# Attractors and Bifurcation Diagrams in Complex Climate Models

###### Abstract

The climate is a complex non-equilibrium dynamical system that relaxes toward
a steady state under the continuous input of solar radiation and dissipative
mechanisms. The steady state is not necessarily unique. A useful tool to
describe the possible steady states under different forcing is the bifurcation
diagram, that reveals the regions of multi-stability, the position of tipping
points, and the range of stability of each steady state. However, its
construction is highly time consuming in climate models with a dynamical deep
ocean, interactive ice sheets or carbon cycle, where the relaxation time
becomes larger than thousand years. Using a coupled setup of MITgcm, we test
two techniques with complementary advantages. The first is based on the
introduction of random fluctuations in the forcing and permits to explore a
wide part of phase space. The second reconstructs the stable branches and is
more precise in finding the position of tipping points.

Group of Applied Physics and Institute for Environmental Sciences, University
of Geneva, Bd. Carl-Vogt 66, CH-1205 Geneva, Switzerland

Maura<EMAIL_ADDRESS>
Bifurcation diagrams are crucial to find multistability regions, position of
B- and N-tipping, and the range of stability of each attractor

We propose two methods for their construction in general circulation models,
with and without random fluctuations in the forcing

The computational cost is much reduced with respect to the standard method
where convergence to the attractor is required

## Plain Language Summary

The behaviour of the climate system under a changing force can be
reconstructed through the so-called bifurcation diagrams. Such graphs allow
one to infer the critical values of the forcing giving rise to abrupt and
possibly irreversible change of the climate. The construction of bifurcation
diagrams requires to run long simulations from a huge ensemble of initial
conditions until convergence to a steady state is attained (standard method).
This procedure has prohibitive computational costs in general circulation
models of the climate that include deep ocean dynamics relaxing on timescales
of the order of thousand years, or other feedback mechanisms that act on even
longer time scales, like continental ice or carbon cycle. We propose two
methods which require lower computational costs. We test them in a numerical
setup that includes deep ocean dynamics and we compare the resulting
bifurcation diagrams with the one obtained with the standard method.

##  1 Introduction

The climate is a dynamical complex system that is fuelled by the incoming
solar radiation and reaches a steady state under the effect of dissipation
over a multitude of temporal and spatial scales [Ghil Lucarini (2020)]. Under
a given forcing (such as astronomical Milankovitch cycles, or the increasing
atmospheric CO2 content due to volcanism or present-day anthropogenic
emissions) the system can be driven out of the steady state. In such
conditions, the system can reach critical thresholds (or tipping points) where
its properties abruptly change, often in an irreversible way. Such behaviour
can be illustrated by the so-called bifurcation diagram (BD), where a state
variable (for example, the mean surface air temperature) is plotted as a
function of the driving force, as schematically shown in Figure 1 (second
row), together with the corresponding potential curve (first row) for the
different types of tipping mechanism [Ashwin . (2012), Vanselow . (2022)]: 1)
B-tipping, that occurs when the deterministic system dynamics reach a
bifurcation; 2) N-tipping, when the internal variability increases up to
exceed the height of a critical barrier separating two basins of attraction,
so that the system can access another dynamical solution; 3) S-tipping,
related to sudden shocks that induce the passage from one state to another, as
happened when a huge asteroid presumably caused the Cretaceous-Tertiary
extinction 66 million years ago; 4) R-tipping, when the rate of change of the
forcing or some internal parameter exceed a critical value, allowing the
system to cross, for example, a moving basin boundary (note, however, that
R-tipping can be induced by different mechanisms and does not necessarily
requires multistability [Vanselow . (2022)]).

Figure 1: Tipping mechanisms in terms of potential landscape (first row) and
corresponding bifurcation diagram (second row). Blue circles: state of the
system; red/black arrows: evolution of landscape/state.

BDs are easily obtained using energy balance models [Budyko (1969), Sellers
(1969), Ghil (1976)], the simplest in the hierarchy of climate models. In
intermediate complexity models [Lucarini Bódai (2017)] or low resolution
general circulation models [Rose (2015), Zhu Rose (2022)], BDs can be still
constructed with a reasonable computational cost. As the complexity of the
model increases, the amount of CPU time needed to perform series of
simulations that explore a huge range of driving force and initial conditions
toward steady states becomes prohibitive. Indeed, the standard method requires
convergence toward the steady state (or attractor) to obtain the BD, that
means to perform simulations over time scales of the order of several times
the relaxation time of the included climatic components [McGuffie Henderson-
Sellers (2005)], which can be $10^{3}$ years for the deep ocean, $10^{4}$
years for the carbon equilibration between atmosphere and ocean [Zhu Rose
(2022)], and even higher for dynamical ice sheets.

Using a general circulation model (thus, at the top of the hierarchy) in a
simplified configuration (i. e. a planet entirely covered by the ocean),
ragon2022 obtained the bifurcation diagram using the standard method over time
scales of thousand years, thus excluding nonlinear feedbacks on longer time
scales such as those induced by ice sheets and vegetation. The MIT general
circulation model [Marshall, Adcroft . (1997), Marshall, Hill . (1997)]
(MITgcm) was used in its coupled atmosphere-ocean-sea ice configuration with
2.8∘ horizontal resolution, 15 levels in an ocean depth of 3 km and 5 pressure
layers in the atmospheric column. The atmospheric module is based on SPEEDY
[Molteni (2003)] that provides a rather realistic representation of the
radiative scheme despite the coarse vertical resolution, with the advantage of
requiring less computer resources than state-of-the-art atmospheric modules.
On the other hand, the ocean dynamics is very accurate and this is crucial to
include nonlinear feedbacks on millennial time scales. Using this setup, it is
possible to run nearly 200 years in 1 day on clusters at the University of
Geneva. The same MITgcm setup, with an additional land module, has been
succesfully applied to investigate the ocean dynamics during the Jurassic
[Brunetti . (2015)], the present-day climate [Brunetti Vérard (2018)], and the
climatic oscillations in the Early Triassic.

Here we compare the standard technique to construct the BD with two additional
methods which require lower computational costs. We will describe such
techniques in Section 2, the resulting BDs in Section 3 and we draw our
conclusions on pros and cons for each method, and future developments in
Section 4.

##  2 Construction of Bifurcation Diagrams

We first describe the standard technique and the corresponding BD. Since, in
this case, convergence to the dynamical attractors is required over time
scales comparable to the longest feedback process in the setup (in our case,
until deep ocean equilibrium is reached), this BD is used for comparisons with
those obtained with less computationally expensive techniques that will be
described in the following subsections.

