# An Exhaustive Study of Using Commercial LTE Network for UAV Communication in
Rural Areas

Mohammed Gharib1, Shashidhar Nandadapu1, Fatemeh Afghah1  1School of
Informatics, Computing and Cyber Systems, Northern Arizona University,
Flagstaff, AZ, USA
E-mail:{mohammed.algharib,sn644<EMAIL_ADDRESS>
###### Abstract

Unmanned aerial vehicles (UAVs) have been increasingly used in a wide area of
military and civilian applications such as data collection and monitoring. A
reliable network for command and control, communication, and data transfer is
crucial, not only for mission purposes but also for safety concerns. The
already deployed cellular networks are appropriate candidates for UAV
communication given the solid security and wide coverage of these networks.
However, the reliability of such networks needs a comprehensive investigation.
In this paper, we use the long-term evolution (LTE) network as the
infrastructure for drone communication and data transfer, in a rural area. We
study the communication characteristics of an LTE-connected drone during low-
altitude flights, for different altitudes and UAV speeds. We show that, in
such areas, the higher elevation benefits from a better signal quality and
experiences a fewer number of handover processes. Higher speed flights also
slightly degrade the communication performance111This material is based upon
work supported by the Air Force Office of Scientific Research under award
number FA9550- 20-1-0090 and the National Science Foundation under Grant
Numbers CNS-2034218, CNS-2039026 and ECCS- 2030047..

###### Index Terms:

LTE, cellular-connected UAVs, field test measurement, RSRP, SINR, throughput.

##  I Introduction

Drones have been increasingly used in both military and civilian applications.
Ease of deployment, highly dynamic 3D movement ability, low price, and the
availability of drones make them find their way into various applications.
While border surveillance, target tracking and strike are some military
applications, traffic management, package delivery, search and rescue,
disaster relief, and post-disaster imagery, and area monitoring are some
civilian applications to be named [1, 2]. Reliable communication is of
paramount importance for command and control to guarantee the safety of
drones, people and infrastructure. It is also crucial to guarantee reliable
data transfer in applications such as monitoring and surveillance.

Exploiting the existing cellular networks for UAV communications is a cost-
effective solution to provide a reliable, wide-coverage, and secure
communication for drones. However, the cellular networks are designed to cover
the near-ground areas and might not be able to offer an optimal service to
aerial users due to several concerns. The first concern is that the current
cellular networks are managed to minimize the interference among different
cells’ signals. The obstacles such as trees, houses, and buildings block the
signals at a certain level, and hence the signal powers are managed based on
the current on-ground physical situations. In higher elevations, there are no
such obstacles and most of the antennas are located in the line-of-sight of
the user. While the line-of-sight communication might have stronger signal, it
may cause considerable interference and make it infeasible to use such
networks for drone communication.

The second concern is about the coverage area in the higher elevations. The
tilt angle of the cellular network antennas are such that the antennas face
the ground to best cover the terrestrial users. Thus, the coverage area is
investigated only for the two-dimension maps. In order to utilize cellular
towers for drone communications, we need to investigate the coverage area
above the ground and for different elevations, to find the three-dimensional
coverage map. The third concern is the effect of the aerial users on the
quality of the communications of the main terrestrial users. To the best of
our knowledge, there is not enough information on the interference caused by
aerial cellular users on the terrestrial users in different geographical
distributions and urbanization.

In this paper, we present the results of field measurements on using LTE for
drone communication in a rural area. We gather data for different low-altitude
flight’s elevations, $40$, $80$, and $120~{}m$, and different speeds of $30$
and $60$ kmph. In our measurements, we consider the call quality, and maximum
achievable up-link and down-link throughput, altogether. The experimental
measurements are performed using a commercial drone in rural area by attaching
a smart phone to the drone. TEMS pocket [3] application is installed on the
smart phone supported with a dedicated script on it to gather the required
data. The script makes a voice call, download a large file, and upload a
stream of data to a server, using the Verizon commercial LTE network in its
1700 MHz band. All tests are done in a rural forest area close to Flagstaff
city, Arizona, US. The collected log files are processed by TEMS discovery
[3]. We then extracted two types of information, those of the serving cell
signal, and those of comparing the best signals with one another including the
serving cell signal and the neighboring cells’ signals.

In our analysis, we measured the reference signal received power (RSRP),
reference signal received quality (RSRQ), reference signal strength indicator
(RSSI), signal to interference and noise ratio (SINR), up-link throughput,
down-link throughput, and the number of handover processes. We show in our
analysis that, in low-altitude flights over rural areas, the lower elevation
results in the worst case performance, since the signal is attenuated with the
obstacles and multi-path fading. We show that the higher elevation results in,
in most cases, the best choice for the voice calls and small-size data
communication. However, the moderate elevation, 80 m in our measurements,
reaches the highest possible throughput in uploading and downloading. We
further show that although the speed has a negligible impact on the signal
quality, in most cases, the lower speed flights have slightly better
performance. Furthermore, the moderate flight’s elevation leads to the largest
number of handovers, whereas in the highest elevation, we see the lowest
number of handover processes. The higher speed flight also shows slightly less
often handover processes for all the tested scenarios. Generally, we find that
there are always a couple of signals with enough strength to keep the call
live, and the interference caused by the line-of-sight signals does not have a
significant effect on the serving cell signal quality, in the understudy
settings.

The rest of this paper is organized as follows. We review the literature works
in Section (II). Then, we present the processes of data collection, cleanup,
and information extraction in Section (III). We analyze the data and represent
the results in Section (IV). Finally, we conclude the paper and mention the
future directions in Section (V).

##  II Related Work

Using the existing cellular networks for drone communications can facilitate
the wide deployment of drone technology in a secure way without the need for
substantial investments to establish new communication networks, however, the
aerial coverage of cellular networks and the interference caused by the aerial
users on terrestrial cellular users need to be thoroughly investigated. The
third generation partnership project (3GPP) made a valuable effort in its
release 15 [4] to discover the support of enhanced long term evaluation (LTE)
for aerial vehicles, and release 17 [5] to support the 5G enhancement for
UAVs. Van Der Bergh et al. [6] studied the impact of interference and path
loss on the connected drones via LTE network. They found that the signal
strength is decreased rapidly by the increment in the altitude until the line-
of-sight propagation is established. Concurrently, the signal quality is
decreased because of the decrements in the SINR.

Lin et al. [7] shared some of their measurements data for low altitude drone
connected to commercial LTE network, gathered in Finland. They found that the
already deployed LTE networks can support the low-altitude aerial
communication, but the interference and mobility may cause challenges. Amorim
et al. [8] made several measurements to model the radio channel for aerial use
of LTE network in Denmark. Their results showed better radio clearance as the
aerial vehicle increases its elevation, in the low-latitude flights. However,
they did not measure the SINR and left it for their future works.

