††thanks: These authors contributed equally††thanks: These authors contributed
equally††thanks: These authors contributed equally

# A universal interatomic potential for perovskite oxides

Jing Wu (gbsn武静)  Key Laboratory for Quantum Materials of Zhejiang Province,
Department of Physics, School of Science and Research Center for Industries of
the Future, Westlake University, Hangzhou, Zhejiang 310030, China  Jiyuan Yang
(gbsn杨季元)  Key Laboratory for Quantum Materials of Zhejiang Province,
Department of Physics, School of Science and Research Center for Industries of
the Future, Westlake University, Hangzhou, Zhejiang 310030, China  Yuan-
Jinsheng Liu (gbsn刘袁今生)  Key Laboratory for Quantum Materials of Zhejiang
Province, Department of Physics, School of Science and Research Center for
Industries of the Future, Westlake University, Hangzhou, Zhejiang 310030,
China  Duo Zhang (gbsn张铎)  DP Technology, Beijing 100080, China  AI for
Science Institute, Beijing 100080, China  Yudi Yang (gbsn杨雨迪)  Key Laboratory
for Quantum Materials of Zhejiang Province, Department of Physics, School of
Science and Research Center for Industries of the Future, Westlake University,
Hangzhou, Zhejiang 310030, China  Yuzhi Zhang (gbsn张与之)  DP Technology,
Beijing 100080, China  AI for Science Institute, Beijing 100080, China
Linfeng Zhang (gbsn张林峰)  DP Technology, Beijing 100080, China  AI for Science
Institute, Beijing 100080, China  Shi Liu (gbsn刘仕<EMAIL_ADDRESS>Key
Laboratory for Quantum Materials of Zhejiang Province, Department of Physics,
School of Science and Research Center for Industries of the Future, Westlake
University, Hangzhou, Zhejiang 310030, China  Institute of Natural Sciences,
Westlake Institute for Advanced Study, Hangzhou, Zhejiang 310024, China

###### Abstract

With their celebrated structural and chemical flexibility, perovskite oxides
have served as a highly adaptable material platform for exploring emergent
phenomena arising from the interplay between different degrees of freedom.
Molecular dynamics (MD) simulations leveraging classical force fields,
commonly depicted as parameterized analytical functions, have made significant
contributions in elucidating the atomistic dynamics and structural properties
of crystalline solids including perovskite oxides. However, the force fields
currently available for solids are rather specific and offer limited
transferability, making it time-consuming to use MD to study new materials
systems since a new force field must be parameterized and tested first. The
lack of a generalized force field applicable to a broad spectrum of solid
materials hinders the facile deployment of MD in computer-aided materials
discovery (CAMD). Here, by utilizing a deep-neural network with a self-
attention scheme, we have developed a unified force field (UniPero) that
enables MD simulations of perovskite oxides involving 14 metal elements and
conceivably their solid solutions with arbitrary compositions. Notably,
isobaric-isothermal ensemble MD simulations with this model potential
accurately predict the experimental temperature-driven phase transition
sequences for several markedly different ferroelectric oxides, including a
6-element ternary solid solution, Pb(In1/2Nb1/2)O3–Pb(Mg1/3Nb2/3)O3–PbTiO3. We
believe the universal interatomic potential along with the training database,
proposed regression tests, and the auto-testing workflow, all released
publicly, will pave the way for a systematic improvement and extension of a
unified force field for solids, potentially heralding a new era in CAMD.

Classical force fields, including CHARMM [1], AMBER [2], and GROMOS [3], which
are capable of accurately describing the interatomic interactions within key
biological molecules such as amino acids and nucleic acids, have played a
pivotal role in expanding the use of molecular dynamics (MD) simulations in
computer-aided drug design (CADD). These force fields enabled large-scale
simulations of atomic and molecular movements in biological systems with
remarkable efficiency and precision, greatly facilitating detailed studies of
drug-receptor interactions [4], protein conformational changes [5, 6], binding
affinities [7], and the search for lead compounds [8]; they are now
indispensable tools in the pursuit of new drugs and therapeutics [9, 10]. In
contrast, although MD simulations have been successful in elucidating the
properties of various types of solids by offering atomic-level insights, their
integration into computer-aided materials discovery (CAMD) has not yet
achieved the same level of industry-wide adoption as in CADD. This disparity
is primarily due to the lack of a generalized force field applicable to a wide
range of solid materials.

Many force fields for crystalline solids are tailored for a specific material
or a narrow group of materials, covering a limited chemical space [11, 12, 13,
14, 15, 16, 17, 18]. While the reactive force-field (ReaxFF) interatomic
potential allows for element-specific parameterization and is commonly viewed
as possessing strong transferability [19], its developers still caution
against arbitrarily combining parameter sets in anticipation of consistent
predictive power [20]. Moreover, the usage of a highly sophisticated energy
function containing a large number of empirical parameters (as seen in ReaxFF)
makes it challenging to parameterize force fields for new materials. In this
work, taking $AB$O3-type perovskite oxides as an example, we demonstrate that
a deep neural network-based model, namely deep potential (DP) [21] with a
novel attention mechanism, referred to as DPA-1 [22], exhibits the required
representability and transferability for the development of a universal
interatomic potential. By adopting the strategy of “modular development of
deep potential” (ModDP) [23] and relying solely on ab initio training data, we
have achieved a unified force field that enables MD simulations for diverse
perovskite oxides comprising 14 different metal elements and is conceivably
adaptable for their solid solutions of any composition. For instance, we
report the first large-scale MD simulations of a 6-element ternary solid
solution, Pb(In1/2Nb1/2)O3–Pb(Mg1/3Nb2/3)O3–PbTiO3 (PIN-PMN-PT), which is a
relaxor ferroelectric possessing superior piezoelectric properties [24, 25].

The $AB$O3-type perovskite oxide is chosen as the model system because of its
role as an exceptionally versatile material platform. The adaptability in
chemical, structural, and compositional variations grants a unique capability
to fine-tune interactions such as electron-phonon coupling, crystal field, and
spin-orbit interaction, the competitions among which often dictate the
functional properties of perovskite oxides [26, 27]. Given these intricacies,
it becomes challenging, if not outright impractical, to rely on a force field
with a fixed analytical structure and a limited set of parameters to create a
generalized force field for perovskite oxides spanning a broad chemical and
compositional range. To address this issue, we adopt the DP model and its
extension augmented with a self-attention scheme.

The DP model, based on a deep neural network with the number of learnable
parameters on the order of 106, offers a robust mathematical structure to
represent highly nonlinear and complex interatomic interactions while
bypassing the need to handcraft descriptors that represent local atomic
environments [21]. Specifically, the DP model features a symmetry-preserving
embedding network that maps an atom’s local environment to inputs for a
fitting neural network which then outputs the atomic energy; the sum of atomic
energies yields the total energy. This approach is a manifestation of the
embedded atom concept, capturing the many-body character of interatomic
interactions. Therefore, the DP model trained using DFT total energies
implicitly accounts for various types of interactions including the long-range
interactions. More recently, the DPA-1 model [22] has introduced an element-
type embedding net and integrated a self-attention mechanism. This mechanism
excels in modeling the importance of neighboring atoms and reweighting the
interactions among them based on both distance and angular information, thus
allowing for the mixing and communication within the latent space of the
embedding parameters of elements and structures. As a result, the DPA-1 model
achieves satisfactory transferability to unseen systems with varied elemental
compositions [22], fulfilling the essential criteria for constructing a
universal force field applicable to solids comprising multiple elements.

To develop a DPA-1 model capable of representing diverse element types and
compositions that the perovskite structure can support, the construction of a
comprehensive training database becomes pivotal. For simple materials systems
comprising two or three elements, the DP-GEN scheme [28, 29] for data
generation is streamlined and largely autonomous, requiring little human
involvement. This concurrent learning procedure iteratively explores the
configuration space (by running MD simulations) using one of the four models
trained with existing data. It then labels only those MD-sampled
configurations that exhibit high uncertainty levels, determined by the maximum
standard deviation of predictions from the four models. The energies and
forces of these newly labeled configurations, computed using first-principles
density functional theory (DFT), are subsequently incorporated into the
training database for the next learning cycle. The iteration stops when all
configurations sampled from MD simulations meet a certain standard of accuracy
across all four models. More details regarding DP-GEN can be found in previous
studies [29, 30, 31]. However, the robustness of the DP-GEN scheme is limited
when constructing the training database for complex oxides with diverse
element types and compositions.

