{+150}_{-100}$ | 0.06${}^{+0.03}_{-0.02}$ | 5.9 | 1.44 | NSBH |  | 148.45 | 28.01 | 6.29 | No | 0.18
GW200210_092254 | 940${}^{+430}_{-340}$ | 0.19${}^{+0.08}_{-0.06}$ | 24.1 | 2.83 | NSBH |  | 40.68 | 10.61 | 2.33 | No | 0.38

Table 2: AstroSat-CZTI results for GW events, that are binary black holes (BBHs) (M${}_{2}>3\mathrm{M}_{\odot}$), from the first three observing runs of the LIGO-Virgo GW detectors. The column labeled Dl, z, M1, and M2 correspond to the luminosity distance, redshift, mass of the more massive component of the binary and the mass of the less massive component in the binary, respectively. The Merger column corresponds to whether both the components of the binary are neutron stars (BNS) or just a neutron star (NSBH), based on the 3 $\mathrm{M}_{\odot}$mass threshold. The columns — FL0.1, FL1.0, FL10.0 are the flux upper-limits for the three search time bins 0.1 s, 1.0 s, and 10.0 s respectively. The column labeled SAA shows the status of AstroSat being in the South Atlantic Anomaly (SAA), and the last column labeled Coverage corresponds to the probability covered by AstroSat for each GW event.  Name | DL | z | M1 | M2 | Merger |  | FUL ($10^{-7}~{}\mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$) | SAA | Coverage
---|---|---|---|---|---|---|---|---|---
| (Mpc) |  | ($\mathrm{M}_{\odot}$) | ($\mathrm{M}_{\odot}$) |  |  | 0.1 s | 1.0 s | 10.0s |  |
GW150914 | 440${}^{+150}_{-170}$ | 0.09${}^{+0.03}_{-0.03}$ | 35.6 | 30.6 | BBH |  |  $\cdots$ |  $\cdots$ |  $\cdots$ | No |  $\cdots$
GW151012 | 1080${}^{+550}_{-490}$ | 0.21${}^{+0.09}_{-0.09}$ | 23.2 | 13.6 | BBH |  |  $\cdots$ |  $\cdots$ |  $\cdots$ | No |  $\cdots$
GW151226 | 450${}^{+180}_{-190}$ | 0.09${}^{+0.04}_{-0.04}$ | 13.7 | 7.7 | BBH |  | 14.97 | 5.31 | 1.23 | No | 0.96
GW170104 | 990${}^{+440}_{-430}$ | 0.20${}^{+0.08}_{-0.08}$ | 30.8 | 20.0 | BBH |  | 19.07 | 3.54 | 0.79 | No | 1.00
GW170608 | 320${}^{+120}_{-110}$ | 0.07${}^{+0.02}_{-0.02}$ | 11.0 | 7.6 | BBH |  | 37.34 | 9.50 | 2.07 | No | 0.24
GW170729 | 2840${}^{+1400}_{-1360}$ | 0.49${}^{+0.19}_{-0.21}$ | 50.2 | 34.0 | BBH |  | 14.16 | 3.87 | 1.33 | No | 0.72
GW170809 | 1030${}^{+320}_{-390}$ | 0.20${}^{+0.05}_{-0.07}$ | 35.0 | 23.8 | BBH |  |  $\cdots$ |  $\cdots$ |  $\cdots$ | Yes |  $\cdots$
GW170814 | 600${}^{+150}_{-220}$ | 0.12${}^{+0.03}_{-0.04}$ | 30.6 | 25.2 | BBH |  | 20.83 | 5.32 | 1.16 | No | 0.21
GW170818 | 1060${}^{+420}_{-380}$ | 0.21${}^{+0.07}_{-0.07}$ | 35.4 | 26.7 | BBH |  | 15.53 | 3.86 | 0.83 | No | 0.95
GW170823 | 1940${}^{+970}_{-900}$ | 0.35${}^{+0.15}_{-0.15}$ | 39.5 | 29.0 | BBH |  | 16.96 | 3.33 | 0.73 | No | 0.50
GW190403_051519 | 8280${}^{+6720}_{-4290}$ | 1.18${}^{+0.73}_{-0.53}$ | 85.0 | 20.0 | BBH |  | 14.62 | 6.60 | 0.95 | No | 0.81
GW190408_181802 | 1540${}^{+440}_{-620}$ | 0.29${}^{+0.07}_{-0.11}$ | 24.8 | 18.5 | BBH |  | 136.08 | 34.89 | 7.71 | No | 1.00
GW190412 | 720${}^{+240}_{-220}$ | 0.15${}^{+0.04}_{-0.04}$ | 27.7 | 9.0 | BBH |  | 13.66 | 2.58 | 0.57 | No | 0.01
GW190413_052954 | 3320${}^{+1910}_{-1400}$ | 0.56${}^{+0.25}_{-0.21}$ | 33.7 | 24.2 | BBH |  | 39.57 | 10.41 | 2.34 | No | 0.58
GW190413_134308 | 3800${}^{+2480}_{-1830}$ | 0.62${}^{+0.32}_{-0.26}$ | 51.3 | 30.4 | BBH |  | 45.57 | 11.91 | 2.68 | No | 0.09
GW190421_213856 | 2590${}^{+1490}_{-1240}$ | 0.45${}^{+0.21}_{-0.19}$ | 42.0 | 32.0 | BBH |  | 14.52 | 3.70 | 0.79 | No | 0.44
GW190426_190642 | 4580${}^{+3400}_{-2280}$ | 0.73${}^{+0.41}_{-0.32}$ | 105.5 | 76.0 | BBH |  |  $\cdots$ |  $\cdots$ |  $\cdots$ | Yes |  $\cdots$
GW190503_185404 | 1520${}^{+630}_{-600}$ | 0.29${}^{+0.10}_{-0.10}$ | 41.3 | 28.3 | BBH |  | 19.82 | 3.76 | 0.83 | No | 0.17
GW190512_180714 | 1460${}^{+510}_{-590}$ | 0.28${}^{+0.08}_{-0.10}$ | 23.2 | 12.5 | BBH |  | 19.79 | 3.73 | 0.82 | No | 0.01
GW190513_205428 | 2210${}^{+990}_{-810}$ | 0.40${}^{+0.14}_{-0.13}$ | 36.0 | 18.3 | BBH |  | 14.57 | 3.66 | 0.61 | No | 0.16
GW190514_065416 | 3890${}^{+2610}_{-2070}$ | 0.64${}^{+0.33}_{-0.30}$ | 40.9 | 28.4 | BBH |  | 15.84 | 5.46 | 0.86 | No | 0.81