In [9], authors performed measurements for aerial users of LTE network at
elevations which do not exceed 40 m. They targeted an urban area with the
highest building height of 15 m and compared their results to those of 3GPP
[4]. They found that the measured metrics for three different frequencies of
800, 1800, and 2600 MHz lead to similar results. Khawaja et al. [10] found in
their measurements that the signal strength mostly follows a two-path ray
propagation model for higher altitudes. Al-Hourani et al. [11] provided a
cellular to aerial channel model in terms of path loss and shadowing, based on
the real experiments performed in a suburban environment.

Hayat et al. [12] experimentally evaluated the LTE network to measure the
signal to interference ratio and the downlink throughput in different
elevations, in a suburban area. They showed that the throughput at 150 m
altitude outperforms all lower elevation throughput. However, the throughput
of 50 m elevation is much higher than that of 100 m. Kovacs et al. [13]
analyzed their measurements on aerial connectivity with an LTE network in a
rural area. They characterized the radio channel behaviour in terms of
downlink and uplink metrics, and estimated the gains of interference
mitigation techniques.

Marques et al. [14] performed an experimental study for using LTE as the
communicating network for aerial vehicles, in rural areas. They measured the
uplink and downlink throughput for different low-altitude elevations. They
found that the 25 m elevation outperforms all other elevations in terms of
uplink and downlink throughput. Muzaffar et al. [15] performed an experimental
test for connecting the aerial user to a 5G network and compared its results
with that of 4G network. They found that the 5G network generally outperform
the 4G network. Overall, there is a significant need for exhaustive studies to
show the usability of already deployed cellular networks for aerial
communication. Most of the mentioned works performed valuable measurements and
analysis. However, none of them presented a sound and complete study, covering
all area types, different elevations, and all the metrics studied in this
paper, altogether. Hence, we aim at studying this problem for more metrics and
different elevations as well as different UAV speeds.

##  III Data Collection and Processing

In this section, we describe the field flight tests, as well as the data
processing procedure to extract communication parameters from the collected
log-files. We use a commercial drone, DJI Matrice 200, in a rural area near
Flagstaff, Arizona, US. Flagstaff lies at approximately 2100 m elevation above
sea level. We performed our tests in the Arboretum Garden, one of the
southwest experimental garden array (SEGA) sites. The test location is covered
by Ponderosa pine trees and surrounded by several Base Transceiver Stations
(BTSs). We performed our tests using the commercial Verizon LTE network on its
1700 MHz band. Fig. (1) shows the area map. Fig. (1a) shows the area with all
of the base stations, where Fig. (1b) shows only the base stations in which we
received their signal in our tests. In this figure, the oval represents the
exact test location. Fig. (1b) shows also the two-dimension coverage area of
the network. This 2D coverage map has been created using multiple drive tests.
As shown in this map, the terrestrial user, in the best case, can receive an
strong signal from only one base station (the eNB with the identification
number 22158), which confirms the weak coverage in this rural area. We put a
Samsung S20 phone on the drone with TEMS pocket app [3] installed on. We flew
the drone at three different elevations of 40 m, 80 m, and 120 m, with two
different speeds 30 kmph and 60 kmph. While the elevation is limited by the
federal aviation administration (FAA) to not exceed 400 feet, i.e. 120 m, the
maximum speed is limited by our drone. In each flight instance, the drone
take-off from the starting point, flies for 500 m in a straight line and
returns back the same way to the take-off point.

(a) Overall view

(b) The captured signals

(c) The test map

Figure 1: A map of the test area.

For the measurement process, as we mentioned earlier, we use the TEMS pocket
application version 22.1.2 [3] installed on the attached smart phone. To be
able to investigate the communication quality and uplink and downlink data
transmission performance, we design a script to call another phone, upload a
file to a server, and download a file from another server, all at the same
time. We record the RSRP, RSRQ, RSSI, SINR, and uplink and downlink throughput
as the performance evaluation metrics. We record the mentioned metrics for the
signal of the neighboring cells as well. We also keep the data for all
handover processes. We use TEMS discovery to process the files and extract the
desired information.

The collected data, i.e. the output of the TEMS discovery, contains some
unnecessary data such as the take-off and landing data when the drone
increases or decreases its elevation. Before we use the extracted information,
we carefully clean up the data from the unnecessary information. Furthermore,
the granularity of TEMS discovery in managing the data is 2 seconds. We
represent the data based on the location of drone, starting from the starting
point, going in the straight line for 500 m, in steps of 50 m. To calculate
the information at the exact locations of interest, we interpolated the
required data points, using Lagrange interpolation.

##  IV Experimental Results

In this section, we first represent and analyze the measured metrics regarding
the serving cell signal. The measured and analyzed metrics are RSRP, RSRQ,
RSSI, SINR, downlink throughput, and uplink throughput. We then compare the
RSRP and RSRQ between the serving cell signal and the signals of the
neighboring cells. The Reference signal received power (RSRP) is the first
investigated metric for the serving cell signal, in our measurement and
analysis. According to 3GPP, RSRP is defined as the ”linear average over the
power contributions of the resource elements that carry cell-specific
reference signals within the considered measurement frequency bandwidth” [16].
In a simpler form, the RSRP represents the power of the reference signal at
the receiver in a LTE network, excluding the noise and interference from
neighboring cells. It is measured in dBm, and the signal is considered
excellent if $(RSRP\geq-80dBm)$, good if $(-90dBm\leq RSRP\leq-80dBm)$, fair
to poor if $(-100dBm\leq RSRP\leq-90dBm)$, and no signal if
$(RSRP\leq-100dBm)$. Fig. (2) shows the RSRP for the combination of different
speeds and different elevations and for both ways of going from the starting
point to 500 m away, and returning back to the starting point. The highlighted
intervals represent a handover process happened at that interval.

Figures (2a and b) show the RSRP for the fixed speed of 60 kmph and three
different elevations of 40 m, 80 m, and 120 m, for both directions. As it is
clear from this figure, the higher elevation helps the network node to get a
signal with higher power, as the higher elevation leads to more areas with a
dominant line-of-sight signal. The number of handover processes shows also
obvious decrements at the elevation of 120 m. Fig. (2c,d) show the same
results for the elevation of 120m and different speeds of 30 kmph and 60 kmph.
While the two speeds represent results with an almost identical pattern, the
lower speed shows slightly better performance. The RSRP shows almost the same
pattern of results for the back and forward directions. Thus, for the other
metrics, we represent the figures only for one way.

(a) Different elev. (speed=60kmph) (b) Different elev. opposite direction
(speed=60kmph)

(c) Different speeds (elev=120m) (d) Different speeds opposite
direction(elev=120m)

Figure 2: A comparison of RSRP for different elevations and different UAV
speeds.