In previous work, we demonstrated that the ModDP protocol facilitates a
systematic development and improvement of DP models for complex oxide solid
solutions [23]. At its core, ModDP embodies the concepts of “data reusing” and
“divide and conquer”: the converged training database associated with an end-
member material is treated as an independent module, and is then reused to
train the DP model of solid solutions derived from these end-members.
Following the same spirit, we devise a procedure that progressively introduces
perovskites with increasing complexity (number of element types) to enhance
the capability of DPA-1 level by level. The workflow of the force field
development procedure that leads to a universal interatomic potential is
depicted in Fig. 1a. The initial database contains $\approx$1000
configurations of 26 different types of perovskites, involving $\approx$200
compositions and 14 metal elements. It is noted that some data for PbTiO3 and
SrTiO3 were taken from a published database [23], and we did not intentionally
design the initial database, aiming to minimize human intervention. We first
use the standard DP-GEN scheme to converge a DPA-1 model suited for 3-element
(including oxygen) perovskite oxides such as PbTiO3, BaTiO3, CaTiO3, SrTiO3,
and NaNbO3. Subsequently, the converged training database serves as the
starting point for DP-GEN to improve DPA-1 for 4-element perovskite systems
like PbZr1-xTixO3 and Pb(Mg1/3Nb${}_{2/3})$O3. Ultimately, we achieve a
converged DPA-1 model for 6-element perovskite systems, including the ternary
solid solution of PIN-PMN-PT. As detailed below, the final DPA-1 model serves
as a universal interatomic potential (named as UniPero), capable of modeling a
wide variety of perovskites with MD simulations.

All DFT calculations are performed with Atomic-orbital Based Ab-initio
Computation at UStc (ABACUS) package [32, 33] using numerical atomic orbitals
(NAOs) and SG15 optimized norm-conserving Vanderbilt (ONCV) pseudopotentials
[34]. The Perdew-Burke-Ernzerhof revised for solids (PBEsol) [35] within the
generalized gradient approximation is chosen as the exchange-correlation
functional, and the double-$\zeta$ plus polarization functions (DZP) with a
plane-wave cutoff energy of 100 Ry is employed as the NAO basis set. The
energies and forces are computed with a 0.1 Bohr-1 $k$-point spacing and an
energy convergence threshold of 1.0$\times 10^{-7}$ Ry. All isobaric-
isothermal ($NPT$) ensemble MD simulations are performed using LAMMPS [36],
with a time step of 2 fs and the temperature controlled via the Nosé-Hoover
thermostat and the pressure controlled by the Parrinello-Rahman barostat. When
simulating temperature-driven phase transitions, the perovskite systems are
modeled with a 10$\times$10$\times$10 supercell (assuming a 5-atom unit cell)
consisting of 5,000 atoms, except for PIN-PMN-PT which is modeled with a
20$\times$20$\times$20 supercell consisting of 40,000 atoms. The smooth
edition of the DP model (with a cutoff radius of 6 Å and the smoothing
starting at 0.5 Å) with an attention scheme, DPA-1, is adopted. The DEEPMD-KIT
package [37] is used in the training cycle of DP-GEN. The basic networks of
DPA-1 comprise the embedding and fitting networks, both of which utilize
ResNet architectures with default settings. Additionally, DPA-1 also
incorporates a one-hot type embedding network with a size of 8, as well as a
two-layer attention mechanism possessing 128 hidden dimensions during the
scaled-dot product attention computation. To enhance reproducibility, we have
made our final training database and hyperparameters accessible via a public
repository [38].

The final training database contains 19288 configurations. We note that DPA-1
aims to reproduce DFT total energies, which inherently vary significantly
between different materials. For configurations in the database, the DFT total
energies span a range of an astonishing $\approx$400 eV/atom (including an
element-specific constant energy shift). Impressively, the DPA-1 model has a
mean absolute error (MAE) of 1.75 meV/atom for the total energy, highlighting
its remarkable representability. To better illustrate the fitting performance,
we compare the energy differences ($\Delta E$) predicted by DPA-1 with DFT
values. For a specific composition, a configuration is (arbitrarily) chosen as
the reference; the $\Delta E$ values are then computed relative to the total
energy of this chosen reference configuration. As shown in Fig. 1b, DPA-1 well
reproduces DFT values of $\Delta E$. The predictions from DPA-1 for atomic
force are in satisfactory agreement with DFT results as well, achieving a MAE
of 0.054 eV/Å (Fig. 1c).

Employing a concurrent learning approach like DP-GEN allows the training
database to be updated efficiently and iteratively. As a result, we do not
have the typical validation or testing datasets commonly associated with
conventional supervised learning. We propose two sets of tasks to evaluate
DPA-1’s applicability for perovskite oxides. For the first set of testing jobs
(denoted as Task I), AIMD simulations are performed at three specific
temperatures (300, 450, 900 K) for a few picoseconds. DPA-1 is then employed
to evaluate the energies for configurations along these AIMD trajectories. The
objective is to ascertain DPA-1’s accuracy in predicting energies for
configurations sampled by AIMD at finite temperatures. As illustrated in Fig.
2a which uses PIN-PMN-PT as an example, the energy evolution trajectory
obtained with DPA-1 closely follows that from AIMD at three different
temperatures, confirmed by Pearson correlation coefficients of 0.93, 0.95, and
0.99, respectively. The second set of evaluations (Task II) is to generate
classical DPMD trajectories using DPA-1 at varying temperatures, the energies
of configurations along which are subsequently determined with DFT. This task
is designed to validate whether classical MD trajectories from DPA-1 properly
sample the relevant configuration spaces at finite temperatures. The
applicability of DPA-1 is gauged by the MAEs of energies and atomic forces for
all the configurations generated from these two sets of tasks. We have
selected 10 representative perovskite oxides for testing: BaTiO3, PbTiO3,
SrTiO3, Ba0.5Sr0.5TiO3, K0.5Na0.5NbO3, NaNbO3–BaTiO3, PbZr0.5Ti0.5O3,
BaZr0.2Ti0.8O3–Ba0.3Ca0.7TiO3, PMN–PT, and PIN–PMN–PT. It is evident from Fig.
2c (for Task I) and Fig. 2d (for Task II) that the DPA-1 model well reproduces
the energy evolution from AIMD and performs appropriate sampling at finite
temperatures for markedly different perovskite systems. The MAE for energy is
lower than 1.45 meV/atom, and that for atomic force is below 0.052 eV/Å.

With DPA-1’s capability to faithfully capture the complex and highly nonlinear
potential energy surfaces involving diverse element types, we expect
developing a DPA-1 model with further enhanced transferability and accuracy
can be achieved by introducing more data from new materials systems into the
current training database. Therefore, the proposed tasks, which are readily
automatable, serve as suitable regression tests (a common practice in software
development) for the continuous improvement of the DPA-1 model. We have
implemented an auto-testing workflow based on an online platform (Bohrium
Notebook) [38] that allows real-time access to our DPA-1 model, cloud-based
computing resources, and data analysis scripts. This initiative aims to
streamline force field development and foster collaborative contributions from
the community.