GW190517_055101 | 1790${}^{+1750}_{-880}$ | 0.33${}^{+0.26}_{-0.15}$ | 39.2 | 24.0 | BBH |  | 12.77 | 2.43 | 0.54 | No | 0.99
GW190519_153544 | 2600${}^{+1720}_{-960}$ | 0.45${}^{+0.24}_{-0.15}$ | 65.1 | 40.8 | BBH |  | 12.50 | 3.28 | 0.54 | No | 0.61
GW190521 | 3310${}^{+2790}_{-1800}$ | 0.56${}^{+0.36}_{-0.27}$ | 98.4 | 57.2 | BBH |  | 84.64 | 21.15 | 3.49 | No | 0.73
GW190521_074359 | 1080${}^{+580}_{-530}$ | 0.21${}^{+0.10}_{-0.10}$ | 43.4 | 33.4 | BBH |  |  $\cdots$ |  $\cdots$ |  $\cdots$ | Yes |  $\cdots$
GW190527_092055 | 2520${}^{+2080}_{-1230}$ | 0.44${}^{+0.29}_{-0.19}$ | 35.6 | 22.2 | BBH |  | 12.02 | 3.76 | 1.09 | No | 0.65
GW190602_175927 | 2840${}^{+1930}_{-1280}$ | 0.49${}^{+0.26}_{-0.20}$ | 71.8 | 44.8 | BBH |  | 23.93 | 6.14 | 1.37 | No | 0.36
GW190620_030421 | 2910${}^{+1710}_{-1320}$ | 0.50${}^{+0.23}_{-0.20}$ | 58.0 | 35.0 | BBH |  | 19.04 | 7.97 | 2.17 | No | 0.88
GW190630_185205 | 870${}^{+530}_{-360}$ | 0.18${}^{+0.09}_{-0.07}$ | 35.1 | 24.0 | BBH |  | 16.54 | 4.63 | 1.00 | No | 0.77
GW190701_203306 | 2090${}^{+770}_{-740}$ | 0.38${}^{+0.11}_{-0.12}$ | 54.1 | 40.5 | BBH |  | 30.02 | 5.68 | 1.71 | No | 1.00
GW190706_222641 | 3630${}^{+2600}_{-2000}$ | 0.60${}^{+0.33}_{-0.29}$ | 74.0 | 39.4 | BBH |  | 45.90 | 11.85 | 2.57 | No | 0.13
GW190707_093326 | 850${}^{+340}_{-400}$ | 0.17${}^{+0.06}_{-0.08}$ | 12.1 | 7.9 | BBH |  | 28.30 | 5.46 | 1.23 | No | 0.82
GW190708_232457 | 930${}^{+310}_{-390}$ | 0.19${}^{+0.06}_{-0.07}$ | 19.8 | 11.6 | BBH |  | 32.83 | 8.49 | 1.87 | No | 0.45
GW190719_215514 | 3730${}^{+3120}_{-2070}$ | 0.61${}^{+0.39}_{-0.30}$ | 36.6 | 19.9 | BBH |  | 142.12 | 27.03 | 6.24 | No | 0.81
GW190720_000836 | 770${}^{+650}_{-260}$ | 0.16${}^{+0.11}_{-0.05}$ | 14.2 | 7.5 | BBH |  | 13.50 | 3.43 | 0.79 | No | 0.94
GW190725_174728 | 1030${}^{+520}_{-430}$ | 0.20${}^{+0.09}_{-0.08}$ | 11.8 | 6.3 | BBH |  | 14.47 | 3.78 | 0.86 | No | 0.91
GW190727_060333 | 3070${}^{+1300}_{-1230}$ | 0.52${}^{+0.18}_{-0.18}$ | 38.9 | 30.2 | BBH |  | 83.74 | 21.02 | 4.68 | No | 0.92
GW190728_064510 | 880${}^{+260}_{-380}$ | 0.18${}^{+0.05}_{-0.07}$ | 12.5 | 8.0 | BBH |  | 48.74 | 12.26 | 2.76 | No | 0.25
GW190731_140936 | 3330${}^{+2350}_{-1770}$ | 0.56${}^{+0.31}_{-0.26}$ | 41.8 | 29.0 | BBH |  |  $\cdots$ |  $\cdots$ |  $\cdots$ | Yes |  $\cdots$
GW190803_022701 | 3190${}^{+1630}_{-1470}$ | 0.54${}^{+0.22}_{-0.22}$ | 37.7 | 27.6 | BBH |  | 69.74 | 18.13 | 3.96 | No | 1.00
GW190805_211137 | 6130${}^{+3720}_{-3080}$ | 0.92${}^{+0.43}_{-0.40}$ | 46.2 | 30.6 | BBH |  | 34.72 | 8.79 | 1.96 | No | 0.20
GW190828_063405 | 2070${}^{+650}_{-920}$ | 0.38${}^{+0.10}_{-0.15}$ | 31.9 | 25.8 | BBH |  | 87.81 | 16.82 | 3.82 | No | 0.73
GW190828_065509 | 1540${}^{+690}_{-650}$ | 0.29${}^{+0.11}_{-0.11}$ | 23.7 | 10.4 | BBH |  | 58.75 | 11.25 | 2.56 | No | 0.35
GW190910_112807 | 1520${}^{+1090}_{-630}$ | 0.29${}^{+0.17}_{-0.11}$ | 43.8 | 34.2 | BBH |  | 26.98 | 6.95 | 1.53 | No | 0.64
GW190915_235702 | 1750${}^{+710}_{-650}$ | 0.32${}^{+0.11}_{-0.11}$ | 32.6 | 24.5 | BBH |  | 31.97 | 6.07 | 1.31 | No | 1.00
GW190916_200658 | 4940${}^{+3710}_{-2380}$ | 0.77${}^{+0.45}_{-0.32}$ | 43.8 | 23.3 | BBH |  |  $\cdots$ |  $\cdots$ |  $\cdots$ | Yes |  $\cdots$
GW190924_021846 | 550${}^{+220}_{-220}$ | 0.11${}^{+0.04}_{-0.04}$ | 8.8 | 5.1 | BBH |  |  $\cdots$ |  $\cdots$ |  $\cdots$ | Yes |  $\cdots$
GW190925_232845 | 930${}^{+460}_{-350}$ | 0.19${}^{+0.08}_{-0.07}$ | 20.8 | 15.5 | BBH |  |  $\cdots$ |  $\cdots$ |  $\cdots$ | Yes |  $\cdots$
GW190926_050336 | 3280${}^{+3400}_{-1730}$ | 0.55${}^{+0.44}_{-0.26}$ | 41.1 | 20.4 | BBH |  | 15.96 | 3.10 | 0.68 | No | 0.75
GW190929_012149 | 3130${}^{+2510}_{-1370}$ | 0.53${}^{+0.33}_{-0.20}$ | 66.3 | 26.8 | BBH |  | 32.58 | 6.06 | 1.36 | No | 0.55
GW190930_133541 | 770${}^{+320}_{-320}$ | 0.16${}^{+0.06}_{-0.06}$ | 14.2 | 6.9 | BBH |  | 12.34 | 3.26 | 0.51 | No | 0.95
GW191103_012549 | 990${}^{+500}_{-470}$ | 0.20${}^{+0.09}_{-0.09}$ | 11.8 | 7.9 | BBH |  | 32.74 | 8.34 | 1.81 | No | 0.75