Reference signal received quality (RSRQ) is the next investigated metric. The
RSRQ represents the quality of received signal at the user equipment and it is
measured in dB. While the RSRP is the main metric for decision-making on
handover and cell reselection, it can provide additional information when RSRP
is insufficient. The signal is considered as excellent if $(RSRQ\geq-10dB)$,
good if $(-15dB\leq RSRQ\leq-10dB)$, fair to poor if $(-20dB\leq
RSRQ\leq-15dB)$, and no signal if ($RSRQ\leq-20dB$). Fig. (3) shows the
serving signal RSRQ for the combination of different elevations and different
speeds. Looking at the range of variation of this metric, we find that there
is not a significant improvement among different settings. However, the higher
elevation and lower speed show slightly better performance in terms of RSRQ.

(a) Different elev. (speed=60 kmph) (b) Different speeds (elev=120 m)

Figure 3: A comparison of RSRQ for different elevations and different UAV
speeds.

The Received signal strength indicator (RSSI) is the next measured metric. The
RSSI represents the strength of the received signal, considering the noise and
interference, and it is measured in dBm. Fig. (4) shows the RSSI for different
elevations and different speeds. Again the superiority of the higher
elevations is significant for this metric. As we discussed earlier, the test
is done in a rural area covered by trees. That is why when the flight altitude
is much larger than the trees’ height, the signal strength remains almost the
same in the low-altitude tests. However, we note that for the same elevation,
the speed has no obvious effect on this parameter.

(a) Different elev. (speed=60 kmph) (b) Different speeds (elev=120 m)

Figure 4: A comparison of RSSI for different elevations and different UAV
speeds.

Signal to interference and noise ratio (SINR), measured in dB, is the next
measured metric. This metric is defined as the RSRP divided by the sum of the
interference power from the neighboring cells and noise power. SINR is
important as a quantifying metric of the relationship between the radio
frequency and achievable throughput. Fig. (5) shows the SINR results for
different elevations and different UAV speeds. While the higher speed as well
as the lowest elevation, in most cases, lead to lower SINR, for the elevations
of 80 m and 120 m there is no significant superiority in terms of SINR.

(a) Different elev. (speed=60 kmph) (b) Different speeds (elev=120 m)

Figure 5: A comparison of SINR for different elevations and different UAV
speeds.

The last investigated metrics are the achievable Transmission Control Protocol
(TCP) downlink and uplink throughput. To measure the throughput, in the script
which is running on the smartphone, we start uploading a large file to a
server with a bandwidth much higher than that of cellular network. At the same
time, we download a large enough file, again from a server with a bandwidth
much more than that of cellular network. In this case, the limiting parameter
is the maximum achievable throughput at the user equipment side, which is the
network throughput.

Fig. (6 and 7) show the throughput for a combination of different elevations
and different drone speeds for downlink and uplink, respectively. In Fig. (6),
the variation in the elevation does not show a significant change in the
achievable downlink throughput as the lower speed shows its superiority.
However, the elevation of 80 m shows slightly better downlink throughput, in
most points. One possible reason could be the higher interference at the
higher elevation and the absence of line of sight signals at the lower
elevation. In Fig. (7), we see that the uplink shows an obvious superiority at
the higher elevation and the lower speed flight.

(a) Different elev. (speed=60 kmph) (b) Different speeds (elev=120 m)

Figure 6: A comparison of downlink throughput for different elevations and
different UAV speeds.

(a) Different elev. (speed=60 kmph) (b) Different speeds (elev=120 m)

Figure 7: A comparison of uplink throughput for different elevations and
different UAV speeds.

Now, we compare the RSRP and RSRQ of the best available signals including the
serving cell’s signal and the signal of the three of its neighboring cells.
Fig. (8) shows the empirical cumulative distribution function (CDF) for the
RSRP of the signals with the highest RSRP for three different elevations and a
fixed speed of 60 kmph. In Fig. (8a), the signal ’a’ always has the close RSRP
to the other signals ’b’ and ’c’ but was never been choosen as the serving
cell. In Fig. (8b), signals ’b’, ’c’, and ’d’ serve as the main signal, where
signal ’a’ always was from a neighboring cell. In Fig. (8b), signal ’d’, in
almost all times, was the serving cell signal.

(a) Elevation= 40 m  (b) Elevation= 80m (c) Elevation= 120 m

Figure 8: A comparison of RSRP CDF for signals with the highest RSRP in
different elevations and a fixed speed of 60 kmph.

Fig. (9) shows the empirical CDF for the RSRP of the signals with the highest
RSRP for two different speeds and an elevation of 120 m. In both figures of
(9\- a and 9\- b) signal ’d’ is chosen as the serving cell signal. While in
Fig. (9a), it is always the signal with the highest RSRP, in Fig. (9b),
signals ’b’ and ’c’ somewhere act better but never have been chosen as the
serving cell signals. It is worthy to mention that we did not experience any
call drop during the tests. As it is clear from Fig. (9 and 10), most of the
times, there are at least a couple of signals with enough power to keep the
call alive. However, we see a different number of handovers in different test
settings. Table (I) compares the number of handover processes for different
test settings. This table shows that at the elevation of 120 m, we experience
less number of handovers in comparison with other elevations, where the
elevation of 80 m leads to the highest number of handover processes.
Generally, we find that the highest speed leads to a slightly less number of
handover processes. It seems that at the elevation of 40 m, there are a couple
of handover processes due to the environmental obstacles. At the elevation of
80 m, the interference of other signals seems to have its highest effect.
However, at the elevation of 120 m, the effect of interference from other
cells’ signals has its lowest effect on the serving signal, which leads to a
less often handovers.

(a) Speed= 30 kmph  (b) Speed= 60 kmph

Figure 9: A comparison of RSRP CDF for signals with the highest RSRP in
different speeds and a fixed elevation of 120m. TABLE I: Number of handovers
Number of Handovers
---
Elevation (m) | Speed (kmph)
40 | 80 | 120 | 30 | 60
2 | 4 | 1 | 1 | 0

Finally, we show the empirical CDF of RSRQ of the serving cell’s signal and
its neighboring cells’ signal for different elevations and different speeds in
Fig. (10 and 11), respectively. We used the same naming of the signals in
these figures as those of Fig. (8 and 9). Thus, the serving signals are the
same as those of previous figures.

(a) Elevation= 40 m  (b) Elevation= 80 m (c) Elevation= 120 m

Figure 10: A comparison of RSRQ CDF for signals with the highest RSRQ in
different elevations and a fixed speed of 60 kmph.

(a) Speed= 30 kmph  (b) Speed= 60 kmph

Figure 11: A comparison of RSRQ CDF for signals with the highest RSRQ in
different speeds and a fixed elevation of 120m.