A definitive test of the applicability of the developed DPA-1-based universal
interatomic potential is to examine temperature-driven phase transitions in
ferroelectric perovskites. As summarized Fig. 3, a single DPA-1 model, without
any tuning, successfully reproduces the experimental sequences of phase
transitions for lead-based ferroelectrics (PbTiO3, Pb0.5Sr0.5TiO3, and
PbZr0.5Ti0.5O3), lead-free ferroelectrics (BaTiO3, KNbO3, and K0.5Na0.5NbO3),
quantum paraelectric SrTiO3, and tertiary solid solutions
(0.29PIN–0.45PMN–0.26PT and 0.36PIN–0.36PMN–0.28PT). For example, the
temperature-dependent lattice constants obtained from $NPT$ MD simulations
reveal a tetragonal ($T$, space group $P4mm$) to cubic ($C$, space group
$Pm\bar{3}m$) transition at $\approx$550 K in PbTiO3 (Fig. 3a), whereas the
transition temperature ($T_{c}$) drops to $\approx$280 K in the $A$-site doped
solid solution, Pb0.5Sr0.5TiO3 (Fig. 3d). In comparison, MD simulations using
DPA-1 predict that the $B$-site doped solid solution, PbZr0.5Ti0.5O3, adopts a
rhombohedral phase ($R$, space group $R3m$) at low temperatures, and it
undergoes rhombohedral-tetragonal-cubic ($R$-$T$-$C$) phase transitions as the
temperature increases (Fig. 3g), consistent with experimental observations
[39, 40]. SrTiO3 is a well-known quantum paraelectric that exhibits a
temperature-driven tetragonal-cubic transition ($I4/mcm\rightarrow
Pm\bar{3}m$) characterized by a decrease in the TiO6 octahedral tilt angle.
This phase transition, corresponding to a tiny energy barrier of $\approx$1
meV/atom, is reproduced by DPA-1 (Fig. 3b), and the theoretical value of
$T_{c}$ is 150 K, which is comparable to the experimental value of 105 K [41].
The same model also correctly predicts the phase transition sequence,
rhombohedral-orthorhombic-tetragonal-cubic ($R$-$O$-$T$-$C$), in BaTiO3,
KNbO3, and K0.5Na0.5NbO3 (see Fig. 3c, e, and h). Moreover, even for the
highly complex 6-element tertiary solid solutions such as PIN-PMN-PT, DPA-1
not only reproduces the $T$–$C$ phase transition observed in experiments, but
also predicts the correct trend in the change of $T_{c}$ due to the
compositional variation. As shown in Fig. 3c and d, the $T_{c}$ rises by about
50 K from 0.29PIN–0.45PMN–0.26PT to 0.36PIN–0.36PMN–0.28PT, similar to the
$T_{c}$ change of $\approx$60 K found in experiments due to the increased
percentage of PIN from 29% to 36% [42, 43]. It is observed that theoretical
$T_{C}$ values for ferroelectric transitions are typically lower than
experimental values by 100-200 K. Such discrepancies appear to be a common
feature for force fields developed by fitting to results computed with the
PBEsol density functional [44, 23, 45]. Given the high fidelity of DPA-1 as
demonstrated by its excellent fitting to DFT results, the underestimation of
$T_{C}$ is more likely a reflection of PBEsol’s limitations in forecasting
finite-temperature properties rather than an issue with DPA-1’s precision.

The universal interatomic potential developed here can serve as a readily
deployable tool to reveal the local structures and lattice dynamics of
perovskite oxides with atomic spatial resolution. Using three 50-ps
equilibrium DPMD trajectories for 0.29PIN–0.45PMN–0.26PT, each with different
$B$-site cation arrangements and modeled with a 40,000-atom supercell, we
analyze the distributions of local displacements of cations with respect to
the center of their surrounding oxygen cages. For describing inversion
symmetry breaking in ferroelectric materials, polarization is often the
natural order parameter. Given a configuration from MD simulations, the
polarization for simple perovskite systems such as PbTiO3 can be estimated
using the following formula,

$\mathbf{P}^{m}(t)=\frac{1}{V_{\rm
uc}}\left[\frac{1}{8}\mathbf{Z}_{\mathrm{Pb}}^{*}\sum_{i=1}^{8}\mathbf{r}_{\mathrm{Pb},i}^{m}(t)+\mathbf{Z}_{\mathrm{Ti}}^{*}\mathbf{r}_{\mathrm{Ti}}^{m}(t)+\frac{1}{2}\mathbf{Z}_{\mathrm{O}}^{*}\sum_{i=1}^{6}\mathbf{r}_{\mathrm{O},i}^{m}(t)\right]$

, where $\mathbf{P}^{m}(t)$ is the polarization of unit cell $m$ at time $t$,
$V_{\rm uc}$ is the volume of the unit cell,
$\mathbf{Z}_{\mathrm{Pb}}^{*},\mathbf{Z}_{\mathrm{Ti}}^{*}$, and
$\mathbf{Z}_{\mathrm{O}}^{*}$ are the Born effective charges of Pb, Ti and O
atoms, $\mathbf{r}_{\mathrm{Pb},i}^{m}(t),\mathbf{r}_{\mathrm{Ti},i}^{m}(t)$,
and $\mathbf{r}_{\mathrm{O},i}^{m}(t)$ are the instantaneous atomic positions.
The polarization is thus the dipole moment per unit volume. However, PIN-PMN-
PT is considered a charge-frustrated system where the $B$ sites host cations
of different charges: Mg2+, In3+, Ti4+, and Nb5+. It becomes problematic to
define a local electric dipole for a unit cell that is not locally charge
neutral (e. g., a local PbInO3 unit cell in PIN-PMN-PT solid solutions).
Therefore, we use local atomic displacements of cations to measure the degree
of inversion symmetry breaking at the unit cell level.

In Fig. 4a-c, the probability density distributions of cation displacements
($d_{x}$, $d_{y}$, $d_{z}$) along the Cartesian axes are shown as a function
of temperature. At a low temperature of 200 K, the distributions of $d_{x}$
and $d_{y}$ are mostly Gaussian-like and centered around zero for all cations.
In contrast, the peak position for $d_{z}$ of Pb deviates the most away from
zero, followed by Ti, Nb, In, and Mg in sequence. This deviation in the
$d_{z}$ distribution, compared to $d_{x}$ and $d_{y}$, aligns with the global
tetragonal symmetry indicated by the lattice constants. Importantly, these
distributions highlight a distinct difference in the local structures between
relaxor ferroelectric PIN–PMN–PT and prototypical ferroelectrics like PbTiO3.
In PIN–PMN–PT, the distributions of cation displacements are more dispersive
and isotropic, spanning both positive and negative values along all three
Cartesian directions. Conversely, both Pb and Ti cations in ferroelectric
PbTiO3 displace along the same polar direction (e. g., [001]) [44]. At an
elevated temperature of 300 K, the peak positions of the $d_{z}$ distributions
for all cations shift toward values lower than those at 200 K. When
temperature rises above $T_{C}$, distributions become Gaussian, all centering
at zero.

Snapshots of instantaneous cation displacements in configurations sampled from
MD are displayed in Fig. 4d-e. We find that the magnitude of local Pb
displacements exhibits less temperature sensitivity compared to the $B$-site
cations. Even at 400 K which is above the theoretical $T_{C}$, many Pb atoms
remain locally displaced; their displacements grow more chaotic, manifesting
diminished long-range order. The displacements of $B$-site cations, however,
display more vulnerability: some are already nonpolar at 200 K, and an
increasing percentage becomes nonpolar as temperature increases. Notably, we
do not observe any polar nanoregions (PNRs) embedded in a nonpolar matrix, as
the majority of Pb atoms are displaced, in agreement with previous MD
simulations of PMN–PT [46]. The markedly different behaviors between Pb and
$B$-site cations could be important for understanding the piezoelectric
response of lead-based relaxor ferroelectrics. These findings underscore the
quantitative nature of the DPA-1 model.

We briefly comment on the main limitations and challenges of machine-learning
force fields like DPA-1, which center around interpretability and
computational efficiency. While the large number of learnable parameters of
DPA-1 facilitates the representation of complex interactions, it
simultaneously hinders our understanding of these very interactions. In
essence, each of these parameters adjusts itself during the learning process
to minimize prediction errors, but the collective behavior of such a high
number of parameters can be enigmatic. This contrasts with traditional force
fields which, through their explicit functional forms, often allow for clearer
physical interpretation. Although the DPA-1 model demonstrates a clear speed
advantage over first-principles methods like DFT, it still lags behind
conventional force fields. For instance, the bond-valence model [44] typically
runs two orders of magnitude faster than the DPA-1 model. Solutions to the
aforementioned challenges exist. Principal component analysis can be performed
to analyze parameters of the DPA-1 model to explore the information within the
latent space. For example, in a recent work of DPA-1 [22], the visualizations
of the element-related parameters learned by the model coincide with the
periodic table, partially demonstrating the model’s interpretability. As for
computational efficiency, model compression [47] is set to alleviate this
problem by using a small number of polynomial coefficients to fit the learned
parameters in the model. This approach can significantly reduce the
computational cost while maintaining accuracy.

