# Search for dark matter mediator in the production of three and four top
quarks.

###### Abstract

In the context of simplified models of dark matter, the contributions of a
scalar mediator to top quark production processes are considered. Tree-level
and one-loop contributions of the mediator’s decay into a top-antitop quark
pair in two-, three-, and four-top quarks production processes are calculated.
A significant contribution from diagrams involving the dark matter mediator is
demonstrated in the total cross section for three- and four-top quark
production processes, taking into account current experimental limits on model
parameters. The perspective of searching for dark matter mediators in the
processes under consideration is determined by the ability to reconstruct the
final state with modern collider detectors and the experimental sensitivity
that has already been achieved for such rare events.

a Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University,
Leninskie gory, GSP-1
Moscow 119991, Russian Federation
b Sarov Branch of Lomonosov Moscow State University,
Sarov 607328, Russian Federation

PACS: 14.65.Ha; 12.60.$-$i

## Introduction

Convincing astrophysical and cosmological observations, such as studies of
galaxy rotation curves [1] and gravitational lensing in galaxy cluster
collisions [2], indicate the presence of a large amount of hidden matter in
the Universe, known as "dark matter" (DM). This dark matter can reveals itself
through gravitational interaction and weak interactions with Standard Model
(SM) fields. Based on the assumption that DM has a particle nature, various
extensions of the SM are created to explain the origin of dark matter and its
possible detection in experiments.

Direct searches for dark matter can be carried out through the interaction of
cosmic DM particles with ground-based detectors [3]. At the same time, the
Large Hadron Collider (LHC) provides a unique opportunity to observe and study
the formation of DM particles under laboratory conditions. This opportunity is
based on calculations of the DM particle annihilation cross-section, which is
close to the weak interaction cross-section at LHC energies [4]. Since DM
particles cannot be directly observed by LHC detectors, the main method of
their detection is the joint production of dark sector particles and SM
particles. In such a case, the DM particle in the final state carries away
energy, which is observed as a "missing transverse energy" in the detector.

Possible interactions between DM particles and SM particles can occur through
an intermediate particle known as a mediator. Depending on the interaction
mechanism between the DM particles and SM particles, the mediator is endowed
with various properties. One type of interaction could be a contact
interaction involving a quark-antiquark pair or two gluons and two DM
particles. In this case, the magnitude and distribution of the missing energy
signal are determined by the nature and mass of the DM particles and the
Lorentz structure of the interaction; the cross-section of the process is the
main free parameter that can be measured. To model such scenarios, a basis of
contact interaction operators can be introduced in effective field theory [5]
to calculate the magnitude of possible signals.

Effective field theory is used under the assumption that the DM mediator is
very heavy. In the case of a presumably lighter mediator, models that
explicitly include mediators are necessary. Such models, known as simplified
models [6, 7], include particles and interactions beyond the SM. These
simplified models [8, 9] can be used in experimental searches for DM particles
at the LHC. Although there are many different ways to construct simplified
models, currently, those in which DM particles interact with SM particles via
new scalar, pseudoscalar, or vector mediators appear theoretically attractive.

Under the assumption of minimal flavor violation (MFV) [10, 11, 12, 13],
third-generation quarks can play a significant role [14] in such interactions.
Therefore, it is particularly interesting to consider the processes of DM
production at colliders in association with a top quark, as it has the largest
mass. This idea has motivated experimental searches for events in which DM
particles are produced in association with a single top quark ($t/\bar{t}$ \+
DM) and a pair of top quarks ($t\bar{t}$ \+ DM) [15, 16]. No significant
excess over SM predictions has been found in LHC experiments, and the issue of
improving the analysis efficiency remains relevant.

In simplified DM models, it is assumed that dark matter particles $\chi$ are
Dirac fermions interacting with the SM and DM sectors via either a massive
electrically neutral scalar particle $\Phi$ (scalar mediator) or a
pseudoscalar particle $A$ (pseudoscalar mediator) [17]. The interaction
Lagrangians of scalar and pseudoscalar particles with SM and DM fermions are
as follows:

$\begin{gathered}L_{\Phi}=g_{\chi}\Phi\bar{\chi}\chi+\frac{g_{f}\Phi}{\sqrt{2}}\sum_{f}\left(y_{f}\bar{f}f\right)\\\
L_{A}=ig_{\chi}A\bar{\chi}\gamma^{5}\chi+i\frac{g_{f}A}{\sqrt{2}}\sum_{f}\left(y_{f}\bar{f}\gamma^{5}f\right)\end{gathered}$

Here, summation is done over all SM fermions, denoted as $f$. The parameters
$y_{f}=\sqrt{2}m_{f}/v$ are Yukawa coupling constants with the vacuum
expectation value of the Higgs field being 246 GeV. $g_{\chi}$ is the coupling
constant of DM fermions with the mediator and $g_{f}$ is the coupling constant
of SM fermions with the mediator. To reduce the number of model parameters, it
is usually assumed that the value of $g_{\chi}$ is the same for all fermion
flavors. Under the assumption of minimal flavor violation, such a simplified
model contains a minimal set of four free parameters: the mass of dark matter
particles $m_{\chi}$, the mass of the mediator $m_{\Phi/A}$, $g_{f}$, and
$g_{\chi}$.

According to the recommendations of the LHC Dark Matter Working Group [18], in
our study the coupling parameter values are taken to be $g_{f}=g_{\chi}=1$,
and the DM particle mass $m_{\chi}=1$ GeV. For these parameter values, scalar
and pseudoscalar mediator masses up to about 400 GeV are currently excluded
[19, 16]. In most cases, the DM mediator decays into a pair of DM particles.
With the above assumptions the branching ratio for the decay of a 400 GeV
scalar mediator into a pair of DM particles is 84%. However, the DM mediator
can also decay into a pair of top and antitop quarks, with a branching ratio
of 16% for a 400 GeV scalar mediator. Accordingly, one can assume that the DM
contribution to the production processes of top quark pairs, 3-top, and 4-top
quarks can be quite significant. In this work, we explore the possibility of
detecting DM in such processes within a simplified model with a scalar
mediator. The pseudoscalar or vector mediator leads to almost the same or even
better sensitivity to DM contribution.

The article is organized as follows. Section 1 analyzes the contribution of
the dark matter mediator to the dominant top quark pair production process
$pp\to\bar{t}t$. Section 2 investigates the process $pp\to\bar{t}t\bar{t}t$,
and Section 3 considers the production of three top quarks. Given the
objective difficulties of distinguishing the four-top quark production process
from the three-top quark process with next-to-leading-order (NLO) corrections,
Section 4 examines the process $pp\to t\bar{t}tW^{-}\bar{b}$, which includes
both three-top and four-top quark processes.

The modeling and numerical calculations were performed using the CompHEP4.6rc1
[20, 21] and MadGraph5 [22] computational packages. The models used in
MadGraph5 are DMScalar, which includes an effective vertex $gg\Phi$ [23], and
DMSimp, which facilitates NLO calculations [24]. All calculations are
presented for proton-proton collision energies of 14 TeV for the LHC, with
parton distribution functions being $\mathrm{NNPDF23\\_nlo\\_as\\_0118}$ from
the LHAPDF6 package [25]. Since at this stage of research it is only important
to compare generators and estimate the possible contribution of DM diagrams,
only Monte Carlo uncertainties are given for the cross-sections.

##  1 Pair production of $t\bar{t}$ in a Simplified DM Model with a Scalar
Mediator

The dominant process involving the vertex $\Phi\to t\bar{t}$ at the LHC is the
process $pp\to\bar{t}t$. Therefore, it makes sense to start analyzing the
contribution of the DM mediator with this process. The renormalization and
factorization scale in the MC calculations for pair production is fixed at
$m_{t}=172.5$ GeV.

The leading order (LO) cross-section for top quark pair production
$pp\to\bar{t}t$ within the SM, calculated using the CompHEP software package,
is $617.9\pm 0.2$ pb for a top quark mass $m_{t}=172.5$ GeV. Pair production
of top quarks occurs either in the collision of a light quark-antiquark pair
with the exchange of a virtual gluon or in the collision of two gluons from
two colliding hadrons.

At the LHC, the process of $t\bar{t}$ pair production in the collision of two
gluons dominates, with a cross-section of $535.6\pm 0.2$ pb.

In the framework of a simplified DM model with a scalar mediator, diagrams
with the production of a DM mediator and its subsequent decay into a
$t\bar{t}$ pair also contribute to the processes of top quark pair production.
The cross-section for the process $pp\to\bar{t}t$ within this model,
calculated using the CompHEP software package, is $618.26\pm 0.06$ pb for a
top quark mass $m_{t}=172.5$ GeV and a scalar DM mediator mass $m_{\Phi}=400$
GeV.

The diagrams for the DM contribution to the subprocess $q\bar{q}\to t\bar{t}$
are shown in Fig. 1(a). The cross-section for these processes is $(4.282\pm
0.002)\cdot 10^{-3}$ pb, which is about 5 orders of magnitude smaller than the
cross-section for the process $pp\to\bar{t}t$ within the SM.

Figure 1: Feynman diagrams describing the production of a top-antitop quark
pair in the collision of a light quark-antiquark pair (a) and in the collision
of two gluons from two colliding hadrons (b) within the framework of a
simplified DM model with a scalar mediator.

In the fusion of two gluons $gg\to t\bar{t}$, the main diagram is the
production of a scalar mediator and its subsequent decay into a top quark pair
– this is a loop diagram, shown in Fig. 1(b). The cross-section for this
process, calculated with the same parameters in the CompHEP and MadGraph
packages, is almost three orders of magnitude larger than the cross-section
for processes with initial state quarks, being $0.917\pm 0.025$ pb and
$0.884\pm 0.001$ pb, respectively. However, this cross-section is small
compared to the total cross-section for the process $pp\to\bar{t}t$ within the
SM, which does not allow using this process for DM searches. All the
aforementioned cross-sections are presented in Table 1.

Process | Cross-section (pb)
---|---
$pp\to t\bar{t}$ (SM)∗ | $617.93\pm 0.23$
$gg\to t\bar{t}$ (SM)∗ | $535.63\pm 0.23$
$pp\to t\bar{t}$ (SM + DM)∗ | $618.26\pm 0.06$
$q\bar{q}\to t\bar{t}$ (tree diagrams DM)∗ | $(4.282\pm 0.002)\cdot 10^{-3}$
$gg\to t\bar{t}$ (loop diagram DM)∗ | $0.917\pm 0.025$
$gg\to t\bar{t}$ (loop diagram DM)∗∗ | $0.884\pm 0.001$
Table 1: Cross-sections for top quark pair production processes within the SM
and within a simplified DM model with a scalar mediator, calculated using
CompHEP (${*}$) and MadGraph(${**}$).