##  V Conclusion

Reusing the already deployed cellular network for low-altitude aerial user
communication sounds promising, due to the cost efficiency, wide-coverage,
high data transmission rate, and security of data transmission. However, the
feasibility of such a communication, as well as its performance, need
comprehensive investigations. In this paper, we gathered data for an aerial
LTE user in a rural area. We showed that for the low-altitude aerial users,
the higher altitude leads to higher signal power, where the moderate altitude
leads to slightly higher throughput. We did not experience any call drop
during our tests which is a good sign for the feasibility of using the pre-
deployed cellular network by the aerial users. The number of handover
processes is very low at the highest altitude and the increment in the speed
slightly decreases this number. While we tried to exhaustively study the reuse
of the existed LTE network for aerial users, our study covered only a rural
area. As future work, we aim at performing a similar study for suburban and
urban areas. We can also compare the results for aerial users with that of
terrestrial user via drive-test.

##  VI acknowledgment

The authors acknowledge the financial and technical support provided by
Infovista in using TEMS POCKET and TEMS Discovery in this project. The authors
also acknowledge the cooperation of SEGA garden management.

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# Almost $n$-ary and almost $n$-aritizable theories111The work of the author
was carried out in the framework of the State Contract of the Sobolev
Institute of Mathematics, Project No. FWNF-2022-0012 (Section 2), and of
Russian Scientific Foundation, Project No. 22-21-00044 (Sections 3, 4).

S.V. Sudoplatov

###### Abstract

We study possibilities for almost $n$-ary and $n$-aritizable theories. Their
dynamics both in general case, for $\omega$-categorical theories, and with
respect to operations for theories are described.

Key words: elementary theory, almost $n$-ary theory, almost $n$-aritizable
theory.

We continue to study arities of theories and of their expansions [1]. In the
present paper we introduce natural notions of almost $n$-ary and almost
$n$-aritizable theories, and describe their dynamics both in general case, for
$\omega$-categorical theories, and with respect to operations for theories.

##  1 Preliminaries

Recall a series of notions related to arities and aritizabilities of theories.

Definition [2]. A theory $T$ is said to be _$\Delta$ -based_, where $\Delta$
is some set of formulae without parameters, if any formula of $T$ is
equivalent in $T$ to a Boolean combination of formulae in $\Delta$.

For $\Delta$-based theories $T$, it is also said that $T$ has quantifier
elimination or quantifier reduction up to $\Delta$.

Definition [2, 3]. Let $\Delta$ be a set of formulae of a theory $T$, and
$p(\bar{x})$ a type of $T$ lying in $S(T)$. The type $p(\bar{x})$ is said to
be _$\Delta$ -based_ if $p(\bar{x})$ is isolated by a set of formulas
$\varphi^{\delta}\in p$, where $\varphi\in\Delta$, $\delta\in\\{0,1\\}$.

The following lemma, being a corollary of Compactness Theorem, noticed in [2].

###### Lemma 1.1

A theory $T$ is $\Delta$-based if and only if, for any tuple $\bar{a}$ of any
(some) weakly saturated model of $T$, the type ${\rm tp}(\bar{a})$ is
$\Delta$-based.

Definition [1]. An elementary theory $T$ is called unary, or $1$-ary, if any
$T$-formula $\varphi(\overline{x})$ is $T$-equivalent to a Boolean combination
of $T$-formulas, each of which is of one free variable, and of formulas of
form $x\approx y$.

For a natural number $n\geq 1$, a formula $\varphi(\overline{x})$ of a theory
$T$ is called $n$-ary, or an $n$-formula, if $\varphi(\overline{x})$ is
$T$-equivalent to a Boolean combination of $T$-formulas, each of which is of
$n$ free variables.

For a natural number $n\geq 2$, an elementary theory $T$ is called $n$-ary, or
an $n$-theory, if any $T$-formula $\varphi(\overline{x})$ is $n$-ary.

A theory $T$ is called binary if $T$ is $2$-ary, it is called ternary if $T$
is $3$-ary, etc.

We will admit the case $n=0$ for $n$-formulae $\varphi(\overline{x})$. In such
a case $\varphi(\overline{x})$ is just $T$-equivalent to a sentence
$\forall\overline{x}\varphi(\overline{x})$.

If $T$ is a theory such that $T$ is $n$-ary and not $(n-1)$-ary then the value
$n$ is called the arity of $T$ and it is denoted by ${\rm ar}(T)$. If $T$ does
not have any arity we put ${\rm ar}(T)=\infty$.

Similarly, for a formula $\varphi$ of a theory $T$ we denote by ${\rm
ar}_{T}(\varphi)$ the natural value $n$ if $\varphi$ is $n$-ary and not
$(n-1)$-ary. If $\varphi$ does not any arity we put ${\rm
ar}_{T}(\varphi)=\infty$. If a theory $T$ is fixed we write ${\rm
ar}(\varphi)$ instead of ${\rm ar}_{T}(\varphi)$.

The following example illustrates the notions above, and it will be used
below.

###### Example 1.2

Recall [4, 5, 6] that a circular, or cyclic order relation is described by a
ternary relation $K_{3}$ satisfying the following conditions:
(co1) $\forall x\forall y\forall z(K_{3}(x,y,z)\to K_{3}(y,z,x));$
(co2) $\forall x\forall y\forall z(K_{3}(x,y,z)\land
K_{3}(y,x,z)\leftrightarrow x=y\lor y=z\lor z=x);$
(co3) $\forall x\forall y\forall z(K_{3}(x,y,z)\to\forall t[K_{3}(x,y,t)\lor
K_{3}(t,y,z)]);$
(co4) $\forall x\forall y\forall z(K_{3}(x,y,z)\lor K_{3}(y,x,z)).$

Clearly, ${\rm ar}(K_{3}(x,y,z))=3$ if the relation has at least three element
domain. Hence, theories with infinite circular order relations are at least
$3$-ary.