The current state of DPA-1 for perovskite oxides can be viewed as a pre-
trained model, which is already sufficiently accurate for many perovskite
oxides as discussed above. Drawing parallels from the development history of
ChatGPT [48] in natural language processing, such a pre-trained model can be
fine-tuned for specific downstream tasks with minimal extra effort using few-
shot or zero-shot learning techniques [49]. As the community continues to
amass high-quality electronic structure data, it is conceivable that this pre-
trained model will progressively expand its coverage, encompassing an ever
broader range of chemical and structural spaces.

In summary, using the $AB$O3-type perovskite oxide as a model system, our
results demonstrate that a deep neural network, augmented with an attention
scheme, can deliver the requisite representability and transferability to
establish a universal interatomic potential for solid materials spanning
diverse element types and compositions, at least for materials systems with a
fixed crystal structure. Even with the complex nature of target systems
involving 15 elements, the final training database remains impressively
concise, containing no more than 20,000 configurations; the associated
computational cost is manageable, requiring fewer than 200,000 CPU hours. The
DPA-1-based force field, UniPero, developed in this work already exhibits
satisfactory accuracy, proving suitable for large-scale MD simulations of
distinct ferroelectric perovskites and reproducing the correct phase
transition sequences driven by temperature. By publicly releasing the training
database and the automated testing workflow, we anticipate our initiative will
facilitate a systematic improvement and extension of a generalized force field
with an even larger scope (e. g., free surfaces and defects). This, in turn,
hopefully could foster the routine and efficient use of molecular dynamics for
simulating solid materials, ultimately providing valuable insights for
computer-aided materials discovery.

###### Acknowledgements.

This work is supported by Natural Science Foundation of Zhejiang Province
(2022XHSJJ006), National Natural Science Foundation of China (12074319), and
Westlake Education Foundation. The computational resource is provided by
Westlake HPC Center and Bohrium platform.

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Figure 1: (a) Workflow for the development of a universal force field of
perovskite oxides following the ModDP protocol. Comparison of (b) relative
energies ($\Delta E$) and (c) atomic forces predicted using the DPA-1 model
with reference DFT results for $\approx$20,000 configurations in the final
training database. The inset shows the distribution of the absolute error.

Figure 2: Proposed tasks to evaluate the applicability of DPA-1 using
PIN–PMN–PT as an example. (a) Task I. The energies of configurations sampled
by AIMD at three different temperatures are checked by DPA-1. (b) Task II. The
energies of configurations sampled by classical MD with DPA-1 are checked by
DFT. Comparison of (c) relative energies and (d) atomic forces for
configurations of 10 representative perovskite oxides generated from Task I
and Task II.

Figure 3: Temperature-dependent lattice constants simulated with $NPT$ DPMD
using the universal interatomic potential for (a) PbTiO3, (b) SrTiO3, (c)
BaTiO3, (d) Pb0.5Sr0.5TiO3, (e) KNbO3, (f) 0.29PIN–0.45PMN–0.26PT, (g)
PbZr0.5Ti0.5O3, (h) K0.5Na0.5NbO3, (i) 0.36PIN–0.36PMN–0.28PT.

Figure 4: Probability density distributions of cation displacements along the
(a) $x$-axis, (b) $y$-axis, (c) $z$-axis as temperature increases in
0.29PIN–0.45PMN–0.26PT modeled with a 40,000-atom supercell. Snapshots of (d)
$A$-site and (e) $B$-site cation displacements at different temperatures
obtained from MD simulations. Each white arrow indicates the local atomic
displacement within a unit cell, with its background color representing the
direction. Note that if the magnitude of the local displacement is less than
0.1 Å, the background is colored white.


# Semi-generalized co-Bassian groups

Andrey R. Chekhlov  Department of Mathematics and Mechanics, Regional
Scientific and Educational Mathematical Center, Tomsk State University, 634050
Tomsk, Russia<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Peter V. Danchev
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113
Sofia, Bulgaria<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Patrick W. Keef
Department of Mathematics, Whitman College, Walla Walla, WA 99362, USA
<EMAIL_ADDRESS>
###### Abstract.

As a common non-trivial generalization of the notion of a generalized co-
Bassian group, recently defined by the third author, we introduce the notion
of a semi-generalized co-Bassian group and initiate its comprehensive study.
Specifically, we give a complete characterization of these groups in the cases
of $p$-torsion groups and groups of finite torsion-free rank by showing that
these groups can be completely determined in terms of generalized finite
$p$-ranks and also depends on their quotients modulo the maximal torsion
subgroup. Surprisingly, for $p$-primary groups, the concept of a semi-
generalized co-Bassian group is closely related to that of a generalized co-
Bassian group.

###### Key words and phrases:

Bassian groups, (generalized) Bassian groups, (generalized) co-Bassian groups,
semi-generalized co-Bassian groups

###### 2010 Mathematics Subject Classification:

20K10, 20K20, 20K30

##  1\. Introduction and Motivation

Throughout the rest of the paper, unless specified something else, all groups
will be additively written Abelian groups. We will primarily use the notation
and terminology of [6, 7, 8], but we will follow somewhat those from [10] and
[9] as well. We just recall that an arbitrary subgroup $H$ of a group $G$ is
essential in $G$ if, for any non-zero subgroup $S$ of $G$, the intersection
between $H$ and $S$ is also non-zero. It is an elementary exercise to see that
every subgroup will be essential as a subgroup of itself and thus, in
particular, the zero subgroup $\\{0\\}$ is essential in itself too.

We begin with a brief review of some of the most important concepts which
motivate our writing of the present article.

Mimicking [1], a group $G$ is said to be Bassian if the existence of an
injective homomorphism $\phi:G\to G/N$ for some subgroup $N$ of $G$ implies
that $N=\\{0\\}$. More generally, imitating [2], if this injection $\phi$
implies that $N$ is a direct summand of $G$, then $G$ is said to be
generalized Bassian.

Note that Bassian groups were completely characterized in [1]. A crucial
example of a generalized Bassian group which is not Bassian is the infinite
elementary $p$-group that is an arbitrary infinite direct sum of the cyclic
group $\mathbb{Z}_{p}$ for some fixed prime $p$. Unfortunately, the class of
generalized Bassian groups is not fully characterized due to the unsettled at
this stage problem of whether or not a subgroup of a generalized Bassian group
remains so. On this vein, for some other interesting properties of subgroups
of (generalized) Bassian groups, we refer the interested readers to [4].

Taking into account all of the information we have so far, and in order to
refine the group property of being generalized Bassian, we introduced in [3] a
group $G$ to be semi-generalized Bassian if, for any its subgroup $N$, the
injective homomorphism $\phi:G\to G/N$ implies that $N$ is essential in a
direct summand of $G$. Clearly, any generalized Bassian group is semi-
generalized Bassian. In difference to the case of generalized Bassian groups,
it is quite curious that semi-generalized Bassian groups were totally
classified in the cases where $G$ is not truly mixed. In fact, we succeeded to
classify torsion, torsion-free and splitting mixed semi-generalized Bassian
groups.

Reciprocally, in [11], was defined the so-called (generalized) co-Bassian
groups in the following manner: A group $G$ is termed co-Bassian if, for all
subgroups $H\leq G$, whenever $\varphi:G\to G/H$ is an injective homomorphism,
then $\varphi(G)=G/H$. In general, if $\varphi(G)$ is a direct summand of
$G/H$, the group $G$ is termed generalized co-Bassian. Fortunately, these two
classes of groups were completely described.