###  2.1 Standard Technique

The standard method to construct the BD is based on the theory of dynamical
systems. The phase space can be divided in basins of attraction, i. e the
minimal invariant closed sets attracting an open set of initial conditions as
time goes to infinity [Strogatz (1994)] for given values of the internal
parameters (such viscosity, diffusion coefficients, albedo of different
surfaces, …) and external forcing. Note that under a constant forcing, the
system is ergodic [Tél . (2020)], i. e sufficiently long temporal averages
correspond to large ensemble averages.

Thus, the first step is to consider a huge ensemble of initial conditions
spanning a large part of phase space. Starting from such initial conditions,
the system then evolves in time until the attractors are reached.
Statistically steady-state conditions are realized within each attractor in
the climate system when its mean annual surface energy balance $F_{s}$, i. e.
the sum of sensible, latent, net solar and longwave radiation fluxes at the
surface, becomes nearly zero. In general, simulation runs over several times
$n$ the relaxation time $t_{relax}$ are needed to guarantee convergence on the
attractor [Drótos . (2017)]. $t_{relax}$ depends on the nonlinear feedbacks
implemented in the numerical simulation. In an aquaplanet, there are no ice
sheets (characterised by a time scale of the order of 105 yr), and vegetation
(102-103 yr). An active carbon cycle between ocean and atmosphere (104 yr) is
also excluded, thus the deep-ocean dynamics is the process with the largest
relaxation time of the order of 103 yr. In our setup, it turns out to be
between 500 and 2000 yr, depending on the attractor (see Figure 2b in
brunetti2019).

Five steady climates have been found [Brunetti . (2019)] under the same
forcing, represented by the same amount of incoming solar radiation
($S_{0}=342$ W m-2) and atmospheric CO2 content (fixed at 326 ppm): snowball
(where ice covers the entire surface), waterbelt (where an ocean belt survives
near the equator), cold state (with an ice cap extending to 43 degree
latitude), warm state (with an ice cap comparable to the present one, up to 60
degree) and hot state (a planet without ice). Simulations are stopped when the
surface energy imbalance $F_{\rm s}$ becomes lower than 0.2 W m-2 in absolute
value in brunetti2019, corresponding to an ocean temperature drift $dT_{\rm
o}/dt=F_{s}/(c_{p}\rho h)$ (Marshall and Plumb 2008, p. 229) lower than 0.058
∘C per century, with $c_{\rm p}=4000$ J K-1 kg-1 being the specific heat
capacity, $\rho=1023$ kg m-3 the seawater density, and $h=3000$ m the ocean
depth. Indeed, under such conditions, there is essentially no drift in the
annual averages of the climatic observables.

The second step is to slightly vary the forcing, starting from each attractor,
and let the system relax again [Drótos . (2017)]. In this way, stable branches
of the BD can be constructed. In practise, in the aquaplanet simulations,
ragon2022 performed the following: the simulations are initialized from the
five attractors found in Brunetti et al. (2019) at $S_{0}=342$ W m-2 with
slightly different values of the incoming solar radiation and same CO2 content
until convergence. This procedure is repeated until a tipping point is
reached, that is where a sudden shift of the state variable to a different
attractor is observed.

Since the attractors are complex dynamical objects living in a high
dimensional space [Brunetti . (2019), Falasca Bracco (2022)], the projection
of their invariant measure on a given state variable, as the surface air
temperature $T$, is arbitrary [Faranda . (2019), Tél . (2020)]. It turns out
that different quantities are maximised in each attractor, as found in
ragon2022, i. e. total precipitation and surface temperature in hot state,
intensity of the Lorenz energy cycle in warm state, heat transport in cold
state, available potential and kinetic energies in waterbelt, while snowball
minimises all the above quantities. Thus, each climatic state would be better
represented by a different projection. However, in the present study, the
projection is performed, as commonly done in the literature, in terms of the
surface air temperature which spans an interval of more than 70 ∘C from the
snowball to the hot state, thus differentiating well all the attractors.

The BD obtained in such a way is shown in Figure 1 of ragon2022, where it can
be seen that the range of stability of the warm state is very small in
comparison to the others, while the snowball is stable over all the range of
forcing that has been explored, from 334 to 350 W m-2. The positions of
B-tipping are also evident, bracketing either end of the stable ranges for
warm and cold states, and the cold end of waterbelt and hot state. Note that
the warm state may become unstable when carbon exchanges between atmosphere
and ocean are included, as shown in zhurose2022 using a setup with horizontal
resolution of 3.75∘.

Such BD has been obtained using a very computationally expensive technique.
The total simulation time $t_{tot}$ can be estimated as $t_{tot}\sim
n(N+M)\,t_{relax}$, where $n\sim 10$ to reach convergence, $N\sim 40$ is the
number of initial conditions in the first step, $M\sim 40$ is the number of
points on the stable branches of the bifurcation diagram (see Figure 1 in
ragon2022), giving approximately $t_{tot}\sim 800\,t_{relax}\sim 8\cdot
10^{5}$ yr. Of course, the effective time can be optimized by launching $N$
simulations in parallel with different initial conditions, and by considering
separately each stable branch. Note, however, that the second term in
$t_{tot}$ cannot be reduced further, since the initial conditions at one
forcing value are the final state of the previous one.

###  2.2 Method I: Random Fluctuations in the Forcing

This method is based on the idea described in
LucariniBodai2020,margazoglou2020 of studying noise induced transitions
between basins of attraction. In order to explore the entire phase space of
the system, and not only the single basin of attraction allowed in nonlinear
deterministic dynamics, random fluctuations of the incoming solar radiation
are introduced around a given value ($S_{0}=342$ W/m2 in our setup) at
discrete times. Even if noise directly affects only a small fraction of
degrees of freedom, it propagates to all, since scales are interconnected as
well as dynamical and thermodynamical fields in the coupled climate model
[Lucarini Bódai (2020)].

If $\eta$ is a random number taken from a normal distribution with zero mean
and standard deviation $\sigma$, a new value of the incoming solar radiation
$S$ is prescribed as $S=S_{0}(1+\eta)$ at regular temporal intervals $\Delta
t_{1}$. We have tested several values of the standard deviation, ranging from
$\sigma=0.01$ to 0.1, and of the temporal interval, from $\Delta t_{1}=1$ yr
to 100 yr. The smaller $\Delta t_{1}$, the more uniformly the phase space is
filled, since the system cannot relax toward the attractor, and thus the
reconstruction of the position of the attractors becomes unfeasible. On the
contrary, too large values of $\Delta t_{1}$ are computational expensive. On
the other hand, the standard deviation $\sigma$ regulates the range of forcing
values being explored and the portion of phase space than can be accessed, a
weaker noise corresponding to rarer transitions between the basins of
attraction [Margazoglou . (2021)].