The following generalization of circular order produces a $n$-ball, or
$n$-spherical, or $n$-circular order relation, for $n\geq 4$, which is
described by a $n$-ary relation $K_{n}$ satisfying the following conditions:
(nbo1) $\forall x_{1},\ldots,x_{n}(K_{n}(x_{1},x_{2},\ldots,x_{n})\to
K_{n}(x_{2},\ldots,x_{n},x_{1}));$
(nbo2) $\forall
x_{1},\ldots,x_{n}\biggm{(}K_{n}(x_{1},\ldots,x_{i},x_{i+1},\ldots,x_{n})\land$

$\land
K_{n}(x_{1},\ldots,x_{i+1},x_{i},\ldots,x_{n})\leftrightarrow\bigvee\limits_{i=1}^{n-1}x_{i}=x_{i+1}\biggm{)};$

(nbo3) $\forall x_{1},\ldots,x_{n}(K_{n}(x_{1},\ldots,x_{n})\to\forall
t[K_{n}(x_{1},\ldots,x_{n-1},t)\lor K_{n}(t,x_{2},\ldots,x_{n})]);$
(nbo4) $\forall
x_{1},\ldots,x_{n}(K_{n}(x_{1},\ldots,x_{i},x_{i+1},\ldots,x_{n})\lor
K_{n}(x_{1},\ldots,x_{i+1},x_{i},\ldots,x_{n})),$ $i<n.$

Clearly, ${\rm ar}(K_{n}(x_{1},\ldots,x_{n}))=n$ if the relation has at least
$n$-element domain. Thus, theories with infinite $n$-ball order relations are
at least $n$-ary.

Definition [1]. A $T$-formula $\varphi(\overline{x})$ is called
$n$-expansible, or $n$-arizable, or $n$-aritizable, if $T$ has an expansion
$T^{\prime}$ such that $\varphi(\overline{x})$ is $T^{\prime}$-equivalent to a
Boolean combination of $T^{\prime}$-formulas with $n$ free variables.

A theory $T$ is called $n$-expansible, or $n$-arizable, or $n$-aritizable, if
there is an $n$-ary expansion $T^{\prime}$ of $T$.

A theory $T$ is called arizable or aritizable, if $T$ is $n$-aritizable for
some $n$.

A $1$-aritizable theory is called unary-able, or unary-tizable. A
$2$-aritizable theory is called binary-tizable or binarizable, a
$3$-aritizable theory is called ternary-tizable or ternarizable, etc.

Definition. [8] The disjoint union
$\bigsqcup\limits_{n\in\omega}\mathcal{M}_{n}$ of pairwise disjoint structures
$\mathcal{M}_{n}$ for pairwise disjoint predicate languages $\Sigma_{n}$,
$n\in\omega$, is the structure of language
$\bigcup\limits_{n\in\omega}\Sigma_{n}\cup\\{P^{(1)}_{n}\mid n\in\omega\\}$
with the universe $\bigsqcup\limits_{n\in\omega}M_{n}$, $P_{n}=M_{n}$, and
interpretations of predicate symbols in $\Sigma_{n}$ coinciding with their
interpretations in $\mathcal{M}_{n}$, $n\in\omega$. The disjoint union of
theories $T_{n}$ for pairwise disjoint languages $\Sigma_{n}$ accordingly,
$n\in\omega$, is the theory

$\bigsqcup\limits_{n\in\omega}T_{n}\rightleftharpoons{\rm
Th}\left(\bigsqcup\limits_{n\in\omega}\mathcal{M}_{n}\right),$

where $\mathcal{M}_{n}\models T_{n}$, $n\in\omega$. Taking empty sets instead
of some structures $\mathcal{M}_{k}$ we obtain disjoint unions of finitely
many structures and theories. In particular, we have the disjoint unions
$\mathcal{M}_{0}\sqcup\ldots\sqcup\mathcal{M}_{n}$ and their theories
$T_{0}\sqcup\ldots\sqcup T_{n}$.

###### Theorem 1.3

[1]. $1.$ For any theories $T_{m}$, $m\in\omega$, and their disjoint union
$\bigsqcup\limits_{m\in\omega}T_{m}$, all $T_{m}$ are $n$-theories iff
$\bigsqcup\limits_{m\in\omega}T_{m}$ is an $n$-theory, moreover,

${\rm ar}\left(\bigsqcup\limits_{m\in\omega}T_{m}\right)={\rm max}\\{{\rm
ar}(T_{m})\mid m\in\omega\\}.$

$2.$ For any theories $T_{m}$, $m\in\omega$, and their disjoint union
$\bigsqcup\limits_{m\in\omega}T_{m}$, all $T_{m}$ are $n$-aritizable iff
$\bigsqcup\limits_{m\in\omega}T_{m}$ is $n$-aritizable.

Definition [7]. Let $\mathcal{M}$ and $\mathcal{N}$ be structures of
relational languages $\Sigma_{\mathcal{M}}$ and $\Sigma_{\mathcal{N}}$
respectively. We define the composition $\mathcal{M}[\mathcal{N}]$ of
$\mathcal{M}$ and $\mathcal{N}$ satisfying the following conditions:

1)
$\Sigma_{\mathcal{M}[\mathcal{N}]}=\Sigma_{\mathcal{M}}\cup\Sigma_{\mathcal{N}}$;

2) $M[N]=M\times N$, where $M[N]$, $M$, $N$ are universes of
$\mathcal{M}[\mathcal{N}]$, $\mathcal{M}$, and $\mathcal{N}$ respectively;

3) if $R\in\Sigma_{\mathcal{M}}\setminus\Sigma_{\mathcal{N}}$, $\mu(R)=n$,
then $((a_{1},b_{1}),\ldots,(a_{n},b_{n}))\in R_{\mathcal{M}[\mathcal{N}]}$ if
and only if $(a_{1},\ldots,a_{n})\in R_{\mathcal{M}}$;

4) if $R\in\Sigma_{\mathcal{N}}\setminus\Sigma_{\mathcal{M}}$, $\mu(R)=n$,
then $((a_{1},b_{1}),\ldots,(a_{n},b_{n}))\in R_{\mathcal{M}[\mathcal{N}]}$ if
and only if $a_{1}=\ldots=a_{n}$ and $(b_{1},\ldots,b_{n})\in
R_{\mathcal{N}}$;

5) if $R\in\Sigma_{\mathcal{M}}\cap\Sigma_{\mathcal{N}}$, $\mu(R)=n$, then
$((a_{1},b_{1}),\ldots,(a_{n},b_{n}))\in R_{\mathcal{M}[\mathcal{N}]}$ if and
only if $(a_{1},\ldots,a_{n})\in R_{\mathcal{M}}$, or $a_{1}=\ldots=a_{n}$ and
$(b_{1},\ldots,b_{n})\in R_{\mathcal{N}}$.

The composition $\mathcal{M}[\mathcal{N}]$ is called $e$-definable, or equ-
definable, if $\mathcal{M}[\mathcal{N}]$ has an $\emptyset$-definable
equivalence relation $E$ whose $E$-classes are universes of the copies of
$\mathcal{N}$ forming $\mathcal{M}[\mathcal{N}]$. If the equivalence relation
$E$ is fixed, the $e$-definable composition is called $E$-definable.