Furthermore, expanding generalized co-Bassian groups in the way of semi-
generalized Bassian groups, we shall say that a group $G$ is semi-generalized
co-Bassian if, for all $H\leq G$, the injection $\varphi:G\to G/H$ forces that
$\varphi(G)$ is essential in a direct summand of $G/H$.

Our motivating tool is to explore the structure of the so-defined semi-
generalized co-Bassian groups and to show that their full description is
rather more complicated than the case of generalized-Bassian groups. The
essence of our claim is the situation of having such groups with infinite
torsion-free rank and, especially, the lack of workable idea and approach how
to prove or even to disprove that if $G$ is a group whose torsion subgroup $T$
is a direct sum of a divisible group and an elementary group such that the
quotient $G/T$ is divisible, then $G$ is semi-generalized co-Bassian group or
not.

Our next work is structured thus: In the present first section, we gave a
short retrospection of the basic notions. In the subsequent second section, we
formulate our chief results and provide their complete proofs. Precisely, the
paper’s main goal is to characterize in detail semi-generalized co-Bassian
$p$-groups as well as semi-generalized co-Bassian groups having finite
torsion-free rank. Concretely, the most important of them state like this: A
$p$-group $G$ is semi-generalized co-Bassian if, and only if, its subgroup
$pG$ has generalized finite $p$-rank (see Theorem 2.7); Suppose $G$ is a group
of finite torsion-free rank. Then, $G$ is semi-generalized co-Bassian if, and
only if, for each prime $p$, the $p$-component of torsion $T_{p}$ is semi-
generalized co-Bassian group such that either (a) $T_{p}$ possesses finite
$p$-rank, or (b) $T_{p}$ is divisible, or (c) the quotient-group $G/T$ is
$p$-divisible (see Theorem 2.9). We also examine the case of groups with
infinite torsion-free rank proving that If a group $G$ is semi-generalized co-
Bassian of infinite torsion-free rank, then the factor-group $G/T$ is
divisible and the torsion part $T$ is a direct sum of a divisible group and an
elementary group (see Proposition 2.11). Finally, we end our work with certain
comments and also state several open problems which quite logically arise and,
hopefully, will stimulate a further exploration on the subject.

##  2\. Main Results and Their Proofs

We first begin with the following elementary but useful observation, which was
not stated and proved in [11], that shows to what extent the classes of co-
Bassian and generalized co-Bassian groups differ each other.

######  Proposition 2.1.

If $G$ is a Hopfian group, then $G$ is generalized co-Bassian if, and only if,
$G$ is co-Bassian.

###### Proof.

One direction being trivial, we deal with the opposite one. In fact, assume in
a way of a contradiction that $G$ is generalized co-Bassian but, however, not
co-Bassian. Then, there exists a subgroup $N$ of $G$ and an embedding $f:G\to
G/N$. Since $G$ is generalized co-Bassian, one has that $f(G)$ is a direct
summand of $G/N$, say $G/N=f(G)\oplus H/N$ for some
$H/N\neq\\{\overline{0}\\}$. Now, there is an isomorphism

$G/H\cong(G/N)/(H/N)\cong f(G),$

say $\varphi:G/H\to f(G)$ and, moreover, if $\pi:G\to G/H$ is the canonical
epimorphism, then the composition $\varphi^{-1}\pi$ is obviously an
epimorphism but a non-isomorphism of $G$ as
$\ker(\varphi^{-1}\pi)=H\neq\\{0\\}$, contrary to the condition that $G$ is
Hopfian. So, $G$ is really co-Bassian, as required. ∎

Our next incidental assertion curiously states as follows.

######  Proposition 2.2.

The free group $G$ is semi-generalized co-Bassian if, and only if, $G$ is
Bassian.

###### Proof.

If we assume for a moment that $G$ has infinite rank, then there exists a
subgroup $N\leq G$ such that the quotient-group $G/N$ is a torsion-free
indecomposable group with ${\rm rank}(G)={\rm rank}(G/N)$. Thus, there is an
injection $f:G\to G/N$ such that $f(G)$ is not essential in $G/N$, whence for
the purification of $f(G)$ we have $\langle f(G)\rangle_{*}\neq G/N$ and,
consequently, in view of indecomposability of $G/N$ the group $G$ is not semi-
generalized co-Bassian, as expected. Therefore, $G$ has a finite rank and
hence applying [1] the group $G$ must be Bassian, as claimed. ∎

We continue our work with two critical constructions as follows:

######  Example 2.3.

(i) Every divisible group is semi-generalized co-Bassian.

(ii) The group $G=\bigoplus_{\alpha}\mathbb{Z}_{p^{n}}$ is semi-generalized
co-Bassian for all ordinals $\alpha$ and integers $n\geq 1$.

###### Proof.

(i) This follows automatically, because the factor-group of a divisible group
is also divisible and, for each divisible group $D$, any its subgroup is
essential in some direct summand of $D$ (see [7, Theorem 24.4]), as needed.

(ii) Since, for every injection $f:G\to G/H$, the image $f(G)$ is a direct sum
of cyclic groups of order exactly $p^{n}$ and $G/H$ is a direct sum of cyclic
groups of order $\leq p^{n}$, we can infer that $f(G)$ is a direct summand in
$G/H$ (see [7, Proposition 27.1]), as required. ∎

The following technicality is helpful for our presentation below.

######  Lemma 2.4.

The class of semi-generalized co-Bassian groups is closed under taking direct
summands.

###### Proof.

Suppose $G$ is a semi-generalized co-Bassian group and $H$ is a summand of
$G$; say $G=H\oplus G^{\prime}$ for some $G^{\prime}\leq G$. Assume now $N$ is
a subgroup of $H$ and $f:H\to H/N:=\overline{H}$ is an injective homomorphism.
Extending $f$ to $f^{\prime}:G\to G/N:=\overline{G}\cong\overline{H}\oplus
G^{\prime}$ by letting it equal to the identity on $G^{\prime}$, it remains an
injective homomorphism.

We, therefore, have $\overline{G}=K\oplus A$, with $A$ containing
$f^{\prime}(G)=f(H)\oplus G^{\prime}$ as an essential subgroup. It readily
follows that $A\cap\overline{H}\subseteq\overline{H}$ contains $f(H)$ as an
essential subgroup. And since $A=(A\cap\overline{H})\oplus G^{\prime}$, we can
also conclude that $A\cap\overline{H}$ is a summand of $\overline{G}$, and
hence of $\overline{H}$, as required. ∎

We now continue with a series of statements necessary for the successful
establishment of our two main results presented in what follows.

Before doing that, we need to review some notation, used in [11]: As above,
when $G$ is some group, $T$ will always be its torsion subgroup and $T_{p}$
the $p$-component of $T$. If we have occasion to refer to some other group
$A$, we will denote its torsion subgroup by $T_{A}$. For the group $G$,
suppose $\phi,$ and $\pi$ are, respectively, injective and surjective
homomorphism $G\to\overline{G}$; when necessary, we may assume
$\overline{G}=G/N$ for some $N\leq G$ and $\pi$ is the usual epimorphism. if
$A$ is any subgroup of $G$, we will let $\hat{A}=\phi(A)$ and
$\overline{A}=\pi(A)$.

######  Lemma 2.5.

If $p$ is a prime and $T_{p}$ is a reduced unbounded $p$-group, then $G$ is
not a semi-generalized co-Bassian group. In particular, any reduced unbounded
$p$-group $G$ is not a semi-generalized co-Bassian group.

###### Proof.

Suppose $T_{p}=\langle b\rangle\oplus T^{\prime}_{b}$, where $b$ has order at
least $p^{2}$, and $G=\langle b\rangle\oplus G^{\prime}$, where
$T^{\prime}_{p}\subseteq G^{\prime}$. If $D$ is a divisible hull for
$T^{\prime}_{p}$ and $Z:=\mathbb{Z}_{p^{\infty}}$, then there is a surjective
homomorphism $T_{p}^{\prime}\to Z\oplus D$ which extends to a surjective
homomorphism $\tau:G^{\prime}\to Z\oplus D$. This, in turn, extends to a
surjective homomorphism

$\pi:G=\langle b\rangle\oplus G^{\prime}\to\overline{G}:=\langle
b\rangle\oplus Z\oplus D\oplus(G^{\prime}/T^{\prime}_{p})$

by setting it equal to the identity on $\langle b\rangle$ and, for $x\in
G^{\prime}$, letting $\pi(x)=\tau(x)+(x+T^{\prime}_{p})$.