In our MITgcm setup, the values that reproduce better the BD obtained with the
standard method in a reasonable amount of time are $\sigma=0.025$ and $\Delta
t_{1}=10$ yr. Such values are model- and setup-dependent. For example, in the
Planet Simulator (PlaSim), an open-source intermediate complexity climate
model with slab ocean [Fraedrich . (2005)], the relaxation time is governed by
upper ocean processes and is of the order of several decades for 5.6∘ of
horizontal resolution and 10 atmospheric levels [Margazoglou . (2021)]. In
this case, the reconstruction of the invariant measure is obtained using
$\sigma=0.18$ and $\Delta t_{1}=1$ yr [Margazoglou . (2021)]. We find that a
good reconstruction occurs after a total simulation time of the order of
$t_{tot}\sim 10^{4}$ yr. Using different seeds and/or initial conditions for
the random variable, such total time can be easily partitioned over different
set of runs.

###  2.3 Method II: Reconstruction of Stable Branches

The first step coincides with that in the standard technique: the attractors
at a given forcing are needed as starting points and are obtained by scanning
a large ensemble of initial conditions [Brunetti . (2019)]. However, the
second step differs. We require the change in forcing to be small enough so
that the invariant measure on the attractor remains nearly the same [Tél .
(2020)], in particular its mean and standard deviation. In other words, while
the forcing is changed by $\Delta S$, the corresponding variation on the
attractor is small enough that the surface energy imbalance remains nearly
zero. This excludes for the presence of R-tipping and guarantees quasi-
ergodicity of the system. In practise, starting from each attractor found
during the first step, the forcing is changed by $+\Delta S$ ($-\Delta S$)
each $\Delta t_{2}=N_{2}$ yr in order to determine the upper (lower) stable
branch in the bifurcation diagram, until a tipping point is reached.

The tipping point is attained when one of the three following criteria is
satisfied: (i) the standard deviation within the $N_{2}$ points for each
forcing value becomes larger than the internal variability on the attractor
(an early warning of critical slowing down of the dynamics [Scheffer .
(2011)]); (ii) the $N_{2}$ points turn out to be ordered in time, pointing
from the original attractor toward a new one; (iii) the surface energy
imbalance $F_{\rm s}$ becomes much larger than zero (in our setup, larger than
0.5 W m-2 in absolute value, which is slightly larger than the threshold 0.2 W
m-2 used to characterise the convergence to an attractor). When one of these
criteria is satisfied, the system is no more on the initial dynamical
attractor and is approaching the unstable branch where a shift toward a new
basin of attraction takes place.

We have tested different values of $\Delta S$ from 0.1 to 0.5 W m-2 and
$N_{2}$ from 10 to 100. Like in the previous case, the choice is model
dependent. We have checked that in our setup good agreement with the
bifurcation diagram from the standard technique (that is very accurate albeit
highly time consuming) is obtained when $\Delta S=0.25$ W m-2 and $N_{2}\geq
20$. With less points, the criterium based on the standard deviation is not
applicable (low statistics). With larger $\Delta S$ the position of B-tipping
is less accurate and the requirement of remaining on the same original
attractor cannot be satisfied, while too small values require large CPU time.
The total simulation time can be estimated as
$t_{tot}=nN\,t_{relax}+M_{2}\Delta t_{2}$, where $M_{2}$ is the number of
points in the forcing along all the branches. The second term is smaller than
the analogous in the standard model since $\Delta t_{2}\ll t_{relax}$.

##  3 Results

We take advantage of the BD obtained with the standard (computational
expensive) technique to test the other two methods described in Sections 2.2
and 2.3, and understand under which conditions they can be applied.

###  3.1 Method I

Figure 2: Invariant measure obtained with Method I by adding random
fluctuations with standard deviation $\sigma$ = 0.025 to the incoming solar
radiation $S_{0}$ at regular temporal intervals $\Delta t_{1}=10$ yr. The
diverging colormap goes from low (blue) to high density (red).

We construct the bifurcation diagram by plotting the normalised 2-dimensional
histogram (invariant measure) in terms of the global annual surface air
temperature and the forcing, as shown in Fig. 2 using a standard deviation of
the normal distribution equal to $\sigma=0.025$ and time interval $\Delta
t_{1}=10$ yr. Other BDs obtained for different values of $\sigma$ and $\Delta
t_{1}$ can be found in Supporting Information (Fig. S1). As can be seen, the
overall structure of the phase plane corresponds quite well to that in the
standard BD (see Fig. 1 in ragon2022). The main attractors (hot, cold,
waterbelt and snowball) can be easily recognised and correspond to regions of
high density of points. However, the uncertainty in the exact position of the
attractors is large, so that, for example, the warm state, with a short stable
branch, cannot be distinguished from the hot climate and, in general, it is
not possible to precisely infer the edges of the stable branches.

Moreover, a region with increased density different from the ones that
correspond to the five attractors appears at $T\sim-20$ ∘C that may correspond
to an additional steady state. This can be checked by performing simulations
without noise starting from initial conditions in such region of high density,
and let the system relaxes towards a steady state under fixed forcing, i. e.
until the surface energy imbalance becomes negligible. It turns out that the
feature at $T\sim-20$ ∘C is only transient, as shown in Supporting Information
(Figure S2): the system remains near such value of temperature for nearly 100
yr with a small surface imbalance but eventually is attracted to the
waterbelt.

###  3.2 Method II

Starting from the five attractors found in brunetti2019 at $S_{0}=342$ W/m2,
the stable branches are derived by changing the forcing by $\Delta S$ each
$N_{2}$ years. The resulting BD is shown in Fig. 3 using $\Delta S=0.1$ W m-2
and $N_{2}=100$ (20 for waterbelt upper branch). As can be seen, the five
stable branches can be precisely recovered. If $\Delta S$ is too large, the
edges of the stable branches are not correctly reproduced, as shown in
Supporting Information (Figure S3). Moreover, by applying the criteria listed
in Section 2.3 to determine the position of B-tipping, we check that they
correspond well to those found with the standard technique (see Figure 3a and
its enlargements (panels b, c, d, e, f), where the red arrows correspond to
B-tipping and colors correspond to the time ordering of the $N_{2}$ points).
In Figure 4, two examples of evolution (hot to colder climates and cold to
snowball) show when the system abandons the attractor at the point where the
standard deviation of the temperature becomes larger than the internal
variability on the attractor and/ or $|F_{\rm s}|>0.5$ W m-2 (following the
criteria (i) and (iii) in Section 2.3, respectively).

Figure 3: (a) Bifurcation diagram obtained with Method II; red arrows show the
positions of B-tipping and panels b-f their enlargements: (b) cold to hot; (c)
cold to snowball; (d) hot to colder climates; (e) warm to hot and cold; (f)
waterbelt to hot.

