# Tests of gravity theories with pulsar timing

Paulo C. C. Freire  Max-Planck-Institut für Radioastronomie, 69 Auf dem Hügel,
D-53121, Bonn, Germany

###### Abstract

Over the last few years, a set of new results from pulsar timing has
introduced much tighter constraints on violations of the strong equivalence
principle (SEP), either via a direct verification of the universality of free
fall for a pulsar in a triple star system, or from tests of the nature of
gravitational waves, in particular a search for dipolar gravitational wave
emission in a variety of binary pulsars with different masses. No deviations
from the SEP have been detected in our experiments. These results introduce
some of the most stringent constraints on several classes of alternative
theories of gravity and complement recent results from the ground-based
gravitational wave detectors.

##  1 Timing binary pulsars

The 1974 discovery of the first binary pulsar (PSR B1913+16) by Russell Hulse
and Joseph Taylor [1] offered us a powerful new tool for understanding gravity
and spacetime. The system consists of two neutron stars (NSs), one of them a
radio pulsar with a spin period of 59 ms, its orbital period is 7h 45m and
eccentricity is 0.617. This high eccentrivity means that the longitude of
periastron, $\omega$ \- the point where the pulsar is closer to its companion
- can be measured precisely. Subsequent timing of this system measured three
relativistic effects on the orbit: the increase in $\omega$ with time,
$\dot{\omega}$ (which is analogous to the relativistic perihelion shift in
Mercury), the so-called “Einstein delay”, a slow-down of the pulsar spin near
periastron caused by special relativistic time dilation and gravitational
redshift ($\gamma_{\rm E}$), and the decay of the orbit, caused by loss of
orbital energy ($\dot{P}_{\rm b}$). Each effect depends on the two unknown
masses of the system, the dependence is specified by a particular theory of
gravity. The self-consistency of the masses obtained from the measured
relativistic effects using the equations of general relativity (GR) means that
this theory passed the test posed by the measurement of $\dot{\omega}$,
$\gamma_{\rm E}$ and $\dot{P}_{\rm b}$ in the PSR B1913+16 system. The loss of
orbital energy associated with the latter parameter is caused by the emission
of gravitational waves (GW); the fact that the measured $\dot{P}_{\rm b}$
agrees with GR’s predictions for the PSR B1913+16 system confirmed
experimentally the existence of GWs [2, 3, 4, 5]. This was a major impulse to
the start of the era of GW astronomy.

##  2 Recent results

Currently, we know more than 3300 pulsars in our Galaxy, the associated
globular cluster system and also in the Magellanic clouds [6]. Of these, 10%
are in binary systems. Despite this, measurements of relativistic parameters
are rare: only 40 of them have had the component masses measured to a relative
precison better than 15%, with an additional 17 systems where the total system
mass has been measured via $\dot{\omega}$. Of all these binaries, 17 systems
(including B1913+16) have been confirmed as double neutron star systems (DNSs)
from strong stellar evolution arguments, with an additional eight
candidates111For a complete, up-to-date lists on binary pulsar mass
measurements and a list of pulsars in DNS systems see https://www3.mpifr-
bonn.mpg.de/staff/pfreire/NS_masses.html..

###  2.1 The Amazing “Double Pulsar” system

One discovery, that of PSR J0737$-$3039A [7], was an extremely lucky find, for
several reasons. First, the system happened to be, at the time, the most
compact DNS known (orbital period of 2h 27m, $e=0.0878$); thus all
relativistic effects observed in PSR B1913+16 ($\dot{\omega}$, $\gamma_{\rm
E}$ and $\dot{P}_{\rm b}$) can also be measured very precisely in this system.
Second, the system happens to have the highest inclination known for any
binary pulsar, $89.35^{\circ}$! This means that further independent and
precise mass constraints can be obtained from the Shapiro delay in the arrival
times of the pulses. Third, the NS companion has also been detected as a radio
pulsar, PSR J0737$-$3039B; the measurement of the sizes of both orbits yields
the NS mass ratio [8].

This system now includes seven independent and precise strong-field tests of
GR, which the theory passes with flying colours [9]. Some highlights:

  * •

The most precise test of the GR quadrupole formula for GW emission: the
measured orbital decay ($-$39.3770 $\pm$ 0.0025 microseconds per year, the
uncertainty is a 68.3% confidence limit) agrees with the GR prediction within
the tiny relative uncertainty of $6.3\times 10^{-5}$. This is 25 times more
precise than for the Hulse-Taylor binary!

  * •

They also include detection, for the first time, the detection of the effect
of light bending in the timing of the pulsar. This experiment probes the
propagation of radiation in a spacetime that has $10^{6}$ times or more the
curvature of any non-pulsar experiments.

  * •

In addition, there are also important second-order effects on the periastron
advance. One of these is the Lense-Thirring effect, the frame dragging caused
by the rotation of pulsar A. The latter effect will, when measured more
precisely, be important for measuring the moment of inertia of pulsar A.

The precise measurement of the mass and moment of inertia of a neutron star
will represent powerful constraints on the largely unknown properties of
super-dense matter in the centres of neutron stars, in particular the equation
of state of dense matter (EOS) [10, 9].

###  2.2 Asymmetric binaries

The strong equivalence principle (SEP) is the assertion that the outcome of
any local experiment, including any gravitational experiments, in a freely
falling laboratory is independent of the velocity of the laboratory and its
location in spacetime. There are suggestive arguments that GR is the only
theory in four spacetime dimensions that fully embodies the SEP [11, 12], with
all other viable alternatives violating it at some level.

Such a violation would have observable consequences, chief among these a
violation of the universality of free fall (UFF). One of the resulting effects
would be the emission of dipolar GWs (DGW) [13]. Detecting DGW emission would
falsity GR, because in GR GWs are, to leading order, quadrupolar; not
detecting it will necessarily constrain many alternative theories of gravity.

DGW emission would be observable in a binary pulsar as an orbital decay that
is faster than that predicted by GR. Although the double pulsar system yields
a very precise test of the GR quadrupole formula, it does not by itself
exclude DGW emission. This happens for two reasons: first, the latter is,
within certain gravity theories, enhanced by the difference in compactness of
the two components in the system; for this reason, pulsar - WD systems can be
of great value for this enterprise. Second, DGW emission can be greatly
enhanced for NS masses larger than those of the components of the double
pulsar system [14].

For these reasons, we have over the years conducted orbital decay experiments
for a variety of binary systems: PSR J1738+0333 [15], PSR J0348+0432 [16], PSR
J1909$-$3744 [17], PSR J1012+5307 [18], and PSR J2222$-$0137 [19], all pulsar
- WD systems. We have also been timing PSR J1913+1102, an asymmetric DNS [20].
Their timing yields tight limits on DGW emission that are valid for the
observered range of NS masses and are independent of the EOS.

###  2.3 The pulsar in a triple star system

In 2014, the first pulsar in a triple star system, PSR J0337+1715, was
discovered [21]. Such systems were expected to provide some of the most
stringent tests of the UFF because, if NSs and WDs accelerate differently in
the gravitational field of the outer WD, the inner PSR - WD pair in the system
would show “orbital polarization” [22] known as the Nordtvedt effect [23].

The suitability of PSR J0337+1715 for this test was demonstrated in 2018, with
the finding that NSs and WDs fall with the same acceleration within 2.6 parts
in $10^{6}$ [24]. A more recent study, using a completely independent data set
(from the Nançay telescope) and an improved timing analysis and theoretical
treatment of the three-body problem in alternative theories of gravity has
found a lower level of systematic noise, and that NSs and WDs fall at the same
rate within 2 parts in $10^{6}$; this is statistical 95% confidence limit
[25]. An update on the timing of this system is provided by G. Voisin in these
proceedings.

##  3 Consequences of these experiments

Figure 1:  Left: Comparison between the cumulative shift in periastron time
predicted by GR (in red) and our measurements (in blue), which are consistent
with the former within measurement uncertainty. Right: Limits on a set of
parameters that quantify deviations from the GR prediction for GW emission at
successive “Post-Newtonian” orders, derived from the double pulsar timing
(red) and different LIGO-Virgo observations (blue and black). The results from
the double pulsar are three orders of magnitude more constraining for the
leading order term. From Kramer et al. (2021).

Figure 2: Left: Constraints on the Damour - Esposito-Farèse scalar-tensor
theories of gravity (DEF gravity) [14]. The double pulsar and triple system
introduce the tightest limits in different regions of the DEF parameter space.
In the right plot, we see a mass-mass diagram obtained with a member of
Bekenstein’s 2004 family of Scalar-Vector-Tensor theories of gravity. The
theory cannot produce a pair of masses that explains all observed relativistic
effects in the double pulsar system and at the same time produce the MOND
phenomenon in the rotation of galaxies; the theory is therefore excluded. From
Kramer et al. (2021).

Figure 3: Limits on scalar coupling for different pulsars (triangles) and for
DEF gravity, using different EOSs (coloured lines). The double pulsar does not
constrain the possibility that more massive NSs might scalarise very
significantly; this was excluded by timing systems with more massive NSs. From
Zhao et al. (2022).

In all experiments mentioned above, we have tested GR to very high precision,
especially in the case of the double pulsar. This can be seen in Fig. 1, where
we see in the left panel the comparison between the cumulative shift in
periastron time predicted by GR and our measurements, and in the right panel a
comparison of constraints on deviations of GW emission from the GR prediction
at successive PN orders[26] for the double pulsar and LIGO/Virgo observations
[9]. Clearly, these experiments provide complementary information, with the
double pulsar experiment providing more constraining limits (by three orders
of magnitude) for the leading order term!

We have not detected any of the consequences of UFF violation, either DGW
emission or the Nordtvedt effect for the pulsar in the triple star system. As
we can see in Fig. 2, these non-detections result in tight constrains on
several classes of alternative gravity theories [9].

Within some of these theories, there is a phenomenon called “spontaneous
scalarization”, where within a range of NS masses, the NS can acquire a very
significant scalar “charge” [14], greatly enhancing DGW emission. This
phenomenon cannot be fully probed by the double pulsar test, because of the
low mass of the NSs in that system (see left panel of Fig. 3). Since DGW
emission is not observed for any of the binaries we have been timing, which
cover the full range of observed NS masses, we can put tight constraints on
spontaneous scalarisation for all observed NS masses that are independent of
the EOS of dense matter, at least for some specific classes of gravity
theories [27] (see right panel in Fig. 3).

## Acknowledgments

I thank the Max-Planck-Gesellschaft and the European Research Council for
their support, and especially Michael Kramer and Norbert Wex for the wonderful
collaboration over the years.

## References

## References

  * [1] R. Hulse & J. H. Taylor The Astrophysical Journal 195, L51 (1975).
  * [2] J. H. Taylor & J. M. Weisberg The Astrophysical Journal 253, 908 (1982).
  * [3] J. H. Taylor & J. M. Weisberg The Astrophysical Journal 435, 434 (1989).
  * [4] T. Damour & J. H. Taylor Phys. Rev. D 45, 1840 (1992).
  * [5] J. M. Weisberg & Y. Huang The Astrophysical Journal 829, 55 (2016).
  * [6] R. N. Manchester, G. B. Hobbs, A. Teoh, M. Hobbs The Astronomical Journal 129, 1993 (2005).
  * [7] M. Burgay, N. D’Amico, A. Possenti, et al. Nature 426, 531 (2003).
  * [8] A. G. Lyne, M. Burgay, M. Kramer et al. Science 303, 1153 (2004).
  * [9] M. Kramer, I. H. Stairs, R. N. Manchester et al. Phys. Rev. X 11, 041050 (2021).
  * [10] H. Hu, M. Kramer, N. Wex et al. Month. Not. Roy. Ast. Soc. 497, 3118 (2020).
  * [11] E. Di Casola, S. Liberati & S. Sonego S. American Journal of Physics 83, 39 (2015).
  * [12] C. Will in Theory and experiment in gravitational physics, (Cambridge University press, Cambridge, UK, 2018).
  * [13] D. M. Eardley The Astrophysical Journal Letters 196, L59 (1975).
  * [14] T. Damour & G. Esposito-Farèse Phys. Rev. Lett. 70, 2220 (1993).
  * [15] P. C. C. Freire, N. Wex, G. Esposito-Farèse et al. Month. Not. Roy. Ast. Soc. 423, 3328 (2012).
  * [16] J. Antoniadis, P. C. C. Freire, N. Wex et al. Science 340, 448 (2013).
  * [17] K. Liu, L. Guillemot, A. G. Istrate et al. Month. Not. Roy. Ast. Soc. 499, 2276 (2020).
  * [18] H. Ding, A. T. Deller, P. C. C. Freire et al. The Astrophysical Journal 896, 85 (2020).
  * [19] Y. J. Guo, P. C. C. Freire. L. Guillemot et al. Astronomy & Astrophysics 654, A16 (2021).
  * [20] R. D. Ferdman, P. C. C. Freire, B. B. P. Perera et al. Nature 583, 211 (2020).
  * [21] S. M. Ransom, I. H. Stairs, A. M. Archibald et al. Nature 505, 520 (2014).
  * [22] P. C. C. Freire, M. Kramer & N. Wex Class. Quant. Gravity 29, 184007 (2012).
  * [23] K. Nordtvedt Physical Review 169, 1014 (1968).
  * [24] A. M. Archibald, N. V. , J. H. T. Hessels et al. Nature 559, 73 (2018)
  * [25] G. Voisin, I. Cognard, P. C. C. Freire et al. Astronomy & Astrophysics 638, A24 (2020)
  * [26] B. P. Abbott, R. Abbott, T. Abbott et al. Phys. Rev. Lett. 116, 221101 (2016)
  * [27] J. Zhao, P. C. C. Freire, M. Kramer et al. Class. Quant. Gravity , accepted, see arXiv:2201.03771 (2022).


# Effective Mass of $\alpha$-Cluster in 12C Nucleus

I. Filikhin  A. Karoui  and B. Vlahovic  CREST, North Carolina Central
University, Durham, NC 27707, USA
<EMAIL_ADDRESS>
(Day Month Year; Day Month Year)

###### Abstract

Based on the effective-mass concept, we perform the Faddeev calculations for a
low-lying spectrum of 3$\alpha$ states in 12C nucleus. A three-body potential
is used to describe the known breaking of the 3$\alpha$-cluster structure in
the nucleus. We show that the contribution of the three-body potential to the
Hamiltonian can be compensated by increasing/decreasing the $\alpha$-particle
free mass. The effective-mass values are adjusted so that to reproduce the
experimental data for the 12C nucleus. The energy dependence of the effective
mass and the correlation to a three-body potential are discussed. We show that
the coupling between the $0^{+}$ ($2^{+}$) levels forms a specific picture of
anti-crossing on the energy/effective-mass plane.

###### keywords:

Cluster models; Effective mass; Faddeev equation

PACS numbers: 21.60.G, 71.18.+y, 11.80.Jy

##  1  Introduction

The effective-mass approach for three-particle systems was proposed in Ref.
[1] based on the effect of a mass-energy compensation that takes place for a
three-body Hamiltonian. This compensation manifests the mass-energy
equivalence in bound nuclear systems. It was shown in Ref. [1] that a decrease
in kinetic energy can be compensated by an increase in the attractive
contribution of potential energy. The corresponding calculations for 3H and
3He nuclei have been performed using the Faddeev method in the configuration
space. The nucleon effective mass was defined to compensate for a contribution
of three-body nuclear potential. The potential is required to reproduce 3$N$
experimental data using phenomenological nucleon-nucleon potentials. The
simulations have shown that the results of this efficient model can be
comparable to those based on sequential consideration of the three-body force
(3BF). The evaluations for kinetic energy and pair-potential terms of the
Hamiltonian have been compared to ones of different authors [2, 3, 4, 5] for
3H nucleus applying the Argonne AV14 potential and different three-body
potentials. It was found that the nucleon effective mass have slightly
increased from the free-mass value that the ratio $m^{*}/m$ of effective mass
$m^{*}$ and free mass $m$ is 1.017. This relatively small correction reflects
the contribution of a three-body force in the 3$N$ Hamiltonian.