Let $c\in Z$ have the same order as $b$. The inclusion
$T^{\prime}_{p}\subseteq D$ extends to a homomorphism $\sigma:G^{\prime}\to
D$. Define a homomorphism $\phi:G\to\overline{G}$ as follows: If $x\in
G^{\prime}$, let $\phi(x)=\sigma(x)+(x+T^{\prime}_{p})$, and $\phi(b)=pb+c$.
It is readily checked that $\phi$ is injective.

If $G$ were semi-generalized co-Bassian, it would follow that
$\hat{G}=\phi(G)$ is essential in a summand $S$ of $\overline{G}$. Since every
element of $\hat{G}[p]=Z[p]\oplus D[p]$ has infinite height in $\overline{G}$,
the $p$-torsion subgroup of $S$ is a divisible subgroup of $\overline{G}$.
However, $\phi(b)=pb+c$ is such a $p$-torsion element of $S$ and its
$p$-height satisfies $|pb+c|_{S}=|pb+c|_{\overline{G}}=1$. With this
contradiction, we can deduce that $G$ is not semi-generalized co-Bassian, as
stated.

The second part is immediate. ∎

######  Proposition 2.6.

Suppose $m>1$ is an integer and $\alpha$ is either $\infty$ or a positive
integer with $m<\alpha$. If $G$ is a group with a summand of the form
$\mathbb{Z}_{p^{m}}\oplus\mathbb{Z}_{p^{\alpha}}^{(\kappa)}$, where $\kappa$
is infinite, then $G$ is not semi-generalized Bassian.

###### Proof.

Utilizing Lemma 2.4, there is no loss of generality in assuming that
$G=\mathbb{Z}_{p^{m}}\oplus\mathbb{Z}_{p^{\alpha}}^{(\kappa)}$. Suppose the
first summand is $\langle b\rangle$. There is clearly a decomposition
$\mathbb{Z}_{p^{\alpha}}^{(\kappa)}=\mathbb{Z}_{p^{\alpha}}\oplus Z$ with an
isomorphism $\sigma:\mathbb{Z}_{p^{\alpha}}^{(\kappa)}\to Z$. Let $c$ be an
element of the first term of this decomposition of order $p^{m}$; in
particular, $pc\neq 0$ and $|pc|_{p}>1$.

Let $N=\langle pb\rangle$ and

$\pi:G\to\overline{G}:=(\langle
b\rangle/N)\oplus\mathbb{Z}_{p^{\alpha}}^{(\kappa)}=(\langle
b\rangle/N)\oplus\mathbb{Z}_{p^{\alpha}}\oplus Z$

be the obvious surjection.

Define now the map $\phi:G\to\overline{G}$ as follows: $\phi(b)=(b+N)+c$ and
$\phi$ restricted to $\mathbb{Z}_{p^{\alpha}}$ agrees with $\sigma$. It is
readily checked that $\phi$ is injective. If, however, $\hat{G}$ were an
essential subgroup of a summand of $\overline{G}$, it would easily follow that
$\langle\phi(b)\rangle$ would be an essential subgroup of a summand of
$(\langle b\rangle/N)\oplus\mathbb{Z}_{p^{\alpha}}$. Since this summand would
have to be cyclic, this cannot be true, because $|\phi(b)|_{p}=0$,
$p\phi(b)=pc\neq 0$ and $|p\phi(b)|_{p}=|pc|_{p}>1$, as required. ∎

Following [11], the $p$-group $G$ is said to have _generalized finite
$p$-rank_ if

$G\cong\mathbb{Z}^{(\rho_{1})}_{p^{\sigma_{1}}}\oplus\dots\oplus\mathbb{Z}^{(\rho_{n})}_{p^{\sigma_{n}}},$

where all ordinals of the increasing sequence $\sigma_{1}<\dots<\sigma_{n}$
are in $\omega\cup\\{\infty\\}$, and $\rho_{1},\dots,\rho_{n}$ are cardinals
with $\rho_{j}$ finite whenever $j>1$.

We are now prepared to prove our first major assertion, which sounds quite
surprising.

######  Theorem 2.7.

The $p$-group $G$ is semi-generalized co-Bassian if, and only if, $pG$ is
generalized co-Bassian, i.e., $pG$ has generalized finite $p$-rank.

###### Proof.

It is easily verified that $G$ satisfies this condition exactly when there is
a decomposition $G=E\oplus H$, where $E$ is a $p$-high subgroup of $G$ and $H$
is a group that has generalized finite $p$-rank with no elementary summands.

Firstly, suppose $G$ is semi-generalized co-Bassian. Using Lemmas 2.4 and 2.5,
$G$ must be of the form:

$G=\mathbb{Z}_{p^{\alpha_{1}}}^{(\kappa_{1})}\oplus\mathbb{Z}_{p^{\alpha_{2}}}^{(\kappa_{2})}\oplus\cdots\oplus\mathbb{Z}_{p^{\alpha_{k}}}^{(\kappa_{k})},$

where the $\kappa$s are all (non-zero) cardinals and
$\alpha_{1}<\alpha_{2}<\cdots<\alpha_{k}$ are either positive integers or
$\infty$. The result, therefore, follows directly from the above Proposition
2.6.

Conversely, suppose that $G=E\oplus H$ possesses the above form. By
definition, $H\cong\mathbb{Z}_{p^{\alpha}}^{(\kappa)}\oplus F$, where
$\alpha>1$ is either an integer or $\infty$, $\kappa$ is a cardinal, $F$ has
finite $p$-rank and $F[p]=p^{\alpha}H[p]$ when $\alpha$ is an integer and
$F=0$ when $\alpha=\infty$. If $\kappa$ is finite, then $G$ itself is
generalized co-Bassian, and hence semi-generalized co-Bassian, as desired. So,
we may assume hereafter that $\kappa$ is infinite.

Let $N$ be a subgroup of $G$, and set $\overline{G}:=G/N$ and
$\phi:G\to\overline{G}$ is an injective homomorphism. As in [10], for any
subgroup $A$ of $G$, we let $\hat{A}=\phi(A)$ and $\overline{A}$ be the image
of $A$ under the canonical homomorphism $\pi:G\to\overline{G}$. We need to
show $\hat{G}=\phi(G)$ is an essential subgroup of a summand of
$\overline{G}$.

To that goal, suppose first that $\alpha=\infty$; so,
$H=\mathbb{Z}_{p^{\infty}}^{(\kappa)}$ is the maximal divisible subgroup of
$G$ and $F=0$. Thus, $\overline{H}$ is divisible and
$p(\overline{G}/\overline{H})=0$, so that
$\overline{G}=E^{\prime}\oplus\overline{H}$, where $E^{\prime}$ is also
elementary.

Clearly, $\hat{H}$ will also be divisible, so that
$\overline{H}=D\oplus\hat{H}$, where $D$ is divisible. If

$E^{\prime\prime}:=\hat{G}\cap(E^{\prime}\oplus D)\subseteq(E^{\prime}\oplus
D)[p],$

it follows that $\hat{G}=E^{\prime\prime}\oplus\hat{H}$. But it also plainly
follows that any subsocle of a direct sum of an elementary group and a
divisible group is supported by a summand of the containing group (see [7] as
well). In particular, this means that $\hat{G}=E^{\prime\prime}\oplus\hat{H}$
is an essential subgroup of a summand of $G$, as required.