Figure 4: Temporal evolution of mean global surface energy imbalance $F_{s}$
(a,c) and surface air temperature $T$ (b,d), with superposed incoming solar
radiation (right axes) for (top) hot to colder climates; (bottom) cold to
snowball. The strip in panels (a,c) corresponds to an imbalance of $\pm 0.5$ W
m-2. Shadow areas in (b,d) correspond to the standard deviation in temperature
within $N_{2}=100$ points.

Another interesting feature in Figures 3a,c is the transition from the cold
state to colder climates. When the system looses stability on the cold branch,
it is attracted first by transient structures, like the one at $T=-20$ ∘C.
Then, instead of relaxing to the waterbelt, it is directly attracted to the
snowball. This can be understood by remembering that the climate state lives
on a high dimensional space that we are arbitrarily projecting to a single
state variable (the surface air temperature). In a higher dimensional space,
the two climatic trajectories (the transition from cold to snowball and the
waterbelt stable branch) would not cross each other. This behavior and the
fact that an analogous crossing occurs in the transition from hot to snowball
(see Figure 3a) show that the waterbelt climate is not well described by a
single state variable.

Finally, the dependence of the invariant measure can be investigated, as done
here, in terms of the incoming solar radiation $S$ for a fixed atmospheric CO2
content. The opposite can also be considered: different $CO_{2}$ for fixed $S$
[Zhu Rose (2022)]. An example is shown in Supporting Information (Figure S4)
where, however, the MITgcm setup does not include the feedback of the
atmospheric carbon with the ocean.

##  4 Summary and Conclusions

We have proposed two methods to construct BDs in general circulation models
(GCM), where the number and timescale of nonlinear feedback mechanisms make
the computational costs of the standard method prohibitive. Such diagrams
store crucial information about the nonlinear structure of the climate system,
like the regions of multistability, the kind of tipping, the amplitude of
climatic oscillations or the intensity of forcing necessary to give rise to a
climatic shift.

In the first method, random fluctuations of the forcing allow to explore the
entire phase space of the climate system [Margazoglou . (2021)]. The choice of
the parameters for such method, i. e. the standard deviation of the Gaussian
fluctuations and the temporal interval after which the forcing randomly
changes its value, is model dependent. However, once the parameters fixed,
such method gives an overall picture of the phase space projected on a given
state variable, revealing the number of attractors and their position. The
disadvantage is that the attractors with small stable branches may be blurred
and the position of B-tippings is not accurate.

The second method reconstructs the stable branches of the attractors by
changing the forcing by a small amount that guarantees the system to remain on
the same attractor until a B-tipping is reached. The signature of a shift is
given by increased variance of the internal variability, non-null surface
energy imbalance or an ordered temporal sequence of points toward another
climatic state. The advantage of such a method is that the reconstruction of
B-tipping is quite accurate with much lower computational costs with respect
to the standard method.

The proposed methods have been described by projecting the high-dimensional
space of the attractors to a single state variable (global annual mean of
surface air temperature). Of course, they can be applied using other
projections, and averages over different temporal and spatial scales. This is
particularly useful in order to develop early warning indicators that are
based not only on temporal series but also on spatial information [Nijp .
(2019)]. From the BD one can infer the forcing that induces a tipping and
perform a detailed analysis in space to identify where the main changes take
place (gridded data in general circulation models and remote sensing permit
such kind of spatial studies). Spatial early warning indicators do not need a
long temporal record to get a meaningful signal, thus playing a crucial role
to identify tipping mechanisms from datasets with irregular or infrequent
temporal resolution [Génin . (2018)].

Apart from the fundamental and practical interest of obtaining the BD for our
present-day climate, there are many aspects that are worth to be analysed. An
open question is, for example, whether the attractors and the BD are model
dependent. The robustness of the results presented here should be tested
against other climatic models. For the comparison, analogous numerical setups
should be chosen at the level of horizontal/vertical resolutions and included
nonlinear feedback mechanisms. Important information on the description of the
nonlinear processes and their interplay can be gained from such comparison of
BDs produced by different models, that can be used to improve algorithms and
to correct biases. This is why we suggest including the comparison of BDs in
coarse-resolution GCMs with simplified configurations in the Tipping Point
Model Intercomparison Project (TipMIP) beside that of Earth System Models in
the present-day configuration.

##  5 Open Research

The data that support the findings of this study were generated by the MIT
general circulation model that is openly available on GitHub
(http://mitgcm.org/, https://github.com/MITgcm/MITgcm, version c67f).

###### Acknowledgements.

We are grateful to Sacha Medaer and Enzo Samy Ferrao for running some of the
MITgcm simulations with noise. We thank Alexis Gomel and Jérôme Kasparian for
useful discussions. The computations were performed on the Baobab and
Yggdrasil clusters at University of Geneva. We acknowledge the financial
support from the Swiss National Science Foundation (Sinergia Project
CRSII5_180253).

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# On certain three algebras generated by binary algebras

H. Ahmed1,3, M.A.A. Ahmed2,3, Sh.K. Said Husain1,2, Witriany Basri1
1Department of Math., Faculty of Science, UPM, Selangor, Malaysia  2Institute
for Mathematical Research (INSPEM), UPM, Serdang, Selangor, Malaysia  3Depart.
of Math., Faculty of Science, Taiz University, Taiz, Yemen

###### Abstract.

This paper’s central theme is to prove the existence of an n-algebra whose
multiplication cannot be expressed employing any binary operation.
Furthermore, to prove if two algebras are not isomorphic, this property does
not hold for $3$-algebras corresponding to these two algebras. The proof
drives applying some results gotten early applying a new approach for the
classification algebras problem, introduced recently, which showed great
success in solving many classification algebras problems.

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##  1\. Introduction

In 1969 [11], Kurosh introduced the notion of multilinear operator algebra. It
is known that such algebraic structures are attractive for their applications
to problems of modern mathematical physics. In 1973 [14], Nambu proposed an
exciting generalization of classical Hamiltonian mechanics; the Nambu bracket
is a generalization of the classical Poisson bracket.

Indeed, the advance of theoretical physics of quantum mechanics and the
discovery of Nambu mechanics (see [14]), together with Okubo’s work on the
Yang-Baxter equation (see [15]), gave impetus to significant development on
triple algebra ($3$-algebras).

Furthermore, Carlsson, Lister, and Loos have studied triple algebra of
associative type (see [9, 12, 13]). Hestenes provided the typical and founding
example of totally associative triple algebra (see [10]).

In this article, we give basic definitions and examples related to general
$n$-algebras, and we shall focus our attention on $3$-algebras structures
generated by binary algebras presented recently in [1]. Then, we introduce the
definition of totally associative $3$-algebras with examples, which show.
Owing to the large size of the matrices involved in our computations of
totally associative $3$-algebras, we present only Mathematica’s results.