We have to note that the idea of effective mass is widely applied in solid-
state physics [8]. For example, in Ref. [9], the effective mass of an electron
in InAs is 0.024$m_{0}$, while in GaAs is 0.064$m_{0}$, where $m_{0}$ is the
mass of a free electron. In nuclear physics and astrophysics, the effective
mass method is used in the study of neutron stars [10] within the more general
one-particle method of many-body physics [11]. According to the consideration,
an individual particle (nucleon) is in a complex interaction with the
environment (other nucleons in nuclei), which affects the particle dynamics
[12, 13], and the one-particle potential is considered to be energy dependent
[14].

Applying the effective mass method, we consider a 3$\alpha$-cluster system
simulating low-lying spectral states of the 12C nucleus. The $\alpha$-particle
model has been proposed almost a century ago (see, for example, [15]) based on
the molecular properties of low-lying 12C states. This molecular aspect has
been studied within the framework of several variational approaches [16]. In
Refs. [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33,
34], the cluster approach has been used to treat the 3$\alpha$ problem with
different inter-cluster potentials, phenomenologically obtained. However, the
cluster model has failed in a detailed description of the 12C spectrum. The
main problem is that this approach does not reproduce the ground state energy
of the 12C nucleus taking into account only $\alpha$-$\alpha$ potential. To
reach an admissible description of experimental data via cluster calculations,
one needs to add a three-body potential [26, 23, 27, 29]. However, the
potential would be different for different spectral levels of the 3$\alpha$
system. Moreover, a three-body potential can be repulsive [24], [26, 30] or
attractive one [23, 33]. That depends on the choosing of a two-body potential
model. For example, a repulsive 3$\alpha$ potential was phenomenologically
introduced in Ref. [32] with the dependence on the total angular momentum
$J$=$0^{+}$, $2^{+}$, and $4^{+}$. The potential has the Gaussian form with
the depths $V(0^{+})$=31.7 MeV, $V(2^{+})$=63.0 MeV, and $V(4^{+})$=150.0 MeV
[34], respectively. An attractive three-body potential has been proposed in
Ref. [31]. In corresponding 3$\alpha$-cluster model, the phenomenological pair
potential having $s$, $d$, and $g$ partial components was used from Ref. [35].

In the present work, we implement the cluster model from Ref. [31]. However,
the current model does not include the three-body potential. We introduce an
$\alpha$-cluster effective mass that effectively substitutes the contribution
of the three-body potential. In our calculations, an adjustment of effective-
mass values has to reproduce the experimental energy for considering levels.
The effective mass is assumed to be a function of energy and orbital momentum
of the 3$\alpha$ state. To solve the corresponding three-body problem, we
apply the method of the Faddeev equations in configuration space [36]. That
allows us to take into account the Coulomb interaction in a rigorous manner
[31].

##  2 Model

###  2.1 Mass-energy compensation: $3N$ system

In this section, we consider the $AAA$ model for the 3H nucleus, where the
masses of protons and neutrons are the same, and the particles in the system
are identical. The averaged nucleon mass $m$ was chosen as in Ref. [4] as
$m_{N}$=938.9 MeV (or $\hbar^{2}/m_{N}$ = 41.473 MeV fm2). Our calculations
for the ground state were performed using the charge-independent AV14
potential [6]. The calculated value for the 3H binding energy is equal to
-7.684 MeV. Without three-body potential, the 3$N$ system is weakly bound, and
the difference between our AV14 results and experimental values $\Delta
E=E^{Cal.}-E_{exp.}$ is 0.798 MeV. One can find many examples of corresponding
calculations in the literature [3, 4, 5] given without and with a three-body
potential. The three-body Hamiltonian reads

$H=H_{0}+V_{2bf}+V_{3bf},$ (1)

where $H_{0}$ is three-body kinetic operator, $V_{2bf}$ ($V_{3bf}$) is two-
body (three-body) interactions. Here, we assume that the three-body force term
is a perturbation. Other factors of the nucleon-nucleon interaction like the
charge symmetry breaking or electro-magnetics terms (see Ref. [7]) are not
considered in this model. Following Ref. [1], we define the effective nucleon
mass $m^{*}$ in a nucleus. The effective mass can be obtained as a correction
to the averaged free nucleon mass to reproduce the experimental value of the
ground state energy. The possibility of the procedure was shown in Ref. [1]
due to the the mass-energy compensation effect takes a place for the three-
body Hamiltonian. To explain this effect, we consider the Hamiltonian written
in following form:

$H=\beta H_{0}+\alpha V_{2bf},$ (2)

where $\beta,\alpha>0$. We can change the mass and the depth of potential
simultaneously by using parameters $\alpha$ and $\beta$ defining the scaling
for the nucleon mass by the relation $m^{*}=m_{N}/\beta$. The corresponding
energy changes are oppositely directed and can be compensated by each other.
Similar compensation is possible for a three-body force as a part of the
potential term.

Fig. 1 illustrates the calculations for the ground state using the Hamiltonian
(2). Increasing the nucleon mass leads to an increase in the binding energy of
the three-body system. The energy $E(\alpha,\beta)$ is a linear function near
the cross point $\alpha=\beta=1$. The compensation has been interpreted in
Ref. [1] as a manifestation of the mass and energy equivalence of applied to
three-nucleon system.

Figure 1:  The ground state energy $E$ of 3H as a function of the parameter
$\alpha$ (the line $E(\alpha$)) or $\beta$ (the line $E(\beta$)) calculated
with the AV14 $NN$ potential and corresponding to the Hamiltonian $H=\beta
H_{0}+\alpha V$, where $\beta,\alpha>0$, and $m\to m/\beta$ in the $H_{0}$.
The linear fits for the calculated values are shown. The effective mass of
nucleon $m^{*}_{N}/m_{N}$ is equal to 1.017 for $\alpha=\beta=1$.

The dependence of the 3H ground state energy on the effective mass $m^{*}/m$
could be applied to an evaluate the nucleon effective mass. We calculate the
energy using the AV14 $NN$ potential. The results of the calculations fitted
by linear functions are presented in Fig. 2. One can obtain the same linear
dependence using the results published in Ref. [37] for the $NN$ CB-Bonn
potential scaled by a parameter. Alternatively, such energy dependence on the
potential scaling one can repeat by nucleon mass scaling [1]. In this way, the
effective mass of nucleon $m^{*}/m$ is about 1.017. That corresponds to the
experimental value for the ground state energy of about -8.48 MeV. In Fig. 2,
we show this correspondence by the vertical and horizontal dashed lines.

It has to note that the result of the calculation for the 3$N$ wave function
within the proposed effective approach is comparable with the results of 3$N$
realistic models from the literature.

Figure 2:  The ground state energy $E$ of 3H as a function of the effective
mass $m^{*}/m$. The solid circles are the results of the calculations (the
AV14 $NN$ potential). The fine red line corresponds to the linear fit for the
calculated values. The dashed lines correspond to the effective mass of
nucleon $m^{*}/m$ about 1.017 and the experimental value for the ground state
energy about -8.48 MeV.

###  2.2 $\alpha$-cluster model for 12C: effective-mass extension

The $\alpha$-cluster model describes the 12C nucleus as a three-body system of
$\alpha$-particles. The particles interact with pair and three-body nuclear
potentials and the Coulomb force. In the present work, we use the formalism of
Ref. [31] based on the Faddeev equations in configuration space. Also, we
apply phenomenological Ali-Bodmer (AB) $\alpha$-$\alpha$ potential [35] having
different $s$, $d$ and $g$ partial components which are written as two-range
Gaussian:

$V(x)=V_{1}\exp(-\frac{x^{2}}{\beta_{1}^{2}})+V_{2}\exp(-\frac{x^{2}}{\beta_{2}^{2}})$

where the strength and range parameters are $V_{1}$=500 MeV, $\beta_{1}$=1.43
fm, $V_{2}$=-130 MeV, $\beta_{2}$=2.11 fm for $s$-wave; $V_{1}$=320 MeV,
$\beta_{1}$=1.43 fm, $V_{2}$=-130 MeV, $\beta_{2}$=2.11 fm for $d$-wave;
$V_{1}$=0, $V_{2}$=-130 MeV, $\beta_{2}$=2.11 fm for $g$-wave. The attractive
$s$-wave component of the potential simulates the Pauli blocking.

The starting point of our approach is related to the kinetic, 2-body, and
3-body potentials terms of the 3-$\alpha$ Hamiltonian. The 2-body potential
must reproduce two-body experimental data. Three-body potential can be found
by 3-$\alpha$ data. The Ali-Bodmer-type potentials satisfy this request[31].
The potential has simple coordinate dependence with a repulsive core in $s$
partial wave. The combination AB +3$\alpha$ potentials obtained in Ref. [31]
describes the low-lying 12C spectrum. The dependence of the 3-$\alpha$
potentials on total orbital momentum was detected. Thus, an additional
adjustment for the 3BF potential will reproduce the spectrum with good
accuracy. Note here that our goal is to present the general scheme of the
effective mass definition for the three-body system. The main assumption of
the approach is that a three-body potential acts in the system. Our choice for
the pair potential is based on the model proposed in Ref. [31] by two
coauthors of the presented work.

The angular momentum configurations {$(l,\lambda)$}, where ${\bf J}={\bf
l}+{\bf{\lambda}}$ is total orbital momentum, ${\bf l}$ is momentum of a pair
of particles, ${\bf{\lambda}}$ is the relative momentum of third particle. One
may choose the quantum number sets[31] $(0,0)(2,2)(4,4)$ for the $0^{+}$ band
and $(2,0)(0,2)(2,2)(2,4)(4,2)(4,4)$ for the $2^{+}$ band. The
$(4,0)(0,4)(2,2)(2,4)(4,2)(4,4)$ configuration of angular momenta for the
$4^{+}$ states and the $(0,1)(2,1)(2,3)(4,3)(4,5)$ configuration for $1^{-}$
state.

The known breaking of cluster structure is described by a three-body potential
which acts at short distances [38]. According to the effective-mass concept
presented in the previous section, the contribution of the three-particle
potential can be compensated by a variation of free mass of
$\alpha$-particles, $m_{\alpha}$. One achieves compensation when the
calculated energy is equal to the experimental value. The effective mass
$m^{*}_{\alpha}$ can be smaller or larger than the free mass depending on the
energy level. Obviously, there is a correlation between depth of three-body
potential and effective mass value.

It may be noted that one of the reasons [42, 43] for the violation of the
cluster structure of 12C is the significant mixing of the $p_{3/2}$ subshell
closure component[42] to the pure 3$\alpha$-cluster wave function. This mixing
generates additional terms in the Hamiltonian. We consider the additional
terms as a form of three-body force. In Ref.[42], such breaking of the
$\alpha$-cluster structure affects all considered states of the $0^{+}$ and
$2^{+}$ bands like a three-body force. Particularly, in Ref.[42], a coupling
between the $0^{+}$ states was discussed. Below we present the evidence for
such coupling, obtained in the framework of the effective-mass approach.

The attractive three-body potential $V_{3}(\rho)$ having the Gaussian form was
defined in Ref. [31]:

$V_{3}(\rho)=V\exp(-\frac{\rho^{2}}{2\beta^{2}}),$ (3)

where $\frac{1}{2}\rho^{2}=\sum_{i=1}^{i=3}{\bf r}_{i}^{2}$ and $\bf{r}_{i}$
is the position vector of ${i}$-th $\alpha$ particle relative to the center of
mass of the system, $V$=-31.935483 MeV, $\beta$ = 3.315 fm. We use this
potential as an auxiliary one for calculations of resonances. It should be
emphasized that the potential (3) does not affect the two-particle threshold
of three-particle systems [31].

##  3 Numerical results

Based on the effective-mass approach, we calculate the low-lying spectrum of a
3$\alpha$-cluster system using the $\alpha$-$\alpha$ Ali-Bodmer potential of
the ”d” version [35]. The results of our calculations are presented in Table.
1. Here, the effective mass $m_{\alpha}^{*}$ is adjusted to reproduce the
experimental energy of the first two levels of the $0^{+}$, $2^{+}$, $4^{+}$
bands, and $3_{1}^{-}$ and $1_{1}^{-}$ levels. The bound states are the states
$0^{+}_{1}$ and $2^{+}_{1}$, and the remaining states are resonances. In order
to calculate the energy of resonances, we used the analog [39] of the method
of continuation in coupled constant [40, 41]. In our study, the effective mass
was considered as a coupled constant. To estimate the energy of the
resonances, we calculate a set of bound state energies obtained with different
effective-mass values.

This evaluation procedure is shown in Fig. 2 for the 3H bound state. The
simple linear interpolation to the positive energy region allows us to
evaluate effective mass by correspondence to the experimental value for
energy.

Table 1: The effective mass $m^{*}_{\alpha}/m_{\alpha}$ with relation to the low-lying levels of the 12C. The calculated energies for each level is shown. The energy is measured from the three-body $\alpha+\alpha+\alpha$ threshold.  12C | $0^{+}_{1}$ | $0^{+}_{2}$ | $2^{+}_{1}$ | $2^{+}_{2}$ | $4^{+}_{1}$ | $4^{+}_{2}$ | $3^{-}_{1}$ | $1^{-}_{1}$
---|---|---|---|---|---|---|---|---
$E$ (MeV) | -7.275 | 0.286 | -2.93 | 2.25 | 6.05 | 6.81 | 2.4 | 3.6
$m^{*}_{\alpha}/m_{\alpha}$ | 1.290 | 1.176 | 1.15 | 1.11 | 0.92 | 1.17 | 1.05 | 1.19

In the case of the 3$\alpha$-cluster system, the binding energy and effective
mass correspondence can be more complicated. The results of our calculations
for the first levels of the $0^{+}$ and $2^{+}$ bands are resented in Fig. 3.
One can see that the energy/effective-mass dependences are non-linear. We
applied a polynomial function to interpolate and extrapolate to negative and
positive energies. The physical reason for the non-linear behavior is the
coupling between the levels. Note that the effective mass adjusted by the
0${}^{+}_{1}$ ground state energy ($m_{\alpha}^{*}/m_{\alpha}$=1.29) does not
lead to the experimental energy for the $0^{+}_{2}$ state. This situation is
shown in Fig. 3 (Upper panel). To obtain the experimental $0^{+}_{2}$ value,
one must reduce the effective mass adjusted for the 0${}^{+}_{1}$ energy to
1.176. The same situation exists for the $2^{+}$ states: the effective mass of
the $2^{+}_{2}$ state is less than one for the $2^{+}_{1}$ state. We explain
this statement in Fig. 3 (Lower panel).

Figure 3:  The energy $E$ of the states $J^{+}_{1}$ (red solid circles) and
$J^{+}_{2}$ (blue solid circles), $J=0,2$ and $0^{+}_{3}$ (open circles) of
$3\alpha$-cluster system along of the $\alpha$-particle effective mass
$m_{\alpha}^{*}/m_{\alpha}$. The left panel (right panel) corresponds to
$0^{+}$ ($2^{+}$ ) band. The experimental data are shown by the dashed
horizontal lines. The vertical lines represent the corresponding values of the
effective masses. The energy is measured from a three-body
$\alpha+\alpha+\alpha$ threshold.

