This research was supported by the National Institutes of Health under Award
Number 4R37HD058305-27. Correspondence concerning this article should be
addressed to Dandan Tang, Department of Psychology, University of Virginia.
Email<EMAIL_ADDRESS>
# Are the Signs of Factor Loadings Arbitrary in Confirmatory Factor Analysis?
Problems and Solutions

Dandan Tang  Steven M. Boker  Xin Tong  Department of Psychology, University
of Virginia

###### Abstract

The replication crisis in social and behavioral sciences has raised concerns
about the reliability and validity of empirical studies. While research in the
literature has explored contributing factors to this crisis, the issues
related to analytical tools have received less attention. This study focuses
on a widely used analytical tool - confirmatory factor analysis (CFA) - and
investigates one issue that is typically overlooked in practice: accurately
estimating factor-loading signs. Incorrect loading signs can distort the
relationship between observed variables and latent factors, leading to
unreliable or invalid results in subsequent analyses. Our study aims to
investigate and address the estimation problem of factor-loading signs in CFA
models. Based on an empirical demonstration and Monte Carlo simulation
studies, we found current methods have drawbacks in estimating loading signs.
To address this problem, three solutions are proposed and proven to work
effectively. The applications of these solutions are discussed and elaborated.

###### keywords:

replication crisis, factor loading signs, confirmatory factor analysis, fixing
methods

##  1 Introduction

The replication crisis in psychology has aroused great attention to the lack
of reproducibility in empirical studies across social and behavioral sciences,
as well as in other scientific disciplines. This raises concerns about the
reliability and validity of empirical research within these fields (Cockburn
et al., 2020; Oberauer & Lewandowsky, 2019). Traditionally, replication, the
practice of repeating a study to confirm its finding, is the bedrock of
scientific validation. Recently, however, more and more previously accepted
findings have proven difficult or impossible to replicate (Camerer et al.,
2018; Open Science Collaboration, 2015; Youyou et al., 2023). Open Science
Collaboration (2015) found a replication rate of 36% among 97 experiments from
papers published in 2008 in three high-ranking psychology journals. Nosek et
al. (2022) found only 64% of 307 experiments replicated. Even some famous
studies, like the social psychological study ”elderly-walking” conducted by
social psychologist John Bargh and colleagues, and its relevant studies failed
to replicate, casting doubts on the theory of the goal priming effect (Bargh,
Chen, & Burrows, 1996; Harris et al., 2013; Muthukrishna & Henrich, 2019;
Pashler et al., 2012).

Many studies have investigated the problem of replication, identifying factors
that may contribute to it, including p-hacking or Cherry-Picking results,
publication bias, low statistical power, questionable research practices,
inadequate research training, failure to share data and methods, pressure to
publish, complexity of scientific research, etc. For instance, $p$-hacking,
manipulating data of statistical analyses to achieve a significant result
($p<.05$; Crane, 2018), may lead to non-replicable research outcomes. A
researcher may analyze data in various ways but only report a significant
result or stop collecting data as soon as a significant result is found. Klein
et al. (2018) discovered that among 28 classic and contemporary published
findings, 54% of the replications had statistically significant results with a
significance level of p ¡ .05. Publication bias means that journals tend to
publish papers with significant/positive results rather than non-
significant/negative results (Wagner III, 2022). This leads researchers to
pursue significant outcomes, sometimes regardless of rigorous methodology.
Franco, Malhotra, and Simonovits (2014) found only 20% of the studies
published in social sciences journals reported non-significant results and 60%
of the studies discovered non-significant results but never reported them. In
practice, although a study may yield significant findings, the associated
statistical power could be low (Anderson & Maxwell, 2017), meaning there is a
high probability of failing to reject a false null hypothesis in the future. A
meta-analysis of 44 reviews of statistical power observed a mean statistical
power of 0.24 to detect a small effect size (d = .20) with a type I error rate
of $\alpha=.05$ (Smaldino & McElreath 2016).