##  2\. Preliminaries

Let $\mathbb{F}$ be any field and the product $A\otimes B$ is the Kronecker
product which stands for the matrix with blocks $(a_{ij}B),$ where
$A=(a_{ij})$ and $B$ are matrices over $\mathbb{F}$.

######  Definition 2.1.

A vector space $\mathbb{A}$ over $\mathbb{F}$ with multiplication
$\cdot:\mathbb{A}\times\mathbb{A}\rightarrow\mathbb{A}$ given by
$(\mathbf{u},\mathbf{v})\mapsto\mathbf{u}\cdot\mathbf{v}$ such that

  * •

$(\alpha\mathbf{u}+\beta\mathbf{v})\cdot\mathbf{w}=\alpha(\mathbf{u}\cdot\mathbf{w})+\beta(\mathbf{v}\cdot\mathbf{w}),$

  * •

$\mathbf{w}\cdot(\alpha\mathbf{u}+\beta\mathbf{v})=\alpha(\mathbf{w}\cdot\mathbf{u})+\beta(\mathbf{w}\cdot\mathbf{v}),$

whenever $\mathbf{u},\mathbf{v},\mathbf{w}\in\mathbb{A}$ and
$\alpha,\beta\in\mathbb{F}$, is said to be an algebra.

######  Definition 2.2.

Two algebras $\mathbb{A}$ and $\mathbb{B}$ are called isomorphic if there is
an invertible linear map $\mathbf{f}:\mathbb{A}\rightarrow\mathbb{B}$ such
that

(2.1)
$\mathbf{f}(\mathbf{u}\cdot_{\mathbb{A}}\mathbf{v})=\mathbf{f}(\mathbf{u})\cdot_{\mathbb{B}}\mathbf{f}(\mathbf{v})\mbox{
whenever }\mathbf{u},\mathbf{v}\in\mathbb{A}.$

######  Definition 2.3.

A vector space $V$ over $\mathbb{F}$ equipped by a multilinear map
$f:\underbrace{V\times V\times...\times V}_{n-times}\longrightarrow V$ is said
to be a $n$-algebra, that means:

  * •

$f(x_{1},x_{2},...,x_{i}+x^{\prime}_{i},...,x_{n})=f(x_{1},x_{2},...,x_{i},...,x_{n})+f(x_{1},x_{2},...,x^{\prime}_{i},...,x_{n})$

  * •

$f(x_{1},x_{2},...,\lambda x_{i},...,x_{n})=\lambda
f(x_{1},x_{2},...,x_{i},...,x_{n})$

where $(x_{1},x_{2},...,x_{i},...,x_{n})\in V.$

######  Example 2.4.

Let $\mathbb{A}=(V,\mu)$ be an algebra over a field $\mathbb{F}$: Then
multilinear map

(2.2)
$f(x_{1},x_{2},…,x_{n})=\underbrace{\mu(x_{1},\mu(x_{2},...,\mu(x_{n-1},x_{n})...))}_{(n-1)-times}$

defines an $n$-algebra structure on $V$.

##  3\. Classification approach of $m$-dimensional $3$-algebras

Let $\mathbb{A}$ be $m$-dimensional $3$-algebra over $\mathbb{F}$ and
$\mathbf{e}=(e^{1},e^{2},...,e^{m})$ its basis. Then the multilinear map
$\cdot$ is represented by a matrix $A=(A^{l}_{ijk})\in M(m\times
m^{3};\mathbb{F})$ as follows

(3.1) $\mathbf{u}\cdot\mathbf{v}\cdot\mathbf{w}=\mathbf{e}A(u\otimes v\otimes
w),$

for $\mathbf{u}=\mathbf{e}u,\mathbf{v}=\mathbf{e}v,\mathbf{w}=\mathbf{e}w,$
where $u=(u_{1},u_{2},...,u_{m})^{T},$ $v=(v_{1},v_{2},...,v_{m})^{T}$ and
$w=(w_{1},w_{2},...,w_{m})^{T}$ are column coordinate vectors of $\mathbf{u},$
$\mathbf{v},$ and $\mathbf{w},$ respectively. The matrix $A\in M(m\times
m^{3};\mathbb{F})$ defined above is called the matrix of structural constants
(MSC) of $\mathbb{A}$ with respect to the basis $\mathbf{e}$. Further we
assume that a basis $\mathbf{e}$ is fixed and we do not make a difference
between the algebra $\mathbb{A}$ and its MSC $A$.

If $\mathbf{e}^{\prime}=(e^{\prime 1},e^{\prime 2},...,e^{\prime m})$ is
another basis of $\mathbb{A}$, $\mathbf{e}^{\prime}g=\mathbf{e}$ with $g\in
G=GL(m;\mathbb{F})$, and $A^{\prime}$ is MSC of $\mathbb{A}$ with respect to
$\mathbf{e}^{\prime}$ then it is known that

(3.2) $A^{\prime}=gA(g^{-1})^{\otimes 3}$

is valid (see [8]). Thus, we can reformulate the isomorphism of algebras as
follows.

######  Definition 3.1.

Two $m$-dimensional $3$-algebras $\mathbb{A}$, $\mathbb{B}$ over $\mathbb{F}$,
given by their matrices of structure constants $A$, $B$, are said to be
isomorphic if $B=gA(g^{-1})^{\otimes 3}$ holds true for some $g\in
GL(m;\mathbb{F})$.

Further we consider only the case $m=2$ then $\mathbb{A}$ can be represented
by its matrix of structural constants (MSC) $A=(\gamma_{ijk}^{l})\in M(2\times
2^{3};F)$ where $i,j,k,l=1,2$ and $\gamma_{ijk}^{l}\in\mathbb{F},$ as follows:

$A=\left(\begin{array}[]{cccccccc}\gamma_{111}^{1}&\gamma_{112}^{1}&\gamma_{121}^{1}&\gamma_{122}^{1}&\gamma_{211}^{1}&\gamma_{212}^{1}&\gamma_{221}^{1}&\gamma_{222}^{1}\\\
\gamma_{111}^{2}&\gamma_{112}^{2}&\gamma_{121}^{2}&\gamma_{122}^{2}&\gamma_{211}^{2}&\gamma_{212}^{2}&\gamma_{221}^{2}&\gamma_{222}^{2}\\\
\end{array}\right)$

(for more information refer to [2]).