The effective mass and three-body potential depth are correlated due to the
initial assumption of our approach. Thus, one can assume a dependence of
effective-mass value on total orbital momentum. In particular, the effect of
the 3$\alpha$ potential in the $2^{+}$ states must be smaller than one in the
$0^{+}$ states [34]. For example, the three-body potential proposed in Ref.
[31] reproduces well low-lying $0^{+}$ states of the nucleus. However, the
same potential applied to the calculations of bound $2^{+}_{1}$ and the
resonance $2^{+}_{2}$ states makes overbounds for the energies of the levels.
An additional adjustment for the parameters of 3BF is needed for 2+ states.
Thus, the hypothesis of the dependence of 3BF on the total orbital momentum
can be considered a probable one.

Comparing the effective mass for $0^{+}_{1}$ and $2^{+}_{1}$ states, one may
detect an energy dependence or an orbital momentum dependence of the effective
mass. This comparison is illustrated by Fig. 4, where we show the effective
mass ($m^{*}/m$) variations for the $J=0\to 2$ vector using data from Tabl. 1.
The mass variation can be represented by the form: $\Delta
m^{*}=({\delta}m^{*}/{\delta}J)\Delta J+({\delta}m^{*}/{\delta}E)\Delta E$
assuming a dependence of effective mass on orbital momentum $J$ and energy
$E$. One can show that $({\delta}m^{*}/{\delta}J)=0$ and the effective mass
depends on the energy only. The variations of the effective mass for different
energies when the orbital momentum is fixed $J^{+}_{1}$ ($J^{+}_{2}$), $J=0,2$
(see Fig. 3) can be described by the form: $\Delta
m^{*}=({\delta}m^{*}/{\delta}E)\Delta E$. In Fig. 4, the corresponding results
are shown by open circles and squares. The results obtained for both
representations coincide. Thus, the effective mass does not have essential
orbital momentum dependence for $0^{+}_{n}$ and $2^{+}_{n}$ states when
$n=1,2$.

Figure 4:  The effective mass ($m^{*}/m$) (solid lines) as a function of
energy of levels given for the vector $J=0\to 2$. The red and blue lines
correspond to lower and upper members of the coupled states ($J^{+}_{1}$ and
$J^{+}_{2}$). The open circles (open squares) show the results of our
calculations for the states $J^{+}_{1}$ (red) ($J^{+}_{2}$, (blue)), $J=0,2$
(see Fig. 3). The energy is measured from the three-body
$\alpha+\alpha+\alpha$ threshold.

In addition, it can be seen that the dependence of the effective mass on
energy can be represented as a linear function. For states $J^{+}_{n}$, where
$J=0.2$ and $n=1.2$, the slopes of the functions are approximately equal. At
the same time, the linear functions are different in constant for the states
$n=1$ and $n=2$.

Figure 5:  The energy $E$ of the low lying levels $J^{+}_{1}$ (red solid and
open circles) and $J^{+}_{2}$ (blue solid and open squares), $J=0,2$ of the
$3\alpha$-cluster system along of the $\alpha$-particle effective mass
$m_{\alpha}^{*}/m_{\alpha}$. The Upper panel (Lower panel) corresponds to
$0^{+}$ ($2^{+}$ ) band. The assumed level crossing is schematically shown by
intersecting thin lines for the case when there is no coupling between the
levels. The three-body potential Eq. (3) was added to the model. The energy is
measured from the $\alpha+\alpha+\alpha$ threshold.

The results shown in Fig. 3 can be interpreted in the terminology of the two-
level theory as anti-crossings of levels [44]. The $n=1$ and $n=2$ states are
the lower and upper states of a quasi-doublet formed by the anti-crossed
levels. To clarify this assumption, we performed calculations with the three-
body potential (3). This potential does not affect the two-body
$\alpha-\alpha$ threshold, and we can make an extension for allowable energy
and effective-mass scopes to apply the numerical procedure proposed above for
bound states. The results of the calculation are given in Fig. 5. The three-
body potential increases the depth of the potential well where are the bound
states $J^{+}_{1}$, $J^{+}_{2}$, $J^{+}_{3}$, $J^{+}_{4}$, $J=0,2$. One can
see that the anti-crossing of the levels occurs for the pairs $0^{+}_{1}$ and
$0^{+}_{2}$, $0^{+}_{2}$ and $0^{+}_{3}$, $0^{+}_{3}$ and $0^{+}_{4}$. In the
$2^{+}$ states, the similar situation is visible for the pairs $2^{+}_{1}$ and
$2^{+}_{2}$, $2^{+}_{3}$ and $2^{+}_{4}$. According to the two-level system
theory, we can definite upper member and lower member of the corresponding
quasi-doublets. In the absence of a coupling between levels, the levels may
cross. The assumed crossing of the levels is schematically presented by
crossing fine lines in Fig. 5. One can assume that the level anti-crossings
will be saved in the continuous spectrum of the nucleus. Thus, our study is
extended to the states of the continuous spectrum.

Figure 6:  The energy $E$ of the 12C low lying levels and the
$\alpha$-particle effective mass $m_{\alpha}^{*}/m_{\alpha}$. (Upper panel)
The results for the levels $4^{+}_{1}$ and $4^{+}_{2}$ (solid and open
circles) are given. (Lower panel) The results for $3^{-}_{1}$ (solid circles)
and $1^{-}_{1}$ (solid squares) states are presented. The experimental data
are shown by the solid/dashed horizontal lines. The vertical lines represent
corresponding values of the effective masses. The fine solid lines are given
for eyes to show a linear extrapolation. The energy is measured from a three-
body $\alpha+\alpha+\alpha$ threshold.

We can conclude that there is tunneling coupling between the $0^{+}$ levels
and between the $2^{+}$ levels. At the same time, the levels with different
$J$ are not coupled due to the different rotation symmetry of these states.
The coupling parameter depends on overlapping wave functions[45, 46] and is
small for these states. The $4^{+}_{1}$ and $4^{+}_{2}$ levels do not
demonstrate also a coupling. The results of our calculations for the levels
$4^{+}_{1}$ and $4^{+}_{2}$ are given in Fig. 6(Upper panel). The effective
masses for these levels have a significant difference, 0.92$m_{\alpha}$ and
1.17$m_{\alpha}$. That gives different contributions of the kinetic energy to
the Hamiltonians and provides the difference in coordinate behavior of the
wave functions. One can assume that the last difference arises from the effect
of coordinate scaling due to the mass difference. The asymptotic behavior of
these functions differs only in the mass coefficient due to the closeness of
the energies of the levels.

The energy/effective mass dependence is a simple linear function with
different slopes for the $4^{+}_{1}$ and $4^{+}_{2}$ levels. One can interpret
that the corresponding three-body potential must have different depths in the
$4^{+}_{1}$ and $4^{+}_{2}$ states to reproduce the experimental data for
these levels. Thus, the results of [34], in which the three-body potential
acts with the same depth for different $4^{+}$ levels, can be explained within
the framework of the effective-mass approach.

The results for the states $3^{-}_{1}$ and $1^{-}_{1}$ are shown in Fig.
6(Lower panel). Also, a linear dependence of energy on the effective mass has
been found for these states. There is no coupling between the levels due to
different rotational symmetries and significant differences in the effective
mass.

##  4 Conclusions

The imperfectness of $\alpha$-cluster structure of the 12C nucleus can be
described with a three-body force. Based on the effect of mass-energy
compensation, the three-body force can be taken into account in a
phenomenological manner using the effective-mass approach.

We have presented the results of the Faddeev calculations for the bound 3$N$
system to demonstrate the mechanism of mass-energy compensation for a three-
body Hamiltonian. Based on this compensation effect, we have proposed the
effective-mass approach which was initially applied for a bound few-body
system. In the presented work, this approach is assumed to be applicable to
describe the resonances states. Based on this assumption, we have defined the
effective mass of the $\alpha$-cluster in the 12C. The corresponding mass
adjustment was performed to reproduce the experimental data for several low-
lying states available with the Ali-Bodmer $\alpha$-$\alpha$ potential.

For the $0^{+}$ (and $2^{+}$) band, we have found that there is a coupling
between pairs of nearest levels. We describe these coupled levels as quasi-
doublets following the terminology of the theory of two-level systems [44].
The anti-crossings picture, which was obtained on the energy/effective-mass
plane, is interpreted within this theory. According to this interpretation,
the coupling matrix elements of the quasi-doublets are non-zero due to
overlapping the wave functions of the nearest levels. The last one is
possible, for example, when these levels have the same orbital momentum and
similar rotation symmetry. We have found that the coupling takes place for the
pairs of the levels $0^{+}_{1}$ and $0^{+}_{2}$, $0^{+}_{2}$ and $0^{+}_{3}$,
$0^{+}_{3}$ and $0^{+}_{4}$. Also, we have found that the structure of the
$2^{+}$ band repeats the one for the $0^{+}$ band. Along with that, the levels
with different orbital momenta like the $0^{+}$ and $2^{+}$ levels are not
coupled. However, the levels $4^{+}_{1}$ and $4^{+}_{2}$ are not coupled also
despite the same orbital momenta. Here, the orbital momentum similarity of
wave functions is broken by differences in the kinetic term in the Hamiltonian
due to the significant difference in the effective masses. The similar
explanation one can give for $3^{-}_{1}$ and $1^{-}_{1}$ resonances. There is
no coupling due to different rotation symmetry and significant differences in
the effective masses.

Within our approach, an effective-mass value is correlated with the strength
of a three-body potential. However, this approach does not support the
assumption about the direct dependence of the effective mass (and 3BF) on only
the total orbital momentum used previously within the approximations [31, 34]
for three-body potentials. We conclude that 3BF has to be separately defined
for each energy level as it was earlier concluded in Ref. [47]. Note that, the
effective mass of the $0^{+}_{1}$ ground state is the largest in the
considered 3$\alpha$ spectrum. In other states, the mass of the
$\alpha$-cluster increases or decreases relative to the value of free mass.

## Acknowledgments

This work is supported by the National Science Foundation Centers of Research
Excellence in Science and Technology NSF CREST 1647022 and, the Department of
Energy/National Nuclear Security Administration award NA0003979, NSF EiR award
number DMR-2101220.

## References

  * [1] I. Filikhin, V. M. Suslov, B. Vlahovic, arXiv:2112.13827 [nucl-th]
  * [2] M. Kamimura and H. Kameyama, Nuclear Physics A 508 (1990) 17.
  * [3] R. Chen, G. L. Payne, J. L. Friar and B. F. Gibson, Phys. Rev. C 31 (1985) 2266 .
  * [4] A. Nogga, D. Hüber, H Kamada and W. Glöckle, Phys. Lett. B 409 (1997) 19.
  * [5] A. Kievsky, M. Viviani, S. Rossati, Nucl. Phys. A 577 (1994) 511.
  * [6] R.B. Wiringa, R.A. Smith, T. L. Ainsworth, Phys. Rev. C 29 (1984) 1207.
  * [7] A. Nogga, A. Kievsky, H. Kamada, W. Glöckle, L. E. Marcucci,S. Rosati, and M. Viviani, Phys. Rev. C 67 (2003) 034004.
  * [8] C. Kittel, Introduction to Solid State Physics (8th Edition, Wiley, 2004).
  * [9] I. Filikhin, V. M. Suslov, B. Vlahovic, Phys. Rev. B 73 (2006) 205332.
  * [10] Li B-An, Cai B-Jun, Chen L-Wen, J. Xu, Progress in Particle and Nuclear Physics 99 (2018) 29.
  * [11] P. Coleman, Introduction to Many-Body Physics (Cambridge: Cambridge University Press, 2015).
  * [12] L. D. Landau, S. I. Pekar, Zh. Eksp. Teor. Fiz. 18 (1948) 419.
  * [13] K. A. Brueckner, Phys. Rev.  97 (1955) 1353.
  * [14] J. P. Jeukenne, A. Lejeune, C. Mahaux, Phys. Rep. 25 (1976) 83.
  * [15] J. A. Wheeler, Phys. Rev. 52 (1937) 1083.
  * [16] R. Bijker, F. Iachello, Nucl. Phys. A 957 (2017) 154.
  * [17] D. R. Harrington, Phys. Rev. 147 (1966) 685.
  * [18] Y. Fujiwara and R. Tamagaki, Prog. Theor. Phys. 56 (1976) 1503.
  * [19] M. Vallieres, H. T. Coelho and T. K. Das, Nucl. Phys. A 271 (1976) 95.
  * [20] O. Portilho, S. A. Coon, Z. Physik A 290 (1979) 93.
  * [21] M. Kamimura, Nucl. Phys. A 351 (1981) 456.
  * [22] O. Portilho, D. A. Agrello, S. A. Coon, Phys. Rev. C 27 (1983) 2923.
  * [23] D. V. Fedorov, A. S. Jensen, Phys. Lett. B 389 (1996) 631.
  * [24] H. Kamada H and S. Oryu, Prog. Theor. Phys. 76 (1986) 1260.
  * [25] R. Pichter, H. Oberhummer, A. Csoto, S. A. Moszkowski, Nucl. Phys. A  618 (1997) 55.
  * [26] E. Hiyama, M. Kamimura, T. Motoba, T. Yamada, Y. Yamomoto, Progr. Theor. Phys. 97 (1997) 881.
  * [27] I. Filikhin, Phys. Atom. Nucl. 63 (2000) 1527.
  * [28] Y. Fujiwara, K. Miyagawa, M. Kohno, Y. Suzuki, D. Baye, J.-M. Sparenberg, Phys. Rev. C 70 (2004) 024002.
  * [29] S. I. Fedotov, O. I. Kartavtzev, V. I. Kochkin, A. V. Malykh, Phys. Rev. C 70 (2004) 014006.
  * [30] C. Kurokawa and K. Kato, Phys. Rev. C 76 (2005) 021301-1.
  * [31] I. Filikhin, V. M. Suslov and B. Vlahovic, J. Phys. G: Nucl. Part. Phys. 31 (2005) 1207.
  * [32] C. Kurokawa and K. Kato, Nucl. Phys. A 792 (2007) 82.
  * [33] S. Ishikawa, Journal of Physics: Conference Series 312 (2011) 042010.
  * [34] Sh-I. Ohtsubo, Y. Fukushima, M. Kamimura, and E. Hiyama, Prog. Theor. Exp. Phys. 2013 (2013) 073D02.
  * [35] S. Ali, A. R. Bodmer, Nucl. Phys. 80 (1966) 99.
  * [36] S. P. Merkuriev, L. D. Faddeev, Quantum Scattering Theory for Several Particle Systems (Kluwer Academic, Dordrecht, 1993).
  * [37] H. Witala, J. Golak, R. Skibiński, K. Topolnicki, E. Epelbaum, H. Krebs, and P. Reinert, Phys. Rev. C 104 (2021) 014002.
  * [38] S. I. Fedotov, O. I. Kartavtseva, and A. V. Malykh, Eur. Phys. J. A 26 (2005) 201.
  * [39] I. Filikhin, V. M. Suslov and B. Vlahovic, FIZ. B (Zagreb) 20 (2011) 189.
  * [40] V. I. Kukulin and V. M. Krasnopol’sky, J. Phys. A: Math. Gen. 10 (1977) L33.
  * [41] V. I. Kukulin, V. M. Krasnopolsky and J. Horacek, Theory of Resonances (Kluwer, Dordrecht, 1989).
  * [42] T. Suhara, Y. Kanada-En’yo, Phys. Rev. C 91 (2015) 024315.
  * [43] M. Freer, H. Horiuchi, Y. Kanada-En’yo, D. Lee and Ulf-G. Meißner, Rev. Mod. Phys. 90 (2018) 035004.
  * [44] C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics, Vol. 1 (Wiley VCH, 1992), ISBN-10: 0471569526.
  * [45] I. Filikhin, S. G. Matinyan, and B. Vlahovic, Sens. Transducers  183 (2014) 116.
  * [46] I. Filikhin, S. G. Matinyan, and B. Vlahovic, Rev. Theor. Sci. 3 (2015) 1.
  * [47] D. V. Fedorov, private communication.