Using a nice basedness of $E$-definable compositions $T_{1}[T_{2}]$ (see [7])
till the formulas of form $E(x,y)$ and generating formulas for $T_{1}$ and
$T_{2}$ we have the following:

###### Theorem 1.4

[1]. $1.$ For any theories $T_{1}$ and $T_{2}$ and their $E$-definable
composition $T_{1}[T_{2}]$, $T_{1}$ and $T_{2}$ are $n$-theories, for $n\geq
2$, iff $T_{1}[T_{2}]$ is an $n$-theory, moreover, ${\rm
ar}(T_{1}[T_{2}])={\rm max}\\{{\rm ar}(T_{1}),{\rm ar}(T_{2})\\}$, if models
of $T_{1}$ and of $T_{2}$ have at least two elements, and ${\rm
ar}(T_{1}[T_{2}])={\rm max}\\{{\rm ar}(T_{1}),{\rm ar}(T_{2}),2\\}$, if a
model of $T_{1}$ or $T_{2}$ is a singleton.

$2.$ For any theories $T_{1}$ and $T_{2}$ and their $E$-definable composition
$T_{1}[T_{2}]$, $T_{1}$ and $T_{2}$ are $n$-aritizable iff $T_{1}[T_{2}]$ is
$n$-aritizable.

##  2 Almost $n$-ary and $n$-aritizable theories, their dynamics

Definition. (Cf. [5, 6]) A theory $T$ is called almost $n$-ary if there are
finitely many formulae
$\varphi_{1}(\overline{x}),\ldots,\varphi_{m}(\overline{x})$ such that each
$T$-formula is $T$-equivalent to a Boolean combination of $n$-formulae and
formulae obtained by substitutions of free variables in
$\varphi_{1}(\overline{x}),\ldots,\varphi_{m}(\overline{x})$.

In such a case we say that the formulae
$\varphi_{1}(\overline{x}),\ldots,\varphi_{m}(\overline{x})$ witness that $T$
is almost $n$-ary.

Almost $1$-ary theories are called almost unary, almost $2$-ary theories are
called almost binary, almost $3$-ary theories are called almost ternary, etc.

A theory $T$ is called almost $n$-aritizable if some expansion $T^{\prime}$ of
$T$ is almost $n$-ary.

Almost $1$-aritizable theories are called almost unary-tizable, almost
$2$-aritizable theories are called almost binarizable, almost $3$-aritizable
theories are called almost ternarizable, etc.

The following properties are obvious.

1\. Any $n$-ary (respectively, $n$-aritizable) theory is almost $n$-ary
(almost $n$-aritizable).

2\. Any almost $n$-ary (respectively, $n$-aritizable) theory is almost $k$-ary
(almost $k$-aritizable) for any $k\geq n$.

3\. Any theory of a finite structure is almost unary.

Families of weakly circularly minimal structures produce examples of almost
binary theories which are not binary [5, 6]. Similarly natural generalizations
of weakly circularly minimal structures till $n$-circular orders give examples
of almost $(n-1)$-ary theories $T_{n}$ with ${\rm ar}(T_{n})=n$, $n\geq 4$.

Assuming that the witnessing set
$\\{\varphi_{1}(\overline{x}),\ldots,\varphi_{m}(\overline{x})\\}$ is minimal
for the almost $n$-ary theory $T$ we have either $m=0$ of $l(\overline{x})>n$.

Thus we have two minimal characteristics witnessing the almost $n$-arity of
$T$: $m$ and $l(\overline{x})$. The pair $(m,l(\overline{x}))$ is called the
degree of the almost $n$-arity of $T$, or the aar-degree of $T$, denoted by
${\rm deg}_{\rm aar}(T)$. Here we assume that $n$ is minimal with almost
$n$-arity of $T$, this $n$ is denoted by ${\rm aar}(T)$. Clearly, ${\rm
aar}(T)\leq{\rm ar}(T)$, and if $m=0$, i.e., $n={\rm ar}(T)={\rm aar}(T)$ then
it is supposed that $l(\overline{x})=0$, too.

We have ${\rm aar}(T)\in\omega$ if and only if ${\rm ar}(T)\in\omega$. So if
${\rm ar}(T)=\infty$ then it is natural to put ${\rm aar}(T)=\infty$.

Besides, $n={\rm ar}(T)={\rm aar}(T)\in\omega$ means that the set
$\Delta_{n}(T)$ of $T$-formulae with $n$ free variables allows to express all
$\emptyset$-definable sets for $T$ by Boolean combinations and taking any set
$\Delta_{k}(T)$ of $T$-formulae with $k<n$ free variables all
$\emptyset$-definable sets for $T$ can be expressed by Boolean combinations of
formulae in $\Delta_{k}$ and substitutions of infinitely many formulae
$\varphi(\overline{x})$ only, where $l(\overline{x})=n$.

By the definition if ${\rm aar}(T)=n$ then

${\rm deg}_{\rm ar}(T)\in\\{(0,0)\\}\cup\\{(m,r)\mid
m\in\omega\setminus\\{0\\},r\in\omega,r>n\\}.$ (1)

The described pairs in the relation (1) are called admissible.

###### Theorem 2.1

For any $m,n\in\omega\setminus\\{0\\}$ with $m\leq n$ there is a theory
$T_{mn}$ with ${\rm aar}(T{mn})=m$ and ${\rm ar}(T_{mn})=n$.

Proof. We use disjoint unions of dense $n$-spherically ordered theories
$T_{n}$ with infinite orders, i.e., theories, generated by $n$-spherical
orders $K_{n}(x_{1},x_{2},x_{3},\ldots,x_{n})$ satisfying the axioms

$\forall x_{1},x_{2},\ldots
x_{n}\Bigg{(}K_{n}(x_{1},x_{2},x_{3},\ldots,x_{n})\wedge\bigwedge_{i\neq
j}\neg x_{i}\approx x_{j}\to$ $\to\exists y\Bigg{(}\bigwedge_{i\leq n}\neg
x_{i}\approx y\wedge K_{n}(x_{1},y,x_{3},\ldots,x_{n})\Bigg{)}\Bigg{)}.$

For $n=2$ we take the theory $T_{2}$ of dense linear order
$K_{2}(x_{1},x_{2})$ without endpoints, having ${\rm ar}(T_{2})=2$, and for
$n=1$ — the theory $T_{1}$ of the empty languages, having ${\rm ar}(T_{1})=1$.

Similarly to dense linear orders and dense circular orders, dense
$n$-spherical orders produce quantifier eliminations with ${\rm ar}(T_{n})=n$,
$n\in\omega\setminus\\{0\\}$.

Using Theorem 1.3 we obtain $m={\rm ar}(T)={\rm aar}(T)$ taking
$T_{mm}=\bigsqcup\limits_{r\in\omega}T^{r}_{m}$, where $T^{r}_{m}$ are copies
of $T_{m}$ in disjoint languages $\\{K^{r}_{m}\\}$, $r\in\omega$. Indeed,
disjoint predicates $K^{r}_{m}$ producing ${\rm ar}(T_{m})=m$ witness that
${\rm ar}(T_{mm})=m$. Finitely many these predicates can not define all
definable sets for $T_{mm}$ since there are infinitely many of them. Thus,
${\rm aar}(T_{mm})=m$, too.