Although many factors leading to the replication crisis have been
investigated, few studies have considered the issues with analytical tools
(Van Lissa et al., 2021). Analytical tools, comprising statistical methods and
software, are the backbone of empirical research, allowing scientists to
discover patterns, make inferences, and establish relationships among
variables based on the collected data (Ali & Bhaskar, 2016; Wen et al., 2018).
Reliable and consistent use of analytical tools ensures robust research
outcomes. If a study uses flawed or inappropriate statistical methods, its
findings could be artifacts of the analysis rather than true reflections of
underlying phenomena (Tang & Wen 2020; Wen et al., 2019).

In this study, we focus on one of the most popular analytical tools in social
and behavioral sciences, confirmatory factor analysis (CFA), which has been
implemented in every structural equation modeling (SEM) program, such as
Mplus, lavaan, and OpenMx (Asparouhov & Muthen, 2007; Boker et al., 2021;
Rosseel, 2012). This technique is particularly important in the development
and validation of measurement instruments, such as psychological tests and
surveys because it can test theoretical expectations about the relationships
between variables and validate the structure of a test or survey (Lin et al.,
2020). Based on CFA models, we can test measurement invariance to check
whether the measurement of the psychological construct is invariant across
groups or times (Vandenberg & Lance, 2000). CFA is broadly applied in
practice, as reflected in the growing number of studies in databases. A search
for ”confirmatory factor analysis” in the PsychINFO database yields 456
articles from 1961-1990, 8820 articles from 1991-2010, and 23,652 articles
from 2011 to now.

Despite CFA’s popularity and well-established methodological foundations,
there are still potential problems with this analytical tool. When testing the
measurement invariance using CFA, one important step is to check factor
loading invariance, which requires that factor loadings are the same across
groups or time. For instance, it is discussed in the literature that the
Children’s Depression Inventory (CDI) is valid to have a one-factor structure
with CFA (Stumper et al., 2019) among adolescents, and the factor loadings of
the one-factor structure are invariant between 13-year-old and 16-year-old
adolescents. However, when we replicated the study to test whether the
structure of CDI is invariant in the same data but without missingness, the
factor loadings in the 13-year-old group were all negative, but in the
16-year-old group, they were all positive. The absolute values of the
corresponding loadings between the two groups were very close. However,
according to the definition, we were not able to conclude that these loadings
are invariant between the two groups. The natural question is, do factor
loading signs matter or not? This question serves as the motivation for our
study.

In CFA, factor loadings represent the strength and direction of the
relationship between observed variables and latent factors. A positive loading
indicates that the observed score increases as the latent factor score
increases. A negative loading indicates that the observed variable score
decreases as the latent factor score increases. Thus, an incorrect sign can
misinterpret the relationship between an observed variable and the latent
factor. If the loading sign in a CFA model is inconsistent or ”incorrect”
(e.g., not in line with theoretical expectations or prior findings),
subsequent analyses based on this model can produce unreliable or invalid
results. Furthermore, suppose other researchers attempt to replicate the study
and find different signs of the factor loadings or can’t reproduce the
results, they may not trust the original findings.

This article aims to explore and address the problem of factor loading signs
in CFA. The outline of this article is as follows. We begin with a real-data
example to demonstrate the factor loading sign problem in CFA models. Next, we
will briefly introduce the estimation method for CFA models, investigate the
reasons for the sign reversal problem, and propose solutions. Thereafter, two
simulations will be conducted to evaluate the impact of the problem and the
effectiveness of these solutions. Then, the empirical example will be re-
analyzed to illustrate these solutions in practice. Last, the article
discusses the prevalence of the factor loading sign problem and provides
practical recommendations for researchers.