# Inductive limits of ideals

Adam Kwela  Institute of Mathematics
Faculty of Mathematics
Physics and Informatics
University of Gdańsk
ul. Wita Stwosza 57
80-308 Gdańsk
Poland<EMAIL_ADDRESS>http://kwela.strony.ug.edu.pl/

###### Abstract.

G. Debs and J. Saint Raymond in 2009 defined the Borel separation rank of an
analytic ideal $\mathcal{I}$ ($\text{rk}(\mathcal{I})$) as minimal ordinal
$\alpha<\omega_{1}$ such that there is
$\mathcal{S}\in\bf{\Sigma^{0}_{1+\alpha}}$ with
$\mathcal{I}\subseteq\mathcal{S}$ and
$\mathcal{I}^{\star}\cap\mathcal{S}=\emptyset$, where $\mathcal{I}^{\star}$ is
the filter dual to the ideal $\mathcal{I}$ (actually, the authors use the dual
notion of filters instead of ideals). Moreover, they introduced ideals
$\text{Fin}_{\alpha}$, for all $\alpha<\omega_{1}$, and conjectured that
$\text{rk}(\mathcal{I})\geq\alpha$ if and only if $\mathcal{I}$ contains an
isomorphic copy of $\text{Fin}_{\alpha}$
($\text{Fin}_{\alpha}\sqsubseteq\mathcal{I}$). To define $\text{Fin}_{\alpha}$
in the case of limit ordinals $0<\alpha<\omega_{1}$, G. Debs and J. Saint
Raymond introduced inductive limits of ideals.

We show that the above conjecture is false in the case of $\alpha=\omega$ by
constructing an ideal $\text{Fin}^{\prime}_{\omega}$ of rank $\omega$ such
that $\text{Fin}_{\omega}\not\sqsubseteq\text{Fin}^{\prime}_{\omega}$.
However, we show that $\text{Fin}^{\prime}_{\omega}\sqsubseteq\mathcal{I}$ is
equivalent to $\forall_{n\in\omega}\text{Fin}_{n}\sqsubseteq\mathcal{I}$. We
discuss (indicated by the above result) possible modification of the original
conjecture for limit ordinals.

###### Key words and phrases:

Ideals, filters, inductive limits of ideals, inductive limits of filters, rank
of analytic ideals, rank of analytic filters, Katětov order of ideals

###### 2010 Mathematics Subject Classification:

Primary: 03E05, 03E15.

##  1\. Introduction

Following G. Debs and J. Saint Raymond [3], for an analytic ideal
$\mathcal{I}$ we define its Borel separation rank by:

${\rm rk}(\mathcal{I})=\min\left\\{\alpha<\omega_{1}:\textrm{ there is
}\mathcal{S}\in\bf{\Sigma^{0}_{1+\alpha}}\textrm{ such that
}\mathcal{I}\subseteq\mathcal{S}\textrm{ and
}\mathcal{I}^{\star}\cap\mathcal{S}=\emptyset\right\\},$

where $\mathcal{I}^{\star}$ is the filter dual to the ideal $\mathcal{I}$
(actually, authors of [3] use the dual notion of filters instead of ideals).
In this paper by rank of $\mathcal{I}$ we mean ${\rm rk}(\mathcal{I})$.

In [3] and [8] it is shown that ranks of analytic ideals are important for
studying ideal pointwise limits of sequences of continuous functions: ${\rm
rk}(\mathcal{I})=\alpha$ if and only if the family of all
$\mathcal{I}$-pointwise limits of continuous real-valued functions defined on
a given zero-dimensional Polish space is equal to the family of all functions
of Borel class $\alpha$ (the definition of $\mathcal{I}$-pointwise limit can
be found in [3] or [8]). For more on ranks of ideals see [1], [2] or [6].

G. Debs and J. Saint Raymond defined ideals $\text{Fin}_{\alpha}$, for
$0<\alpha<\omega_{1}$ (for definition see Subsection 2.4). To define
$\text{Fin}_{\alpha}$ in the case of limit ordinals $0<\alpha<\omega_{1}$,
they introduced inductive limits of ideals. The main motivation of this paper
is the conjecture of G. Debs and J. Saint Raymond from 2009:

######  Conjecture 1.1 ([3, Conjecture 7.8]).

Let $\mathcal{I}$ be an analytic ideal. Then
$\text{rk}(\mathcal{I})\geq\alpha$ if and only if $\mathcal{I}$ contains an
isomorphic copy of $\text{Fin}_{\alpha}$.

It is known that this is true for $\alpha=1,2$ ([3, Theorem 7.5] and [8,
Theorem 4]). Moreover, the implication $\Leftarrow$ is true in general.

Let us point out a connection of the above conjecture to descriptive
complexity of ideals. Each ideal $\mathrm{Fin}_{n}$, for $n\in\omega$, is
$\bf{\Sigma^{0}_{2n}}$. By a result of G. Debs and J. Saint Raymond from [2],
any ideal $\mathcal{I}$ containing an isomorphic copy of $\mathrm{Fin}_{n}$
cannot be $\bf{\Pi^{0}_{2n}}$. Thus, Conjecture 1.1 would lead to an
interesting result, by giving lower estimate of Borel complexity of ideals of
a given finite rank.

What is more, Conjecture 1.1 would imply that we can skip the assumption about
zero-dimensionality in the result mentioned in the second paragraph of this
Introduction and obtain the following result: ${\rm rk}(\mathcal{I})=\alpha$
if and only if the family of all $\mathcal{I}$-pointwise limits of continuous
real-valued functions defined on a given Polish space is equal to the family
of all functions of Borel class $\alpha$ (cf. [3, Corollary 7.6]).

In this paper we show that Conjecture 1.1 is false in the case of
$\alpha=\omega$ by constructing an ideal $\text{Fin}^{\prime}_{\omega}$ of
rank $\omega$ which does not contain an isomorphic copy of
$\text{Fin}_{\omega}$. However, we discuss a way of modifying the original
conjecture for limit ordinals.

The paper is organized as follows. In Section 2 we collect some definitions
needed in our further considerations. Section 3 contains a short study of the
problem when an inductive limit of ideals is an ideal – we show that
$\mathrm{Fin}_{\alpha}$, for all limit ordinals $0<\alpha<\omega_{1}$, are
ideals despite the fact that [3, Proposition-Definition 5.4], which defines
them, is false. Our main result showing that Conjecture 1.1 is false for
$\alpha=\omega$ is proved in Section 4. Finally, in Section 5 we show that an
ideal $\mathcal{I}$ contains an isomorphic copy of
$\mathrm{Fin}^{\prime}_{\omega}$ if and only if $\mathcal{I}$ contains
isomorphic copies of all $\mathrm{Fin}_{n}$, for $n\in\omega$. This result
enables us to discuss a possible modification of Conjecture 1.1.

##  2\. Preliminaries

###  2.1. Basic notions

A collection $\mathcal{I}$ of subsets of a set $X$ is called an ideal on $X$
if it is closed under subsets and finite unions of its elements. In this case
we denote the set $X$ by $\text{dom}(\mathcal{I})$. All ideals considered in
this paper are defined on infinite countable sets. We assume additionally that
$\mathcal{P}(X)$ is not an ideal, and that every ideal contains all finite
subsets of $X$. By $\mathrm{Fin}$ we denote the ideal of all finite subsets of
$\omega$.

We treat the power set $\mathcal{P}(X)$ as the space $2^{X}$ of all functions
$f:X\rightarrow 2$ (equipped with the product topology, where each space
$2=\left\\{0,1\right\\}$ carries the discrete topology) by identifying subsets
of $X$ with their characteristic functions. Thus, we can talk about
descriptive complexity of subsets of $\mathcal{P}(X)$ (in particular, ideals
on $X$).

For $\mathcal{A}\subseteq\mathcal{P}(X)$ we write

$\mathcal{A}^{\star}=\\{X\setminus A:\ A\in\mathcal{A}\\}.$

If $\mathcal{I}$ is an ideal on $X$ then $\mathcal{I}^{\star}$ is a filter
(i.e., a family closed under supersets and finite intersections of its
elements). We call it the dual filter of $\mathcal{I}$. Observe that the map
$A\mapsto X\setminus A$ is continuous. Thus, $\mathcal{I}$ and
$\mathcal{I}^{\star}$ have the same descriptive complexity. The above
observation enables us to transfer the results of [3] into the language of
ideals.

If $\mathcal{I}$ and $\mathcal{J}$ are ideals, then:

  * •

we say that $\mathcal{J}$ and $\mathcal{I}$ are isomorphic, if there is a
bijection $f:\text{dom}(\mathcal{J})\to\text{dom}(\mathcal{I})$ such that

$\forall_{A\subseteq\text{dom}(\mathcal{I})}\ f^{-1}[A]\in\mathcal{J}\
\Longleftrightarrow\ A\in\mathcal{I};$

  * •

we say that $\mathcal{J}$ contains an isomorphic copy of $\mathcal{I}$ and
write $\mathcal{I}\sqsubseteq\mathcal{J}$, if there is a bijection
$f:\text{dom}(\mathcal{J})\to\text{dom}(\mathcal{I})$ such that
$f^{-1}[A]\in\mathcal{J}$ for each $A\in\mathcal{I}$;

  * •

we say that $\mathcal{J}$ is above $\mathcal{I}$ in the Katětov order and
write $\mathcal{I}\leq_{K}\mathcal{J}$, if there is
$f:\text{dom}(\mathcal{J})\to\text{dom}(\mathcal{I})$ (not necessary
bijection) such that $f^{-1}[A]\in\mathcal{J}$ for each $A\in\mathcal{I}$.

We say that an ideal $\mathcal{I}$ on $X$ is generated by the family
$\mathcal{F}\subseteq\mathcal{P}(X)$ if

$\mathcal{I}=\\{A\subseteq X:\ \exists_{n\in\omega}\
\exists_{F_{0},\ldots,F_{n}\in\mathcal{F}}\ A\subseteq F_{0}\cup\ldots\cup
F_{n}\\}.$

If $Y\notin\mathcal{I}$ then the restriction of $\mathcal{I}$ to $Y$ is an
ideal on $Y$ given by $\mathcal{I}|Y=\\{A\cap Y:\ A\in\mathcal{I}\\}$.

###  2.2. Fubini sums of ideals

If $\left(X_{i}\right)_{i\in I}$ is a family of sets, then by $\sum_{i\in
I}X_{i}$ we denote its disjoint sum, i.e., the set of all pairs $(i,x)$ where
$i\in I$ and $x\in X_{i}$.

Following [3], if $\mathcal{I}$ and $\mathcal{J}_{i}$, for
$i\in\text{dom}(\mathcal{I})$, are ideals, then the $\mathcal{I}$-Fubini sum
of $(\mathcal{J}_{i})_{i\in\text{dom}(\mathcal{I})}$ is an ideal on
$\sum_{i\in\text{dom}(\mathcal{I})}\text{dom}(\mathcal{J}_{i})$ defined by:

$\mathcal{I}-\sum_{i\in\text{dom}(\mathcal{I})}\mathcal{J}_{i}=\left\\{\sum_{i\in\text{dom}(\mathcal{I})}A_{i}:\
\left\\{i\in\text{dom}(\mathcal{I}):\
A_{i}\notin\mathcal{J}_{i}\right\\}\in\mathcal{I}\right\\}.$

If $\mathcal{J}_{i}=\mathcal{J}$ for all $i\in\text{dom}(\mathcal{I})$ then
$\mathcal{I}\otimes\mathcal{J}=\mathcal{I}-\sum_{i\in\text{dom}(\mathcal{I})}\mathcal{J}_{i}$
is an ideal on $\text{dom}(\mathcal{I})\times\text{dom}(\mathcal{J})$ called
the Fubini product of ideal $\mathcal{I}$ and $\mathcal{J}$.

In [5, p. 240] M. Katětov defined ideals $\mathrm{Fin}^{\alpha}$, for
$0<\alpha<\omega_{1}$ by:

  * •

$\mathrm{Fin}^{1}=\mathrm{Fin}$, $\text{dom}(\mathrm{Fin}^{1})=\omega$;

  * •

$\mathrm{Fin}^{\alpha+1}=\mathrm{Fin}\otimes\mathrm{Fin}^{\alpha}$,
$\text{dom}(\mathrm{Fin}^{\alpha+1})=\sum_{i\in\omega}\text{dom}(\mathrm{Fin}^{\alpha})$;

  * •

if $0<\lambda<\omega_{1}$ is a limit ordinal then
$\mathrm{Fin}^{\lambda}=\mathcal{I}_{\lambda}-\sum_{0<\alpha<\lambda}\mathrm{Fin}^{\alpha}$
and
$\text{dom}(\mathrm{Fin}^{\lambda})=\sum_{0<\alpha<\lambda}\text{dom}(\mathrm{Fin}^{\alpha})$,
where $\mathcal{I}_{\lambda}=\\{A\subseteq\lambda\setminus\\{0\\}:\
\exists_{\alpha<\lambda}\ A\subseteq\alpha\\}$ is an ideal on
$\lambda\setminus\\{0\\}$;

(actually, paper [5] concerns filters instead of ideals). In particular, if
$n\in\omega$ then $\mathrm{Fin}^{n+1}=\mathrm{Fin}\otimes\mathrm{Fin}^{n}$ is
an ideal on $\omega^{n+1}$. The ideals $\mathrm{Fin}^{\alpha}$ will be useful
in Subsection 2.4 where we define ideals $\mathrm{Fin}_{\alpha}$ (note the
role of subscript and superscript in this notation).

###  2.3. Inductive limits of ideals

This subsection is a short summary of notions defined in [3, Section 5], which
we need to introduce in order to define the ideal $\mathrm{Fin}_{\omega}$.

Following [3, Definition 5.1], a quasi-homomorphism from $\mathcal{J}$ to
$\mathcal{I}$ is a mapping $f:M\to{\rm dom}\left(\mathcal{I}\right)$, where
$M\in\mathcal{J}^{\star}$, such that for each $A\in\mathcal{I}$ its preimage
$f^{-1}[A]$ belongs to $\mathcal{J}$. It is easy to see that
$\mathcal{I}\leq_{K}\mathcal{J}$ is equivalent to existence of a quasi-
homomorphism from $\mathcal{J}$ to $\mathcal{I}$.

For a directed set $\left(I,\leq\right)$ and a family of ideals
$\left(\mathcal{I}_{i}\right)_{i\in I}$, the family
$\left(\pi_{i,j}\right)_{\stackrel{{\scriptstyle i\leq j}}{{i,j\in I}}}$,
where each $\pi_{i,j}$ is a quasi-homomorphism from $\mathcal{I}_{j}$ to
$\mathcal{I}_{i}$, is called a system of quasi-homomorphisms for
$\left(\mathcal{I}_{i}\right)_{i\in I}$.