##  4\. $3$-algebras generated by binary algebras

Due to [1] we have the following classification theorems according to
$Char(\mathbb{F})\neq 2,3.$

######  Theorem 4.1.

Over an algebraically closed field $\mathbb{F}$ $(Char(\mathbb{F})\neq 2$ and
$3)$, any non-trivial $2$-dimensional algebra is isomorphic to only one of the
following algebras listed by their matrices of structure constants:

  * •

$A_{1}(\mathbf{c})=\left(\begin{array}[]{cccc}\alpha_{1}&\alpha_{2}&\alpha_{2}+1&\alpha_{4}\\\
\beta_{1}&-\alpha_{1}&-\alpha_{1}+1&-\alpha_{2}\end{array}\right),\
\mbox{where}\
\mathbf{c}=(\alpha_{1},\alpha_{2},\alpha_{4},\beta_{1})\in\mathbb{F}^{4},$

  * •

$A_{2}(\mathbf{c})=\left(\begin{array}[]{cccc}\alpha_{1}&0&0&1\\\
\beta_{1}&\beta_{2}&1-\alpha_{1}&0\end{array}\right)\simeq\left(\begin{array}[]{cccc}\alpha_{1}&0&0&1\\\
-\beta_{1}&\beta_{2}&1-\alpha_{1}&0\end{array}\right),\ \mbox{where}\
\mathbf{c}=(\alpha_{1},\beta_{1},\beta_{2})\in\mathbb{F}^{3},$

  * •

$A_{3}(\mathbf{c})=\left(\begin{array}[]{cccc}0&1&1&0\\\
\beta_{1}&\beta_{2}&1&-1\end{array}\right),\ \mbox{where}\
\mathbf{c}=(\beta_{1},\beta_{2})\in\mathbb{F}^{2},$

  * •

$A_{4}(\mathbf{c})=\left(\begin{array}[]{cccc}\alpha_{1}&0&0&0\\\
0&\beta_{2}&1-\alpha_{1}&0\end{array}\right),\ \mbox{where}\
\mathbf{c}=(\alpha_{1},\beta_{2})\in\mathbb{F}^{2},$

  * •

$A_{5}(\mathbf{c})=\left(\begin{array}[]{cccc}\alpha_{1}&0&0&0\\\
1&2\alpha_{1}-1&1-\alpha_{1}&0\end{array}\right),\ \mbox{where}\
\mathbf{c}=\alpha_{1}\in\mathbb{F},$

  * •

$A_{6}(\mathbf{c})=\left(\begin{array}[]{cccc}\alpha_{1}&0&0&1\\\
\beta_{1}&1-\alpha_{1}&-\alpha_{1}&0\end{array}\right)\simeq\left(\begin{array}[]{cccc}\alpha_{1}&0&0&1\\\
-\beta_{1}&1-\alpha_{1}&-\alpha_{1}&0\end{array}\right),\ \mbox{where}\
\mathbf{c}=(\alpha_{1},\beta_{1})\in\mathbb{F}^{2},$

  * •

$A_{7}(\mathbf{c})=\left(\begin{array}[]{cccc}0&1&1&0\\\
\beta_{1}&1&0&-1\end{array}\right),\ \mbox{where}\
\mathbf{c}=\beta_{1}\in\mathbb{F},$

  * •

$A_{8}(\mathbf{c})=\left(\begin{array}[]{cccc}\alpha_{1}&0&0&0\\\
0&1-\alpha_{1}&-\alpha_{1}&0\end{array}\right),\ \mbox{where}\
\mathbf{c}=\alpha_{1}\in\mathbb{F},$

  * •

$A_{9}=\left(\begin{array}[]{cccc}\frac{1}{3}&0&0&0\\\
1&\frac{2}{3}&-\frac{1}{3}&0\end{array}\right),$

  * •

$A_{10}=\left(\begin{array}[]{cccc}0&1&1&0\\\ 0&0&0&-1\end{array}\right),$

  * •

$A_{11}=\left(\begin{array}[]{cccc}0&1&1&0\\\ 1&0&0&-1\end{array}\right),$

  * •

$A_{12}=\left(\begin{array}[]{cccc}0&0&0&0\\\ 1&0&0&0\end{array}\right).$

######  Example 4.2.

In example 2.4, if $V$ is $2$-dimensional vector space with a fixed basis
$\\{e_{1},e_{2}\\}$ and $(V,f)$ is $3$-algebra structure on $V.$ Then $f$ and
$\mu$ can be expressed by their structure constants as follows:
$f(e_{i},e_{j},e_{k})=\gamma_{ijk}^{1}e_{1}+\gamma_{ijk}^{2}e_{2}$ and
$\mu(e_{r},e_{s})=\eta_{rs}^{1}e_{1}+\eta_{rs}^{2}e_{2}.$ Then due to (2.2) we
get the system of equations

(4.1)
$\left.\begin{array}[]{cc}\gamma_{ijk}^{1}=&\eta_{jk}^{1}\eta_{i1}^{1}+\eta_{jk}^{2}\eta_{i2}^{1}\\\
\gamma_{ijk}^{2}=&\eta_{jk}^{1}\eta_{i1}^{2}+\eta_{jk}^{2}\eta_{i2}^{2}\end{array}\right\\}$

we get $2^{4}=16$ equations for $2^{3}=8$ unknowns (the coefficients
$\eta_{ij}^{k}$), which cannot be solved in general, except maybe for some
very special cases.
Indeed, using (4.1), we can find the $3$-algebras corresponding to all
algebras presented in [1] under this procedure (see Table 1).