GW191105_143521 | 1150${}^{+430}_{-480}$ | 0.23${}^{+0.07}_{-0.09}$ | 10.7 | 7.7 | BBH |  | 13.62 | 3.47 | 0.76 | No | 0.25
GW191109_010717 | 1290${}^{+1130}_{-650}$ | 0.25${}^{+0.18}_{-0.12}$ | 65.0 | 47.0 | BBH |  | 38.43 | 10.05 | 2.19 | No | 0.62
GW191113_071753 | 1370${}^{+1150}_{-620}$ | 0.26${}^{+0.18}_{-0.11}$ | 29.0 | 5.9 | BBH |  | 29.13 | 5.55 | 1.60 | No | 0.42
GW191126_115259 | 1620${}^{+740}_{-740}$ | 0.30${}^{+0.12}_{-0.13}$ | 12.1 | 8.3 | BBH |  | 15.14 | 3.77 | 0.84 | No | 0.79
GW191127_050227 | 3400${}^{+3100}_{-1900}$ | 0.57${}^{+0.40}_{-0.29}$ | 53.0 | 24.0 | BBH |  | 40.41 | 7.52 | 1.66 | No | 0.99
GW191129_134029 | 790${}^{+260}_{-330}$ | 0.16${}^{+0.05}_{-0.06}$ | 10.7 | 6.7 | BBH |  |  $\cdots$ |  $\cdots$ |  $\cdots$ | Yes |  $\cdots$
GW191204_110529 | 1800${}^{+1700}_{-1100}$ | 0.34${}^{+0.25}_{-0.18}$ | 27.3 | 19.3 | BBH |  | 70.74 | 12.82 | 2.83 | No | 0.77
GW191204_171526 | 650${}^{+190}_{-250}$ | 0.13${}^{+0.04}_{-0.05}$ | 11.9 | 8.2 | BBH |  |  $\cdots$ |  $\cdots$ |  $\cdots$ | Yes |  $\cdots$
GW191215_223052 | 1930${}^{+890}_{-860}$ | 0.35${}^{+0.13}_{-0.14}$ | 24.9 | 18.1 | BBH |  | 13.50 | 3.43 | 0.74 | No | 0.85
GW191216_213338 | 340${}^{+120}_{-130}$ | 0.07${}^{+0.02}_{-0.03}$ | 12.1 | 7.7 | BBH |  | 89.91 | 23.00 | 4.99 | No | 1.00
GW191222_033537 | 3000${}^{+1700}_{-1700}$ | 0.51${}^{+0.23}_{-0.26}$ | 45.1 | 34.7 | BBH |  | 16.87 | 4.22 | 0.92 | No | 0.89
GW191230_180458 | 4300${}^{+2100}_{-1900}$ | 0.69${}^{+0.26}_{-0.27}$ | 49.4 | 37.0 | BBH |  | 27.34 | 7.03 | 1.49 | No | 0.17
GW200112_155838 | 1250${}^{+430}_{-460}$ | 0.24${}^{+0.07}_{-0.08}$ | 35.6 | 28.3 | BBH |  | 54.39 | 10.51 | 2.22 | No | 0.82
GW200128_022011 | 3400${}^{+2100}_{-1800}$ | 0.56${}^{+0.28}_{-0.28}$ | 42.2 | 32.6 | BBH |  | 19.77 | 3.67 | 0.85 | No | 0.64
GW200129_065458 | 900${}^{+290}_{-380}$ | 0.18${}^{+0.05}_{-0.07}$ | 34.5 | 28.9 | BBH |  | 29.94 | 7.55 | 1.67 | No | 0.00
GW200202_154313 | 410${}^{+150}_{-160}$ | 0.09${}^{+0.03}_{-0.03}$ | 10.1 | 7.3 | BBH |  | 63.97 | 17.05 | 3.65 | No | 0.00
GW200208_130117 | 2230${}^{+1000}_{-850}$ | 0.40${}^{+0.15}_{-0.14}$ | 37.8 | 27.4 | BBH |  |  $\cdots$ |  $\cdots$ |  $\cdots$ | Yes |  $\cdots$
GW200208_222617 | 4100${}^{+4400}_{-1900}$ | 0.66${}^{+0.54}_{-0.28}$ | 51.0 | 12.3 | BBH |  | 27.76 | 5.63 | 1.20 | No | 0.30
GW200209_085452 | 3400${}^{+1900}_{-1800}$ | 0.57${}^{+0.25}_{-0.26}$ | 35.6 | 27.1 | BBH |  |  $\cdots$ |  $\cdots$ |  $\cdots$ | No |  $\cdots$
GW200216_220804 | 3800${}^{+3000}_{-2000}$ | 0.63${}^{+0.37}_{-0.29}$ | 51.0 | 30.0 | BBH |  | 34.58 | 9.41 | 1.46 | No | 1.00
GW200219_094415 | 3400${}^{+1700}_{-1500}$ | 0.57${}^{+0.22}_{-0.22}$ | 37.5 | 27.9 | BBH |  | 16.62 | 3.27 | 0.99 | No | 0.35
GW200220_061928 | 6000${}^{+4800}_{-3100}$ | 0.90${}^{+0.55}_{-0.40}$ | 87.0 | 61.0 | BBH |  | 18.08 | 4.97 | 1.04 | No | 0.34
GW200220_124850 | 4000${}^{+2800}_{-2200}$ | 0.66${}^{+0.36}_{-0.31}$ | 38.9 | 27.9 | BBH |  | 54.52 | 14.94 | 3.14 | No | 0.59
GW200224_222234 | 1710${}^{+490}_{-640}$ | 0.32${}^{+0.08}_{-0.11}$ | 40.0 | 32.5 | BBH |  | 17.68 | 3.34 | 0.72 | No | 0.00
GW200225_060421 | 1150${}^{+510}_{-530}$ | 0.22${}^{+0.09}_{-0.10}$ | 19.3 | 14.0 | BBH |  | 42.91 | 8.18 | 1.80 | No | 0.28
GW200302_015811 | 1480${}^{+1020}_{-700}$ | 0.28${}^{+0.16}_{-0.12}$ | 37.8 | 20.0 | BBH |  | 35.62 | 9.17 | 2.08 | No | 0.78
GW200306_093714 | 2100${}^{+1700}_{-1100}$ | 0.38${}^{+0.24}_{-0.18}$ | 28.3 | 14.8 | BBH |  | 31.47 | 7.96 | 1.70 | No | 0.37
GW200308_173609 | 5400${}^{+2700}_{-2600}$ | 0.83${}^{+0.32}_{-0.35}$ | 36.4 | 13.8 | BBH |  | 21.93 | 5.57 | 0.91 | No | 0.69
GW200311_115853 | 1170${}^{+280}_{-400}$ | 0.23${}^{+0.05}_{-0.07}$ | 34.2 | 27.7 | BBH |  | 60.51 | 14.76 | 3.28 | No | 1.00
GW200316_215756 | 1120${}^{+470}_{-440}$ | 0.22${}^{+0.08}_{-0.08}$ | 13.1 | 7.8 | BBH |  |  $\cdots$ |  $\cdots$ |  $\cdots$ | Yes |  $\cdots$
GW200322_091133 | 3600${}^{+7000}_{-2000}$ | 0.60${}^{+0.84}_{-0.30}$ | 34.0 | 14.0 | BBH |  | 47.76 | 12.39 | 2.88 | No | 0.73