The proof where $\alpha$ is a positive integer is similar; for clarity, denote
$\alpha$ by $n$. Since $p^{n}G=p^{n}H$ is co-Bassian and $\phi$ and $\pi$
restrict to an injection and surjection, respectively, on $p^{n}G$, it follows
that $p^{n}\hat{H}=p^{n}\hat{G}=p^{n}\overline{G}$. Since $H$ has no summands
isomorphic to $\mathbb{Z}_{p^{m}}$ for $m<n$, this readily implies that
$\overline{G}=\hat{H}\oplus K$, where $p^{n}K=0$. It follows that $\hat{G}\cap
K\subseteq K[p]$. And since $K$ is bounded, $\hat{G}\cap K$ supports a summand
$K^{\prime}$ of $K$. Consequently, $K^{\prime}\oplus\hat{H}$ will be a summand
of $\overline{G}$ containing $\hat{G}$ as an essential subgroup, as wanted. ∎

The pivotal instrument, needed to establish our second major assertion listed
below, is the following.

######  Proposition 2.8.

If $G$ is a semi-generalized co-Bassian group and $p$ is a prime, then $T_{p}$
is semi-generalized co-Bassian.

###### Proof.

Suppose $G$ is a semi-generalized co-Bassian group, $p$ is a prime and
$T_{p}=R\oplus D$, where $R$ is reduced and $D$ is divisible. If $D$ has
infinite $p$-rank, then by Proposition 2.6 and Lemma 2.4, $R=\\{0\\}$ so that
$T_{p}$ is semi-generalized co-Bassian. So, we may assume that $D$ has finite
rank. If $G=G^{\prime}\oplus D$, where $R=T_{G^{\prime}}$, then by Lemma 2.4,
$G^{\prime}$ is semi-generalized co-Bassian, which by Lemma 2.5 implies that
$R$ is bounded. The fact that $T_{p}$ is semi-generalized co-Bassian then
quickly follows from Theorem 2.7, Proposition 2.6 and Lemma 2.4. ∎

We next present a full characterization of the semi-generalized co-Bassian
group of finite torsion-free rank.

######  Theorem 2.9.

If $G$ has finite torsion-free rank, then $G$ is semi-generalized co-Bassian
if, and only if, for every prime $p$, $T_{p}$ is semi-generalized co-Bassian
and either (a) $T_{p}$ has finite $p$-rank, or (b) $T_{p}$ is divisible, or
(c) $G/T$ is $p$-divisible.

###### Proof.

Suppose $G$ is semi-generalized co-Bassian and $p$ is a prime. By Proposition
2.8, $T_{p}$ is semi-generalized co-Bassian. Assume $G/T$ is not $p$-divisible
and $T_{p}$ has infinite $p$-rank; we need to show that $T_{p}$ is divisible.
Assume not, so that it has a non-zero cyclic summand $C=\langle c\rangle$ of
minimal order $p^{k}$. It is straightforward to verify that there is a
decomposition $T_{p}=C\oplus K$ and an injection $\phi:T_{p}\to K$ such that
$\phi(T_{p}[p])=K[p]$.

Since $T_{p}$ is the direct sum of a bounded and a divisible group, there is a
decomposition $G=T_{p}\oplus A=C\oplus K\oplus A$; we are therefore assuming
that multiplication by $p$ is an injective, but not surjective, endomorphism
on $A$. It follows that $A/p^{k}A$ is isomorphic to a direct sum of copies of
$\mathbb{Z}_{p^{k}}$. So, if $a\in A$ has $p$-height 0, then $a+p^{k}A$ will
have order $p^{k}$ in $A/p^{k}A$, and hence will generate a summand. Thus,
there is a composite of natural homomorphisms

$\gamma:A\to A/p^{k}A\to\langle a+p^{k}A\rangle\to C$

such that $\gamma(a)=c$. Extend $\phi$ to an injection $\phi:G\to G$ by
setting, for all $x\in A$, $\phi(x)=\gamma(x)+px$.

Letting $\pi:G\to\overline{G}:=G$ be the identity, then since $G$ is semi-
generalized co-Bassian, we can conclude that $\hat{G}=\phi(G)$ is an essential
subgroup of some summand $S$ of $G$. Since $\hat{G}[p]=K[p]$, $p^{k-1}c$ does
not represent an element of the Ulm factor $U_{k-1}(S)\subseteq U_{k-1}(G)$.
On the other hand, considering $p$-heights,

$|p^{k-1}\phi(a)|=|p^{k-1}c+p^{k}a|=k-1$

and

$|p^{k}\phi(a)|=|p^{k+1}a|=k+1,$

so there is an element $s\in S$ such that $|s|=k$ and $ps=p^{k}\phi(a)$.
Therefore, $p^{k-1}\phi(a)-s$ does represent an element of $U_{k-1}(S)$. But
since

$|(p^{k-1}\phi(a)-s)-p^{k-1}c|=|p^{k}a-s|\geq k,$

we can conclude that $p^{k-1}\phi(a)-s$ and $p^{k-1}c$ represent the same
element of $U_{k-1}(G)$, giving a contradiction. Therefore, $G$ must be
divisible, completing the proof of sufficiency.

Conversely, suppose that for all primes $p$, $T_{p}$ is semi-generalized co-
Bassian and if $G/T$ is not $p$-divisible, then either $G$ has finite $p$-rank
or $T_{p}$ is divisible. To show $G$ is semi-generalized co-Bassian, suppose
$\pi$ and $\phi$ are, respectively, surjective and injective homomorphisms
$G\to\overline{G}$.

Let $\mathcal{P}_{1}$ be the collection of primes such that $G/T$ is
$p$-divisible, and $\mathcal{P}_{2}$ be those primes $p\not\in\mathcal{P}_{1}$
for which $T_{p}$ has finite $p$-rank and $\mathcal{P}_{3}$ be the remaining
primes; so if $p\in\mathcal{P}_{3}$, then $G/T$ is not $p$-divisible and
$T_{p}$ is divisible (of infinite rank).

If $N$ is the kernel of $\pi$, then the fact that $G$ has finite torsion-free
rank implies that $N\subseteq T$ and $T_{\overline{G}}=\overline{T}$.
Therefore, for each prime $p$, $\pi$ and $\phi$ restrict to surjective and
injective homomorphisms $T_{p}\to\overline{T}_{p}$. Since we are assuming
$T_{p}$ is semi-generalized co-Bassian, we can conclude that there is a
decomposition $\overline{T}_{p}=L_{p}\oplus B_{p}$, where $L_{p}$ contains
$\hat{T}_{p}$ as an essential subgroup. Note that if $p\in\mathcal{P}_{2}$,
then $L_{p}=\hat{T}_{p}=\overline{T}_{p}$ and $B_{p}=0$; and if
$p\in\mathcal{P}_{3}$, then both $L_{p}=\hat{T}_{p}$ and $B_{p}$ are
divisible. Letting $L=\oplus_{p}L_{p}$ and $B=\oplus_{p}B_{p}$, then $\hat{T}$
is essential in $L$ and $\overline{T}=L\oplus B$.

Observe that there are isomorphisms of torsion-free finite rank groups
$\hat{G}/\hat{T}\cong G/T\cong\overline{G}/\overline{T}$. Therefore, since
$\hat{G}/\hat{T}$ clearly embeds in $\overline{G}/\overline{T}$ and they are
isomorphic,

$None$
$\overline{G}/[\overline{T}+\hat{G}]\cong(\overline{G}/\overline{T})/(\hat{G}/\hat{T}):=X$

is a finite group. And since both groups in this quotient are $p$-divisible
for all primes $p\in\mathcal{P}_{1}$, it follows that $X$ has no $p$-torsion
whenever $p\in\mathcal{P}_{1}$.

Suppose $x_{0},\dots x_{k}$ in $\overline{G}$ project onto a linearly
independent generating set for $X$. If $n_{i}$ is the order of the image of
$x_{i}$ in $X$, then $n_{i}$ is not divisible by any prime from
$\mathcal{P}_{1}$. So, there is a finite set of primes
$F\subseteq\mathcal{P}_{2}\cup\mathcal{P}_{3}$, such that each $n_{i}$ is only
divisible by primes from $F$.