######  Definition 2.1 (see [3, Proposition-Definition 5.4]).

Let $\left(\pi_{i,j}\right)_{\stackrel{{\scriptstyle i\leq j}}{{i,j\in I}}}$
be a system of quasi-homomorphisms for $\left(\mathcal{I}_{i}\right)_{i\in
I}$. The set

$\underleftarrow{\lim}\left(\mathcal{I}_{i},\pi_{i,j}\right)_{\stackrel{{\scriptstyle
i\leq j}}{{i,j\in I}}}=\left\\{M\subseteq\sum_{i\in
I}\text{dom}\left(\mathcal{I}_{i}\right):\ \exists_{i\in I}\
\exists_{P\in\mathcal{I}_{i}^{\star}}\ M\cap\left(\sum_{j\geq
i}\pi_{i,j}^{-1}\left[P\right]\right)=\emptyset\right\\}$

is called the inductive limit of
$\left(\mathcal{I}_{i},\pi_{i,j}\right)_{\stackrel{{\scriptstyle i\leq
j}}{{i,j\in I}}}$. For simplicity, we write just
$\underleftarrow{\lim}\mathcal{I}_{i}$ instead of
$\underleftarrow{\lim}\left(\mathcal{I}_{i},\pi_{i,j}\right)_{\stackrel{{\scriptstyle
i\leq j}}{{i,j\in I}}}$ whenever the system of quasi-homomorphisms is evident
from the context.

Inductive limits of ideals were introduced by G. Debs and J. Saint Raymond in
order to define ideals $\mathrm{Fin}_{\alpha}$ for limit ordinals
$0<\alpha<\omega_{1}$ (see Subsection 2.4).

###  2.4. Canonical ideals

In this subsection we define ideals $\mathrm{Fin}_{\alpha}$ for successor
ordinals $0<\alpha<\omega_{1}$ and for $\alpha=\omega$.

Observe that $\left(\pi_{i,j}\right)_{0<i\leq j<\omega}$, where
$\text{dom}(\pi_{i,j})=\text{dom}(\mathrm{Fin}^{j})=\omega^{j}$ and
$\pi_{i,j}$ is the projection onto the last $i$ coordinates, i.e.,

$\pi_{i,j}(x_{0},x_{1},\ldots,x_{j-1})=(x_{j-i},x_{j-i+1},\ldots,x_{j-1}),$

for all $(x_{0},x_{1},\ldots,x_{j-1})\in\omega^{j}$), is a system of quasi-
homomorphisms for $\left(\mathrm{Fin}^{i}\right)_{i\in\omega}$.

Following [3, Section 6.2], we define:

  * •

$\mathrm{Fin}_{1}=\mathrm{Fin}^{1}=\mathrm{Fin}$;

  * •

$\mathrm{Fin}_{\alpha+1}=\mathrm{Fin}^{1+\alpha}$;

  * •

$\mathrm{Fin}_{\omega}=\underleftarrow{\lim}(\mathrm{Fin}^{n})_{n\in\omega}$.

Obviously, $\mathrm{Fin}_{n}=\mathrm{Fin}^{n}$ for all $n\in\omega$ and
$\mathrm{Fin}_{\alpha+1}=\mathrm{Fin}^{\alpha}$ for all
$\omega\leq\alpha<\omega_{1}$.

We do not introduce ideals $\mathrm{Fin}_{\lambda}$ for limit ordinals
$\omega<\lambda<\omega_{1}$. Except the facts that they are complicated
objects and we will not use them in our considerations, the reason for
omitting their definition is that those ideals were defined ambiguously in [3,
Section 6.2]. Actually, that is also the case of $\mathrm{Fin}_{\omega}$. In
[3, Section 6.2] it is only written that since each $\mathrm{Fin}^{n+1}$ is
obtained as the $\mathrm{Fin}$-Fubini sum of $\mathrm{Fin}^{n}$, applying [3,
Proposition 5.8] inductively one gets a system of quasi-homomorphisms for
$(\mathrm{Fin}^{n})_{n\in\omega}$. While from [3, Proposition 5.8] (and its
proof) we see that any quasi-inductive system of ideals
$\left(\mathcal{I}_{i},\nu_{i,j}\right)_{\stackrel{{\scriptstyle i\leq
j}}{{i,j\in\omega}}}$ can be extended to
$\left(\mathcal{I}_{i},\nu_{i,j}\right)_{\stackrel{{\scriptstyle i\leq
j}}{{i,j\in\omega+1}}}$, where $\mathcal{I}_{\omega}$ is the
$\mathrm{Fin}$-Fubini sum of $(\mathcal{I}_{i})_{i\in\omega}$ and
$\nu_{i,\omega}=\sum_{j\geq i,j\in\omega}\nu_{i,j}$ is the disjoint union
mapping. Thus, in order to get a quasi-homomorphism $\pi_{1,2}$ from
$\mathrm{Fin}^{2}$ to $\mathrm{Fin}$ we need to put
$\mathcal{I}_{i}=\mathrm{Fin}$ and can guess that $\pi_{i,j}(x)=x$, for all
$i,j\in\omega$, $i\leq j$ and $x\in\omega$. However, the problem is that
applying [3, Proposition 5.8] we get infinitely many quasi-homomorphism from
$\mathrm{Fin}^{2}$ to $\mathrm{Fin}$ – each of them is given by the projection
onto the second coordinate, but their domains differ (and are of the form
$(\omega\setminus i)\times\omega$). Nothing is mentioned about which of those
quasi-homomorphism we should choose. In the above definition of
$\mathrm{Fin}_{\omega}$ we always choose the quasi-homomorphism with the
largest domain (which seems to be the most obvious choice). Fortunately,
regardless of those issues, it seems that each version of
$\mathrm{Fin}_{\lambda}$, for a limit ordinals $0<\lambda<\omega_{1}$,
possesses all the properties required in [3] (provided that it is an ideal –
see the next section).

##  3\. When an inductive limit of ideals is an ideal?

In this section we show that $\mathrm{Fin}_{\omega}$ is an ideal despite the
fact that [3, Proposition-Definition 5.4], which defines it, is false.

######  Definition 3.1 ([3, Definition 5.3]).

A system of quasi-homomorphisms $\left(\pi_{i,j}\right)_{i\leq j}$ for a
family of ideals $\left(\mathcal{I}_{i}\right)_{i\in I}$ is coherent provided
that for all $i,j,k\in I$ with $i\leq j\leq k$ the following condition is
satisfied:

$\pi_{i,k}(a)=\pi_{i,j}\left(\pi_{j,k}(a)\right)$

for every
$a\in\text{dom}\left(\pi_{i,k}\right)\cap\text{dom}\left(\pi_{j,k}\right)\cap\pi_{j,k}^{-1}\left(\text{dom}\left(\pi_{i,j}\right)\right)$.
In this case we call
$\left(\mathcal{I}_{i},\pi_{i,j}\right)_{\stackrel{{\scriptstyle i\leq
j}}{{i,j\in I}}}$ a coherent quasi-inductive system of ideals.

Next example shows that an inductive limit of a coherent quasi-inductive
system of ideals does not have to be an ideal. This means that [3,
Proposition-Definition 5.4] is false.

######  Example 3.2.

Put $I=\omega\setminus\\{0\\}$ and consider the ideals $\mathrm{Fin}^{i}$ for
$i\in I$ (i.e., $\mathrm{Fin}^{1}=\mathrm{Fin}$,
$\mathrm{Fin}^{2}=\mathrm{Fin}\otimes\mathrm{Fin}$ and
$\mathrm{Fin}^{i+1}=\mathrm{Fin}\otimes\mathrm{Fin}^{i}$). For each $i>1$ let
$F_{i}=\omega^{i}\setminus\left(\left\\{0\right\\}\times\omega^{i-1}\right)\in(\mathrm{Fin}^{i})^{\star}$
and define a quasi-homomorphism $\pi_{1,i}:F_{i}\to\omega$ from
$\mathrm{Fin}^{i}$ to $(\mathrm{Fin}^{1})^{\star}$ by
$\pi_{1,i}(x_{0},\ldots,x_{i-1})=x_{i-1}$ for all $(x_{0},\ldots,x_{i-1})\in
F_{i}$ (i.e., $\pi_{1,i}$ is the projection onto the last coordinate). For
indices $1<i\leq j$ let $\pi_{i,j}:\omega^{j}\to\omega^{i}$ be the projection
onto the last $i$ coordinates. Then $\left(\pi_{i,j}\right)_{i\leq j}$ is a
coherent system of quasi-homomorphisms for the family of ideals
$\left(\mathrm{Fin}^{i}\right)_{i\in I}$. However, we will show that its
inductive limit, let us denote it by $\mathcal{J}$, is not an ideal.

Consider $A=\sum_{j\geq 1}\pi_{1,j}^{-1}[\omega]$ and $B=\sum_{j\geq
2}\pi_{2,j}^{-1}[\omega^{2}]$. Obviously, $A^{c},B^{c}\in\mathcal{J}$. We
claim that

$A^{c}\cup B^{c}=(A\cap B)^{c}=\left(\Sigma_{j>1}F_{j}\right)^{c}$

does not belong to $\mathcal{J}$. Indeed, if $i>1$ and
$P\in(\mathrm{Fin}^{i})^{\star}$, then $\sum_{j\geq
i}\pi_{i,j}^{-1}[P]\not\subseteq\Sigma_{j>1}F_{j}$ as for any
$(x_{0},\ldots,x_{i-1})\in P\subseteq\omega^{i}$ we have
$\pi_{i,i+1}((0,x_{0},\ldots,x_{i-1}))\in P$ and
$(0,x_{0},\ldots,x_{i-1})\notin F_{i+1}$. On the other hand, if $i=1$ then
$\sum_{j\geq i}\pi_{i,j}^{-1}[P]\not\subseteq A\cap B$ as $(\\{1\\}\times
P)\subseteq\sum_{j\geq i}\pi_{i,j}^{-1}[P]$ and $(\\{1\\}\times
P)\cap\left(\Sigma_{j>1}F_{j}\right)=\emptyset$. This shows that $\mathcal{J}$
is not an ideal.

Now we want to focus on showing that $\mathrm{Fin}_{\omega}$ is an ideal. We
will need the following notion.

######  Definition 3.3.

We say that a quasi-inductive system
$\left(\mathcal{I}_{i},\pi_{i,j}\right)_{\stackrel{{\scriptstyle i\leq
j}}{{i,j\in I}}}$ satisfies condition (C) if it is coherent and

$\forall_{k\geq j}\ \pi^{-1}_{j,k}\left({\rm
dom}(\pi_{i,j})\right)\subseteq{\rm dom}(\pi_{i,k})$

for all indices $i\leq j$.

######  Proposition 3.4.

Let $\left(\mathcal{I}_{i},\pi_{i,j}\right)_{\stackrel{{\scriptstyle i\leq
j}}{{i,j\in I}}}$ be a quasi-inductive system satisfying condition (C). Then
$\underleftarrow{\lim}\mathcal{I}_{i}$ is an ideal.

###### Proof.

Define ideals

$\overline{\mathcal{I}}_{i}=\left\\{M\subseteq\Sigma_{k\in I}{\rm
dom}\left(\mathcal{I}_{k}\right):\ \exists_{P\in\mathcal{I}^{\star}_{i}}\
M\cap\overline{P}=\emptyset\right\\},$

where for $i\in I$ and $P\in\mathcal{I}^{\star}_{i}$ we denote
$\overline{P}=\sum_{j\geq i}\pi_{i,j}^{-1}[P]$.

Observe that

$\underleftarrow{\lim}\mathcal{I}_{i}=\bigcup_{i\in
I}\overline{\mathcal{I}}_{i}.$

We will show that
$\overline{\mathcal{I}}_{i}\subseteq\overline{\mathcal{I}}_{j}$ provided that
$i\leq j$. It will follow that $\underleftarrow{\lim}\mathcal{I}_{i}$ is an
ideal as a union of an increasing family of ideals.

To show that $\overline{\mathcal{I}}_{i}\subseteq\overline{\mathcal{I}}_{j}$
whenever $i\leq j$, fix $M\in\overline{\mathcal{I}}_{i}$. Then there is
$P\in\mathcal{I}^{\star}_{i}$ such that $M\cap\overline{P}=\emptyset$. Set
$P^{\prime}=\pi_{i,j}^{-1}\left[P\right]\in\mathcal{I}^{\star}_{j}$. We will
show that $\overline{P}^{\prime}\subseteq\overline{P}$ (which will end the
proof as $\overline{P}^{\prime}\cap M\subseteq\overline{P}\cap M=\emptyset$).
If $k\geq j$ and $x\in\pi_{j,k}^{-1}\left[P^{\prime}\right]$ (so $(k,x)$ is an
element of $\overline{P}^{\prime}$) then $x\in\pi^{-1}_{j,k}\left({\rm
dom}(\pi_{i,j})\right)$ (as $P^{\prime}\subseteq{\rm dom}(\pi_{i,j})$). By
condition (C) we know that $x\in{\rm dom}(\pi_{i,k})$. Now, using the fact
that $\left(\mathcal{I}_{i},\pi_{i,j}\right)_{\stackrel{{\scriptstyle i\leq
j}}{{i,j\in I}}}$ is coherent, we get $\pi_{i,k}(x)=\pi_{i,j}(\pi_{j,k}(x))\in
P$. Thus, $x$ is an element of $\pi_{i,k}^{-1}\left[P\right]$ and $(k,x)$ is
an element of $\overline{P}$. This finishes the proof. ∎

######  Corollary 3.5.

$\mathrm{Fin}_{\omega}$ is an ideal.

###### Proof.

It is easy to show that $\left(\pi_{i,j}\right)_{0<i\leq j<\omega}$ defined in
Subsection 2.4 is a system of quasi-homomorphisms for
$\left(\mathrm{Fin}^{i}\right)_{i\in\omega}$ satisfying condition (C). Hence,
the previous result implies that $\mathrm{Fin}_{\omega}$ is an ideal. ∎

We end this section with a modification of [3, Proposition 5.8].

######  Proposition 3.6.

Let $\left(\mathcal{I}_{i},\pi_{i,j}\right)_{\stackrel{{\scriptstyle i\leq
j}}{{i,j\in I}}}$ be a quasi-inductive system satisfying condition (C),
$\mathcal{J}$ be the ideal on $I$ generated by the family $\\{\\{j\in I:\
j\leq i\\}:\ i\in I\\}$, and $\mathcal{I}_{\infty}$ denote the
$\mathcal{J}$-Fubini sum of $\left(\mathcal{I}_{i}\right)_{i\in I}$. If
$\pi_{i,\infty}:\sum_{k\geq i}{\rm dom}(\pi_{i,k})\to{\rm
dom}(\mathcal{I}_{i})$ are given by $\pi_{i,\infty}=\sum_{j\geq i}\pi_{i,j}$,
then $\left(\mathcal{I}_{i},\pi_{i,j}\right)_{\stackrel{{\scriptstyle i\leq
j}}{{i,j\in I\cup\left\\{\infty\right\\}}}}$ is also a quasi-inductive system
satisfying condition (C).

###### Proof.

In [3, Proposition 5.8] it is shown that
$\left(\mathcal{I}_{i},\pi_{i,j}\right)_{\stackrel{{\scriptstyle i\leq
j}}{{i,j\in I\cup\left\\{\infty\right\\}}}}$ is a coherent system. We need to
show that

$\forall_{k\geq j}\ \pi^{-1}_{j,k}\left({\rm
dom}(\pi_{i,j})\right)\subseteq{\rm dom}(\pi_{i,k})$

for all indices $i\leq j$.

Set $i\leq j$, $i,j\in I$. Observe that

$\pi^{-1}_{j,\infty}\left({\rm dom}(\pi_{i,j})\right)=\sum_{k\geq
j}\pi^{-1}_{j,k}\left({\rm dom}(\pi_{i,j})\right).$

Applying condition (C) we obtain that

$\sum_{k\geq j}\pi^{-1}_{j,k}\left({\rm
dom}(\pi_{i,j})\right)\subseteq\sum_{k\geq j}{\rm
dom}(\pi_{i,k})\subseteq\sum_{k\geq i}{\rm dom}(\pi_{i,k})={\rm
dom}(\pi_{i,\infty}),$

which finishes the proof. ∎

##  4\. The counterexample

This section is devoted to showing that Conjecture 1.1 is false in the case of
$\alpha=\omega$. Our counterexample is defined as follows.