# On $m$-bi-interior ideals of an ordered semigroup

Susmita Mallick
Department of Mathematics, Visva-Bharati University,
Santiniketan, Bolpur - 731235, West Bengal, India
<EMAIL_ADDRESS>
###### Abstract

In this article, we concurrently explore the class of $m$-bi-ideals as well as
the class of $m$-interior ideals in an ordered semigroup. We call these class
of ideals as $m$-bi-interior ideals. Here, $m$-simple, $m$-regular ordered
semigroups and their various subclasses get characterized in diverse manners
as per their $m$-bi-interior ideals.

Keywords and Phrases: $m$-interior ideal, $m$-bi-ideal, $m$-bi-interior ideal,
$m$-bi-interior-simple, $m$-regular.
2010 Mathematics Subject Classification: 20M10;06F05.

##  1 Introduction and preliminaries

A semigroup $(S,\cdot)$ with a partial order that is compatible with the
semigroup operation such that for all $a,b\ \textrm{and}\;x\in S$, $a\leq b$
implies $xa\leq xb$ and $ax\leq bx$, is called an ordered semigroup and
denoted by $(S,\cdot,\leq)$. For an ordered semigroup $S$ and $H\subseteq S$,
denote $(H]:=\\{t\in H:t\leq h,\textrm{for some}\;h\in H\\}$. The concept of
bi-ideals of semigroups were first presented by R. A. Good and D. R. Hughes[1]
while S. Lajos[5] and G. Szasz [9],[10] have since developed the idea of
interior ideals in semigroup theory. The bi-interior ideal in semigroups was a
term introduced by M. M. K. Rao[6] to generalize these two types of ideals. M.
Munir[7] established the idea of $m$-left and $m$-right ideals in semirings,
later this concepts are proposed for the semigroups by A. Shafiq and M.
Munir[8]. In [11], we have studied $m$-ideals and $m$-regularity in an ordered
semigroup.