For each $i$, there is a $y_{i}\in\hat{G}$ and $z_{i}\in\overline{T}$ such
that $n_{i}x_{i}=y_{i}+z_{i}$. For each $i$ there is an $m_{i}$ not divisible
by any prime of $F$ such that $m_{i}z_{i}$ has order divisible only by primes
in $F$. Clearly, the $m_{i}x_{i}$ still project to a linearly independent
generating set for $X$, so replacing $x_{i},y_{i}$ and $z_{i}$ by
$m_{i}x_{i},m_{i}y_{i}$ and $m_{i}z_{i}$ respectively, we may assume that each
$z_{i}$ is in $\oplus_{p\in F}\overline{T}_{p}=\oplus_{p\in
F}(\hat{T}_{p}\oplus B_{p})$. We may clearly absorb each component of $z_{i}$
in this decomposition that lives in $\hat{T}_{p}$ into the corresponding
element $y_{i}\in\hat{G}$. So, we may assume $z_{i}\in\oplus_{p\in F}B_{p}$.
But since $F\subseteq\mathcal{P}_{2}\cup\mathcal{P}_{3}$, the direct sum
$\oplus_{p\in F}B_{p}$ is divisible. Let $w_{i}\in\oplus_{p\in F}B_{p}$
satisfy $n_{i}w_{i}=z_{i}.$ So, $x_{i}^{\prime}:=x_{i}-w_{i}$ represent the
exact same elements of $X$, but now, $n_{i}x_{i}^{\prime}=y_{i}$.

Replacing $x_{i}$ by $x_{i}^{\prime}$ and letting

$S=\hat{G}+\langle x_{1},\dots,x_{k}\rangle+L\subseteq\overline{G},$

we claim that $T_{S}=L$: The containment $\supseteq$ being obvious, suppose
$s\in T_{S}$; we need to show $s\in L$. By definition, there is $u\in\hat{G}$,
$v\in L$ and integers $j_{0},\dots,j_{k}$ such that

$s=u+j_{0}x_{0}+\cdots+j_{k}x_{k}+v.$

If we map this relation into the quotient $X$, if readily follows that for
each $i$, that $j_{i}x_{i}\in\hat{G}$. Therefore,

$u+j_{0}x_{0}+\cdots+j_{k}x_{k}=s-v\in\hat{G}\cap\overline{T}=\hat{T}\subseteq
L.$

This gives $s=v+(s-v)\in L$, as required.

Notice also that $({\dagger})$ also implies that
$\overline{G}=S+\overline{T}$. We have $\overline{T}=L\oplus B=T_{S}\oplus B$,
so that $S+B=S+\overline{T}=G$ and $S\cap B=T_{S}\cap B=0$. Therefore,
$\overline{G}=S\oplus B$.

And finally, since $\hat{T}$ is essential in $L=T_{S}$ and $S/\hat{G}$ is
torsion, it readily follows that $\hat{G}$ is an essential subgroup of $S$, as
required. ∎

The next result follows immediately from Theorem 2.9.

######  Corollary 2.10.

The finite torsion-free rank group $G$ is semi-generalized co-Bassian if, and
only if, for every prime $p$, its localization $G_{(p)}$ is semi-generalized
co-Bassian.

Having characterized the semi-generalized co-Bassian groups of finite rank, we
turn to a consideration of those with infinite rank. We will say a group is
D+E if it is isomorphic to $D\oplus E$, where $D$ is divisible and $E$ is
elementary.

######  Proposition 2.11.

If $G$ is semi-generalized co-Bassian of infinite torsion-free rank, then
$G/T$ is divisible and $T$ is D+E.

###### Proof.

Let $\kappa$ be the rank of $G$. Since each $T_{p}$ is semi-generalized co-
Bassian, $T$ will be a direct sum of cocyclic groups. This easily implies that
there is a decomposition $G=G^{\prime}\oplus T^{\prime}$, where $T^{\prime}$
is torsion, $G^{\prime}$ has cardinality $\kappa$ and $T$ is D+E if, and only
if, that condition holds for $T_{G^{\prime}}$. So, by Lemma 2.4, there is no
loss of generality in assuming that $G$ has cardinality and rank $\kappa$.

Suppose first that $G/T$ is not $p$-divisible for some $p$. Let $A$ be a
subgroup of $G$ containing $T$ such that $G/A$ has order $p$ and let
$\tau:G\to G/A$ be the usual epimorphism. If $D$ is a divisible hull for $G$,
then there is clearly a surjective homomorphism $A\to D$ which extends to a
homomorphism $\gamma:G\to D$.

The homomorphism $\pi:G\to\overline{G}:=(G/A)\oplus D$ given by
$\pi(x)=(x+A)+\gamma(x)$ is clearly onto. The homomorphism
$\phi:G\to(G/A)\oplus D$ given by $\phi(x)=(x+A)+x$ is clearly injective.

If $G$ were semi-generalized co-Bassian, then $\hat{G}$ would be contained as
an essential subgroup of a summand $S\subseteq\overline{G}$. Since
$S[p]=0\oplus T[p]\subseteq D\subseteq\overline{G}$, we could conclude that
the $p$-torsion subgroup of $S$ is divisible. If $x\not\in A$, then
$|\phi(x)|_{S}=|\phi(x)|_{\overline{G}}=0$, and
$|p\phi(x)|_{S}=|p\phi(x)|_{\overline{G}}=\infty$, which cannot happen when
the $p$-torsion of $S$ is divisible.

Suppose now that $G$ has a cyclic summand, $C=\langle c\rangle$, of order
$p^{k}$, where $k>1$. Let $G=C\oplus A$. If $D$ is a divisible hull for $A$
and $Z\cong\mathbb{Z}_{p^{\infty}}$, then there is clearly a surjection

$\pi:A\to Z\oplus D$

and an injection

$\phi:A\subseteq D\subseteq Z\oplus D.$

Extending $\pi$ to $G$ by setting it equal to the identity on $C$ makes it a
surjection

$G\to\overline{G}:=C\oplus Z\oplus D.$

Choosing $z\in Z$ of order $p^{k}$ and extending $\phi$ by setting
$\phi(c)=pc+z$ turns it into an injection $\phi:G\to\overline{G}$. Again
$\hat{G}[p]\subseteq(Z\oplus D)[p]$, so that all elements of $\hat{G}[p]$ have
infinite height. So, if $\hat{G}$ were contained as an essential subgroup of a
summand $S$, then the $p$-torsion subgroup of $S$ would be divisible. This,
however, contradicts the fact that $|\phi(c)|_{\overline{G}}=1\neq\infty$. ∎

As a direct consequence, we yield:

######  Corollary 2.12.

Suppose $G$ is a semi-generalized co-Bassian group of infinite torsion-free
rank and $G\cong D\oplus R$, where $D$ is divisible and $R$ is reduced. Then,
$T_{R}$ is elementary and $R/T_{R}$ is divisible.

The following is a partial converse to Proposition 2.11.

######  Proposition 2.13.

If the group $G$ itself is D+E, then $G$ is semi-generalized co-Bassian.

###### Proof.

It is easily seen that a homomorphic image of a D+E-group is also D+E, and
that any D+E-subgroup of a D+E-group will be essential in a summand. This will
imply the pursued result. ∎

So, if $G$ is a torsion-splitting group, i.e., $T$ is a summand of $G$, then
the converse of Proposition 2.11 holds. However, it is not clear that this
converse holds when $G$ is not torsion-splitting. In fact, the problem may be
quite difficult and we leave it unsettled at this stage.

Thus, we explicitly pose the following:

######  Problem 2.14.

Does it follow that a group $G$ is semi-generalized co-Bassian, provided its
torsion part $T$ is a direct sum of a divisible group and an elementary group
and the quotient $G/T$ is a divisible group?

Funding: The work of the first-named author A.R. Chekhlov was supported by the
Ministry of Science and Higher Education of Russia (agreement No.
075-02-2023-943). The work of the second-named author P.V. Danchev was
partially supported by the Bulgarian National Science Fund under Grant KP-06
No. 32/1 of December 07, 2019, as well as by the Junta de Andalucía under
Grant FQM 264, and by the BIDEB 2221 of TÜBÍTAK.

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