######  Definition 4.1.

Fix a family $\\{\\{X_{s}:s\in\omega^{n+1}\\}:n\in\omega\\}$ of partitions of
$\omega$ into infinite sets (i.e., for every $n\in\omega$ each $X_{s}$, for
$s\in\omega^{n+1}$, is infinite and $\\{X_{s}:s\in\omega^{n+1}\\}$ is a
partition of $\omega$) such that $\bigcap_{i\in F}X_{s_{i}}$ is infinite for
all $F\in[\omega]^{<\omega}$ and $s_{i}\in\omega^{i+1}$, $i\in F$. Then
$\mathrm{Fin}^{\prime}_{\omega}$ is an ideal on $\omega$ generated by the
family $\bigcup_{n\in\omega}\mathrm{Fin}^{n+2}(\\{X_{s}:s\in\omega^{n+1}\\})$,
where $\mathrm{Fin}^{n+2}(\\{X_{s}:s\in\omega^{n+1}\\})$ is an isomorphic copy
of $\mathrm{Fin}^{n+2}$ on $\omega$ given by any bijection
$\sigma:\omega^{n+2}\to\omega$ such that $\sigma[\\{s\\}\times\omega]=X_{s}$
for all $s\in\omega^{n+1}$ (note that each such bijection defines the same
ideal on $\omega$).

Before proving that $\mathrm{Fin}^{\prime}_{\omega}$ is as needed, we need to
establish some of its properties which will be helpful in our considerations.

For a family $\mathcal{A}\subseteq\mathcal{P}(X)$ we denote
$\mathcal{A}^{\uparrow}=\\{B\subseteq X:\exists_{A\in\mathcal{A}}B\supseteq
A\\}$.

######  Definition 4.2.

Define

$\mathcal{J}_{n}=\left\\{A\subseteq\omega:\
\phi_{n}(A)\in\left(\mathrm{Fin}^{\star}\times\ldots\times(\mathrm{Fin}^{n+1})^{\star}\right)^{\uparrow}\right\\},$

where
$\phi_{n}:\mathcal{P}(\omega)\to\mathcal{P}(\omega\times\omega^{2}\times\ldots\times\omega^{n+1})$
is given by

$\phi_{n}(A)=\left\\{(s_{0},\ldots,s_{n})\in\omega\times\omega^{2}\times\ldots\times\omega^{n+1}:\
A\cap X_{s_{0}}\cap\ldots\cap X_{s_{n}}\in\mathrm{Fin}\right\\}.$

######  Lemma 4.3.

$\mathrm{Fin}^{\prime}_{\omega}=\bigcup_{n\in\omega}\mathcal{J}_{n}$ and
$\mathcal{J}_{n}\subseteq\mathcal{J}_{n+1}$, for each $n\in\omega$.

###### Proof.

The inclusion $\mathcal{J}_{n}\subseteq\mathcal{J}_{n+1}$, for each
$n\in\omega$, is obvious. Thus, we only prove
$\mathrm{Fin}^{\prime}_{\omega}=\bigcup_{n\in\omega}\mathcal{J}_{n}$.

Let $A\in\mathrm{Fin}^{\prime}_{\omega}$. Then there are $n\in\omega$ and sets
$A_{i}$, for $i\leq n$, such that $A=\bigcup_{i\leq n}A_{i}$ and
$A_{i}\in\mathrm{Fin}^{i+2}(\\{X_{s}:s\in\omega^{i+1}\\})$, for all $i\leq n$.
Thus, for each $i\leq n$ there is $B_{i}\in\mathrm{Fin}^{i+1}$ such that
$A_{i}\cap X_{s}\notin\mathrm{Fin}$ if and only if $s\in B_{i}$, for all
$s\in\omega^{i+1}$. Observe that
$(B_{0})^{c}\times\ldots\times(B_{n})^{c}\subseteq\phi_{n}(A)$, since for each
$(s_{0},\ldots,s_{n})\in(B_{0})^{c}\times\ldots\times(B_{n})^{c}$ we have

$A\cap X_{s_{0}}\cap\ldots\cap X_{s_{n}}=\bigcup_{i\leq n}\left(A_{i}\cap
X_{s_{0}}\cap\ldots\cap X_{s_{n}}\right)\subseteq\bigcup_{i\leq n}A_{i}\cap
X_{s_{i}}\in\mathrm{Fin},$

i.e., $(s_{0},\ldots,s_{n})\in\phi_{n}(A)$. As
$(B_{0})^{c}\times\ldots\times(B_{n})^{c}\in\mathrm{Fin}^{\star}\times\ldots\times(\mathrm{Fin}^{n+1})^{\star}$
and
$(\mathrm{Fin}^{\star}\times\ldots\times(\mathrm{Fin}^{n+1})^{\star})^{\uparrow}$
is closed under supersets, we conclude that
$\phi_{n}(A)\in(\mathrm{Fin}^{\star}\times\ldots\times(\mathrm{Fin}^{n+1})^{\star})^{\uparrow}$.

Let $n\in\omega$ and $A\subseteq\omega$ be such that
$\phi_{n}(A)\in(\mathrm{Fin}^{\star}\times\ldots\times(\mathrm{Fin}^{n+1})^{\star})^{\uparrow}$.
Then there are $B_{0},\ldots,B_{n}$ such that
$\phi_{n}(A)\supseteq(B_{0})^{c}\times\ldots\times(B_{n})^{c}$ and
$B_{i}\in\mathrm{Fin}^{i+1}$, for all $i\leq n$. Define $A_{i}=\bigcup_{s\in
B_{i}}X_{s}$ for all $i\leq n$. Obviously,
$A_{i}\in\mathrm{Fin}^{i+2}(\\{X_{s}:s\in\omega^{i+1}\\})$, for all $i\leq n$,
i.e., $A^{\prime}=\bigcup_{i\leq n}A_{i}\in\mathrm{Fin}^{\prime}_{\omega}$.
Moreover, $(A\setminus A^{\prime})\cap X_{s_{0}}\cap\ldots\cap
X_{s_{n}}\in\mathrm{Fin}$ for all
$(s_{0},\ldots,s_{n})\in\omega\times\ldots\times\omega^{n+1}$.

To finish the proof, we will inductively show that for each $B\subseteq\omega$
and $n\in\omega$, if $B\cap X_{s_{0}}\cap\ldots\cap X_{s_{n}}\in\mathrm{Fin}$
for all $(s_{0},\ldots,s_{n})\in\omega\times\ldots\times\omega^{n+1}$, then
$B\in\mathrm{Fin}^{\prime}_{\omega}$.

For $n=0$ this is obvious as $B\cap X_{s_{0}}\in\mathrm{Fin}$ for all
$s_{0}\in\omega$ implies
$B\in\mathrm{Fin}^{2}(\\{X_{s}:s\in\omega\\})\subseteq\mathrm{Fin}^{\prime}_{\omega}$.

Assume that the induction hypothesis is true for all $k<n$ and fix
$B\subseteq\omega$ with $B\cap X_{s_{0}}\cap\ldots\cap
X_{s_{n}}\in\mathrm{Fin}$ for all
$(s_{0},\ldots,s_{n})\in\omega\times\ldots\times\omega^{n+1}$. Fix two
bijections $h:\omega\to\omega^{n+1}$ and
$g:\omega\to\omega\times\ldots\times\omega^{n}$ and for each
$(s_{0},\ldots,s_{n-1})\in\omega\times\ldots\times\omega^{n}$ denote

$Y_{(s_{0},\ldots,s_{n-1})}=X_{s_{0}}\cap\ldots\cap X_{s_{n-1}}$

(note that $(Y_{g(j)})_{j\in\omega}$ is a partition of $\omega$). Define
$C=\bigcup_{k\in\omega}\bigcup_{j\leq k}B\cap X_{h(k)}\cap Y_{g(j)}$ and
$D=\bigcup_{j\in\omega}\bigcup_{k\leq j}B\cap X_{h(k)}\cap Y_{g(j)}$. Then $D$
is in $\mathrm{Fin}^{\prime}_{\omega}$ by the induction hypothesis, as $D\cap
Y_{g(j)}\in\mathrm{Fin}$ for all $j\in\omega$. Similarly,
$C\in\mathrm{Fin}^{n+2}(\\{X_{s}:s\in\omega^{n+1}\\})$ as $C\cap
X_{h(k)}\in\mathrm{Fin}$ for all $k\in\omega$. Finally, $B=C\cup D$ as for
each $x\in B$ we can find a unique
$(s_{0},\ldots,s_{n})\in\omega\times\ldots\times\omega^{n+1}$ with $x\in
X_{s_{0}}\cap\ldots\cap X_{s_{n}}$ and either $h^{-1}(s_{n})\leq
g^{-1}((s_{0},\ldots,s_{n-1}))$ (in this case $x\in D$) or $h^{-1}(s_{n})\geq
g^{-1}((s_{0},\ldots,s_{n-1}))$ (in this case $x\in C$). ∎

######  Lemma 4.4.

Let $n\in\omega$. Then $\mathcal{J}_{n}$ is a $\bf{\Sigma^{0}_{2n+4}}$ ideal.
In particular, $\operatorname{rk}(\mathcal{J}_{n})\leq 2n+4$.

###### Proof.

It is easy to see that $\mathcal{J}_{n}$ is an ideal. The ”’In particular”’
part will follow directly from $\mathcal{J}_{n}\in\bf{\Sigma^{0}_{2n+4}}$ and
the definition of rank of an ideal. Thus, we only need to show that
$\mathcal{J}_{n}\in\bf{\Sigma^{0}_{2n+4}}$.

For this proof, for a given $n\in\omega$ denote by $\mathcal{P}_{n}$ the
family of all $\mathcal{Y}=\\{\\{Y_{s}:s\in\omega^{i+1}\\}:i\leq n\\}$ such
that:

  * •

$Y_{s}$ is a subset of $\omega$, for each $s\in\omega^{i+1}$, $i\leq n$;

  * •

$Y_{s}\cap Y_{t}=\emptyset$ whenever $s,t\in\omega^{i+1}$ for some $i\leq n$
and $s\neq t$.

Moreover, for given $n\in\omega$ and $\mathcal{Y}\in\mathcal{P}_{n}$ we denote
by $\mathcal{J}_{n}(\mathcal{Y})$ the family consisting of all
$A\subseteq\omega$ such that
$\phi_{n}(\mathcal{Y})(A)\in\left(\mathrm{Fin}^{\star}\times\ldots\times(\mathrm{Fin}^{n+1})^{\star}\right)^{\uparrow}$,
where

$\phi_{n}(\mathcal{Y})(A)=\left\\{(s_{0},\ldots,s_{n})\in\omega\times\omega^{2}\times\ldots\times\omega^{n+1}:A\cap
Y_{s_{0}}\cap\ldots\cap Y_{s_{n}}\in\mathrm{Fin}\right\\}.$

We will inductively show that for every $n\in\omega$ and
$\mathcal{Y}\in\mathcal{P}_{n}$ the family $\mathcal{J}_{n}(\mathcal{Y})$ is
$\bf{\Sigma^{0}_{2n+4}}$. This will finish the proof since for each
$n\in\omega$ we have
$\mathcal{J}_{n}=\mathcal{J}_{n}(\\{\\{X_{s}:s\in\omega^{i+1}\\}:i\leq n\\})$,
where the sets $X_{s}$, for $s\in\omega^{i+1}$, $i\leq n$, are as in
Definition 4.1.

Firstly, fix any $\mathcal{Y}=\\{\\{Y_{s}:s\in\omega\\}\\}\in\mathcal{P}_{0}$
and observe that $\mathcal{J}_{0}(\mathcal{Y})$ is $\bf{\Sigma^{0}_{4}}$,
since

$\mathcal{J}_{0}(\mathcal{Y})=\\{A\subseteq\omega:\ \exists_{i\in\omega}\
\forall_{j>i}\ A\cap
Y_{j}\in\mathrm{Fin}\\}=\bigcup_{i\in\omega}\bigcap_{j>i}\bigcup_{F\in[Y_{i}]^{<\omega}}\\{A\subseteq\omega:\
A\cap Y_{i}=F\\}$

and $\\{A\subseteq\omega:\ A\cap Y_{i}=F\\}$ is closed.

Now, fix any $\mathcal{Y}=\\{\\{Y_{s}:s\in\omega^{i+1}\\}:i\leq
1\\}\in\mathcal{P}_{1}$ and observe that:

$\mathcal{J}_{1}(\mathcal{Y})=\left\\{A\subseteq\omega:\ \exists_{i\in\omega}\
\forall_{j>i}\ \exists_{k\in\omega}\ \forall_{l>k}\ A\cap Y_{j}\cap
Y_{(j,l)}\in\mathrm{Fin}\right\\}=$ $=\left\\{A\subseteq\omega:\
\exists_{i\in\omega}\ \forall_{j>i}\ A\cap
Y_{j}\in\mathcal{J}_{0}((Y_{\\{j\\}\times s})_{s\in\omega})\right\\}.$

Since $A\mapsto A\cap X_{j}$ is continuous and
$\mathcal{J}_{0}((Y_{\\{j\\}\times s})_{s\in\omega})\in\bf{\Sigma^{0}_{4}}$,
for each $j\in\omega$, the family $\mathcal{J}_{1}(\mathcal{Y})$ is
$\bf{\Sigma^{0}_{6}}$.

Thus, we have shown the thesis for $n=0$ and $n=1$. Suppose now that
$\mathcal{J}_{m}(\mathcal{Y})$ is $\bf{\Sigma^{0}_{2m+4}}$ for all $m<n$ and
$\mathcal{Y}\in\mathcal{P}_{m}$. Fix any
$\mathcal{Y}=\\{\\{Y_{s}:s\in\omega^{i+1}\\}:i\leq n\\}\in\mathcal{P}_{n}$.
Note that:

$\mathcal{J}_{n}(\mathcal{Y})=\left\\{A\subseteq\omega:\ \exists_{i\in\omega}\
\forall_{j>i}\ A\cap Y_{j}\in\mathcal{J}_{n-1}((Y_{\\{j\\}\times
s})_{s\in\omega},\ldots,(Y_{\\{j\\}\times s})_{s\in\omega^{n}})\right\\}.$

Hence, using the inductive assumption, we get that
$\mathcal{J}_{n}(\mathcal{Y})$ is $\bf{\Sigma^{0}_{2n+4}}$. ∎

We are ready to prove that $\mathrm{Fin}^{\prime}_{\omega}$ is as needed,
i.e., that it is a Borel ideal of rank $\omega$ not containing any isomorphic
copy of $\mathrm{Fin}_{\omega}$.

######  Theorem 4.5.

$\mathrm{Fin}^{\prime}_{\omega}$ is a $\bf{\Sigma^{0}_{\omega}}$ ideal of rank
$\omega$.

###### Proof.

As $\mathrm{Fin}_{n+2}\sqsubseteq\mathrm{Fin}^{\prime}_{\omega}$ for all
$n\in\omega$, $\text{rk}(\mathrm{Fin}^{\prime}_{\omega})\geq n+2$ for all
$n\in\omega$ (by [3, Lemma 7.2]), i.e.,
$\text{rk}(\mathrm{Fin}^{\prime}_{\omega})\geq\omega$. On the other hand,
$\text{rk}(\mathrm{Fin}^{\prime}_{\omega})\leq\omega$ since
$\mathrm{Fin}^{\prime}_{\omega}$ is $\bf{\Sigma^{0}_{\omega}}$ (by Lemmas 4.3
and 4.4). ∎

######  Theorem 4.6.

$\mathrm{Fin}_{\omega}\not\sqsubseteq\mathrm{Fin}^{\prime}_{\omega}$.