Now for any $m<n$ we form
$T_{mn}=T_{n}\sqcup\bigsqcup\limits_{r\in\omega}T^{r}_{m}$. By $T_{n}$ in
$T_{mn}$ and $m<n$ we have ${\rm ar}(T_{mn})=n$. And ${\rm aar}(T_{mn})=m$
since there are infinitely many disjoint predicates of arity $m$, as required.

The following theorem shows that all admissible pairs are realized.

###### Theorem 2.2

For any admissible pair $(m,r)$ and $n\in\omega\setminus\\{0,1\\}$ there is a
theory $T$ with ${\rm aar}(T)=n$ and ${\rm deg}_{\rm ar}(T)=(m,r)$.

Proof. The admissible pair $(0,0)$ with ${\rm ar}(T)=n$ is realized by Theorem
2.1. Now for an admissible pair $(m,r)\neq(0,0)$ we can take a disjoint union
$T$ of countably many dense $n$-spherically ordered theories and of $m$ dense
$r$-spherically ordered theories. Using arguments for Theorem 2.1 we obtain
${\rm aar}(T)=n$ and ${\rm deg}_{\rm ar}(T)=(m,r)$, as required.

###### Proposition 2.3

Any almost $n$-ary theory $T$ is $k$-ary for some $k\geq n$.

Proof. Let $T$ be an almost $n$-ary theory witnessed by the formulae
$\varphi_{1}(\overline{x}),\ldots,$ $\varphi_{m}(\overline{x})$. Then taking
the set $\Delta_{k}$ of all formulae with $k={\rm max}\\{n,l(\overline{x})\\}$
we observe, using
$\varphi_{1}(\overline{x}),\ldots,\varphi_{m}(\overline{x})$, that $T$ is
$\Delta_{k}$-based, i.e., $T$ is $k$-ary, as required.

###### Corollary 2.4

Any theory $T$ is $n$-ary for some $n$ iff $T$ is almost $m$-ary for some $m$.

###### Corollary 2.5

Any theory $T$ is $n$-aritizable for some $n$ iff $T$ is almost $m$-aritizable
for some $m$.

##  3 $\omega$-categorical almost $n$-ary and $n$-aritizable theories

###### Proposition 3.1

If $T$ is an almost $n$-ary $\omega$-categorical theory, for some $n$, then
$T$ is almost $k$-ary for any $k\in\omega\setminus\\{0\\}$, i.e., ${\rm
aar}(T)=1$.

Proof. If $k\geq n$ then $T$ is almost $m$-ary, as noticed above. If $k<n$
then we collect in a set $Z$ all formulae
$\varphi_{1}(\overline{x}),\ldots,\varphi_{m}(\overline{x})$ witnessing the
almost $n$-arity of $T$ and, by Ryll-Nardzewski Theorem, all non-equivalent
formulae $\psi_{1}(\overline{y}),\ldots,\psi_{r}(\overline{y})$ with
$l(\overline{y})=n$. Clearly, the set $Z$ witnesses that $T$ is almost
$k$-ary. Taking $k=1$ we obtain ${\rm aar}(T)=1$, as required.

###### Corollary 3.2

For any $\omega$-categorical theory $T$ either ${\rm aar}(T)=1$ with ${\rm
ar}(T)\in\omega$, or ${\rm aar}(T)=\infty$ with ${\rm ar}(T)=\infty$.

The following example illustrates Corollary 3.2.

###### Example 3.3

Taking a dense linear order $K_{2}$ without endpoint we can step-by-step
extend it to a chain of dense $n$-spherical orders $K_{n}$, $n\geq 2$, in the
following way.

We put $(a,b,c)\in K_{3}$ if $(a,b)\in K_{2}$ and $(b,c)\in K_{2}$, or
$(b,c)\in K_{2}$ and $(c,a)\in K_{2}$, or $(c,a)\in K_{2}$ and $(a,b)\in
K_{2}$. If $K_{n}$, $n\geq 3$, is defined then we put
$(a_{1},\ldots,a_{n+1})\in K_{n+1}$ if $(a_{1},\ldots,a_{n})\in K_{n}$ and
$(a_{2},\ldots,a_{n+1})\in K_{n}$, or $a_{2},\ldots,a_{n+1}\in K_{n}$ and
$(a_{3},\ldots,a_{n+1},a_{1})\in K_{n}$, or $(a_{3},\ldots,a_{n+1},a_{1})\in
K_{n}$ and $(a_{4},\ldots,a_{n+1},a_{1},a_{2})\in K_{n}$. The obtained
structure $\mathcal{M}_{\infty}$ in the language $\Sigma_{\infty}=\\{K_{n}\mid
n\in\omega\setminus\\{0,1\\}\\}$ has an $\omega$-categorical theory
$T_{\infty}$ with ${\rm aar}(T_{\infty})={\rm ar}(T_{\infty})=\infty$, since
each new $K_{n}$ increases the arity. The same characteristics have
restrictions of $T_{\infty}$ to any infinite sublanguages.

At the same time each restriction $\mathcal{M}$ of $\mathcal{M}_{\infty}$ to a
finite nonempty sublanguage $\\{K_{n_{1}},\ldots,$ $K_{n_{m}}\\}$ produces a
theory $T$ with ${\rm aar}(T)=1$ with ${\rm ar}(T)={\rm
max}\\{n_{1},\ldots,n_{m}\\}$.

By Corollary 3.2 and the definition of almost aritizability we immediately
have:

###### Corollary 3.4

Any restriction $T$ of $n$-ary $\omega$-categorical theory $T^{\prime}$ is
almost unary-tizable.

##  4 Operations for almost $n$-ary and $n$-aritizable theories

In this section we consider links for arities of theories with respect to
disjoint unions of theories and $E$-definable compositions of theories.

###### Theorem 4.1

$1.$ For any theories $T_{1}$, $T_{2}$ and their disjoint union $T_{1}\sqcup
T_{2}$, both $T_{1}$ and $T_{2}$ are almost $n$-ary iff $T_{1}\sqcup T_{2}$ is
almost $n$-ary, moreover, ${\rm aar}(T_{1}\sqcup T_{2})={\rm max}\\{{\rm
aar}(T_{1}),{\rm aar}(T_{2})\\}$.

$2.$ For any theories $T_{1}$, $T_{2}$ and their disjoint union $T_{1}\sqcup
T_{2}$, both $T_{1}$ and $T_{2}$ are almost $n$-aritizable iff $T_{1}\sqcup
T_{2}$ is almost is $n$-aritizable.