algebra |  $3$-algebra
---|---
$A_{1}(\mathbf{c})$ | $B_{1}(\mathbf{c})=\left(\begin{array}[]{cccccccc}\alpha_{2}\beta_{1}+\alpha_{1}^{2}&0&\alpha_{1}+\alpha_{2}&\alpha_{1}\alpha_{4}-\alpha_{2}^{2}&\alpha_{4}\beta_{1}+\alpha_{2}\alpha_{1}+\alpha_{1}&\alpha_{2}^{2}-\alpha_{2}-\alpha_{1}\alpha_{4}&\alpha_{2}^{2}+2\alpha_{2}-\alpha_{1}\alpha_{4}+\alpha_{4}+1&\alpha_{4}\\\ 0&\alpha_{2}\beta_{1}+\alpha_{1}^{2}&\alpha_{2}\beta_{1}+\alpha_{1}^{2}-\alpha_{1}+\beta_{1}&\alpha_{4}\beta_{1}+\alpha_{1}\alpha_{2}&-\alpha_{2}\beta_{1}-\alpha_{1}^{2}+\alpha_{1}&\alpha_{2}&1-\alpha_{1}&\alpha_{2}^{2}-\alpha_{1}\alpha_{4}+\alpha_{4}\\\ \end{array}\right)$
$A_{2}(\alpha_{1},\beta_{1},\beta_{2})$ | $B_{2}(\alpha_{1},\beta_{1},\beta_{2})=\left(\begin{array}[]{cccccccc}\alpha_{1}^{2}&0&0&\alpha_{1}&\beta_{1}&\beta_{2}&1-\alpha_{1}&0\\\ \alpha_{1}\beta_{1}+\beta_{2}\beta_{1}&\beta_{2}^{2}&\left(1-\alpha_{1}\right)\beta_{2}&\beta_{1}&\alpha_{1}\left(1-\alpha_{1}\right)&0&0&1-\alpha_{1}\\\ \end{array}\right)$
$A_{3}(\beta_{1},\beta_{2})$ | $B_{3}(\beta_{1},\beta_{2})=\left(\begin{array}[]{cccccccc}\beta_{1}&\beta_{2}&1&-1&0&1&1&0\\\ \beta_{1}\beta_{2}&\beta_{2}^{2}+\beta_{1}&\beta_{1}+\beta_{2}&-\beta_{2}&-\beta_{1}&1-\beta_{2}&0&1\\\ \end{array}\right)$
$A_{4}(\alpha_{1},\beta_{2})$ | $B_{4}(\alpha_{1},\beta_{2})=\left(\begin{array}[]{cccccccc}\alpha_{1}^{2}&0&0&0&0&0&0&0\\\ 0&\beta_{2}^{2}&\left(1-\alpha_{1}\right)\beta_{2}&0&\alpha_{1}\left(1-\alpha_{1}\right)&0&0&0\\\ \end{array}\right)$
$A_{5}(\alpha_{1})$ | $B_{5}(\alpha_{1})=\left(\begin{array}[]{cccccccc}\alpha_{1}^{2}&0&0&0&0&0&0&0\\\ 3\alpha_{1}-1&\left(2\alpha_{1}-1\right){}^{2}&\left(2\alpha_{1}-1\right)\left(1-\alpha_{1}\right)&0&\alpha_{1}\left(1-\alpha_{1}\right)&0&0&0\\\ \end{array}\right)$
$A_{6}(\alpha_{1},\beta_{1})$ | $B_{6}(\alpha_{1},\beta_{1})=\left(\begin{array}[]{cccccccc}\alpha_{1}^{2}&0&0&\alpha_{1}&\beta_{1}&1-\alpha_{1}&-\alpha_{1}&0\\\ \beta_{1}&\left(1-\alpha_{1}\right){}^{2}&-\alpha_{1}\left(1-\alpha_{1}\right)&\beta_{1}&-\alpha_{1}^{2}&0&0&-\alpha_{1}\\\ \end{array}\right)$
$A_{7}(\beta_{1})$ | $B_{7}(\beta_{1})=\left(\begin{array}[]{cccccccc}\beta_{1}&\beta_{1}+1&0&-1&0&1&1&0\\\ \beta_{1}&1&\beta_{1}&-1&-\beta_{1}&-1&0&1\\\ \end{array}\right)$
$A_{8}(\alpha_{1})$ | $B_{8}(\alpha_{1})=\left(\begin{array}[]{cccccccc}\alpha_{1}^{2}&0&0&0&\alpha_{1}^{2}&0&0&0\\\ 0&\left(1-\alpha_{1}\right){}^{2}&-\alpha_{1}\left(1-\alpha_{1}\right)&0&0&0&0&0\\\ \end{array}\right)$
$A_{9}$ | $B_{9}=\left(\begin{array}[]{cccccccc}\frac{1}{9}&0&0&0&0&0&0&0\\\ 1&\frac{4}{9}&-\frac{2}{9}&0&-\frac{1}{9}&0&0&0\\\ \end{array}\right)$
$A_{10}$ | $B_{10}=\left(\begin{array}[]{cccccccc}0&0&0&-1&0&1&1&0\\\ 0&0&0&0&0&0&0&1\\\ \end{array}\right)$
$A_{11}$ | $B_{11}=\left(\begin{array}[]{cccccccc}1&0&0&-1&-1&1&1&0\\\ 0&1&1&0&0&0&0&1\\\ \end{array}\right)$
$A_{12}$ | trivial
Table 1. $2$-dimensional $3$-algebras generated by binary algebras(where
$\mathbf{c}=(\alpha_{1},\alpha_{2},\alpha_{4},\beta_{1})$)

From the table we can see

  * •

$\left(\begin{array}[]{cccccccc}1&0&0&1&0&1&-1&0\\\ 0&-1&1&0&1&0&0&1\\\
\end{array}\right)$ is a $3$-algebra which cannot be expressed by any algebras
in the above theorem and it is not isomorphic to any $B_{i}$ where
$i=1,...,11.$

  * •

On the other hand, $A_{4}(1/3,-1/3)$ and $A_{5}(1/3)$ are not isomorphic
algebras, but from these two non-isomorphic algebras, we get one $3$-algebra
$\left(\begin{array}[]{cccccccc}\frac{1}{9}&0&0&0&0&0&0&0\\\
0&\frac{1}{9}&-\frac{2}{9}&0&\frac{2}{9}&0&0&0\\\ \end{array}\right).$

  * •

Also $A_{4}(1,-1)$ and $A_{4}(1,1)$ are not isomorphic algebras but from these
two algebras we get one $3$-algebra
$\left(\begin{array}[]{cccccccc}1&0&0&0&0&0&0&0\\\ 0&1&0&0&0&0&0&0\\\
\end{array}\right).$

##  5\. Totally associative $3$-algebras

In this section, we introduce the following definition of totally associative
$3$-algebra using its MSC.

######  Definition 5.1.

A $3$-algebra $\mathbb{A}$ is a totally associative $3$-algebra if

(5.1)
$(\mathbf{u}\cdot\mathbf{v}\cdot\mathbf{w})\cdot\mathbf{x}\cdot\mathbf{y}=\mathbf{u}\cdot(\mathbf{v}\cdot\mathbf{w}\cdot\mathbf{x})\cdot\mathbf{y}=\mathbf{u}\cdot\mathbf{v}\cdot(\mathbf{w}\cdot\mathbf{x}\cdot\mathbf{y})$

for all $\mathbf{u},\mathbf{v},\mathbf{w},\mathbf{x},\mathbf{y}\in\mathbb{A}.$

######  Example 5.2.

Let $\\{e_{1},e_{2}\\}$ be a basis of a $2$-dimensional $3$-algebra
$\mathbb{A},$ the multilinear map ”$\cdot$” given by:

$e_{1}\cdot e_{1}\cdot e_{1}=e_{1},\ \ e_{1}\cdot e_{1}\cdot e_{2}=e_{2}$

defines a totally associative $3$-algebra.

According to (5.1) and (3.1), we can reformulate Definition 5.1 as follows.