###### Proof.

Fix any bijection $f:\omega\to\sum_{i>0}\omega^{i}$ and suppose to the
contrary that it witnesses
$\mathrm{Fin}_{\omega}\sqsubseteq\mathrm{Fin}^{\prime}_{\omega}$. For a set
$X\subseteq\omega^{k}$ denote $\overline{X}=\sum_{i\geq k}\pi_{k,i}^{-1}[X]$
(so that $X\in\mathrm{Fin}^{k}$ implies $\overline{X}\in\mathrm{Fin}_{\omega}$
and $f^{-1}[\overline{X}]\in\mathrm{Fin}^{\prime}_{\omega}$).

Since $\\{1\\}\times\omega,\\{2\\}\times\omega^{2}\in\mathrm{Fin}_{\omega}$,
we have
$A=f^{-1}[(\\{1\\}\times\omega)\cup(\\{2\\}\times\omega^{2})]\in\mathrm{Fin}^{\prime}_{\omega}$.
For each $i,j\in\omega$ the set $\\{(i,j)\\}\times\omega$ is in
$\mathrm{Fin}^{3}$, so
$A_{i,j}=f^{-1}\left[\overline{\\{(i,j)\\}\times\omega}\right]\in\mathrm{Fin}^{\prime}_{\omega}$.
Observe that $\omega=A\cup\bigcup_{i,j\in\omega}A_{i,j}$.

We claim that there is $m\in\omega$ such that each $Y\subseteq\omega\setminus
A$ with

$\exists_{h\in\omega^{\omega}}\ Y\subseteq\bigcup_{i\in\omega}\bigcup_{j\leq
h(i)}A_{i,j}$

is in $\mathcal{J}_{m}$. Suppose otherwise. Then for each $k\in\omega$ we
could find $Y_{k}\subseteq\omega\setminus A$ such that
$Y_{k}\subseteq\bigcup_{i\in\omega}\bigcup_{j\leq h_{k}(i)}A_{i,j}$ for some
$h_{k}\in\omega^{\omega}$, but $Y_{k}\notin\mathcal{J}_{k}$. Note that
$B_{i}=\bigcup_{j\in\omega}A_{i,j}\in\mathrm{Fin}^{\prime}_{\omega}$ for each
$i\in\omega$, as $B_{i}=f^{-1}\left[\overline{\\{i\\}\times\omega^{2}}\right]$
and $\\{i\\}\times\omega^{2}$ is in $\mathrm{Fin}^{3}$. Let
$g\in\omega\to\omega$ be any nondecreasing function satisfying
$\lim_{i}g(i)=+\infty$ and $\bigcup_{j\leq g(i)}B_{j}\in\mathcal{J}_{i}$ for
all $i\in\omega$ (recall that $\mathcal{J}_{i}\subseteq\mathcal{J}_{i+1}$, for
all $i\in\omega$, by Lemma 4.3, so such function exists). Define
$Y=\bigcup_{k\in\omega}Y_{k}\setminus(B_{0}\cup\ldots\cup B_{g(k)})$.
Obviously,
$Y\notin\mathrm{Fin}^{\prime}_{\omega}=\bigcup_{k\in\omega}\mathcal{J}_{k}$ as
for each $k\in\omega$ we have $Y_{k}\setminus(B_{0}\cup\ldots\cup
B_{g(k)})\subseteq Y$, $Y_{k}\notin\mathcal{J}_{k}$ and $B_{0}\cup\ldots\cup
B_{g(k)}\in\mathcal{J}_{k}$. On the other hand, denoting
$a_{i}=\max\\{h_{k}(i):k\in g^{-1}(\\{0,\ldots,i-1\\})\\}$ (note that
$g^{-1}(\\{0,\ldots,1\\})$ is finite by $\lim_{i}g(i)=+\infty$), we have:

$Y=\bigcup_{i\in\omega}(B_{i}\cap\bigcup\\{Y_{k}:g(k)<i\\})\subseteq\bigcup_{i\in\omega}\bigcup_{j\leq
a_{i}}A_{i,j}=f^{-1}\left[\overline{\bigcup_{i\in\omega}\bigcup_{j\leq
a_{i}}\\{(i,j)\\}\times\omega}\right].$

Since $\bigcup_{i\in\omega}\bigcup_{j\leq
a_{i}}\\{(i,j)\\}\times\omega\in\mathrm{Fin}^{3}$, we get that
$Y\in\mathrm{Fin}^{\prime}_{\omega}$. This contradiction proves that there is
$m\in\omega$ such that each $Y\subseteq\omega\setminus A$ satisfying
$Y\subseteq\bigcup_{i\in\omega}\bigcup_{j\leq h(i)}A_{i,j}$, for some
$h\in\omega^{\omega}$, is in $\mathcal{J}_{m}$.

We need to introduce some notations. Define $A^{\prime}=f^{-1}[\bigcup_{i\leq
2m+4}\\{i\\}\times\omega^{i}]$ and note that
$A^{\prime}\in\mathrm{Fin}^{\prime}_{\omega}$ as $\bigcup_{i\leq
2m+4}\\{i\\}\times\omega^{i}\in\mathrm{Fin}_{\omega}$. Let
$C^{n}_{s}=f^{-1}\left[\overline{\\{s\\}\times\omega}\right]\setminus
A^{\prime}$ for all $n\leq 2m+4$ and $s\in\omega^{n}$ (i.e.,
$C^{2}_{(i,j)}=A_{i,j}\setminus A^{\prime}$). Consider the following ideals on
$\omega\setminus A^{\prime}$:

$Y\in\overline{\mathrm{Fin}^{2m+5}}\ \Longleftrightarrow\
\\{s\in\omega^{2m+4}:Y\cap
C^{2m+4}_{s}\notin\mathrm{Fin}\\}\in\mathrm{Fin}^{2m+4};$
$Y\in\overline{\mathrm{Fin}^{2m+4}}\ \Longleftrightarrow\
\\{s\in\omega^{2m+3}:Y\cap
C^{2m+3}_{s}\notin\mathrm{Fin}\\}\in\mathrm{Fin}^{2m+3};$
$Y\in\overline{\emptyset^{2}\otimes\mathrm{Fin}^{2m+2}}\ \Longleftrightarrow\
\forall_{t\in\omega^{2}}\ \\{s\in\omega^{2m+1}:Y\cap
C^{2m+3}_{t^{\frown}s}\neq\emptyset\\}\in\mathrm{Fin}^{2m+1};$
$Y\in\overline{(\emptyset\otimes\mathrm{Fin})\otimes\emptyset}\
\Longleftrightarrow\ \\{s\in\omega^{2}:Y\cap
C^{2}_{s}\neq\emptyset\\}\in\emptyset\otimes\mathrm{Fin};$
$Y\in\overline{\mathrm{Fin}\otimes\emptyset}\ \Longleftrightarrow\
\\{n\in\omega:Y\cap C^{1}_{(n)}\neq\emptyset\\}\in\mathrm{Fin}.$

By the choice of $m$, we have
$\overline{(\emptyset\otimes\mathrm{Fin})\otimes\emptyset}\subseteq\mathcal{J}_{m}$.
Moreover,

$\overline{\emptyset^{2}\otimes\mathrm{Fin}^{2m+2}}\cup\overline{(\emptyset\otimes\mathrm{Fin})\otimes\emptyset}\cup\overline{\mathrm{Fin}\otimes\emptyset}\subseteq\overline{\mathrm{Fin}^{2m+4}}\subseteq\overline{\mathrm{Fin}^{2m+5}}$

and for each $Y\in\overline{\mathrm{Fin}^{2m+4}}$ there are
$Y^{0}\in\overline{\emptyset^{2}\otimes\mathrm{Fin}^{2m+2}}$,
$Y^{1}\in\overline{(\emptyset\otimes\mathrm{Fin})\otimes\emptyset}$ and
$Y^{2}\in\overline{\mathrm{Fin}\otimes\emptyset}$ with $Y\subseteq Y^{0}\cup
Y^{1}\cup Y^{2}$.

What is more, for each $n\in\omega$ consider the ideals
$\overline{\mathrm{Fin}^{2m+4}_{n}}$ and
$\overline{\emptyset\otimes\mathrm{Fin}^{2m+3}_{n}}$ on $C^{1}_{(n)}$ given
by:

$Y\in\overline{\mathrm{Fin}^{2m+4}}(n)\ \Longleftrightarrow\
\\{s\in\omega^{2m+3}:Y\cap
C^{2m+4}_{(n)^{\frown}s}\notin\mathrm{Fin}\\}\in\mathrm{Fin}^{2m+3};$
$Y\in\overline{\emptyset\otimes\mathrm{Fin}^{2m+3}}(n)\ \Longleftrightarrow\
\\{s\in\omega^{2m+3}:Y\cap
C^{2m+4}_{(n)^{\frown}s}\notin\mathrm{Fin}\\}\in\emptyset\otimes\mathrm{Fin}^{2m+2}_{n}.$

Then
$\overline{\emptyset\otimes\mathrm{Fin}^{2m+3}}(n)\subseteq\overline{\mathrm{Fin}^{2m+4}}(n)$
and if $Y_{n}\in\overline{\mathrm{Fin}^{2m+4}}(n)$ for all $n\in\omega$ then
$\bigcup_{n\in\omega}Y_{n}\in\overline{\mathrm{Fin}^{2m+5}}$. Observe that
$\overline{\mathrm{Fin}^{2m+5}}$ is isomorphic with $\mathrm{Fin}^{2m+5}$ and
$\overline{\mathrm{Fin}^{2m+4}}$ as well as each
$\overline{\mathrm{Fin}^{2m+4}}(n)$ are isomorphic with $\mathrm{Fin}^{2m+4}$.

We claim that $Y\in\overline{\mathrm{Fin}^{2m+5}}$ implies
$f[Y]\in\mathrm{Fin}_{\omega}$. Indeed, fix
$Y\in\overline{\mathrm{Fin}^{2m+5}}$ (in particular, $Y$ is a subset of
$\omega\setminus A^{\prime}$), denote $S=\\{s\in\omega^{2m+4}:Y\cap
C^{2m+4}_{s}\notin\mathrm{Fin}\\}\in\mathrm{Fin}^{2m+4}$ and observe that
$f[\bigcup_{s\in
S}C^{2m+4}_{s}]=\overline{S\times\omega}\in\mathrm{Fin}_{\omega}$ as
$S\in\mathrm{Fin}^{2m+4}$ and $S\times\omega\in\mathrm{Fin}^{2m+5}$. For each
$x\in Y\setminus(\bigcup_{s\in S}C^{2m+4}_{s})$ let $i_{x}\geq 2m+5$ be the
unique index with $f(x)\in\\{i_{x}\\}\times\omega^{i_{x}}$. Then $|\\{x\in
Y\setminus(\bigcup_{s\in
S}C^{2m+4}_{s}):\pi_{2m+5,i_{x}}(f(x))\in\\{s\\}\times\omega\\}|<\omega$ for
each $s\in\omega^{2m+4}$ and

$f\left[Y\setminus\left(\bigcup_{s\in
S}C^{2m+4}_{s}\right)\right]\subseteq\overline{\left\\{\pi_{2m+5,i_{x}}(f(x)):x\in
Y\setminus\left(\bigcup_{s\in
S}C^{2m+4}_{s}\right)\right\\}}\in\mathrm{Fin}_{\omega}.$

Thus, $f[Y]\subseteq f[\bigcup_{s\in S}C^{2m+4}_{s}]\cup
f[Y\setminus(\bigcup_{s\in S}C^{2m+4}_{s})]\in\mathrm{Fin}_{\omega}$.

By the above observation, in order to finish the proof it suffices to find
$W\in\overline{\mathrm{Fin}^{2m+5}}\setminus\mathrm{Fin}^{\prime}_{\omega}$.
However, at first we need to define inductively sequences
$(j(n))\subseteq\omega$ and $(Z_{n})\subseteq\mathcal{P}(\omega)$ such that
$Z_{n}\subseteq Y(h(n))$ and
$Z_{n}\in\overline{\mathrm{Fin}^{2m+5}}\setminus\mathcal{J}_{m}|(Y(h(n))$ for
all $n\in\omega$, where
$h:\omega\to\bigcup_{n>m+1}(\omega^{m+2}\times\ldots\times\omega^{n})$ is a
fixed bijection and

$Y(t)=\bigcup\\{X_{s_{0}}\cap\ldots\cap X_{s_{m}}\cap X_{t_{0}}\cap\ldots\cap
X_{t_{k}}:\
(s_{0},\ldots,s_{m})\in\omega\times\omega^{2}\times\ldots\times\omega^{m+1}\\},$

for all $k\in\omega$ and
$t=(t_{0},\ldots,t_{k})\in\omega^{m+2}\times\omega^{2}\times\ldots\times\omega^{m+k+2}$
(note that $Y(t)\notin\mathcal{J}_{m}$ as $X_{s_{0}}\cap\ldots\cap
X_{s_{m}}\cap X_{t_{0}}\cap\ldots\cap X_{t_{k}}$ are infinite). The sequence
$(j(n))$ will indicate whether
$Z_{n}\in\overline{\emptyset^{2}\otimes\mathrm{Fin}^{2m+2}}$ (the case of
$j(n)=0$) or $Z_{n}\in\overline{\emptyset\otimes\mathrm{Fin}^{2m+3}}(k)$ (the
case of $j(n)=k+1$).

Pick any $n\in\omega$ and consider the ideal $\mathcal{J}_{m}|Y(h(n))$. Recall
that $\overline{\mathrm{Fin}^{2m+4}}$ is isomorphic with
$\mathrm{Fin}^{2m+4}$, which is homogeneous (see [4] or [7, Remark after
Proposition 2.9]), i.e., either $Y(h(n))\in\overline{\mathrm{Fin}^{2m+4}}$ or
$\overline{\mathrm{Fin}^{2m+4}}|Y(h(n))$ is isomorphic with
$\mathrm{Fin}^{2m+4}$. Moreover, as $\mathcal{J}_{m}$ is
$\bf{\Sigma^{0}_{2m+4}}$ (by Lemma 4.4) and the identity function from
$\mathcal{P}(Y(h(n)))$ to $\mathcal{P}(\omega)$ is continuous,
$\mathcal{J}_{m}|Y(h(n))$ is $\bf{\Sigma^{0}_{2m+4}}$ as well, so
$\text{rk}(\mathcal{J}_{m}|Y(h(n)))\leq 2m+3$. Thus, either
$Y(h(n))\in\overline{\mathrm{Fin}^{2m+4}}$ or
$\overline{\mathrm{Fin}^{2m+4}}|Y(h(n))\not\subseteq\mathcal{J}_{m}|Y(h(n))$
(by [3, Lemma 7.2]). Let $Z\subseteq Y(h(n))$ be such that
$Z\in\overline{\mathrm{Fin}^{2m+4}}\setminus\mathcal{J}_{m}|Y(h(n))$. Then
there are $Z^{0}\in\overline{\emptyset^{2}\otimes\mathrm{Fin}^{2m+2}}$,
$Z^{1}\in\overline{(\emptyset\otimes\mathrm{Fin})\otimes\emptyset}$ and
$Z^{2}\in\overline{\mathrm{Fin}\otimes\emptyset}$ with $Z\subseteq Z^{0}\cup
Z^{1}\cup Z^{2}\subseteq Y(h(n))$ and $Z^{0}\cup Z^{1}\cup
Z^{2}\in\overline{\mathrm{Fin}^{2m+4}}$. Since
$Z^{1}\in\mathcal{J}_{m}|Y(h(n))$ by
$\overline{(\emptyset\otimes\mathrm{Fin})\otimes\emptyset}\subseteq\mathcal{J}_{m}$,
there are two possible cases:

  * •

If $Z^{0}\notin\mathcal{J}_{m}|Y(h(n))$ then put $j(n)=0$ and $Z_{n}=Z^{0}$.