##  2 An Empirical Example

In this section, we provide a detailed illustration of the estimation problem
in a CFA model using a real data example from the study conducted by Stumper
et al. (2019). The factor structure of the CDI was investigated using a sample
of 227 adolescents aged approximately 13 at baseline (T1) and 16 at follow-up
(T2). The CDI has 27 items, scaling 0-2. Stumper et al. (2019) used a weighted
least square mean and variance (WLSMV) estimation method in Mplus (Muthén &
Muthén, 2017) to analyze the data because the observed variables were binary.
To replicate the analysis, we first analyzed the full data, including missing
values, using diagonally weighted least squares (DWLS; Rosseel, 2012) in the
lavaan package in R (see the codes in our GitHub ), which in Mplus is given by
the WLSMV estimators (Asparouhov & Muthen, 2007). As shown in Table 1, the
loadings from the one-factor model at both T1 and T2 were positive. However,
after removing 12 subjects who did not finish the CDI at age 16, the loadings
at T1 became negative, but at T2, they were still positive.

When testing measurement invariance, we could not conclude that these loadings
are invariant between T1 and T2 because, for the data containing missing
values, the loadings are in the same direction, whereas for the data without
missingness, they have different signs. Furthermore, the data with missingness
and without missingness at T1 are from the same population, but the signs of
the loadings are opposite. This indicates that results from a later analysis
may not confirm the previous analysis. Researchers may even suspect that
removing the 12 subjects with missing values changed the sample distribution.

Table 1: Factor loadings for a one-factor model of the CDI at Time 1 and Time
2

Item | With missingness | Without missingness
---|---|---
T1 | T2 | T1 | T2
CDI1 | 0.772 | 0.746 | -0.773 | 0.742
CDI2 | 0.649 | 0.649 | -0.639 | 0.660
CDI3 | 0.618 | 0.659 | -0.614 | 0.655
CDI4 | 0.534 | 0.716 | -0.525 | 0.741
CDI5 | 0.552 | 0.692 | -0.567 | 0.688
CDI6 | 0.477 | 0.544 | -0.464 | 0.538
CDI7 | 0.747 | 0.829 | -0.747 | 0.826
CDI8 | 0.604 | 0.740 | -0.637 | 0.761
CDI9 | 0.675 | 0.741 | -0.690 | 0.738
CDI10 | 0.822 | 0.770 | -0.823 | 0.766
CDI11 | 0.769 | 0.708 | -0.765 | 0.703
CDI12 | 0.571 | 0.551 | -0.577 | 0.546
CDI13 | 0.503 | 0.626 | -0.532 | 0.618
CDI14 | 0.536 | 0.700 | -0.541 | 0.696
CDI15 | 0.451 | 0.576 | -0.457 | 0.580
CDI16 | 0.650 | 0.638 | -0.644 | 0.631
CDI17 | 0.692 | 0.578 | -0.694 | 0.569
CDI18 | 0.416 | 0.554 | -0.404 | 0.559
CDI19 | 0.400 | 0.462 | -0.399 | 0.455
CDI20 | 0.819 | 0.827 | -0.814 | 0.824
⋮ | ⋮ | ⋮ | ⋮ | ⋮
CDI27 | 0.673 | 0.714 | -0.691 | 0.711

  * Note: The table only lists part of the results because of limited space. For complete results, please see our GitHub.

##  3 Estimation Method

We will first introduce the estimation method for CFA models to understand the
problem of the factor loading signs. If the observed variables $\mathbf{x}$
are continuous, a CFA model can be expressed as:

$\mathbf{x}=\mathbf{\tau}+\Lambda\mathbf{\eta}+\mathbf{\epsilon},$

where $\Lambda$ is a vector of factor loadings, $\mathbf{\eta}$ is a vector of
latent factors, $\mathbf{\tau}$ is a vector of latent intercepts, and
$\mathbf{\epsilon}$ is a vector of measurement errors. It is assumed that the
measurement error is normally distributed with mean $\mathbf{0}$ and variance
$\mathbf{\Theta}$, namely,
$\mathbf{\epsilon}\sim\mathcal{N}(\mathbf{0},\mathbf{\Theta})$.