######  Definition 5.3.

A $2$-dimensional $3$-algebra $\mathbb{A}$ with multiplication $\cdot$ over a
field $\mathbb{F}$ is said to be a totally associative $3$-algebra if all of
the following conditions are met:

(5.2a) $\displaystyle A\left(A\otimes I\otimes I-I\otimes A\otimes
I\right)=0,$ (5.2b) $\displaystyle A\left(A\otimes I\otimes I-I\otimes
I\otimes A\right)=0,$ (5.2c) $\displaystyle A\left(I\otimes A\otimes
I-I\otimes I\otimes A\right)=0,$

where $I$ is the identity $2\times 2$ matrix.

Using a computation program (here, we use Mathematica), it is easy to verify
that the $3$-algebras from the list in Table (1) satisfying the system (5.2)
are:

  1. (i)

$B_{2}(\alpha_{1},\beta_{1},\beta_{2})$ when

     * •

$\alpha_{1}=0,\ \beta_{1}=0,\ \beta_{2}=0,$

     * •

$\alpha_{1}=\frac{1}{2},\ \beta_{1}=0,\ \beta_{2}=-\frac{1}{2},$

     * •

$\alpha_{1}=\frac{1}{2},\ \beta_{1}=0,\ \beta_{2}=\frac{1}{2}.$

  2. (ii)

$B_{4}(\alpha_{1},\beta_{2})$ when

     * •

$\alpha_{1}=0,\ \beta_{2}=0,$

     * •

$\alpha_{1}=\frac{1}{2},\ \beta_{2}=-\frac{1}{2},$

     * •

$\alpha_{1}=\frac{1}{2},\ \beta_{2}=0,$

     * •

$\alpha_{1}=\frac{1}{2},\ \beta_{2}=\frac{1}{2},$

     * •

$\alpha_{1}=1,\ \beta_{2}=-1,$

     * •

$\alpha_{1}=1,\ \beta_{2}=0,$

     * •

$\alpha_{1}=1,\ \beta_{2}=1.$

That means from the above list we get the following totally associative
$3$-algebras:

  1. (i)

$B_{2}(0,0,0)=\left(\begin{array}[]{cccccccc}0&0&0&0&0&0&1&0\\\
0&0&0&0&0&0&0&1\\\ \end{array}\right),$

  2. (ii)

$B_{2}(\frac{1}{2},0,-\frac{1}{2})=\left(\begin{array}[]{cccccccc}\frac{1}{4}&0&0&\frac{1}{2}&0&-\frac{1}{2}&\frac{1}{2}&0\\\
0&\frac{1}{4}&\frac{1}{4}&0&\frac{1}{4}&0&0&\frac{1}{2}\\\
\end{array}\right),$

  3. (iii)

$B_{2}(\frac{1}{2},0,\frac{1}{2})=\left(\begin{array}[]{cccccccc}\frac{1}{4}&0&0&\frac{1}{2}&0&\frac{1}{2}&\frac{1}{2}&0\\\
0&\frac{1}{4}&\frac{1}{4}&0&\frac{1}{4}&0&0&\frac{1}{2}\\\ \end{array}\right)$

  4. (iv)

$B_{4}(\frac{1}{2},-\frac{1}{2})=\left(\begin{array}[]{cccccccc}\frac{1}{4}&0&0&0&0&0&0&0\\\
0&\frac{1}{4}&-\frac{1}{4}&0&\frac{1}{4}&0&0&0\\\ \end{array}\right),$

  5. (v)

$B_{4}(\frac{1}{2},0)=\left(\begin{array}[]{cccccccc}\frac{1}{4}&0&0&0&0&0&0&0\\\
0&0&0&0&\frac{1}{4}&0&0&0\\\ \end{array}\right),$

  6. (vi)

$B_{4}(\frac{1}{2},\frac{1}{2})=\left(\begin{array}[]{cccccccc}\frac{1}{4}&0&0&0&0&0&0&0\\\
0&\frac{1}{4}&\frac{1}{4}&0&\frac{1}{4}&0&0&0\\\ \end{array}\right),$

  7. (vii)

$B_{4}(1,-1)=B_{4}(1,1)=\left(\begin{array}[]{cccccccc}1&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0\\\ \end{array}\right),$

  8. (viii)

$B_{4}(1,0)=\left(\begin{array}[]{cccccccc}1&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ \end{array}\right).$

According to ([5]), over an algebraically closed field, $\mathbb{F}$ of
characteristic, not $2,\ 3$ every nontrivial $2$-dimensional associative
algebra is isomorphic to only one algebra listed below by

  1. (i)

$A_{2}\left(\displaystyle\frac{1}{2},0,\displaystyle\frac{1}{2}\right)=\left(\begin{array}[]{cccc}\displaystyle\frac{1}{2}&0&0&1\\\
0&\displaystyle\frac{1}{2}&\displaystyle\frac{1}{2}&0\end{array}\right),$

  2. (ii)

$A_{4}\left(1,0\right)=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&0&0&0\end{array}\right),$

  3. (iii)

$A_{4}\left(\displaystyle\frac{1}{2},\displaystyle\frac{1}{2}\right)=\left(\begin{array}[]{cccc}\displaystyle\frac{1}{2}&0&0&0\\\
0&\displaystyle\frac{1}{2}&\displaystyle\frac{1}{2}&0\end{array}\right),$

  4. (iv)

$A_{4}\left(1,1\right)=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&1&0&0\end{array}\right),$

  5. (v)

$A_{4}\left(\displaystyle\frac{1}{2},0\right)=\left(\begin{array}[]{cccc}\displaystyle\frac{1}{2}&0&0&0\\\
0&0&\displaystyle\frac{1}{2}&0\end{array}\right),$

  6. (vi)

$A_{12}=\left(\begin{array}[]{cccc}0&0&0&0\\\ 1&0&0&0\end{array}\right).$

We conclude that there are totally associative $3$-algebras, $B_{2}(0,0,0),$
$B_{2}(\frac{1}{2},0,-\frac{1}{2}),$ $B_{4}(\frac{1}{2},-\frac{1}{2}),$
$B_{4}(1,-1),$ are generated by non-associative algebras, $A_{2}(0,0,0),$
$A_{2}(\frac{1}{2},0,-\frac{1}{2}),$ $A_{4}(\frac{1}{2},-\frac{1}{2}),$
$A_{4}(1,-1),$ respectively.

Conclusion

Depending on the approach introduced in [8] and applied in [1], one can study
the classification of $n$-algebras and then study some identities of
$n$-algebras (refer to [2, 3, 4, 5, 6, 7]).

Acknowledgement

This research was funded by Grant 01-01-18-2032FR Kementerian Pendidikan
Malaysia.

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