In this paper we introduce the notion of $m$-bi-interior ideals in an ordered
semigroup. Our approach allows one to see a detailed exposition of $m$-bi-
interior ideals in $m$-simple, $m$-regular ordered semigroup and its different
subclasses. we have explored some inter-relations between $m$-bi-interior
ideal and other $m$-ideals.

######  Definition 1.1.

[6] A non-empty subset $B$ of a semigroup $S$ is said to be bi-interior ideal
of $S$ if $BSB\cap SBS\subseteq B$.

######  Definition 1.2.

[11] A subsemigroup $L$ of $S$ is called $m$-left($m$-right) ideal of $S$ if
$S^{m}L\subseteq L$ ($RS^{m}\subseteq R$) and $(L]=L$($(R]=R$), for any
positive integer $m$ not necessarily $1$.

A subsemigroup $I$ of $S$ is called an $m$-two sided ideal or simply an
$m$-ideal of $S$ if it is both $m$-left ideal and $m$-right ideal of $S$.

######  Definition 1.3.

[11] Let $m$ be non-negative integer. An ordered semigroup $S$ is said to be
$m$-left-simple($m$-right-simple) if it does not contain any proper non
trivial $m$-left($m$-right) ideal. An ordered semigroup $S$ is called
$m$-simple if it is both $m$-left and $m$-right simple.

######  Definition 1.4.

[11] Let $S$ be an ordered semigroup and $B$ be subsemigroup of $S$, then $B$
is called $m$-bi-ideal of $S$ if $BS^{m}B\subseteq B$ and $(B]=B$, where
$m\geq 1$ is a positive integer. $m$ is called bipotency of the bi-ideal $B$.

######  Definition 1.5.

A subsemigroup $I$ of an ordered semigroup $S$ is called $m$-interior ideal of
$S$ if $S^{m}IS^{m}\subseteq I$ and $(I]=I$, where $m\geq 1$ is a positive
integer.

######  Definition 1.6.

[11] A subsemigroup $Q$ of an ordered semigroup $S$ is called $m$-quasi-ideal
of $S$ if $(S^{m}Q]\cap(QS^{m}]\subseteq Q$ and $(Q]=Q$, where $m\geq 1$,
called quasipotency of quasi-ideal $Q$.

######  Definition 1.7.

[11] An element $a$ of an ordered semigroup $S$ is called $m$-regular if
$a\leq axa$, for some $x\in S^{m}$. An ordered semigroup $S$ is $m$-regular if
every element of $S$ is $m$-regular that is $m$-regular if $a\in(aS^{m}a]$,
for all $a\in S$.

######  Lemma 1.8.

[4] Let $S$ be an ordered semigroup and $A$ and $B$ be subsets of $S$. Then
the following statements hold:

  1. 1.

$A\subseteq(A]$.

  2. 2.

$((A]]=(A]$.

  3. 3.

If $A\subseteq B$ then $(A]\subseteq(B]$.

  4. 4.

$(A\cap B]\subseteq(A]\cap(B]$.

  5. 5.

$(A\cup B]=(A]\cup(B]$.

  6. 6.

$(A](B]\subseteq(AB]$.

  7. 7.

$((A](B]]=(AB]$.

######  Lemma 1.9.

[11] Let $S$ be an ordered semigroup. $S$ is $m$-regular if and only if for
every $m$-right ideal $R$ and $m$-left ideal $L$ of $S$, $(RL]=R\cap L$.

######  Theorem 1.10.

[11] If $S$ be an $m$-regular ordered semigroup then a non-empty subset $B$ of
$S$ is a $m$-bi-ideal of $S$ if and only if $B=(RL]$ for some $m$-left ideal
$L$ and $m$-right ideal $R$ of $S$.

##  2  $m$-Bi-interior ideals of Ordered Semigroups

In this section, we introduce the notion of $m$-bi-interior ideal as a
generalization of $m$-bi-ideal and $m$-interior ideal of an ordered semigroup
and study the properties of $m$-bi-interior ideals of ordered semigroups,
$m$-simple ordered semigroups and $m$-regular ordered semigroups.

######  Definition 2.1.

A non-empty subset $B$ of an ordered semigroup $S$ is said to be $m$-bi-
interior ideal of $S$ if $(BS^{m}B]\cap(S^{m}BS^{m}]\subseteq B$ and $(B]=B$.

######  Definition 2.2.

An ordered semigroup $S$ is said to be $m$-bi-interior-simple ordered
semigroup if $S$ has no $m$-bi-interior ideals other than $S$ itself.

In the following theorem, we mention some important properties.

######  Proposition 2.3.

Let $S$ be an ordered semigroup. Then the following statements hold:

  1. 1.

Every $m$-left($m$-right) ideal is an $m$-bi-interior ideal of $S$.

  2. 2.

Every $m$-ideal is an $m$-bi-interior ideal.

  3. 3.

If $A$ and $B$ are $m$-bi-interior ideals of $S$ then $A\cap B$(provided the
intersection is non-empty) is an $m$-bi-iterior ideal of $S$.

  4. 4.

Intersection of an $m$-right ideal and an $m$-left ideal of $S$ is an $m$-bi-
interior ideal of $S$.

  5. 5.

Every $m$-quasi-ideal is an $m$-bi-interior ideal of $S$.