Thus, the search for the production of a DM mediator decaying into a top-
antitop quark pair in the processes of top quark pair production is
unpromising.

##  2 Four top quarks production $t\bar{t}t\bar{t}$ in a Simplified DM Model
with a Scalar Mediator

The production of four top quarks $pp\to t\bar{t}t\bar{t}$ is one of the
rarest SM processes observable with current LHC parameters. The cross-section
of this process within the SM, calculated using the MadGraph package, is
$8.88\pm 0.02$ fb. In this section, as well as in sections 3 and 4, the
renormalization and factorization scales in the calculations are set to 350
GeV. The process $pp\to t\bar{t}\Phi\to t\bar{t}t\bar{t}$ is one of the most
promising channels for the production of a DM mediator [26, 27].

The main contribution to the four top-quark production cross-section at the
LHC comes from diagrams with gluons in the initial state. Representative
diagrams are shown in Fig.2. In the framework of a simplified DM model with a
scalar mediator, additional contributions come from diagrams involving the
production of the DM mediator. Examples of such diagrams are also shown in
Fig.2. Table 2 presents the cross-sections of the process $gg\to
t\bar{t}t\bar{t}$ within the SM and within the simplified DM model with a
scalar mediator at leading order (LO) in perturbation theory, calculated in
CompHEP and MadGraph. The cross-section of this process, calculated within the
DM model, significantly exceeds the cross-section calculated within the SM.

Figure 2: Feynman diagrams describing the production of four top quarks in the
collision of two gluons from two colliding hadrons. The top row shows
representative diagrams within the SM, and the bottom row shows diagrams
within the Simplified Dark Matter Model with a scalar mediator.

Table 3 presents the cross-section of the process $gg\to t\bar{t}t\bar{t}$ in
the considered DM model, including loop diagrams, as well as the separate
contributions of LO and loop diagrams involving the DM mediator, calculated in
MadGraph. The contribution of diagrams involving the DM mediator to the cross-
section of the process $gg\to t\bar{t}t\bar{t}$ within the simplified DM model
is approximately 32%, and searching for DM manifestations in four top-quark
production processes may have good prospects. The contribution of loop
diagrams is two orders of magnitude smaller than the contribution of other DM
diagrams, and its influence on the total cross-section is negligible.

Process | CompHEP Cross-section (fb) | MadGraph Cross-section (fb)
---|---|---
$gg\to t\bar{t}t\bar{t}$ (SM) | $7.71\pm 7.55*10^{-3}$ | $7.79\pm 2.3*10^{-2}$
$gg\to t\bar{t}t\bar{t}$ (SM + DM) | $11.15\pm 1.60*10^{-2}$ | $11.41\pm 3.4*10^{-2}$

Table 2: LO Cross-sections of the process $gg\to t\bar{t}t\bar{t}$ within the
SM and within the simplified DM model with a scalar mediator, calculated in
CompHEP and MadGraph

Description of the process $gg\to t\bar{t}t\bar{t}$ (DM) | MadGraph Cross-section (fb)
---|---
Full set of diagrams, including loop diagrams | $11.44\pm 3*10^{-2}$
Contribution of diagrams with a mediator, excluding loop diagrams | $2.89\pm 9.9*10^{-3}$
Contribution of loop diagrams with a mediator | $8.67*10^{-2}\pm 2.74*10^{-4}$

Table 3: Cross-section of the process $gg\to t\bar{t}t\bar{t}$ within the
simplified DM model with a scalar mediator, including loop diagrams;
contribution of diagrams with a mediator, with and without considering loop
diagrams.

##  3 Processes with the final state $t\bar{t}tW^{-}$ within the framework of
a simplified DM model with a scalar mediator

The process of producing three top quarks in association with a W boson is an
even rarer process than the production of four top quarks and has not yet been
experimentally observed.

For the dominant processes of three top-quark production within the SM, it has
been shown that the interference of QCD and electroweak diagrams is quite
large and negative, so it is important to consider the full set of diagrams to
calculate the cross-sections of these processes [28]. A comparison of the
contributions is given in Table 6.

Table 4 describes the results of calculations of various DM contributions to
this process. It is noticeable that the DM contribution is significantly
higher than for the production of four top quarks, with the cross-section of
diagrams involving the DM mediator accounting for 60% of the total cross-
section. As in the production of four top quarks, loop diagrams give cross-
sections several orders of magnitude smaller and can be neglected in this
process. Thus, this process is highly promising for DM searches at colliders.

Process $pp\to t\bar{t}tW^{-}$ | MadGraph Cross-section (fb)
---|---
Full set of diagrams in the SM | $0.62\pm 1.6*10^{-3}$
Full set of diagrams in the DM model, excluding loop diagrams | $1.54\pm 5*10^{-3}$
Full set of diagrams in the DM model, including loop diagrams | $1.53\pm 5*10^{-3}$
Contribution of diagrams with a DM mediator, excluding loop diagrams | $0.90\pm 2.4*10^{-3}$
Contribution of loop diagrams with a DM mediator | $1.20*10^{-3}\pm 3.7*10^{-6}$

Table 4: Cross-sections of the process $pp\to t\bar{t}tW^{-}$ and individual
contributions within the SM and the Simplified DM Model with a Scalar
Mediator.

##  4 Processes with the final state $t\bar{t}tW^{-}\bar{b}$ within the
framework of a simplified DM model with a scalar mediator

The processes of producing three top quarks in association with a W boson at
NLO and the processes of producing four top quarks with the subsequent decay
of one top quark share the same final state $t\bar{t}tW^{-}\bar{b}$.
Separating these contributions is analogous to the task of distinguishing
contributions from $tWb$ and $t\bar{t}$ processes [29], requiring detailed
study due to significant negative interference between these processes.
Studying the complete set of diagrams for the process $pp\to
t\bar{t}tW^{-}\bar{b}$ can provide insight into the DM contributions to both
processes without the need for their separate consideration.

Table 5 presents the cross-sections of the process $gg\to
t\bar{t}tW^{-}\bar{b}$ calculated within the SM and the considered DM model,
with and without loop diagrams included. Specifically calculated and presented
are the contributions of diagrams involving the DM mediator without loop
diagrams and the contribution of loop diagrams. The contribution of diagrams
involving the DM mediator to the cross-section of $gg\to
t\bar{t}tW^{-}\bar{b}$ is comparable to that of $gg\to t\bar{t}t\bar{t}$ due
to its dominant contribution (Table 6), approximately 33%.

Process $gg\to t\bar{t}tW^{-}\bar{b}$ | MadGraph Cross-section (fb)
---|---
Full set of diagrams in the SM | $15.55\pm 4.7*10^{-2}$
Full set of diagrams in the DM model, excluding loop diagrams | $23.02\pm 6.2*10^{-2}$
Full set of diagrams in the DM model, including loop diagrams | $23.16\pm 6*10^{-2}$
Contribution of diagrams with a DM mediator, excluding loop diagrams | $6.03\pm 1.2*10^{-2}$
Contribution of loop diagrams with a DM mediator | $0.17\pm 5*10^{-4}$

Table 5: Cross-sections of the process $gg\to t\bar{t}tW^{-}\bar{b}$ and individual contributions within the SM and the Simplified DM Model with a Scalar Mediator. Process Description | MadGraph Cross-section (fb)
---|---
$pp\to t\bar{t}tW^{-}$, QCD+EW | $0.622\pm 1.6*10^{-3}$
$pp\to t\bar{t}tW^{-}$, QCD | $0.453\pm 1.3*10^{-3}$
$pp\to t\bar{t}tW^{-}$, EW | $0.452\pm 1.3*10^{-3}$
$pp\to t\bar{t}tW^{-}$, QCD+EW+DM | $1.54\pm 5*10^{-3}$
$gg\to t\bar{t}tW^{-}\bar{b}$, QCD+EW | $15.5\pm 4.7*10^{-2}$
$gg\to t\bar{t}tW^{-}\bar{b}$, QCD | $14.0\pm 4.6*10^{-2}$
$gg\to t\bar{t}tW^{-}\bar{b}$, EW | $6.08\pm 1.4*10^{-2}$
$gg\to t\bar{t}tW^{-}\bar{b}$, QCD+EW+DM | $23.02\pm 6*10^{-2}$
Table 6: Comparison of QCD and electroweak contributions to processes
involving the production of 3 and 4 top quarks. Notations are introduced: QCD
- contribution from QCD diagrams only, EW - contribution from electroweak
diagrams (QED==3), QCD+EW - full set of diagrams within the SM (QED=3),
QCD+EW+DM - full set of diagrams in the DM model, excluding loop diagrams.

##  5 Results

The possibility of searching for a dark matter (DM) mediator decaying into a
pair of top-antitop quarks in processes involving the production of two,
three, and four top quarks has been considered. Taking into account modern
experimental constraints and the parameter space recommended for calculations
in the simplified model with a scalar mediator, potential DM contributions in
tree-level and loop-level approximations were computed for the primary
processes accessible to contemporary collider experiments. It has been
demonstrated that for the dominant process of top-antitop quark pair
production, the possible contribution from the DM mediator, including loop
diagrams, amounts to approximately 0.1% and is not of interest for
experimental searches.

Conversely, significantly rarer processes involving the production of four and
three top quarks exhibit substantial potential, where the DM contribution can
reach 32% and 60% of the total cross-section, respectively, for the
recommended parameters of the simplified DM model. Considering the complexity
of experimentally distinguishing between the processes involving three and
four top quarks, the process $pp\to t\bar{t}tW^{-}\bar{b}$ was computed,
encompassing the production of both three and four top quarks and the
resulting interferences. It has been shown that the DM contribution can
constitute 33% of the total cross-section of this process, making it the most
promising from the standpoint of experimental searches in modern collider
experiments.

Similar sensitivity can be expected for simplified models with pseudoscalar
and vector mediators.

## Acknowledgments

This study was conducted within the scientific program of the Russian National
Center for Physics and Mathematics, section 5 «Particle Physics and
Cosmology».

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considered as a proxy for the value of the Sérsic index, $n=3\;{\rm b}/{\rm
t}+1$.

DeepLeGATo exploits neural network models, that for the EMC were trained on a
single Tesla V100 GPU. The output was provided in three separate catalogues of
different minimum S/N, according to the input source lists that were
distributed to the participants; a different model was used for each S/N
section, and the training time was of the order of 2 hours per model. At
inference, it then took a few minutes to “fit” all galaxies in a simulated
field; therefore, concerning computational times DeepLeGATo has by far the
best performance, due to the very different fitting technique.

We stress again that the subdivision of the catalogue into subsamples fitted
with different models caused evident features in the global distribution of
$\delta f$, though the average trends and the final $\mathcal{S}$ values were
generally quite good. In particular, the bright sources in each section were
systematically underestimated while the faint ones were overestimated. There
were also many bright outliers especially in the DS runs, with bulges
typically being underestimated and discs being overestimated; this is likely
to be a consequence of the CNN fitting technique (fixed dimensions of the
stamps, and less training at the bright end). Figure 24 provides an example of
these features.