If the observed variables $\mathbf{x}$ are categorical and have $C$
categories, the CFA model can be expressed as:

$\mathbf{x^{\ast}}=\mathbf{\tau}+\Lambda\mathbf{\eta}+\mathbf{\epsilon},$

where $\mathbf{x^{\ast}}$ is an underlying continuous variable that is related
to $\mathbf{x}$ though a set of $C+1$ thresholds,
$\mathbf{v}=(v_{0},v_{1},...,v_{C+1})$, and $v_{0}=-\infty$ and
$v_{C+1}=\infty$. The probability of $\mathbf{x}=c$ is given as

$p(\mathbf{x}=c)=p(v_{c}\leq x^{\ast}\leq v_{c+1}),$

where $c=0,1,...,C$. The covariance structure of the CFA model is

$\mathbf{\Sigma=\Lambda\Phi\Lambda^{\prime}+\Theta},$ (1)

where $\mathbf{\Sigma}$ is the covariance matrix implied by the CFA model, and
$\Phi$ is the covariance of the latent factors (Liu et al., 2022a).

Maximum Likelihood Estimation (MLE) is often applied to estimate model
parameters when the observed variables are continuous and normally
distributed. MLE aims to find the parameter values that maximize the
likelihood function, meaning the parameter values maximize the probability of
observing the current sample data (Li, 2016; Tang & Tong, 2023). In the CFA
model, the likelihood function is given as

$\mathbf{F}=ln|\mathbf{\Sigma}|-ln|\mathbf{S}|+tr(\mathbf{S}\mathbf{\Sigma}^{-1})-p,$

where $p$ is the number of the observed variables, and $\mathbf{S}$ is the
covariance matrix of the observed variables if the observed variables
$\mathbf{x}$ are continuous (Liu et al., 2022b), or is the covariance matrix
of $\mathbf{x^{\ast}}$ if the observed variables $\mathbf{x}$ are categorical.

When the observed variables are categorical, weighted least square (WLS)
estimation is often applied to estimate model parameters. WLS aims to find the
parameter values that minimize the fit function, meaning the parameter values
minimize the difference between the observed data and the theoretical model
based on a weight matrix (Asparouhov & Muthen, 2007; Li, 2016). In the CFA
model, the fit function can be expressed by

$\mathbf{F_{wls}}=(s-\sigma(\theta)^{\prime})\mathbf{W^{-1}}(s-\sigma(\theta)),$

where $\theta$ is the vector of model parameters, $\mathbf{W}$ is the weight
matrix, $\sigma(\theta)$ is the model-implied vector containing the
nonredundant, vectorized elements of $\mathbf{\Sigma}$, and $s$ is the vector
containing the unique elements of sample statistics (i.e., threshold and
polychoric correlation estimates; Li, 2016). If $\mathbf{W}$ is a diagonal
matrix, where off-diagonal entries are 0, and diagonal entries remain the
same, WLS will become WLSMV (Asparouhov & Muthen, 2007).

###  3.1 Problems and Solutions

Because the latent factor itself does not have a natural scale, Equation (1)
is not identified. To identify it, two approaches are often used in popular
software for CFA, such as lavaan, Mplus, and OpenMx (Asparouhov & Muthen,
2007; Boker et al., 2021; Rosseel, 2012). One approach is to fix the factor
variance $\sigma^{2}_{\eta}$, a diagonal element of the covariance matrix
$\Phi$, to 1 and the factor means $\mu_{\eta}$ to 0, called fixed factor
variance. Another approach is to fix one of the factor loadings on each factor
to 1, called fixed factor loading.

From Equation (1), we obtain

$cov(x_{i},x_{j})\leavevmode\nobreak\ \mbox{or}\leavevmode\nobreak\
cov(x_{i}^{\ast},x_{j}^{\ast})=\lambda_{i}\lambda_{j}\sigma^{2}_{\eta}+\sigma^{2}_{ij},$
(2)

where $i=1,...,p$ and $j=1,...,p$. Without loss of generality, we assume that
when $i=j$, $\sigma^{2}_{ij}=\sigma^{2}_{i}$; otherwise, $\sigma^{2}_{ij}=0$.
When using the first fixing method to identify the model, Equation (2) can be
written as

$cov(x_{i},x_{j})\leavevmode\nobreak\ \mbox{or}\leavevmode\nobreak\
cov(x_{i}^{\ast},x_{j}^{\ast})=\lambda_{i}\lambda_{j}+\sigma^{2}_{ij}.$ (3)