  6. 6.

Every $m$-bi-ideal of $S$ is an $m$-bi-interior ideal of $S$.

  7. 7.

Every $m$-interior ideal of ordered semigroup $S$ is an $m$-bi-interior ideal
of $S$

  8. 8.

If $B$ is an $m$-bi-interior ideal of $S$, then $(BS]$ and $(SB]$ are $m$-bi-
interior ideals of $S$.

  9. 9.

If $T$ be an $m$-right ideal of $S$ and $B$ be an $m$-bi-interior ideal of $S$
then $B\cap T$ is an $m$-bi-interior ideal of $S$.

###### Proof.

$(1)$: Suppose $L$ is an $m$-left ideal of $S$ implies $S^{m}L\subseteq L$ and
$(L]=L$. Now
$(LS^{m}L]\cap(S^{m}LS^{m}]\subseteq(LSL^{m}]\subseteq(SS^{m}L]\subseteq(S^{m+1}L]\subseteq(S^{m}L]\subseteq
L$. Hence $L$ is an $m$-bi-interior ideal of $S$.

Similarly it can be proved that every $m$-right ideal is an $m$-bi-interior
ideal of $S$.

$(2)$: Since every $m$-ideal is both $m$-left and $m$-right ideal of an
ordered semigroup $S$ then from previous property, it follows that every
$m$-ideal of $S$ is an $m$-bi-interior ideal of $S$.

$(3)$: Let $A$ and $B$ be $m_{1}$-bi-interior ideal and $m_{2}$-bi-interior
ideal respectively of an ordered semigroup $S$. Then
$(AS^{m_{1}}A]\cap(S^{m_{1}}AS^{m_{1}}]\subseteq A$ and $(A]=A$,
$(BS^{m_{2}}B]\cap(S^{m_{2}}BS^{m_{2}}]\subseteq B$ and $(B]=B$.Let
$m=max\\{m_{1},m_{2}\\}$ Now $(S^{m}(A\cap
B)S^{m}]\subseteq(S^{m}AS^{m}]\cap(S^{m}BS^{m}]\subseteq(S^{m_{1}}AS^{m_{1}}]\cap(S^{m_{2}}BS^{m_{2}}]$.
Also $((A\cap B)S^{m}(A\cap
B)]\subseteq(AS^{m}A]\cap(BS^{m}B]\subseteq(AS^{m_{1}}A]\cap(BS^{m_{2}}B]$.
Hence $(S^{m}(A\cap B)S^{m}]\cap((A\cap B)S^{m}(A\cap
B)]\subseteq(S^{m_{1}}AS^{m_{1}}]\cap(S^{m_{2}}BS^{m_{2}}]\cap(AS^{m_{1}}A]\cap(BS^{m_{2}}B]\subset
A\cap B$. Now $(A\cap B]\subseteq(A]\cap(B]=A\cap B$ implies $(A\cap B]=A\cap
B$. Hence $A\cap B$ is an $m$-bi-interior ideal of $S$.

$(4)$: It follows from above result.

$(5)$: Suppose that $Q$ is an $m$-quasi-ideal of an ordered semigroup $S$,
that is $(QS^{m}]\cap(S^{m}Q]\subseteq Q$ and $(Q]=Q$. Now
$(QS^{m}Q]\subseteq(QS^{m}]\cap(S^{m}Q]\subseteq Q$ implies
$(QS^{m}Q]\cap(S^{m}QS^{m}]\subseteq(QS^{m}Q]\subseteq Q$. Hence $Q$ is an
$m$-bi-interior ideal of $S$.

$(6)$: Suppose that $B$ is an $m$-bi-ideal of an ordered semigroup $S$ that is
$BS^{m}B\subseteq B$ and $(B]=B$. Now
$(BS^{m}B]\cap(S^{m}BS^{m}]\subseteq(BS^{m}B]\subseteq(B]=B$. Hence $B$ is an
$m$-bi-interior ideal of $S$.

$7$:

Let $I$ is an $m$-interior ideal of $S$. Now
$(IS^{m}I]\cap(S^{m}IS^{m}]\subseteq(S^{m}IS^{m}]\subseteq(I]=I$. Hence $I$ is
an $m$-bi-interior ideal of $S$.

$(8)$: Let $B$ be an $m$-bi-interior ideal of an ordered semigroup $S$. Now
$((BS]S^{m}(BS]]\cap(S^{m}(BS]S^{m}]\subseteq(BSS^{m}SS]\cap(S^{m}BS^{m+1}]\subseteq(BS^{m+3}]\subseteq(BS]$.
Hence $(BS]$ is an $m$-bi-interior ideal of $S$.

Similarly it can be shown that $(SB]$ is an $m$-bi-interior ideal of $S$.

$(9)$: Consider $B$ is an $m$-bi-interior ideal of an ordered semigroup $S$
and $T$ is an $m$-right ideal of $S$. Now $((B\cap T)S^{m}(B\cap
T)]\cap(S^{m}(B\cap
T)S^{m}]\subseteq(BS^{m}B]\cap(TS^{m}T]\cap(S^{m}BS^{m}]\cap(S^{m}TS^{m}]\subseteq(BS^{m}B]\cap(S^{m}BS^{m}]\cap(TT]\subseteq
B\cap(T]\subseteq B\cap T$. Again $(B\cap T]\subseteq(B]\cap(T]=B\cap T$.
Hence $B\cap T$ is an $m$-bi-interior ideal of $S$. ∎

But in general, every $m$-bi-interior ideal is not $m$-bi-ideal. In the next
theorem we state the condition for which $m$-bi-ideals and $m$-bi-interior
ideals coincide.