###  B.2 Galapagos-2

Galapagos-2 employs the down-hill gradient minimisation method of
Galfit/GalfitM, so it is typically fast, and can be reasonably parallelized to
use 16 cores (due to an IDL limitation; ways around this limit exist, as it
can also run several sessions in parallel). For the EMC, the code was run on a
machine with 48 cores (2.3 GHz each) at 96 threads and ample (512 GB) memory,
using 12 threads (at 1.15 GHz) at a time (and 4 sessions in parallel) for
multi-band fits. The run times of the Galfit fits themselves were measured and
stored, and account for the majority of the time used, with the wrapper not
adding significant computational time. The typical GalfitM fitting times were
approximately as follows:

  * •

DS F1 single-band single Sérsic fit, 160 hours (about 5.7 seconds per galaxy);

  * •

DS F1 single-band double Sérsic fit, 150 hours (about 5.4 seconds per galaxy);

  * •

DS F0 multi-band single Sérsic fit, 8000 hours (about 288 seconds per galaxy);

  * •

DS F0 multi-band double Sérsic fit, 10 000 hours (about 360 seconds per
galaxy).

Therefore the multi-band fitting, while providing the results for nine bands
in parallel, took significantly more than the time elapsed to fit individual
bands. However, as shown, it clearly provides better results for the shallower
bands.

The overall results were good, with no evident pathological behaviour. The
performance on $I_{\scriptscriptstyle\rm E}$ in the multi-band fit DS F0, i.e.
obtained simultaneously with the other bands, is significantly worse than
those on other fields, where $I_{\scriptscriptstyle\rm E}$ is measured alone
(see Fig. 25); this is not surprising, because when fitting one band
individually all the parameters get optimised to minimise that band, while
fitting more bands in parallel the structural parameters are minimized
averaging over the wavelengths (and this is particularly true if the band is
the deepest and the one with the higher resolution, as is the case for
$I_{\scriptscriptstyle\rm E}$). However, it should be pointed out that multi-
band fits “fail” less often than the combination of several single-band fits,
which have independent constraints (see Häußler et al. 2013).

The total magnitude estimated with a single Sérsic fit (which is always
provided by the code) is typically better than the one from the double Sérsic
in $I_{\scriptscriptstyle\rm E}$, even on the DS simulation (Fig. 26). On the
contrary, the double Sérsic magnitude estimate is better for the other bands
(Fig. 27). While the latter result is expected, given that the code is fitting
a double profile with two separate analytical functions (and, the shallower
the band the more the multi-band fit should improve the results, so the NIR
and LSST bands should benefit more than $I_{\scriptscriptstyle\rm E}$ from
it), the opposite outcome found on $I_{\scriptscriptstyle\rm E}$ images is
more surprising. We point out that since the other teams did not provide
single Sérsic flux estimates for DS runs, we could not check whether this
particular result is a general one, or only concerns Galapagos-2.

Figure 25: Galapagos-2 results on F0 (top) and F1 (bottom) DS
$I_{\scriptscriptstyle\rm E}$ band (one component fits). The fit on F1,
performed on the $I_{\scriptscriptstyle\rm E}$ image only rather than in
multi-band mode, is more accurate. See text for details.

Figure 26: Galapagos-2 results on F1 DS $I_{\scriptscriptstyle\rm E}$ band,
one (top) and two (bottom; $n_{\rm bulge}$ fixed) component fits. The one-
component (i.e. single Sérsic) fit is more accurate in
$I_{\scriptscriptstyle\rm E}$-only runs. See text for details.

Figure 27: Galapagos-2 results on F0 DS LSST $g$ band, one (top) and two
(bottom; $n_{\rm bulge}$ fixed) component fits. The two-component (double
Sérsic) fit is more accurate for non-$I_{\scriptscriptstyle\rm E}$ bands in
the multi-band simultaneous fit. See text for details.

Figure 28: Test on the Galapagos-2 LSST $i$ band issue. In both sets of
panels, for a chosen badly fitted source (ID 2158 in F0), top to bottom we
show scientific image; Galapagos-2 models; and residuals. Left to right are
LSST $g$, $r$, $i$, and $z$ bands. Top set: reference run including
$I_{\scriptscriptstyle\rm E}$ information in the fit. Bottom set: test run
without including $I_{\scriptscriptstyle\rm E}$ information in the fit. The
residual in the $i$ band, evident in the upper panel, vanishes in the bottom
panel, when $I_{\scriptscriptstyle\rm E}$ is not included. See text for
details. Figure 29: Galapagos-2 results on F0 LSST $i$ band. The evident
double trend is due to the multi-band simultaneous fitting (see Sect. B.2).

Interestingly, the fit of the LSST $i$ band on the DS F0 data set by
Galapagos-2 is substantially worse than the fits provided for the other bands.
After an in-depth analysis, we concluded that this is actually due to the
polynomial fitting method used in the software. In short, Galapagos-2 works
with effective central wavelengths, and when performing a multi-band fit on a
data set including two bands with a very close central wavelength, the
shallower of the two bands might get a sub-optimal fit with a systematic
offset. This is exactly the case here, with the LSST $i$ band having a central
wavelength very close to the $I_{\scriptscriptstyle\rm E}$ one
(${\rm\lambda}_{I_{\scriptscriptstyle\rm E}}=710$ nm and
${\rm\lambda}_{i}=754$ nm), and very different filter widths; see Sect. 2.1
and Fig. 1. Noticeably, this does not affect the whole catalogue, but only a
fraction of the sources. Despite trying to find a correlation with some other
input or output parameter of the data set, we were not able to identify the
criteria resulting in a good or a bad fit for a given source. Nevertheless, we
verified the reliability of our conclusion by picking one source having a bad
fit, and looking at the residual images produced by Galapagos-2 in the full
multi-band run, and in a test run in which the $I_{\scriptscriptstyle\rm E}$
band information was not used to fit the $i$ band. The result of the test is
shown in Fig. 28: while the other LSST bands are well-fitted in both cases,
when the $I_{\scriptscriptstyle\rm E}$ band is used the $i$ band has an
evident residual, which flattens out when excluding $I_{\scriptscriptstyle\rm
E}$. Of course, this cannot be considered a viable, definitive ‘solution’ to
the issue, since $I_{\scriptscriptstyle\rm E}$ is the most important and
deepest band, so it cannot be excluded from the fit to improve the fit in
another band; on the contrary, one might want to exclude the $i$ band from the
fit, although this would imply entirely discarding a band and its information.
Therefore, for the sake of comparison with the other software packages we
agreed to keep the delivered data set including this issue, although it non-
negligibly worsens the statistics. Finally, we note that the issue is more
evident when using the fluxes of the single Sérsic fit, while it is somehow
mitigated if the double Sérsic fit is considered (Galapagos-2 provides both
for the DS realisation, which is the only one including LSST bands).

###  B.3 Morfometryka

Figure 30: Example of diagnostic plot for Morfometryka. This plot refers to
the $I_{\scriptscriptstyle\rm E}$ SS F2 run. The colour coding clearly shows
that the bimodal distribution of points correlates with the input bulge-to-
total fraction, with bulge-dominated galaxies (bluer points) being fitted with
less accuracy than disc-dominated ones (redder points).

Morfometryka runs were performed on an Intel(R) Xeon(R) CPU E5-2640 v3 2.60
GHz shared 32-core workstation, running 16 jobs in parallel at any given time.
The wall time was roughly 920 minutes per field, which is roughly 10 seconds
per source. For each run, Morfometryka spends around 30% of its running time
for the Sérsic profile fits, and the remaining time is dedicated to other
measurements; however, runtimes strongly depend on the input image size.

There is an evident dependence on morphological type in the results. Bulge-
dominated galaxies (having high Sérsic index, which in the SS realisation is
obtained from the bulge-to-total ratio given in the Egg catalogue, see Sect.
2.1) are strongly biased ($\mathcal{B}\simeq 0.15$), and therefore are
considered outliers in the computation of $\mathcal{S}$, thus significantly
affecting the accuracy and resulting in a sub-optimal overall performance (see
Fig. 30).

###  B.4 ProFit

ProFit ran at a mean single-core time of 144 minutes per run, resulting in
about 93 000 hours of total CPU time; the runs were performed on the Magnus
super computer (operated by Pawsey in Western Australia), which is a multi-
node cluster where each node is 24 core (48 thread), comprised of two Intel
Xeon E5-2690 v3 (Haswell) 12-core CPUs.

The typical single Sérsic profile fit took a little under a minute per object,
and a double Sérsic fit around two minutes on average; this included the whole
processing, but in practice the vast majority of the time was spent doing the
ProFit optimisation, with the ProFound detection stage usually taking about
one second. This was deemed enough time to find a good global solution per
fit, but it was not long enough to thoroughly explore uncertainties, as shown
in Sect. 5.5.

Figure 31: Example of diagnostic plot for ProFit. Top to bottom:
$I_{\scriptscriptstyle\rm E}$ SS F2; $I_{\scriptscriptstyle\rm E}$ DS F2
(fixed bulge); and LSST $g$ F0. The $I_{\scriptscriptstyle\rm E}$ SS fit is
substantially better than the DS fit, and there is a ‘U-shaped’ trend in the
LSST bands.

Good results were provided for the SS realisation, but much less so on the DS
and RM ones. The faint end is typically strongly biased (fluxes are
overestimated); LSST bands also have a ‘U-shaped’ trend (see Figs. 31), for
which we were not able to find a simple explanation.

###  B.5 SourceXtractor++

Single-band runs (SS, RM, DS on $I_{\scriptscriptstyle\rm E}$) were performed
on a small cluster at LMU Munich, with 56 cores and 2 GB of RAM per core.
Computational times were of about 50 minutes per field for single Sérsic runs,
and 2 hours per field for double Sérsic runs.

Multi-band runs were performed on a local server at the Institut Astrophysique
de Paris, with 24 AMD EPYC 7402P cores (ignoring hyperthreading) running at
2.8 GHz, using an improved version of the code with increased parallelisation
efficiency. The computational time was 45 hours (it is worth recalling that
the NIR and LSST bands were rebinned to their original pixel scales, so their
sizes were smaller than the $I_{\scriptscriptstyle\rm E}$ one – respectively
12 500 and 8333 pixels per side).

Figure 32: Example of diagnostic plot for SourceXtractor++. These plots refers
to the $I_{\scriptscriptstyle\rm E}$ DS F2 run; the upper panel shows the fit
with $n_{\rm bulge}=4$, while the bottom panel shows the one with $n_{\rm
bulge}$ left free to vary.