If the covariance $cov(x_{i},x_{j})\leavevmode\nobreak\
\mbox{or}\leavevmode\nobreak\ cov(x_{i}^{\ast},x_{j}^{\ast})$ is positive,
Equation (3) indicates that $\lambda_{i}$ and $\lambda_{j}$ should be both
positive or negative. Whether positive or negative, the loadings are correctly
estimated, but the direction of the estimated loadings could be opposite. This
explains why the loadings at T1 from the data without missing values are all
negative in the empirical example. However, when factor loading signs are
opposite, the interpretation of the factor also has opposite meanings.

When using the second fixing method to identify the model, Equation (2) can be
written as

$cov(x_{1},x_{j})\leavevmode\nobreak\ \mbox{or}\leavevmode\nobreak\
cov(x_{1}^{\ast},x_{j}^{\ast})=\lambda_{j}\sigma^{2}_{\eta}+\sigma^{2}_{1j}.$
(4)

Equation (4) indicates that $\lambda_{j}$ should keep the same sign as the
covariance $cov(x_{1},x_{j})\leavevmode\nobreak\ \mbox{or}\leavevmode\nobreak\
cov(x_{1}^{\ast},x_{j}^{\ast})$. However, if the true value of the first
loading is negative but fixed to 1, this will also lead to a direction problem
in the estimation. To solve this estimation problem, we propose to fix a
positive loading to 1, which can help other loadings to be of the expected
sign.

To fit a CFA model using MLE or WLS, an optimization algorithm iteratively
adjusts the parameter estimates to find the values that maximize the
likelihood or minimize the fit function (Kochenderfer & Wheeler, 2019). As
shown by the symmetric curve in Figure 1, the optimal solution can fall in the
negative or positive range of the x-axis. To solve this estimation problem, we
propose to set the lower or upper bounds of the loadings to be larger
(smaller) than 0 when the true values of the loadings are expected to be
larger (smaller) than 0.

Figure 1:  The sample graph of the likelihood and fit function 1

Furthermore, the optimization algorithm needs a starting value for each
parameter. The starting value is an initial guess for the parameter, and then
the estimation procedure iteratively refines this starting guess to converge
toward the parameter’s most optimal value (Kochenderfer & Wheeler, 2019).
Sometimes, algorithms can get stuck in ”local” solutions, which are specific
to the region around the starting values, rather than finding the ”global”
solution, which is the best solution overall (Arora et al., 1995). Figure 2
shows parameter estimates can be negative when algorithms get stuck in ”local”
solutions at the negative x-axis. To solve this estimation problem, we propose
adjusting starting values to obtain factor loading estimates with expected
signs. In addition, a large positive or small negative starting value may help
obtain positive or negative estimates.

Figure 2:  The sample graph of the likelihood and fit function 2 Figure 3:
Path diagram of a confirmatory factor analysis model.

We conducted two Monte Carlo simulations to understand the estimation problem
and proposed solutions comprehensively. The first simulation study will assess
how different estimation methods and computational algorithms affect the signs
and consistency of factor loadings across the above CFA software under various
situations. The second simulation study will further assess our solutions to
the estimation problem.

##  4 Simulation 1: Investigating the Estimation Problem

###  4.1 Simulation Design

We conducted a Monte Carlo simulation study to systematically investigate the
estimation problems with two different methods in CFA for model
identification. Regardless of continuous or categorical observed variables,
the CFA model has the same estimation problem. Thus, the simulation focuses on
continuous observed variables using MLE.

A one-factor CFA model with three continuous indicators was specified as the
population model (see Figure 3) from which the data were generated. In the
population model, the absolute values of all the loadings
$|\lambda_{k}|(k=1,2,3)$ were 0.7, the variance of the indicators
$\sigma^{2}_{k}$ was 1, the variance of the latent factor $\sigma^{2}_{\eta}$
was 1, and the intercepts were set at 0.