######  Theorem 2.4.

Let $S$ be $m$-simple ordered semigroup. Every $m$-bi-interior ideal is an
$m$-bi-ideal of $S$.

###### Proof.

Let $S$ be an $m$-simple ordered semigroup and $B$ be an $m$-bi-interior ideal
of $S$ then $(BS^{m}B]\cap(S^{m}BS^{m}]\subseteq B$ and $(B]=B$. Now it can be
easily check that $(S^{m}BS^{m}]$ is an $m$-ideal of $S$. Now $m$-simplicity
of $S$ gives $(S^{m}BS^{m}]=S$. Hence
$(BS^{m}B]\cap(S^{m}BS^{m}]\subseteq(BS^{m}B]\cap S\subseteq B$ which implies
$(BS^{m}B]\subseteq B$ that is $BS^{m}B\subseteq(BS^{m}B]\subseteq B$ and
$(B]=B$. Hence $B$ is an $m$-bi-ideal of $S$.

∎

######  Theorem 2.5.

Let $S$ be an ordered semigroup then following statements hold :

  1. 1.

Let $A$ and $C$ be subsemigroups of $S$ and $B=(AC]$. If $A$ is an $m$-left
ideal then $B$ is an $m$-bi-interior ideal of $S$.

  2. 2.

Let $A$ and $C$ be subsemigroups of $S$ and $B=(CA]$. If $A$ is an $m$-right
ideal then $B$ is an $m$-bi-interior ideal of $S$.

###### Proof.

$(1)$: Let $A$ and $C$ be subsemigroups of ordered semigroup $S$ and $B=(AC]$.
Suppose $A$ is an $m$-left ideal of $S$ then
$(BS^{m}B]=((AC]S^{m}(AC]]\subseteq(ACS^{m}AC]\subseteq(AC]=B$ implies
$(BS^{m}B]\cap(S^{m}BS^{m}]\subseteq(BS^{m}B]\subseteq B$. Hence $B$ is an
$m$-bi-interior ideal of $S$.

$(2)$: Let $A$ and $C$ be subsemigroups of ordered semigroup $S$ and $B=(CA]$.
Suppose $C$ is an $m$-right ideal of $S$ then
$(BS^{m}B]=((CA]S^{m}(CA]]\subseteq(CAS^{m}CA]\subseteq(CA]=B$ implies
$(BS^{m}B]\cap(S^{m}BS^{m}]\subseteq(BS^{m}B]\subseteq B$. Hence $B$ is an
$m$-bi-interior ideal of $S$. ∎

######  Theorem 2.6.

Let $S$ be an ordered semigroup. $S$ is m-bi-interior simple ordered semigroup
if and only if $(S^{m}aS^{m}]\cap(aS^{m}a]=S$, for all $a\in S$.

###### Proof.

Consider $S$ is an m-bi-interior simple ordered semigroup and $a\in S$. Now
$(S^{m}((S^{m}aS^{m}]\cap(aS^{m}a])S^{m}]\cap(((S^{m}aS^{m}]\cap(aS^{m}a])S^{m}((S^{m}aS^{m}]\cap(aS^{m}a])]\subseteq(S^{m}(S^{m}aS^{m}]S^{m}]\cap((aS^{m}a]S^{m}(aS^{m}a]]\subseteq(S^{2m}aS^{2m}]\cap(aS^{m}SS^{m}SS^{m}a]\subseteq(S^{m}aS^{m}]\cap(aS^{m}a]$.
Hence $(S^{m}aS^{m}]\cap(aS^{m}a]$ is an $m$-bi-interior ideal of $S$ which
contradict the fact that $S$ is $m$-bi-interior simple. Hence
$(S^{m}aS^{m}]\cap(aS^{m}a]=S$.

Conversely, suppose that $(S^{m}aS^{m}]\cap(aS^{m}a]=S$, for all $a\in S$. Let
$B$ be an $m$-bi-interior ideal of $S$ and $a\in B$. Then
$S=(S^{m}aS^{m}]\cap(aS^{m}a]\subseteq(S^{m}BS^{m}]\cap(BS^{m}B]\subseteq B$
which implies $S=B$. Thus $S$ is m-bi-interior simple ordered semigroup.

∎

######  Theorem 2.7.

An ordered semigroup $S$ is $m$-regular if and only if $B\cap I\cap
L\subseteq(BIL]$ for any $m$-bi-interior ideal $B$, $m$-ideal $I$ and $m$-left
ideal $L$ of $S$.

###### Proof.

Suppose $S$ is an $m$-regular ordered semigroup and $B$, $I$, $L$ are $m$-bi-
interior ideal, $m$-ideal and $m$-left ideal of $S$ respectively. Let $a\in
B\cap I\cap L$. . Since $S$ is $m$-regular
$a\in(aS^{m}a]\subseteq(aS^{m}aS^{m}aS^{m}aS^{m}aS^{m}a]\subseteq((aS^{m}aS^{m}a](S^{m}aS^{m}a](S^{m}a]]$.

Now
$(aS^{m}aS^{m}a]\subseteq(BS^{m}BS^{m}B]\subseteq(BS^{m}SS^{m}B]\subseteq(BS^{2m+1}B]\subseteq(BS^{m}B]$.
Again $(aS^{m}aS^{m}a]\subseteq(S^{m}BS^{m}]$. Thus
$(aS^{m}aS^{m}a]\subseteq(BS^{m}B]\cap(S^{m}BS^{m}]\subseteq B$. Again
$(S^{m}aS^{m}a]\subseteq(S^{m}IS^{m}I]\subseteq(I]=I$ and
$(S^{m}a]\subseteq(S^{m}L]\subseteq L$. Hence
$a\in((aS^{m}aS^{m}a](S^{m}aS^{m}a](S^{m}a]]\subseteq(BIL]$ that is $B\cap
I\cap L\subseteq(BIL]$.