The results were generally good, with no evident trends or pathological
behaviour. Among the provided output, SourceXtractor++ typically obtained the
best values for the global diagnostics. Interestingly, in the DS realisation
the runs with free $n_{\rm bulge}$ typically yielded slightly better results
than those with fixed $n_{\rm bulge}=4$, in terms of average bias (see Fig.
32).

As mentioned, the processing pipeline adopted for the EMC included a
substantial pre-processing of the data set, and the priors used for the
measurements were calibrated on the provided samples of true values, albeit
only in a statistical sense (see below).

####  B.5.1 Re-runs on raw data

The NIR and LSST images were rebinned back to their original pixel scales
(while the ones provided in the data set shared with the participants had been
rebinned to the $I_{\scriptscriptstyle\rm E}$ pixel scale of
0.1$\mathrm{\SIUnitSymbolArcsecond}$). Additionally, the PSFs used in the
fitting process were not those provided in the data set; instead, they were
extracted from the images using PSFEx. This did not require any additional
input, and therefore it was not outside the guidelines of the challenge.
Still, we felt it was necessary to check the performance of the software on
the non-processed images and using the official PSFs, for a fairer comparison
with the other participants. To this end, a few additional runs were performed
by the SourceXtractor++ team (using version 0.16 of the code instead of 0.12),
and we compared the results with the ones officially provided for the EMC. The
values of the four diagnostics $\mathcal{C}$, $\mathcal{B}$, $\mathcal{D}$,
and $\mathcal{O}$, and of the global $\mathcal{S}$, for both the EMC runs and
the re-run, are given in Table 3. Some trends for the DSb4 multi-band
realisation are also shown in Fig. 33 as examples (namely the
$I_{\scriptscriptstyle\rm E}$, $H_{\scriptscriptstyle\rm E}$, and $g$ trends,
since the aim was to check the impact of the re-extraction of PSFs and of the
re-rebinning of images with native pixel scale different from the
$I_{\scriptscriptstyle\rm E}$ one). The results do not seem to significantly
improve because of the pre-processing; most of the values and trends look
similar, and in some cases even slightly better in the re-run. The evident
exception is given by the NIR bands, in particular concerning the median bias;
including the weighting factors, the values of $\mathcal{S}$ are significantly
worse in the re-run, albeit still reasonably good (i.e. close to 1).

These re-runs required 40 minutes per field for the single Sérsic runs (SS and
RM), and 1 hour and 20 minutes per field for the double Sérsic
$I_{\scriptscriptstyle\rm E}$ runs (using 28 cores). Finally, the multi-band
re-run was performed on a different set of nodes, with 32 AMD EPYC 7302
16-Cores at 3.2 GHz, and required 67 hours.

Run | $\mathcal{S}$ | $\mathcal{C}$ | $\mathcal{B}$ | $\mathcal{D}$ | $\mathcal{O}$
---|---|---|---|---|---
SS F4 $I_{\scriptscriptstyle\rm E}$ | 0.81 | 0.94 | 0.01 | 1.44 | 0.07
| 0.87 | 0.94 | 0.01 | 1.40 | 0.08
DSb4 F4 $I_{\scriptscriptstyle\rm E}$ | 1.20 | 0.96 | 0.02 | 1.48 | 0.07
| 0.92 | 0.96 | 0.00 | 1.41 | 0.11
DSbf F0 $I_{\scriptscriptstyle\rm E}$ | 0.95 | 0.96 | 0.01 | 1.46 | 0.10
| 0.94 | 0.96 | 0.00 | 1.42 | 0.11
RM F4 $I_{\scriptscriptstyle\rm E}$ | 2.13 | 0.83 | 0.03 | 2.00 | 0.14
| 2.23 | 0.83 | 0.03 | 2.00 | 0.15
DSb4 F0 $Y_{\scriptscriptstyle\rm E}$ | 0.51 | 0.97 | 0.01 | 1.07 | 0.04
| 1.20 | 0.96 | 0.04 | 1.23 | 0.05
DSb4 F0 $J_{\scriptscriptstyle\rm E}$ | 0.58 | 0.97 | 0.01 | 1.12 | 0.05
| 1.18 | 0.96 | 0.03 | 1.26 | 0.06
DSb4 F0 $H_{\scriptscriptstyle\rm E}$ | 0.67 | 0.97 | 0.01 | 1.15 | 0.05
| 1.02 | 0.96 | 0.02 | 1.26 | 0.06
DSb4 F0 $u$ | 1.01 | 0.97 | 0.03 | 1.43 | 0.03
| 0.99 | 0.96 | 0.03 | 1.42 | 0.03
DSb4 F0 $g$ | 1.46 | 0.97 | 0.04 | 1.50 | 0.06
| 1.45 | 0.96 | 0.04 | 1.53 | 0.07
DSb4 F0 $r$ | 1.40 | 0.97 | 0.04 | 1.47 | 0.05
| 1.39 | 0.96 | 0.03 | 1.49 | 0.07
DSb4 F0 $i$ | 1.34 | 0.97 | 0.04 | 1.48 | 0.06
| 1.35 | 0.96 | 0.03 | 1.52 | 0.73
DSb4 F0 $z$ | 1.32 | 0.97 | 0.03 | 1.51 | 0.06
| 1.32 | 0.96 | 0.03 | 1.55 | 0.08
Table 3: Example values of the diagnostics for the SourceXtractor++ runs in
the output catalogues, provided for the EMC (top value in each box) and in
corresponding runs with non-pre-processed images and PSFs (i.e., using those
provided for the Challenge; bottom value in each box). Figure 33:
SourceXtractor++ re-runs on non-pre-processed (top to bottom)
$I_{\scriptscriptstyle\rm E}$, $H_{\scriptscriptstyle\rm E}$, and LSST $g$ F0
DS images (empty symbols, solid lines), compared to output provided for the
EMC, obtained with pre-processing of the images (full symbols, dashed lines).
Note the different limits on the axes.

####  B.5.2 SourceXtractor++ priors

As mentioned, SourceXtractor++ priors were obtained by determining an
appropriate transfer function to map each prior to a Gaussian mimicking the
distributions of the provided samples of the input true catalogues. Each
parameter was calibrated independently, without including covariances; only
the statistical distributions were used (i.e. there was no object-by-object
comparison in the process). We have shown how this choice likely had a
significant impact on the accuracy of the results.

Even though SourceXtractor++ models source ellipticity internally, with the
standard axis ratio $q$ and position angle $\varphi$ variables, it was found
useful to express the priors needed for these two degrees of freedom in terms
of two $e_{1}$, $e_{2}$ Cartesian ellipticity parameters (that usually enter
weak lensing studies). More specifically, in complex notation,
$e=\frac{1-q}{1+q}\exp(2{\rm i}\varphi)\equiv e_{1}+{\rm i}e_{2}$. Hence, a
Normal prior on both $e_{1}$ and $e_{2}$ was set, centred on zero and with
$\sigma=0.3$. This was in broad agreement with the population distribution of
axis ratios in the SS F4 simulation, for which ground truth was provided. A
similar prior was built for the Sérsic index, as well as for the effective
radius.

In multi-component models, priors were set under the assumption that the bulge
and disc of each source share a common position angle, but axis ratios are not
correlated (this is not actually the case, since axis ratios do not vary
across the spectrum in the Egg catalogues). Therefore the default $q$ and
$\varphi$ variables were used instead of the ($e_{1}$, $e_{2}$) ellipticities.
A single fit across all bands was run for bulge and disc effective radii,
ellipticity, and position angle; the amplitudes of the bulge and disc
components were left free to vary. Here again, the fitting procedure prefers
the definition of a set of total magnitudes and a set of ${\rm b}/{\rm t}$
(with their own associated priors), instead of using the flux of each
component in each band as a set; in practice, ${\rm b}/{\rm t}$ is expressed
via the transform $X_{{\rm b}/{\rm t}}=\operatorname{log_{10}}({\rm b}/{\rm
t}+0.01)/(1.01-{\rm b}/{\rm t})$. The priors on $X_{{\rm b}/{\rm t}}$ were set
to be band dependent, but with little variation.


00footnotetext: Support of the research of the authors by the Austrian Science
Fund (FWF), project I 4579-N, and the Czech Science Foundation (GAČR), project
20-09869L, entitled “The many facets of orthomodularity”, is gratefully
acknowledged.

# The logic of quantum mechanics incorporating time dimension

Ivan Chajda and Helmut Länger

###### Abstract

Similarly as classical propositional calculus is based algebraically on
Boolean algebras, the logic of quantum mechanics was based on orthomodular
lattices by G. Birkhoff and J. von Neumann [3] and K. Husimi [14]. However,
this logic does not incorporate time dimension although it is apparent that
the propositions occurring in the logic of quantum mechanics are depending on
time. The aim of the present paper is to show that so-called tense operators
can be introduced also in such a logic for given time set and given time
preference relation. In this case we can introduce these operators in a purely
algebraic way. We derive several important properties of such operators, in
particular we show that they form dynamic pairs and, altogether, a dynamic
algebra. We investigate connections of these operators with logical
connectives conjunction and implication derived from Sasaki projections. Then
we solve the converse problem, namely to find for given time set and given
tense operators a time preference relation in order that the resulting time
frame induces the given operators. We show that the given operators can be
obtained as restrictions of operators induced by a suitable extended time
frame.

AMS Subject Classification: 03G12, 03B46, 06C15

Keywords: Complete orthomodular lattice, event-state system, logic of quantum
mechanics, tense operator, time frame, dynamic pair, dynamic algebra

##  1 Introduction

It is well known that any physical theory determines an event-state system
$(\mathcal{E},\mathcal{S})$ where $\mathcal{E}$ contains the events that may
occur with respect to the given system and $\mathcal{S}$ contains the states
that such a physical system may assume. In quantum physics one usually
identifies $\mathcal{E}$ with the set of projection operators of a Hilbert
space $\mathbf{H}$. This set of operators is in bijective correspondence with
the set $\mathcal{C}(\mathbf{H})$ of closed subspaces of $\mathbf{H}$. The set
$\mathcal{C}(\mathbf{H})$ ordered by inclusion forms a complete orthomodular
lattice. Such lattices were introduced in 1936 by G. Birkhoff and J. von
Neumann [3] and independently in 1937 by K. Husimi [14] as a suitable
algebraic tool for investigating the logical structure underlying physical
theories that, like mentioned quantum mechanics, do not obey the laws of
classical logic. For the theory of orthomodular lattices cf. the monographs
[1] and [15].

However, the logic based on orthomodular lattices does not incorporate the
dimension of time. In order to organize this logic as a so-called tense logic
(or time logic in another terminology, see e.g. [6] and [18]) we have to
introduce the so-called tense operators $P$, $F$, $H$ and $G$. Their meaning
is as follows:

> $P$ … “It has at some time been the case that”,
> $F$ … “It will at some time be the case that”,
> $H$ … “It has always been the case that”,
> $G$ … “It will always be the case that”.