  * •

If $Z^{0}\in\mathcal{J}_{m}|Y(h(n))$ then
$Z^{2}\notin\mathcal{J}_{m}|Y(h(n))$, so there is $k\in\omega$ with
$C^{1}_{(k)}\notin\mathcal{J}_{m}|Y(h(n))$. Put $j(n)=k+1$. Similarly as
above, one can conclude that there is $Z^{\prime}\subseteq Y(h(n))\cap
C^{1}_{(k)}$ with
$Z^{\prime}\in\overline{\mathrm{Fin}^{2m+4}}(k)\setminus\mathcal{J}_{m}|(Y(h(n))\cap
C^{1}_{(k)})$. Actually, since $Z^{\prime}\setminus\bigcup_{j\leq
l}C^{2}_{(k,j)}\notin\mathcal{J}_{m}|Y(h(n))$, for every $l\in\omega$ (by the
choice of $m$), we can find $Z_{n}\subseteq Y(h(n))\cap C^{1}_{(k)}$ with
$Z_{n}\in\overline{\emptyset\otimes\mathrm{Fin}^{2m+3}}(k)\setminus\mathcal{J}_{m}|(Y(h(n))\cap
C^{1}_{(k)})$.

Note that $Z_{n}\setminus\bigcup_{i\in\omega}\bigcup_{j\leq
l}C^{2}_{(i,j)}\notin\mathcal{J}_{m}|Y(h(n))$, for every $l\in\omega$ (by the
choice of $m$).

We are ready to define the required set
$W\in\overline{\mathrm{Fin}^{2m+5}}\setminus\mathrm{Fin}^{\prime}_{\omega}$:

$W=\bigcup_{i\in\omega}\bigcup_{j\in\omega}\left(C^{2}_{(i,j)}\cap\bigcup\left\\{Z_{n}:\
n\leq j,j(n)\in\\{0,i+1\\}\right\\}\right).$

It is easy to see that

$\bigcup_{i\in\omega}\bigcup_{j\in\omega}\left(C^{2}_{(i,j)}\cap\bigcup\left\\{Z_{n}:\
n\leq
j,j(n)=0\right\\}\right)\in\overline{\emptyset^{2}\otimes\mathrm{Fin}^{2m+2}}\subseteq\overline{\mathrm{Fin}^{2m+5}}$

and $\bigcup_{j\in\omega}\left(C^{2}_{(i,j)}\cap\bigcup\left\\{Z_{n}:\ n\leq
j,j(n)=i+1\right\\}\right)$ is in
$\overline{\emptyset\otimes\mathrm{Fin}^{2m+3}}(i)$, for each $i\in\omega$.
Hence,

$\bigcup_{i\in\omega}\bigcup_{j\in\omega}\left(C^{2}_{(i,j)}\cap\bigcup\left\\{Z_{n}:\
n\leq j,j(n)=i+1\right\\}\right)\in\overline{\mathrm{Fin}^{2m+5}},$

which implies that $W\in\overline{\mathrm{Fin}^{2m+5}}$.

To end the proof, we have to explain why
$W\notin\mathrm{Fin}^{\prime}_{\omega}$. Suppose to the contrary that
$W\in\mathrm{Fin}^{\prime}_{\omega}$. By Lemma 4.3, there is $k\in\omega$ with
$W\in\mathcal{J}_{k}$. Since $\mathcal{J}_{n}\subseteq\mathcal{J}_{n+1}$, for
all $n\in\omega$ (by Lemma 4.3), we may assume that $k>m$. By the definition
of $\mathcal{J}_{k}$, $W\in\mathcal{J}_{k}$ means that there are
$M_{0}\subseteq\omega,M_{1}\subseteq\omega^{2},\ldots,M_{k}\subseteq\omega^{k+1}$
with $M_{i}\in(\mathrm{Fin}^{i+1})^{\star}$, for $i=0,\ldots,k$, such that if
$s_{i}\in M_{i}$, for $i=0,\ldots,k$, then $W\cap X_{s_{0}}\cap\ldots\cap
X_{s_{k}}\in\mathrm{Fin}$. Pick any sequence $(s_{m+1},\ldots,s_{k})\in
M_{m+1}\times\ldots\times M_{k}$ and denote
$n=h^{-1}((s_{m+1},\ldots,s_{k}))$. As
$(Z_{n}\setminus\bigcup_{i\in\omega}\bigcup_{j<n}C^{2}_{(i,j)})\subseteq W$
and $Z_{n}\subseteq Y(h(n))$, we get

$W\cap X_{s_{0}}\cap\ldots\cap X_{s_{k}}=W\cap X_{s_{0}}\cap\ldots\cap
X_{s_{m}}\cap
Y(h(n))\supseteq\left(Z_{n}\setminus\bigcup_{i\in\omega}\bigcup_{j<n}C^{2}_{(i,j)}\right)\cap
X_{s_{0}}\cap\ldots\cap X_{s_{m}}.$

Since
$Z_{n}\setminus\bigcup_{i\in\omega}\bigcup_{j<n}C^{2}_{(i,j)}\notin\mathcal{J}_{m}$,
there is $(s_{0},\ldots,s_{m})\in M_{0}\times\ldots\times M_{m}$ such that
$(Z_{n}\setminus\bigcup_{i\in\omega}\bigcup_{j<n}C^{2}_{(i,j)})\cap
X_{s_{0}}\cap\ldots\cap X_{s_{m}}\notin\mathrm{Fin}$. Hence, $W\cap
X_{s_{0}}\cap\ldots\cap X_{s_{k}}\notin\mathrm{Fin}$, which contradicts the
choice of $M_{0},\ldots,M_{k}$ and proves that
$W\notin\mathrm{Fin}^{\prime}_{\omega}$. ∎

##  5\. The modified conjecture

In this section we discuss a possible modification of Conjecture 1.1. However,
we start with a nice feature of the ideal $\mathrm{Fin}^{\prime}_{\omega}$.

######  Theorem 5.1.

The following conditions are equivalent for any ideal $\mathcal{I}$:

  * (a)

$\mathrm{Fin}^{\prime}_{\omega}\sqsubseteq\mathcal{I}$;

  * (b)

$\mathrm{Fin}^{\prime}_{\omega}\leq_{K}\mathcal{I}$;

  * (c)

$\mathrm{Fin}^{n+1}\sqsubseteq\mathcal{I}$ for all $n\in\omega$;

  * (d)

$\mathrm{Fin}^{n+1}\leq_{K}\mathcal{I}$ for all $n\in\omega$.

###### Proof.

(a)$\implies$(b): Obvious.

(b)$\implies$(d): For $n\geq 2$ this follows directly from the definition of
$\mathrm{Fin}^{\prime}_{\omega}$ and for $n=1$ this is satisfied for every
ideal.

(d)$\implies$(c): This is [1, Example 4.1].

(c)$\implies$(a): Without loss of generality we may assume that $\mathcal{I}$
is an ideal on $\omega$. By [1, Lemma 3.3] to prove
$\mathrm{Fin}^{\prime}_{\omega}\sqsubseteq\mathcal{I}$ it suffices to define
an injection $f:\omega\to\omega$ witnessing that
$\mathrm{Fin}^{\prime}_{\omega}\leq_{K}\mathcal{I}$.

For each $n\in\omega$ let $h_{n}:\omega^{n+2}\to\omega$ denote the bijection
witnessing that $\mathrm{Fin}^{n+2}\sqsubseteq\mathcal{I}$. Given
$i,n\in\omega$ let $g_{i}(n)\in\omega^{n+1}$ be the unique index such that
$i\in h_{n}[\\{g_{i}(n)\\}\times\omega]$ (i.e., $g_{i}(n)$ indicates in which
element of the partition $\\{h_{n}[\\{s\\}\times\omega]:s\in\omega^{n+1}\\}$
number $i$ is).

We will define $f$ inductively. At first let $f(0)$ be any element of
$X_{g_{0}(0)}\cap X_{g_{0}(1)}\cap\ldots\cap X_{g_{0}(g_{0}(0))}$. Assume now
that $f(j)$ for all $j<i$ are already defined. Let $f(i)$ be any element of

$X_{g_{i}(0)}\cap X_{g_{i}(1)}\cap\ldots\cap
X_{g_{i}(g_{i}(0))}\setminus\\{f(j):j<i\\}.$

Note that this is possible as $X_{g_{i}(0)}\cap X_{g_{i}(1)}\ldots\cap
X_{g_{i}(g_{i}(0))}$ is infinite.

The important observation is that for each $n\in\omega$ and
$i\in\omega\setminus\bigcup_{j<n}h_{0}[\\{j\\}\times\omega]$ we have
$g_{i}(0)\geq n$, so $f(i)\in X_{g_{i}(0)}\cap\ldots\cap X_{g_{i}(n)}\subseteq
X_{g_{i}(n)}$. Now we will check that $f$ is as needed. Since $\mathcal{I}$ is
an ideal, it suffices to check that $f^{-1}[A]\in\mathcal{I}$ for all
$A\in\mathrm{Fin}^{n+2}(\\{X_{s}:s\in\omega^{n+1}\\})$ and $n\in\omega$.

Fix $n\in\omega$ and $A\in\mathrm{Fin}^{n+2}(\\{X_{s}:s\in\omega^{n+1}\\})$.
Then $A=A^{\prime}\cup A^{\prime\prime}$, where $A^{\prime}\cap
X_{s}\in\mathrm{Fin}$ for all $s\in\omega^{n+1}$ and
$A^{\prime\prime}\subseteq\bigcup_{s\in B}X_{s}$ for some
$B\in\mathrm{Fin}^{n+1}$. Observe that

$f^{-1}[X_{s}]\subseteq
h_{n}[\\{s\\}\times\omega]\cup\bigcup_{j<n}h_{0}[\\{j\\}\times\omega]$

for all $s\in\omega^{n+1}$: for each
$i\notin\bigcup_{j<n}h_{0}[\\{j\\}\times\omega]$ we have $f(i)\in
X_{g_{i}(0)}\cap\ldots\cap X_{g_{i}(n)}$, so $f(i)\in X_{s}$ implies
$s=g_{i}(n)$ and $i\in h_{n}[\\{s\\}\times\omega]$, since
$\\{h_{n}[\\{t\\}\times\omega]:t\in\omega^{n+1}\\}$ is a partition of
$\omega$. Thus,

$f^{-1}[A^{\prime\prime}]\subseteq
h_{n}[B\times\omega]\cup\bigcup_{j<n}h_{0}[\\{j\\}\times\omega].$

Moreover,

$C=f^{-1}[A^{\prime}]\setminus\bigcup_{j<n}h_{0}[\\{j\\}\times\omega]\in\mathcal{I}$

as $C\cap h_{n}[\\{s\\}\times\omega]\in\mathrm{Fin}$ for all
$s\in\omega^{n+1}$. Indeed, if $i\in C\cap h_{n}[\\{s\\}\times\omega]$ then
$s=g_{i}(n)$ (by the definition of $g_{i}(n)$) and using the above observation
we have $f(i)\in A^{\prime}\cap X_{s}$. Since $A^{\prime}\cap
X_{s}\in\mathrm{Fin}$ and $f$ is injective, we conclude that $C\cap
h_{n}[\\{s\\}\times\omega]\in\mathrm{Fin}$.

Therefore,

$f^{-1}[A]=f^{-1}[A^{\prime}]\cup
f^{-1}[A^{\prime\prime}]\subseteq\bigcup_{j<n}h_{0}[\\{j\\}\times\omega]\cup
C\cup h_{n}[B\times\omega]\in\mathcal{I}$

(since $B\times\omega\in\mathrm{Fin}^{n+2}$). ∎

######  Remark.

In the language of [1], equivalence of items (a) and (b) in Theorem 5.1 means
that $\mathrm{Fin}^{\prime}_{\omega}$ has property Kat.

######  Corollary 5.2.

$\mathrm{Fin}^{\prime}_{\omega}\sqsubseteq\mathrm{Fin}_{\omega}$.

###### Proof.

It is easy to check that $\mathrm{Fin}^{n+1}\sqsubseteq\mathrm{Fin}_{\omega}$,
for all $n\in\omega$. Thus, the thesis follows from Theorem 5.1. ∎

In the light of Theorem 5.1, there is a chance of reducing Conjecture 1.1 to
the case of successor ordinals. We believe that the following can be proved
using similar methods to those from this paper.

######  Conjecture 5.3.

For each limit ordinal $0<\alpha<\omega_{1}$ there is a
$\bf{\Sigma^{0}_{\alpha}}$ ideal $\mathrm{Fin}^{\prime}_{\alpha}$ of rank
$\alpha$ such that the following conditions are equivalent for any ideal
$\mathcal{I}$:

  * (a)

$\mathrm{Fin}^{\prime}_{\alpha}\sqsubseteq\mathcal{I}$;

  * (b)

$\mathrm{Fin}^{\prime}_{\alpha}\leq_{K}\mathcal{I}$;

  * (c)

$\mathrm{Fin}_{\beta}\sqsubseteq\mathcal{I}$ for all successor ordinals
$\beta<\alpha$;

  * (d)

$\mathrm{Fin}_{\beta}\leq_{K}\mathcal{I}$ for all successor ordinals
$\beta<\alpha$.

By Theorems 4.5 and 5.1, the above is true for $\alpha=\omega$.

Conjecture 5.3 would reduce Conjecture 1.1 to the case of successor ordinals:

######  Remark.

Assume that Conjecture 5.3 is true. Then Conjecture 1.1 is true for all
successor ordinals if and only if for each analytic ideal $\mathcal{I}$:

  * •

$\text{rk}(\mathcal{I})\geq\alpha$ is equivalent to
$\mathrm{Fin}_{\alpha}\sqsubseteq\mathcal{I}$, for all successor ordinals
$0<\alpha<\omega_{1}$;

  * •

$\text{rk}(\mathcal{I})\geq\alpha$ is equivalent to
$\mathrm{Fin}^{\prime}_{\alpha}\sqsubseteq\mathcal{I}$, for all limit ordinals
$0<\alpha<\omega_{1}$.

## References

  * [1] P. Barbarski, R. Filipów, N. Mrożek, P. Szuca, When does the Katětov order imply that one ideal extends the other?, Colloq. Math., 130 (2013), 91–102.
  * [2] G. Debs, J. Saint Raymond, A combinatorial property of Fréchet iterated filters, Topology Appl., 159 (2012), 2205–2227.
  * [3] G. Debs and J. Saint Raymond, Filter descriptive classes of Borel functions, Fund. Math. 204 (2009), no. 3, 189–213.
  * [4] D. H. Fremlin, Filters of countable type, available at https://www1.essex.ac.uk/maths/people/fremlin/preprints.htm (2007), last accessed February 7th 2021.
  * [5] M. Katětov, On descriptive classification of functions, Proceedings of the Third Prague Topological Symposium, (1972), 235–242.
  * [6] A. Kwela, I. Recław, Ranks of $\mathcal{F}$-limits of filter sequences, J. Math. Anal. Appl., 398 (2013), 872–878.
  * [7] A. Kwela, J. Tryba, Homogeneous ideals on countable sets, Acta Math. Hungar., 151 (2017), 139–161.
  * [8] M. Laczkovich, I. Recław, Ideal limits of sequences of continuous functions, Fund. Math., 203 (2009), 39–46.