In the simulation, we varied the loadings in terms of their signs, creating
four distinct conditions:
Condition 1: All loadings are positive.
Condition 2: All loadings are negative.
Condition 3: One of the three loadings is negative.
Condition 4: Two of the three loadings are negative.

The sample size was set at 200 for the simulation study. For each condition,
500 datasets were generated. We fitted the population model to each dataset in
three popular CFA software packages, lavaan 0.6-16, Mplus 8.10, and OpenMx
2.21.8. These analyses used various fixing techniques. In the fixing factor
variance approach, we used both the default starting values for factor
loadings provided by the software packages and also manually set these
starting values. Therefore, to thoroughly assess the impact of starting values
on the direction of factor loadings, we used the default starting value, a
positive starting value of 1, and a negative starting value of -1. For the
fixed factor loading approach, the common default in most software is to fix
the first-factor loading at 1. As a result, the four methods used in this
simulation are:
Method 1: Fixing factor variance with default starting values of the factor
loadings.
Method 2: Fixing factor variance with starting values of the factor loadings
set to 1.
Method 3: Fixing factor variance with starting values of the factor loadings
set to -1.
Method 4: Fixing the first loading at 1.

###  4.2 Results

To evaluate the estimation problem with the four fixed methods, we calculated
the directional consistency rate (DCR) to measure the possibility that the
direction of the loading estimates is the same as the direction of the true
values among 500 replicates. DCR can be expressed by $DCR=\frac{M}{N}*100\%$,
where $M$ is the number of factor loading estimates with the same directional
sign as the true values, and $N$ is the total number of factor loading
estimates across all replicates. Table 2 presents the directional consistency
rates for four fixing methods under various conditions in lavaan. When using
Method 4, where the first-factor loading is fixed at 1, DCR was not computed
for this loading as it was fixed and cannot be freely estimated. Under various
conditions, the DCR for each loading was either 100% or 0%. A DCR of 100%
implies that across the replicates, the estimated factor loadings have the
same directional sign (either all positive or all negative) as the true factor
loadings, which is complete consistency. Conversely, a DCR of 0% indicates
that each estimated loading is directionally opposite to its true value, which
is a complete lack of consistency.

Method 1 could lead to loading estimates with inconsistent signs. For Methods
2 and 3, if the sign of the starting values was not consistent with true
factor loadings, these loading estimates could have incorrect signs. For
Method 4, the same directional inconsistency could arise when the first
loading was fixed to 1, but its population value was negative. This implies
that fixing a negative loading to 1 could mistakenly estimate positive
loadings as negative or vice versa. Thus, the simulation findings underscore
the potential drawbacks of these four fixing methods, particularly the risk of
obtaining estimated loadings with unexpected signs.

We further validated our findings by running the simulation using Mplus 8.10
and OpenMx 2.21.8. Tables 3 and 4 display results largely consistent with
those obtained using lavaan. Although some difference is observed between
Method 1 and 2, the overall results from Mplus and OpenMx further validate our
findings that these four fixing methods may lead to loading estimates with
inconsistent signs. Additionally, despite the unknown default starting values
in these three software packages, Method 1 still can not guarantee the loading
estimates in the expected directions.

Table 2: Directional consistency rates (%) for simulation 1 with four fixing
methods (lavaan)

Fixing method | Condition 1 | Condition 2 | Condition 3 | Condition 4
---|---|---|---|---
loading | 0.7 | 0.7 | 0.7 | -0.7 | -0.7 | -0.7 | -0.7 | 0.7 | 0.7 | -0.7 | -0.7 | 0.7
Method 1 | 100 | 100 | 100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Method 2 | 100 | 100 | 100 | 0 | 0 | 0 | 100 | 100 | 0 | 0 | 0 | 0
Method 3 | 0 | 0 | 0 | 100 | 100 | 100 | 0 | 0 | 0 | 100 | 100 | 100
Method 4 |  | 100 | 100 |  | 0 | 0 |  | 0 | 0 |  | 0 | 0

Table 3: Directional consistency rates (%) for simulation 1 with four fixing
methods (Mplus)