As the reader may guess, we need a time scale. For this reason a so-called
time frame is introduced. It is a pair $(T,R)$ consisting of a non-empty set
$T$ of time and a non-empty binary relation $R$ on $T$, the so-called relation
of time preference, i.e. for $s,t\in T$ we say that $s\mathrel{R}t$ means $s$
is before $t$ or, equivalently, $t$ is after $s$. For our purposes we will
consider sometimes so-called serial relations $R$ (see [6]), i.e. binary
relations $R$ such that for each $s\in T$ there exist $t,u\in T$ with
$t\mathrel{R}s$ and $s\mathrel{R}u$. Every reflexive binary relation is
serial. In physical theories $R$ is usually considered to be an order or a
quasiorder.

It is worth noticing that our tense operators are in fact special sorts of
modal operators, see e.g. [8] and [16]. The theory of tense logic has its
origin in works by A. N. Prior (cf. [16] and [17]) and in the monographs and
chapters [10], [11], [12] and [13]. For the classical propositional calculus,
these operators were studied in [2], for MV-algebras in [7], for
intuitionistic logic in [8], and for De Morgan algebras in [9]. We hope that
our approach for orthomodular lattices constitute an appropriate new
achievement.

##  2 Preliminaries

First we recall several concepts used in lattice theory.

An antitone involution on a poset $(P,\leq)$ is a mapping ′ from $P$ to $P$
satisfying the following conditions for all $x,y\in P$:

  1. (i)

$x\leq y$ implies $y^{\prime}\leq x^{\prime}$,

  2. (ii)

$x^{\prime\prime}=x$.

A complementation on a bounded poset $(P,\leq,0,1)$ is a mapping ′ from $P$ to
$P$ satisfying $x\vee x^{\prime}=1$ and $x\wedge x^{\prime}=0$ for all $x\in
P$. An orthomodular lattice is an algebra $(L,\vee,\wedge,{}^{\prime},0,1)$ of
type $(2,2,1,0,0)$ such that $(L,\vee,\wedge,0,1)$ is a bounded lattice, ′ is
an antitone involution that is a complementation and the orthomodular law
holds:

$\text{If }x,y\in L\text{ and }x\leq y\text{ then }y=x\vee(y\wedge
x^{\prime}).$

Let us remark that according to the De Morgan’s laws the orthomodular law is
equivalent to the following condition:

$\text{If }x,y\in L\text{ and }x\leq y\text{ then }x=y\wedge(x\vee
y^{\prime}).$

In the following we consider a non-trivial (i.e. not one-element) complete
(orthomodular) lattice $\mathbf{L}=(L,\vee,\wedge,0,1)$
($\mathbf{L}=(L,\vee,\wedge,{}^{\prime},0,1)$) and a given time frame $(T,R)$.
We can define the tense operators as quantifiers over the time frame as
follows:

$\displaystyle P(q)(s)$ $\displaystyle:=\bigvee\\{q(t)\mid t\mathrel{R}s\\},$
$\displaystyle F(q)(s)$ $\displaystyle:=\bigvee\\{q(t)\mid s\mathrel{R}t\\},$
$\displaystyle H(q)(s)$ $\displaystyle:=\bigwedge\\{q(t)\mid
t\mathrel{R}s\\},$ $\displaystyle G(q)(s)$
$\displaystyle:=\bigwedge\\{q(t)\mid s\mathrel{R}t\\}$

for every $q\in L^{T}$ and $s\in T$. In such a case we call the tense
operators $P$, $F$, $H$ and $G$ to be derived from or induced by the time
frame $(T,R)$.

In complete orthomodular lattices there is a close connection between the
tense operators $P$ and $H$ and the tense operators $F$ and $G$. Namely, if
these tense operators are induced by the time frame $(T,R)$ then due to De
Morgan’s laws we have $H(q)=P(q^{\prime})^{\prime}$ and
$G(q)=F(q^{\prime})^{\prime}$ for all $q\in L^{T}$.

######  Example 2.1.

Consider the orthomodular lattice $\mathbf{L}$ depicted in Fig. 1:

$0$$a$$b$$c$$c^{\prime}$$b^{\prime}$$a^{\prime}$$1$$d$$d^{\prime}$Fig. 1

Put $(T,R):=(\\{1,2,3,4,5\\},\leq)$ and define time depending propositions
$p,q\in L^{T}$ as follows:

$\begin{array}[]{l|l|l|l|l|l}t&1&2&3&4&5\\\ \hline\cr
p(t)&c^{\prime}&b^{\prime}&c^{\prime}&a^{\prime}&b^{\prime}\end{array}\quad\quad\quad\begin{array}[]{l|l|l|l|l|l}t&1&2&3&4&5\\\
\hline\cr q(t)&a&b^{\prime}&d&a&a^{\prime}\end{array}$

Then we have

$\begin{array}[]{l|l|l|l|l|l}t&1&2&3&4&5\\\ \hline\cr
P(p)(t)&c^{\prime}&1&1&1&1\\\ F(p)(t)&1&1&1&1&b^{\prime}\\\
H(p)(t)&c^{\prime}&a&a&0&0\\\
G(p)(t)&0&0&0&c&b^{\prime}\end{array}\quad\quad\quad\begin{array}[]{l|l|l|l|l|l}t&1&2&3&4&5\\\
\hline\cr P(q)(t)&a&b^{\prime}&1&1&1\\\ F(q)(t)&1&1&1&1&a^{\prime}\\\
H(q)(t)&a&a&0&0&0\\\ G(q)(t)&0&0&0&0&a^{\prime}\end{array}$

##  3 Dynamic pairs

At first we prove that for tense operators as defined above the pairs $(P,G)$
and $(F,H)$ form so-called dynamic pairs, thus $(\mathbf{L},P,F,H,G)$ is a so-
called dynamic algebra (see [6] for details).

######  Theorem 3.1.

Let $(L,\vee,\wedge,0,1)$ be a complete lattice, $(T,R)$ a time frame with
serial relation $R$, $P$, $F$, $H$ and $G$ denote the tense operators induced
by $(T,R)$ and $p,q\in L^{T}$. Then the following holds:

  1. (i)

$P(0)=F(0)=H(0)=G(0)=0$ and $P(1)=F(1)=H(1)=G(1)=1$,

  2. (ii)

$p\leq q$ implies $P(p)\leq P(q)$, $F(p)\leq F(q)$, $H(p)\leq H(q)$ and
$G(p)\leq G(q)$,

  3. (iii)

$PG(q)\leq q\leq GP(q)$,
$FH(q)\leq q\leq HF(q)$.

###### Proof.

Let $s\in T$.

  1. (i)

Since $R$ is serial we have $P(0)(s)=\bigvee\\{0\mid t\mathrel{R}s\\}=0$ and
$P(1)(s)=\bigvee\\{1\mid t\mathrel{R}s\\}=1$. The situation for $F$, $H$ and
$G$ is analogous.

  2. (ii)

Assume $p\leq q$. Then

$p(t)\leq q(t)\leq\bigvee\\{q(u)\mid u\mathrel{R}s\\}=P(q)(s)$

for all $t\in T$ with $t\mathrel{R}s$ and hence

$P(p)(s)=\bigvee\\{p(t)\mid t\mathrel{R}s\\}\leq P(q)(s).$

This shows $P(p)\leq P(q)$. The inequality $F(p)\leq F(q)$ can be shown
analogously. Moreover,

$H(p)(s)=\bigwedge\\{p(u)\mid u\mathrel{R}s\\}\leq p(t)\leq q(t)$

for all $t\in T$ with $t\mathrel{R}s$ and hence

$H(p)(s)\leq\bigwedge\\{q(t)\mid t\mathrel{R}s\\}=H(q)(s).$

This shows $H(p)\leq H(q)$. The inequality $G(p)\leq G(q)$ can be shown
analogously.

  3. (iii)

The following are equivalent:

$\displaystyle PG(q)(s)$ $\displaystyle\leq q(s),$
$\displaystyle\bigvee\\{G(q)(t)\mid t\mathrel{R}s\\}$ $\displaystyle\leq
q(s),$ $\displaystyle G(q)(t)$ $\displaystyle\leq q(s)\text{ for all }t\in
T\text{ with }t\mathrel{R}s,$ $\displaystyle\bigwedge\\{q(u)\mid
t\mathrel{R}u\\}$ $\displaystyle\leq q(s)\text{ for all }t\in T\text{ with
}t\mathrel{R}s.$

Since the last statement is true, the same holds for the first statement.
Analogously, one can prove $FH(q)\leq q$. Now the following are equivalent:

$\displaystyle q(s)$ $\displaystyle\leq GP(q)(s),$ $\displaystyle q(s)$
$\displaystyle\leq\bigwedge\\{P(q)(t)\mid s\mathrel{R}t\\},$ $\displaystyle
q(s)$ $\displaystyle\leq P(q)(t)\text{ for all }t\in T\text{ with
}s\mathrel{R}t,$ $\displaystyle q(s)$ $\displaystyle\leq\bigvee\\{q(u)\mid
u\mathrel{R}t\\}\text{ for all }t\in T\text{ with }s\mathrel{R}t.$

Since the last statement is true, the same holds for the first statement.
Analogously, one can prove $q\leq HF(q)$.

∎

If the operators $P$, $F$, $H$ and $G$ on the complete lattice $\mathbf{L}$
satisfy (i), (ii) and (iii) of Theorem 3.1 then the quintuple
$(\mathbf{L},P,F,H,G)$ will be referred to as a dynamic algebra.

######  Example 3.2.

For $p$ and $q$ of Example 2.1 we obtain

$\begin{array}[]{l|l|l|l|l|l}t&1&2&3&4&5\\\ \hline\cr
p(t)&c^{\prime}&b^{\prime}&c^{\prime}&a^{\prime}&b^{\prime}\\\
PG(p)(t)&0&0&0&c&b^{\prime}\\\
GP(p)(t)&c^{\prime}&1&1&1&1\end{array}\quad\quad\quad\begin{array}[]{l|l|l|l|l|l}t&1&2&3&4&5\\\
\hline\cr q(t)&a&b^{\prime}&d&a&a^{\prime}\\\ PG(q)(t)&0&0&0&0&a^{\prime}\\\
GP(q)(t)&a&b^{\prime}&1&1&1\end{array}$

showing that $PG(p)\leq p\leq GP(p)$ and $PG(q)\leq q\leq GP(q)$ and,
moreover, that these inequalities are strict.

In the following we establish several properties of tense operators on
complete lattices that are in accordance with the general approach presented
in [6] and [18].