Proof. 1. Let $\Phi_{1}$ and $\Phi_{2}$ be finite sets of formulas witnessing
that $T_{1}$ and $T_{2}$ are almost $n$-ary, respectively. Using the
definition of disjoint union and Theorem 1.3 we obtain that the finite set
$\Phi_{1}\cup\Phi_{2}$ witnesses that $T_{1}\sqcup T_{2}$ is almost $n$-ary.
Conversely, if a finite set $\Phi$ of formulas witnesses that $T_{1}\sqcup
T_{2}$ is almost $n$-ary then $Phi$ witnesses that both $T_{1}$ and $T_{2}$
are almost $n$-ary.

The equality ${\rm aar}(T_{1}\sqcup T_{2})={\rm max}\\{{\rm aar}(T_{1}),{\rm
aar}(T_{2})\\}$ follows from the definition of disjoint union since if ${\rm
aar}(T_{1}\sqcup T_{2})=k$ then the maximal value of ${\rm aar}(T_{1})$ and
${\rm aar}(T_{2})$ is responsible for this equality.

Item 2 follows from Item 1 since expansions of $T_{1}$ and $T_{2}$ correspond
expansions of $T_{1}\sqcup T_{2}$: some expansions of $T^{\prime}_{1}$ and
$T^{\prime}_{2}$ of $T_{1}$ and $T_{2}$, respectively, are almost $n$-ary iff
$T^{\prime}_{1}\sqcup T^{\prime}_{2}$ produces an almost $n$-ary expansion of
$T_{1}\sqcup T_{2}$, as required.

Using induction we obtain the following:

###### Corollary 4.2

$1.$ For any theories $T_{1},T_{2},\ldots,T_{m}$ and their disjoint union
$\bigsqcup\limits_{i=1}^{m}T_{i}$, all $T_{1},T_{2},\ldots,T_{m}$ are almost
$n$-ary iff $\bigsqcup\limits_{i=1}^{m}T_{i}$ is almost $n$-ary, moreover,

${\rm aar}\left(\bigsqcup\limits_{i=1}^{m}T_{i}\right)={\rm max}\\{{\rm
aar}(T_{i})\mid i\leq m\\}.$

$2.$ For any theories $T_{1},T_{2},$ $\ldots,T_{m}$ and their disjoint union
$\bigsqcup\limits_{i=1}^{m}T_{i}$, all $T_{1},T_{2},$ $\ldots,T_{m}$ are
almost $n$-aritizable iff $\bigsqcup\limits_{i=1}^{m}T_{i}$ is almost
$n$-aritizable.

###### Remark 4.3

Both almost $n$-arity and almost $n$-aritizability can fail taking disjoint
unions of infinitely many theories $T_{i}$, $i\in I$. Indeed, each theory
$T_{i}$ can have its own finite set $\Phi_{i}$ of formulas witnessing the
almost $n$-arity/$n$-aritizability, say in disjoint languages, whereas finite
unions $\bigcup\Phi_{i}$ can not witness the almost
$n$-arity/$n$-aritizability for $\bigsqcup\limits_{i\in I}T_{i}$.

Generalizing Theorem 1.4 we obtain:

###### Theorem 4.4

$1.$ For any theories $T_{1}$ and $T_{2}$ and their $E$-definable composition
$T_{1}[T_{2}]$, $T_{1}$ and $T_{2}$ are almost $n$-ary iff $T_{1}[T_{2}]$ is
almost $n$-ary, moreover, ${\rm aar}(T_{1}[T_{2}])={\rm max}\\{{\rm
aar}(T_{1}),{\rm aar}(T_{2}),2\\}$, if models of $T_{1}$ and of $T_{2}$ have
at least two elements, and ${\rm aar}(T_{1}[T_{2}])={\rm max}\\{{\rm
aar}(T_{1}),{\rm aar}(T_{2})\\}$, if a model of $T_{1}$ or $T_{2}$ is a
singleton.

$2.$ For any theories $T_{1}$ and $T_{2}$ and their $E$-definable composition
$T_{1}[T_{2}]$, $T_{1}$ and $T_{2}$ are almost $n$-aritizable iff
$T_{1}[T_{2}]$ is almost $n$-aritizable.

Proof. 1. Let $T_{i}$ be $\Delta_{i}$-based for $i=1,2$. Since $T_{1}[T_{2}]$
is $E$-definable it is $\Delta$-based, where $\Delta$ consists of formulae in
$\Delta_{1}\cup\Delta_{2}$ and $E(x,y)$ [7]. Now assuming that $T_{1}$ and
$T_{2}$ are almost $n$-ary we can choose $\Delta_{i}$ consisting of
$n$-formulae and finitely many formulae forming $\Phi_{i}$, $i=1,2$. Hence
$T_{1}[T_{2}]$ is almost $n$-ary.

Conversely, if $T_{1}[T_{2}]$ is almost $n$-ary and it is witnessed by a set
$\Phi$ of formulae then $\Phi$ witnesses that $T_{1}$ and $T_{2}$ are almost
$n$-ary.

If models of $T_{1}$ and of $T_{2}$ have at least two elements then
$T_{1}[T_{2}]$ is at least binary that witnessed by the formula $E(x,y)$. Thus
since $T_{1}[T_{2}]$ is $\Delta$-based we have ${\rm aar}(T_{1}[T_{2}])={\rm
max}\\{{\rm aar}(T_{1}),{\rm aar}(T_{2}),2\\}$. If $T_{1}$ or $T_{2}$ is a
theory of singleton then $T_{1}[T_{2}]$ is $(\Delta_{1}\cup\Delta_{2})$-based
implying ${\rm aar}(T_{1}[T_{2}])={\rm max}\\{{\rm aar}(T_{1}),{\rm
aar}(T_{2})\\}$.

Item 2 follows from Item 1 repeating the arguments for Item 2 of Theorem 4.1.

Theorem 4.4 immediately implies

###### Corollary 4.5

Any composition of finitely many almost $n$-ary (almost $n$-aritizable)
theories, for $n\geq 2$, is again many almost $n$-ary (almost $n$-aritizable).

##  5 Conclusion

We considered possibilities for almost arities and almost aritizabilities of
theories and their dynamics both in general case, for $\omega$-categorical
theories, and with respect to operations for theories. It would be interesting
to describe values of almost arities and almost aritizabilities for natural
classes of theories.

## References

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Sobolev Institute of Mathematics,
4, Acad. Koptyug avenue, Novosibirsk, 630090, Russia;
Novosibirsk State Technical University,
20, K.Marx avenue, Novosibirsk, 630073, Russia;
Novosibirsk State University,
1, Pirogova street, Novosibirsk, 630090, Russia

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