Fixed methods | Condition 1 | Condition 2 | Condition 3 | Condition 4
---|---|---|---|---
loading | 0.7 | 0.7 | 0.7 | -0.7 | -0.7 | -0.7 | -0.7 | 0.7 | 0.7 | -0.7 | -0.7 | 0.7
Method 1 | 100 | 100 | 100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Method 2 | 100 | 100 | 100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Method 3 | 0 | 0 | 0 | 100 | 100 | 100 | 0 | 0 | 0 | 100 | 100 | 100
Method 4 |  | 100 | 100 |  | 0 | 0 |  | 0 | 0 |  | 0 | 100

Table 4: Directional consistency rates (%) for simulation 1 with four fixing methods (OpenMx) Fixed methods | Condition 1 | Condition 2 | Condition 3 | Condition 4
---|---|---|---|---
loading | 0.7 | 0.7 | 0.7 | -0.7 | -0.7 | -0.7 | -0.7 | 0.7 | 0.7 | -0.7 | -0.7 | 0.7
Method 1 | 100 | 100 | 100 | 0 | 0 | 0 | 100 | 100 | 100 | 0 | 0 | 0
Method 2 | 100 | 100 | 100 | 0 | 0 | 0 | 100 | 100 | 0 | 0 | 0 | 0
Method 3 | 0 | 0 | 0 | 100 | 100 | 100 | 0 | 0 | 0 | 100 | 100 | 100
Method 4 |  | 100 | 100 |  | 0 | 0 |  | 0 | 0 |  | 0 | 0

##  5 Simulation 2: Solutions to the Estimation Problem

###  5.1 Simulation Design

To address the issue of directional inconsistency in CFA factor loading
estimates, we will propose two practical solutions and one potential solution
that can yield estimates in a more interpretable and consistent direction.
Furthermore, a Monte Carlo simulation was conducted to assess the
effectiveness of these three solutions to the estimation problems for CFA.
This simulation used the same population model as Simulation 1, with a sample
size of 200 and 500 datasets generated for each condition.

####  5.1.1 Solution 1: fixing a positive factor loading at 1

Our previous simulation findings, particularly with Method 4, indicated that
fixing a negative loading at 1 led to inconsistencies in the directions of
other loadings. Conversely, fixing a positive loading at 1 aligned other
loadings in the expected directions. Therefore, we used this solution in
lavaan 0.6-16, Mplus 8.10, and OpenMx 2.21.8 for the population model under
Conditions 1, 3, and 4, where not all loadings are negative.

####  5.1.2 Solution 2: setting bounds for factor loadings

This solution is to fix the factor variance and set a lower bound for the
inherently positive loadings and an upper bound for the inherently negative
loadings. For instance, specifying 0 as the lower and upper bounds for
positive and negative loadings, respectively. This solution is uniquely
implementable in the OpenMx R package, which allows setting the same bound for
all loading and a separate bound for each loading. Consequently, we used this
solution in OpenMx 2.21.8, using 0 as the bound under Conditions 1 and 2,
where each condition has the factor loadings with uniform signs. Specifically,
we set the lower bound of 0 for all loadings under Condition 1 and an upper
bound of 0 under Condition 2.

####  5.1.3 Solution 3: adjusting starting values of factor loadings

The potential solution is to fix the factor variance and set the starting
values of the factor loadings in the same direction as the true loadings. As
suggested by our simulation results for Methods 2 and 3, setting a start value
consistent with the direction of true loadings could steer the estimates
toward the desired direction. Therefore, we used this solution in lavaan
0.6-16, Mplus 8.10, and OpenMx 2.21.8 for each dataset under Conditions 1 to
4. Specifically, we set a large positive start value of 1 for a true positive
loading or a small negative start value of -1 for a true negative loading.