######  Theorem 3.3.

Let $(L,\vee,\wedge,0,1)$ be a complete lattice, $(T,R)$ a time frame with
serial relation $R$, $P$, $F$, $H$ and $G$ denote the tense operators induced
by $(T,R)$ and $q\in L^{T}$. Then the following hold:

  1. (i)

$H(q)\leq P(q)$ and $G(q)\leq F(q)$,

  2. (ii)

if $R$ is reflexive then $H(q)\leq q\leq P(q)$ and $G(q)\leq q\leq F(q)$.

###### Proof.

Let $s\in T$.

  1. (i)

Since $R$ is serial there exists some $u\in T$ with $u\mathrel{R}s$ and we
have

$H(q)(s)=\bigwedge\\{q(t)\mid t\mathrel{R}s\\}\leq q(u)\leq\bigvee\\{q(t)\mid
t\mathrel{R}s\\}=P(q)(s).$

The proof for $G$ and $F$ is analogous.

  2. (ii)

We have

$H(q)(s)=\bigwedge\\{q(t)\mid t\mathrel{R}s\\}\leq q(s)\leq\bigvee\\{q(t)\mid
t\mathrel{R}s\\}=P(q)(s).$

The proof for $G$ and $F$ is analogous.

∎

We define $A\leq B$ for $A,B\in\\{P,F,H,G\\}$ by $A(q)\leq B(q)$ for all $q\in
L^{T}$.

######  Theorem 3.4.

Let $(L,\vee,\wedge,0,1)$ be a complete lattice, $(T,R)$ a time frame with
reflexive $R$, $P$, $F$, $H$ and $G$ denote the tense operators induced by
$(T,R)$, $A\in\\{P,F,H,G\\}$, $B\in\\{P,F\\}$ and $C\in\\{H,G\\}$. Then the
following hold:

  1. (i)

$A\leq AB$ and $AC\leq A$,

  2. (ii)

if $R$ is, moreover, transitive then $AA=A$.

###### Proof.

  1. (i)

This follows from Theorems 3.3 and 3.1.

  2. (ii)

According to (i) we have $P\leq PP$. Let $q\in L^{T}$ and $s\in T$. Then the
following are equivalent:

$\displaystyle PP(q)(s)$ $\displaystyle\leq P(q)(s),$
$\displaystyle\bigvee\\{P(q)(t)\mid t\mathrel{R}s\\}$ $\displaystyle\leq
P(q)(s),$ $\displaystyle P(q)(t)$ $\displaystyle\leq P(q)(s)\text{ for all
}t\in T\text{ with }t\mathrel{R}s,$ $\displaystyle\bigvee\\{q(u)\mid
u\mathrel{R}t\\}$ $\displaystyle\leq P(q)(s)\text{ for all }t\in T\text{ with
}t\mathrel{R}s,$ $\displaystyle q(u)$ $\displaystyle\leq\bigvee\\{q(w)\mid
w\mathrel{R}s\\}\text{ for all }t\in T\text{ with }t\mathrel{R}s\text{ and all
}u\in T\text{ with}$ $\displaystyle\hskip 17.07164ptu\mathrel{R}t.$

Since, due to transitivity of $R$, $t\mathrel{R}s$ and $u\mathrel{R}t$
together imply $u\mathrel{R}s$, the last statement is true and hence the same
holds for the first statement. Therefore $PP\leq P$, and together we obtain
$PP=P$. The proof of $FF=F$ is analogous. According to (i) we have $HH\leq H$.
Now the following are equivalent:

$\displaystyle H(q)(s)$ $\displaystyle\leq HH(q)(s),$ $\displaystyle H(q)(s)$
$\displaystyle\leq\bigwedge\\{H(q)(t)\mid t\mathrel{R}s\\},$ $\displaystyle
H(q)(s)$ $\displaystyle\leq H(q)(t)\text{ for all }t\in T\text{ with
}t\mathrel{R}s,$ $\displaystyle H(q)(s)$
$\displaystyle\leq\bigwedge\\{q(u)\mid u\mathrel{R}t\\}\text{ for all }t\in
T\text{ with }t\mathrel{R}s,$ $\displaystyle\bigwedge\\{q(w)\mid
w\mathrel{R}s\\}$ $\displaystyle\leq q(u)\text{ for all }t\in T\text{ with
}t\mathrel{R}s\text{ and all }u\in T\text{ with }u\mathrel{R}t.$

Since, due to transitivity of $R$, $t\mathrel{R}s$ and $u\mathrel{R}t$
together imply $u\mathrel{R}s$, the last statement is true and hence the same
holds for the first statement. This shows $H\leq HH$, and together we obtain
$HH=H$. The proof of $GG=G$ is analogous.

∎

##  4 Connections with logical connectives

Considering the logic based on an orthomodular lattice
$(L,\vee,\wedge,{}^{\prime},0,1)$ one can ask for logical connectives. One way
how to introduce the conjunction $\odot$ and the implication $\rightarrow$ is
based on the so-called Sasaki projections (see [1]). This method was
successfully used by the authors in [4] and [5] for investigating left
adjointness. Let us recall the corresponding definitions:

$\begin{array}[]{l}x\odot y:=(x\vee y^{\prime})\wedge y,\\\ x\rightarrow
y:=(y\wedge x)\vee x^{\prime}\end{array}$ (1)

for all $x,y\in L$. The Sasaki projection $p_{y}$ on $[0,y]$ is given by
$p_{y}(x):=(x\vee y^{\prime})\wedge y$ for all $x\in L$. Hence we have $x\odot
y=p_{y}(x)$ and $x\rightarrow y=\big{(}p_{x}(y^{\prime})\big{)}^{\prime}$ for
all $x,y\in L$.

The following result was proved in [4] and [5].

######  Proposition 4.1.

Let $(L,\vee,\wedge,{}^{\prime},0,1)$ be an orthomodular lattice, $\odot$ and
$\rightarrow$ defined by (1) and $a,b,c\in L$. Then the following holds:

  1. (i)

$a\odot 1=1\odot a=a$,

  2. (ii)

$a\odot b\leq c$ if and only if $a\leq b\rightarrow c$ (left adjointness),

  3. (iii)

$a^{\prime}=a\rightarrow 0$.

The following lemma will be used in the next proof.

######  Lemma 4.2.

Let $(L,\vee,\wedge,{}^{\prime},0,1)$ be an orthomodular lattice, $\odot$ and
$\rightarrow$ defined by (1) and $a,b\in L$. Then the following holds:

  1. (i)

$(a\rightarrow b)\odot a=a\wedge b$,

  2. (ii)

$a\leq b\rightarrow(a\odot b)$.

###### Proof.

  1. (i)

Using the orthomodular law we obtain

$(a\rightarrow b)\odot a=\big{(}(b\wedge a)\vee a^{\prime}\vee
a^{\prime}\big{)}\wedge a=a\wedge\big{(}(a\wedge b)\vee
a^{\prime}\big{)}=a\wedge b.$

  2. (ii)

Using again the orthomodular law we obtain

$a\leq a\vee b^{\prime}=b^{\prime}\vee\big{(}(a\vee b^{\prime})\wedge
b\big{)}=\big{(}(a\vee b^{\prime})\wedge b\wedge b\big{)}\vee
b^{\prime}=b\rightarrow(a\odot b).$

∎

Our next task is to show connections of tense operators with logical
connectives $\odot$ and $\rightarrow$. Using Proposition 4.1 and Lemma 4.2 we
can prove the following theorem.

######  Theorem 4.3.

Let $(L,\vee,\wedge,{}^{\prime},0,1)$ be a complete orthomodular lattice,
$(T,R)$ a time frame, $P$, $F$, $H$ and $G$ denote the tense operators induced
by $(T,R)$ and $A\in\\{P,F,H,G\\}$. Then the following two assertions are
equivalent:

  1. (i)

$A(x)\odot A(y)\leq A(x\odot y)$ for all $x,y\in L^{T}$,

  2. (ii)

$A(x\rightarrow y)\leq A(x)\rightarrow A(y)$ for all $x,y\in L^{T}$.

###### Proof.

Let $p,q\in L^{T}$. First assume (i). According to (i) we have

$A(p\rightarrow q)\odot A(p)\leq A\big{(}(p\rightarrow q)\odot p\big{)}$

Now

$(p\rightarrow q)\odot p=p\wedge q$

because of Lemma 4.2 and hence

$A\big{(}(p\rightarrow q)\odot p\big{)}=A(p\wedge q).$

Applying Theorem 3.1 to $p\wedge q\leq q$ yields

$A(p\wedge q)\leq A(q).$

Altogether, we obtain

$A(p\rightarrow q)\odot A(p)\leq A(q).$

Thus, by Proposition 4.1 we conclude

$A(p\rightarrow q)\leq A(p)\rightarrow A(q)$

showing (ii). Conversely, assume (ii). According to Lemma 4.2 we have

$p\leq q\rightarrow(p\odot q).$

Applying Theorem 3.1 we conclude

$A(p)\leq A\big{(}q\rightarrow(p\odot q)\big{)}.$

Using (ii) we obtain

$A\big{(}q\rightarrow(p\odot q)\big{)}\leq A(q)\rightarrow A(p\odot q).$

Altogether, we have

$A(p)\leq A(q)\rightarrow A(p\odot q).$

Thus, by Proposition 4.1 we conclude

$A(p)\odot A(q)\leq A(p\odot q)$

showing (i). ∎

However, we can prove also further interesting connections between these
operators.

######  Theorem 4.4.

Let $(L,\vee,\wedge,{}^{\prime},0,1)$ be a complete orthomodular lattice,
$(T,R)$ a time frame with reflexive $R$, $P$, $F$, $H$ and $G$ denote the
tense operators induced by $(T,R)$, $A,A_{1},A_{2}\in\\{P,F\\}$,
$B,B_{1},B_{2}\in\\{H,G\\}$ and $p,q\in L^{T}$. Then the following holds:

  1. (i)

$p\leq q\rightarrow A_{1}\big{(}A_{2}(p)\odot q\big{)}$,

  2. (ii)

$B(p\odot q)\leq A(p)\odot q$,

  3. (iii)

$B(p)\leq q\rightarrow A(p\odot q)$,

  4. (iv)

$B_{1}\big{(}B_{2}(p)\odot q\big{)}\leq p\odot q$,

  5. (v)

$p\rightarrow q\leq A_{1}\big{(}p\rightarrow A_{2}(q)\big{)}$,

  6. (vi)

$B(p\rightarrow q)\odot p\leq A(q)$,

  7. (vii)

$p\rightarrow B(q)\leq A(p\rightarrow q)$,

  8. (viii)

$B_{1}\big{(}p\rightarrow B_{2}(q)\big{)}\odot p\leq q$.

###### Proof.

We use Theorems 3.1 and 3.3, (1) and Proposition 4.1.