Conversely, consider $B\cap I\cap L\subseteq(BIL]$, for any $m$-bi-interior
ideal $B$, $m$-ideal $I$ and $m$-left ideal $L$ of $S$. Let $R$, $L$ be an
$m$-right and $m$-left ideal of $S$. Now, by assumption $R\cap L=R\cap
S^{m}\cap L\subseteq(RS^{m}L]\subseteq(RL]$ and $(RL]\subseteq(R\cdot 1\cdot
1\cdot\cdot S]\subseteq(RS^{m}]\subseteq(R]=R$. Dually $(RL]\subseteq L$.
Hence $R\cap L=(RL]$. Thus Lemma1.9 yields that $S$ is an $m$-regular.

∎

######  Theorem 2.8.

Let $S$ be an ordered semigroup. If $S$ is $m$-regular then every $m$-ideals
and $m$-interior ideals co-incide.

###### Proof.

Its very obvious that every $m$-ideal is an $m$-interior ideal.

Now for the converse part, consider an ordered semigroup $S$ such that $S$ is
$m$-regular. Let $I$ be an interior ideal of $S$. Now $m$-regularity of $S$
infers that $I\subseteq(IS^{m}I]$. Now
$IS^{m}\subseteq(IS^{m}I]S^{m}\subseteq(IS^{m}I](S^{m}]\subseteq((IS^{m}I]S^{m}I](S^{m}]\subseteq(S^{m+1}IS^{2m+1}\subseteq(S^{m}IS^{m}]\subseteq(I]=I$.
Similarly $S^{m}I\subseteq I$. Hence $I$ is an $m$-ideal of $S$. ∎

######  Theorem 2.9.

Let $S$ be an ordered semigroup and $B$ is an $m$-bi-interior ideal of $S$.
Then $S$ is $m$-regular if and only if $(BS^{m}B]\cap(S^{m}BS^{m}]=B$.

###### Proof.

First suppose that $S$ is an $m$-regular ordered semigroup and $B$ is an
$m$-bi-interior ideal of $S$. Then $(BS^{m}B]\cap(S^{m}BS^{m}]\subseteq B$ and
$(B]=B$. Let $x\in B$ then $x\in(xS^{m}x]\subseteq(xS^{m}xS^{m}x]$. Hence
$(xS^{m}xS^{m}x]\in(BS^{m}B]\cap(S^{m}BS^{m}]$ which implies
$B\subseteq(BS^{m}B]\cap(S^{m}BS^{m}]$. Hence $(BS^{m}B]\cap(S^{m}BS^{m}]=B$.

Conversely, assume that$(BS^{m}B]\cap(S^{m}BS^{m}]=B$, for all $m$-bi-interior
ideal $B$ of $S$. Let $B=R\cap L$, where $R$ is an $m$-right and $L$ is an
$m$-left ideal of $S$. Then by Proposition 2.3, $B$ is an $m$-bi-interior
ideal of $S$. Now $(BS^{m}B]\cap(S^{m}BS^{m}]=((R\cap L)S^{m}(R\cap
L)]\cap(S^{m}(R\cap L)S^{m}]\subseteq(RS^{m}L]\subseteq(RL]$ and
$(RL]\subseteq R\cap L$. Hence following Lemma 1.9, $S$ is an $m$-regular
ordered semigroup. ∎

######  Theorem 2.10.

Let $B$ be a subsemigroup of an $m$-regular ordered semigroup $S$. Then for
some $m$-right ideal $R$ and $m$-left ideal $L$ of $S$, $B$ can be represented
as $B=(RL]$ if and only if $B$ is an $m$-bi-interior ideal of $S$.

###### Proof.

Let $S$ be an $m$-regular ordered semigroup and $L$ be an $m$-left and $R$ be
an $m$-right ideal of $S$. Now $(BS^{m}B]\subseteq(RLS^{m}RL]\subseteq(RL]=B$.
Hence $(BS^{m}B]\cap(S^{m}BS^{m}]\subseteq(BS^{m}B]\subseteq B$. Evidently
$(B]=(RL]=B$. Thus $B$ is an $m$-bi-interior ideal of $S$.

Conversely suppose that $B$ is an m-bi-interior ideal of $S$. Now by Theorem
2.9, $m$-regularity of $S$ yields $(BS^{m}B]\cap(S^{m}BS^{m}]=B$. Let
$R=(BS^{m}]$ and $L=(S^{m}B]$. It can be easily check that $R$ and $L$ are
$m$-right and $m$-left ideal of $S$ respectively. Consider $x\in B$. Now
$m$-regularity of $S$ gives $x\in(xS^{m}x]\subseteq(BS^{m}B]$ that is
$B\subseteq(BS^{m}B]$. Now
$(RL]\subseteq((BS^{m}](S^{m}B]]\subseteq(BS^{m}B]$. Again
$(RL]\subseteq((BS^{m}](S^{m}B]]\subseteq(BS^{2m}B]=((BS^{m}B]S^{2m}(BS^{m}B]]\subseteq(S^{m}BS^{m}]$.
Hence $(RL]\subseteq(BS^{m}B]\cap(S^{m}BS^{m}]\subseteq B$. Again
$B\subseteq(BS^{m}B]\subseteq(BS^{m}BS^{m}BS^{m}B]\subseteq\\\
((BS^{m}]BS^{m}B(S^{m}B]]\subseteq(RSS^{m}SL]\subseteq(RS^{m}L]\subseteq(RL]$.
Hence $B=(RL]$. ∎

Acknowledgements

The authors would like to thank the funding agency, the University Grant
Commission (UGC) of the Government of India, for providing financial support
for this research in the form of UGC-CSIR NET-JRF.

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