  1. (i)

We have

$\displaystyle p\odot q$ $\displaystyle\leq A_{1}(p\odot q)\text{ by
Theorem~{}\ref{th2}},$ $\displaystyle p$ $\displaystyle\leq A_{2}(p)\text{ by
Theorem~{}\ref{th2}},$ $\displaystyle p\odot q$ $\displaystyle\leq
A_{2}(p)\odot q\text{ by (\ref{equ1})},$ $\displaystyle A_{1}(p\odot q)$
$\displaystyle\leq A_{1}\big{(}A_{2}(p)\odot q\big{)}\text{ by
Theorem~{}\ref{th1}},$ $\displaystyle p\odot q$ $\displaystyle\leq
A_{1}\big{(}A_{2}(p)\odot q\big{)}\text{ by transitivity},$ $\displaystyle p$
$\displaystyle\leq q\rightarrow A_{1}\big{(}A_{2}(p)\odot q\big{)}\text{ by
Proposition~{}\ref{prop1}}.$

The other statements follow in an analogous way.

  2. (ii)

follows from $B(p\odot q)\leq B\big{(}A(p)\odot q\big{)}\leq A(p)\odot q$,

  3. (iii)

follows from $B(p)\odot q\leq A\big{(}B(p)\odot q\big{)}\leq A(p\odot q)$ by
applying Proposition 4.1,

  4. (iv)

follows from $B_{1}\big{(}B_{2}(p)\odot q\big{)}\leq B_{1}(p\odot q)\leq
p\odot q$,

  5. (v)

follows from $p\rightarrow q\leq A_{1}(p\rightarrow q)\leq
A_{1}\big{(}p\rightarrow A_{2}(q)\big{)}$,

  6. (vi)

follows from $B(p\rightarrow q)\leq B\big{(}p\rightarrow A(q)\big{)}\leq
p\rightarrow A(q)$ by applying Proposition 4.1,

  7. (vii)

follows from $p\rightarrow B(q)\leq A\big{(}p\rightarrow B(q)\big{)}\leq
A(p\rightarrow q)$,

  8. (viii)

follows from $B_{1}\big{(}p\rightarrow B_{2}(q)\big{)}\leq B_{1}(p\rightarrow
q)\leq p\rightarrow q$ by applying Proposition 4.1.

∎

##  5 A construction of the time frame

A tense logic is established if for a given logic a time frame $(T,R)$ exists
such that the lattice together with the tense operators forms a dynamic
algebra and the logical connectives are related with tense operators in the
way shown in Section 4. Hence, if such a logic incorporating time dimension is
created, we can define tense operators $P$, $F$, $H$ and $G$. The question is
whether, conversely, for given time set $T$ and given tense operators there
exists a suitable time frame $(T,R)$ such that the given tense operators are
derived from it. In other words, we ask if for given tense operators $P$, $F$,
$H$ and $G$ on a time set $T$ one can find some time preference relation $R$
such that these operators are induced by $(T,R)$. To show that this is
possible is the goal of Section 5.

If $P$, $F$, $H$ and $G$ are tense operators on a complete lattice
$(L,\vee,\wedge,0,1)$ with time set $T$ then the relations

$\displaystyle R_{1}$ $\displaystyle:=\\{(s,t)\in T^{2}\mid q(s)\leq
P(q)(t)\text{ and }q(t)\leq F(q)(s)\text{ for all }q\in L^{T}\\},$
$\displaystyle R_{2}$ $\displaystyle:=\\{(s,t)\in T^{2}\mid H(q)(t)\leq
q(s)\text{ and }G(q)(s)\leq q(t)\text{ for all }q\in L^{T}\\},$ $\displaystyle
R_{3}$ $\displaystyle:=R_{1}\cap R_{2}$

are called the relation induced by $P$ and $F$, the relation induced by $H$
and $G$ and the relation induced by $P$, $F$, $H$ and $G$, respectively.

Observe that whenever tense operators $P$, $F$, $H$ and $G$ on a complete
lattice $\mathbf{L}$ are induced by an arbitrarily given time frame then
$(\mathbf{L},P,F,H,G)$ forms a dynamic algebra, and if, moreover, $\mathbf{L}$
is a complete orthomodular lattice then Theorems 4.3 and 4.4 hold for these
operators.

At first we show the relationship between given tense operators $P$ and $F$
and the corresponding operators $P^{*}$ and $F^{*}$ induced by the time frame
$(T,R)$ where $R$ is induced by $P$ and $F$.

######  Theorem 5.1.

Let $P$ and $F$ be tense operators on a complete lattice $(L,\vee,\wedge,0,1)$
with time set $T$, $R$ denote the relation induced by these operators and
$P^{*}$ and $F^{*}$ denote the tense operators induced by the time frame
$(T,R)$. Then $P^{*}\leq P$ and $F^{*}\leq F$.

###### Proof.

If $q\in L^{T}$ and $s\in T$ then

$\displaystyle P^{*}(q)(s)$ $\displaystyle=\bigvee\\{q(t)\mid
t\mathrel{R}s\\}\leq P(q)(s),$ $\displaystyle F^{*}(q)(s)$
$\displaystyle=\bigvee\\{q(t)\mid s\mathrel{R}t\\}\leq F(q)(s).$

∎

Analogously, one can prove

######  Theorem 5.2.

Let $H$ and $G$ be tense operators on a complete lattice $(L,\vee,\wedge,0,1)$
with time set $T$, $R$ denote the relation induced by these operators and
$H^{*}$ and $G^{*}$ denote the tense operators induced by the time frame
$(T,R)$. Then $H\leq H^{*}$ and $G\leq G^{*}$.

From Theorems 5.1 and 5.2 we obtain

######  Corollary 5.3.

Let $P$, $F$, $H$ and $G$ be tense operators on a complete lattice
$(L,\vee,\wedge,0,1)$ with time set $T$, $R$ denote the relation induced by
these operators and $P^{*}$, $F^{*}$, $H^{*}$ and $G^{*}$ denote the tense
operators induced by the time frame $(T,R)$. Then $P^{*}\leq P$, $F^{*}\leq
F$, $H\leq H^{*}$ and $G\leq G^{*}$.

######  Example 5.4.

Consider the lattice $\mathbf{L}$ and the time set $T=\\{1,2,3,4,5\\}$ from
Example 2.1. Define new tense operators $P$, $F$, $H$ and $G$ as follows:

$P(q)(t):=\left\\{\begin{array}[]{ll}q(t)&\text{if }t=2,\\\
1&\text{otherwise}\end{array}\right.\quad\quad\quad
F(q)(t):=\left\\{\begin{array}[]{ll}q(t)&\text{if }t=1,\\\
1&\text{otherwise}\end{array}\right.$
$H(q)(t):=\left\\{\begin{array}[]{ll}q(t)&\text{if }t=1,\\\
0&\text{otherwise}\end{array}\right.\quad\quad\quad
G(q)(t):=\left\\{\begin{array}[]{ll}q(t)&\text{if }t=2,\\\
0&\text{otherwise}\end{array}\right.$

for all $q\in L^{T}$ and all $t\in T$. Note that these operators satisfy the
conditions

$H(q)\leq q\leq P(q)\text{ and }G(q)\leq q\leq F(q)$

for all $q\in L^{T}$ which were considered in Theorem 3.3. Let $R$ denote the
relation induced by $P$, $F$, $H$ and $G$. Then
$R=\\{1\\}^{2}\cup\\{2\\}^{2}\cup\\{3,4,5\\}^{2}$. This can be seen as
follows: Obviously, $\\{1\\}^{2}\cup\\{2\\}^{2}\cup\\{3,4,5\\}^{2}\subseteq
R$. Now let $(s,t)\in R$.
$s=1\neq t$ would imply $q(t)\leq F(q)(1)=q(1)$ for all $q\in L^{T}$, a
contradiction.
$s=2\neq t$ would imply $q(2)=G(q)(2)\leq q(t)$ for all $q\in L^{T}$, a
contradiction.
$s\neq 1=t$ would imply $q(1)=H(q)(1)\leq q(s)$ for all $q\in L^{T}$, a
contradiction.
$s\neq 2=t$ would imply $q(s)\leq P(q)(2)=q(2)$ for all $q\in L^{T}$, a
contradiction.
This shows $R=\\{1\\}^{2}\cup\\{2\\}^{2}\cup\\{3,4,5\\}^{2}$. For $p$ from
Example 2.1 we have

$\begin{array}[]{l|l|l|l|l|l}t&1&2&3&4&5\\\ \hline\cr
p(t)&c^{\prime}&b^{\prime}&c^{\prime}&a^{\prime}&b^{\prime}\\\
P(p)(t)&1&b^{\prime}&1&1&1\\\ F(p)(t)&c^{\prime}&1&1&1&1\\\
P^{*}(p)(t)&c^{\prime}&b^{\prime}&1&1&1\\\
F^{*}(p)(t)&c^{\prime}&b^{\prime}&1&1&1\end{array}$

showing $P^{*}\leq P$ and $F^{*}\leq F$ in accordance with Corollary 5.3, but
$P^{*}\neq P$ and $F^{*}\neq F$, thus this inequality is strict.

######  Remark 5.5.

Although the new tense operators $P^{*}$, $F^{*}$, $H^{*}$ and $G^{*}$
constructed as shown in Corollary 5.3 satisfy only the inequalities $P^{*}\leq
P$, $F^{*}\leq F$, $H\leq H^{*}$ and $G\leq G^{*}$ and, by Example 5.4, these
inequalities may be strict, it is almost evident from the construction of
these operators that $(\mathbf{L},P^{*},F^{*},H^{*},G^{*})$ forms a dynamic
algebra and that these operators are connected with the logical connectives
$\odot$ and $\rightarrow$ in the way shown in Theorems 4.3 and 4.4 provided
$\mathbf{L}$ is a complete orthomodular lattice.

Conversely, if a time frame $(T,R)$ on a complete lattice is given and we
consider the tense operators $P$, $F$, $H$ and $G$ induced by $(T,R)$ then the
relation induced by these operators coincides with $R$, see the following
result.

######  Theorem 5.6.

Let $(L,\vee,\wedge,0,1)$ be a complete lattice, $(T,R)$ a time frame, $P$,
$F$, $H$ and $G$ denote the tense operators induced by $(T,R)$ and $R^{*}$
denote the relation induced by these operators. Then $R=R^{*}$ and hence the
tense operators induced by the time frame $(T,R^{*})$ coincide with those
induced by the time frame $(T,R)$.

###### Proof.

If $s\mathrel{R}t$ then

$\displaystyle H(q)(t)$ $\displaystyle=\bigwedge\\{q(u)\mid
u\mathrel{R}t\\}\leq q(s)\leq\bigvee\\{q(u)\mid u\mathrel{R}t\\}=P(q)(t),$
$\displaystyle G(q)(s)$ $\displaystyle=\bigwedge\\{q(u)\mid
s\mathrel{R}u\\}\leq q(t)\leq\bigvee\\{q(u)\mid s\mathrel{R}u\\}=F(q)(s)$

for all $q\in L^{T}$ and hence $s\mathrel{R^{*}}t$. This shows $R\subseteq
R^{*}$. Now assume $R\neq R^{*}$. Then there exists some $(s,t)\in
R^{*}\setminus R$. For every $u\in T$ let $q_{u}$ denote the following element
of $L^{T}$:

$q_{u}(t):=\left\\{\begin{array}[]{ll}1&\text{if }t=u,\\\
0&\text{otherwise}\end{array}\right.$

($t\in T$). Now we would obtain

$1=q_{s}(s)\leq P(q_{s})(t)=\bigvee\\{q_{s}(u)\mid
u\mathrel{R}t\\}=\bigvee\\{0\mid u\mathrel{R}t\\}=0,$

a contradiction. This shows $R=R^{*}$. ∎

######  Remark 5.7.

Assume that tense operators $P$, $F$, $H$ and $G$ on a complete lattice with
time set $T$ are given. We want to know if such operators are induced by a
suitable time frame with possibly unknown time preference relation. We
construct the relation $R$ on $T$ induced by these operators and then we
construct the tense operators $P^{*}$, $F^{*}$, $H^{*}$ and $G^{*}$ induced by
the time frame $(T,R)$. Now two cases can happen: Either $P^{*}=P$, $F^{*}=F$,
$H^{*}=H$ and $G^{*}=G$ or at least one of these equalities is violated, it
means it is a proper inequality. In the first case the given operators $P$,
$F$, $H$ and $G$ are induced by the time frame $(T,R)$ whereas in the second
case $P$, $F$, $H$ and $G$ are not induced by any time frame because of
Theorems 5.6.

In Theorem 5.6 we showed that if a complete lattice $(L,\vee,\wedge,0,1)$ and
a time frame $(T,R)$ are given and $P$, $F$, $H$ and $G$ denote the tense
operators induced by this time frame then the relation $R^{*}$ induced by
these operators coincides with $R$. If, conversely, the tense operators $P$
and $F$ are given on a complete lattice with a given time set $T$, we can ask
whether we can construct a relation inducing these operators. In Theorem 5.1
we showed that if $R$ is induced by given tense operators $P$ and $F$ on a
given time set $T$ in the complete lattice $\mathbf{L}$ then the operators
$P^{*}$ and $F^{*}$ induced by the time frame $(T,R)$ need not coincide with
$P$ and $F$, respectively, they satisfy only the inequalities $P^{*}\leq P$
and $F^{*}\leq F$. However, such tense operators $P^{*}$ and $F^{*}$ are still
related with the logical connectives $\odot$ and $\rightarrow$ as shown in
Theorems 4.3 and 4.4 provided the complete lattice $\mathbf{L}$ is
orthomodular. We are going to show that the given time set $T$ can be extended
to some set $\bar{T}$ and $R$ can be extended to some binary relation
$\bar{R}$ on $\bar{T}$ such that the tense operators induced by the time frame
$(\bar{T},\bar{R})$ can be considered in some sense as extensions of the given
tense operators $P$ and $F$, respectively. Put

$\bar{T}:=T_{1}\cup T\cup T_{2}\text{ where }T_{1}:=T\times\\{1\\}\text{ and
}T_{2}:=T\times\\{2\\}.$ (2)

We extend our “world” $L^{T}$ by adding two of its copies, so-called “parallel
worlds”, namely the “past” $L^{T_{1}}$ and the “future” $L^{T_{2}}$. In this
way we obtain our “new world” $L^{\bar{T}}$ over the extended time set
$\bar{T}$. We also extend our time depending propositions $q\in L^{T}$ to
$\bar{q}\in L^{\bar{T}}$ by defining

$\begin{array}[]{l}\bar{q}\big{(}(s,1)\big{)}:=P(q)(s),\\\
\bar{q}(s):=q(s),\\\ \bar{q}\big{(}(s,2)\big{)}:=F(q)(s)\\\ \text{for all
}s\in T.\end{array}$ (3)

Now we show that the given operators $P$ and $F$ can be considered in some
sense as restrictions of the operators $\bar{P}$ and $\bar{F}$ induced by the
time frame $(\bar{T},\bar{R})$, respectively.

######  Theorem 5.8.

Let $P$ and $F$ be tense operators on a complete lattice $(L,\vee,\wedge,0,1)$
with time set $T$ and $R$ denote the relation induced by these operators.
Define $\bar{T}$ by (2), put

$\bar{R}:=\\{\big{(}(s,1),s\big{)}\mid s\in T\\}\cup
R\cup\\{\big{(}s,(s,2)\big{)}\mid s\in T\\}.$

and let $\bar{P}$ and $\bar{F}$ denote the tense operators induced by the time
frame $(\bar{T},\bar{R})$. Moreover, for every $q\in L^{T}$ let $\bar{q}\in
L^{\bar{T}}$ denote the extension of $q$ defined by (3). Then $\bar{R}|T=R$
and

$\big{(}\bar{P}(\bar{q})\big{)}|T=P(q)\text{ and
}\big{(}\bar{F}(\bar{q})\big{)}|T=F(q)$

for all $q\in L^{T}$.

###### Proof.

We have $\bar{R}|T=\bar{R}\cap T^{2}=R$. If $q\in L^{T}$ and $s\in T$ then
$q(t)\leq P(q)(s)$ for all $t\in T$ with $t\mathrel{R}s$ and hence
$\bigvee\\{q(t)\mid t\mathrel{R}s\\}\leq P(q)(s)$ which implies

$\displaystyle\bar{P}(\bar{q})(s)$
$\displaystyle=\bigvee\\{\bar{q}(\bar{t})\mid\bar{t}\bar{R}s\\}=\bar{q}\big{(}(s,1)\big{)}\vee\bigvee\\{\bar{q}(t)\mid
t\mathrel{R}s\\}=P(q)(s)\vee\bigvee\\{q(t)\mid t\mathrel{R}s\\}=$
$\displaystyle=P(q)(s)$

showing $\big{(}\bar{P}(\bar{q})\big{)}|T=P(q)$. Analogously, one can prove
$\big{(}\bar{F}(\bar{q})\big{)}|T=F(q)$. ∎

An analogous result holds for $H$ and $G$ instead of $P$ and $F$,
respectively, but the extensions of $q\in L^{T}$ to $\bar{q}\in L^{\bar{T}}$
must be slightly modified.

######  Theorem 5.9.

Let $H$ and $G$ be tense operators on a complete lattice $(L,\vee,\wedge,0,1)$
with time set $T$ and $R$ denote the relation induced by these operators.
Define $\bar{T}$ by (2), put

$\bar{R}:=\\{\big{(}(s,1),s\big{)}\mid s\in T\\}\cup
R\cup\\{\big{(}s,(s,2)\big{)}\mid s\in T\\}.$

and let $\bar{H}$ and $\bar{G}$ denote the tense operators induced by the time
frame $(\bar{T},\bar{R})$. Moreover, for every $q\in L^{T}$ let $\bar{q}\in
L^{\bar{T}}$ denote the extension of $q$ defined by

$\displaystyle\bar{q}\big{(}(s,1)\big{)}$ $\displaystyle:=H(q)(s),$
$\displaystyle\bar{q}(s)$ $\displaystyle:=q(s),$
$\displaystyle\bar{q}\big{(}(s,2)\big{)}$ $\displaystyle:=G(q)(s)$

for all $s\in T$. Then $\bar{R}|T=R$ and

$\big{(}\bar{H}(\bar{q})\big{)}|T=H(q)\text{ and
}\big{(}\bar{G}(\bar{q})\big{)}|T=G(q)$

for all $q\in L^{T}$.

######  Example 5.10.

Consider the time set $T$, the proposition $p$ and the tense operators $P$ and
$F$ from Example 2.1 and write $ti$ instead of $(t,i)$ for $t\in T$ and
$i=1,2$. Let $R$ denote the relation induced by $P$ and $F$. Then

$\displaystyle R$ $\displaystyle=\\{(s,t)\in\\{1,2,3,4,5\\}^{2}\mid s\leq
t\\},$ $\displaystyle\bar{R}$ $\displaystyle=\\{(s1,s)\mid s\in T\\}\cup
R\cup\\{(s,s2)\mid s\in T\\}.$

Let $\bar{P}$ and $\bar{F}$ denote the tense operators induced by the time
frame $(\bar{T},\bar{R})$. Then we have

$\begin{array}[]{l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l}\bar{t}&11&21&31&41&51&1&2&3&4&5&12&22&32&42&52\\\
\hline\cr
p(t)&&&&&&c^{\prime}&b^{\prime}&c^{\prime}&a^{\prime}&b^{\prime}&&&&&\\\
P(p)(t)&&&&&&c^{\prime}&1&1&1&1&&&&&\\\
F(p)(t)&&&&&&1&1&1&1&b^{\prime}&&&&&\\\
\bar{p}(\bar{t})&c^{\prime}&1&1&1&1&c^{\prime}&b^{\prime}&c^{\prime}&a^{\prime}&b^{\prime}&1&1&1&1&b^{\prime}\\\
\bar{P}(\bar{p})(\bar{t})&0&0&0&0&0&c^{\prime}&1&1&1&1&c^{\prime}&b^{\prime}&c^{\prime}&a^{\prime}&b^{\prime}\\\
\bar{F}(\bar{p})(\bar{t})&c^{\prime}&b^{\prime}&c^{\prime}&a^{\prime}&b^{\prime}&1&1&1&1&b^{\prime}&0&0&0&0&0\end{array}$

where

$T_{1}=\\{11,21,31,41,51\\},T=\\{1,2,3,4,5\\}\text{ and
}T_{2}=\\{12,22,32,42,52\\}.$

Evidently, $\bar{R}|T=R$, $\big{(}\bar{P}(\bar{q})\big{)}|T=P(q)$ and
$\big{(}\bar{F}(\bar{q})\big{)}|T=F(q)$ in accordance with Theorem 5.8.

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Authors’ addresses:

Ivan Chajda
Palacký University Olomouc
Faculty of Science
Department of Algebra and Geometry
17\. listopadu 12
771 46 Olomouc
Czech Republic
<EMAIL_ADDRESS>
Helmut Länger
TU Wien
Faculty of Mathematics and Geoinformation
Institute of Discrete Mathematics and Geometry
Wiedner Hauptstraße 8-10
1040 Vienna
Austria, and
Palacký University Olomouc
Faculty of Science
Department of Algebra and Geometry
17\. listopadu 12
771 46 Olomouc
Czech Republic
<EMAIL_ADDRESS>