###  5.2 Results

The DCR was again computed to evaluate the effectiveness of three Solutions.
Tables 5 to 7 present the DCR results for three solutions under various
conditions. Solution 1, which fixes a positive factor loading at 1,
successfully aligned other loading estimates in their correct direction. In
Solution 2, fixing factor variance with a lower bound of 0 or an upper bound
of 0 for all loadings effectively ensured that all loadings were consistently
positive or negative. Solution 3, fixing the factor variance and setting the
starting values to 1 or -1, proved effective in shifting the loadings toward
the desired direction. From these findings, we concluded that Solutions 1 and
2 can solve the issue of directional inconsistency in factor loading, and
Solution 3 can obtain the preferred direction of factor loadings.

Table 5: Directional consistency rates (%) for Solution 1

Fixing method | Condition 1 | Condition 3 | Condition 4
---|---|---|---
loading | 0.7 | 0.7 | 0.7 | -0.7 | 0.7 | 0.7 | -0.7 | -0.7 | 0.7
Solution 1 (lavaan) | 100 | 100 |  | 100 | 100 |  | 100 | 100 |
Solution 1 (Mplus) | 100 | 100 |  | 100 | 100 |  | 100 | 100 |
Solution 1 (OpenMx) | 100 | 100 |  | 100 | 100 |  | 100 | 100 |

  * Note: Under each condition, the last factor loading was fixed at 1.

Table 6: Directional consistency rates (%) for Solution 2

Fixing method | Condition 1 | Condition 2
---|---|---
loading | 0.7 | 0.7 | 0.7 | -0.7 | -0.7 | -0.7
Solution 2 (OpenMX) | 100 | 100 | 100 | 100 | 100 | 100

Table 7: Directional consistency rates (%) for Solution 3 Fixing method | Condition 1 | Condition 2 | Condition 3 | Condition 4
---|---|---|---|---
loading | 0.7 | 0.7 | 0.7 | -0.7 | -0.7 | -0.7 | -0.7 | 0.7 | 0.7 | -0.7 | -0.7 | 0.7
Solution 3 (lavaan) | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100
Solution 3 (Mplus) | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100
Solution 3 (OpenMX) | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100

##  6 Reanalysis of the Empirical Example with the Solutions

In practice, we can use the three solutions to handle the issue of directional
inconsistency in factor loading estimates. However, not all proposed solutions
can be conducted in popular CFA software packages. Specifically, lavaan and
OpenMx support the first solution for both continuous and categorical
variables, while Mplus only supports continuous variables. The second solution
can only be applied in OpenMx. The third solution can be used in these three
software. However, OpenMx can not handle more than 20 categorical observed
variables. Detailed implementation guidelines and codes are available in the
Appendix 1.

Since there are more than 20 categorical observed variables in the empirical
example, the second solution can not be applied in this example. Only
Solutions 1 and 3 are used through lavaan and Mplus (see the codes in our
hrefhttps://github.com/DandanTang0/sign-of-factor-
loading/tree/main/exampleGitHub ). For Solution 1, the first-factor loading is
fixed at 1 because the first item, Mood (Questions related to feelings of
sadness or happiness) and the latent factor of Children’s Depression have a
positive relationship given substantive psychological theory. For Solution 3,
a positive start value of 1 is applied because all the items have a positive
relationship with the latent factor of Children’s Depression. The re-analysis
results are presented in Table 8. All loadings at T1 in the data without
missingness are positive. This implies that Solutions 1 and 3 work for this
example and the previous non-replicated results were due to how CFA was
implemented.

Table 8: Factor loadings for a one-factor model of the CDI at Time 1 in the
data without missingness

Item | lavaan | Mplus
---|---|---
| Solution 1 | Solution 3 | Solution 1 | Solution 3
CDI1 | 1.000 | 0.773 | 1.000 | 0.775
CDI2 | 0.827 | 0.639 | 0.820 | 0.635
CDI3 | 0.795 | 0.614 | 0.793 | 0.615
CDI4 | 0.679 | 0.525 | 0.675 | 0.523
CDI5 | 0.733 | 0.567 | 0.732 | 0.567
CDI6 | 0.600 | 0.464 | 0.614 | 0.476
CDI7 | 0.967 | 0.747 | 0.898 | 0.696
CDI8 | 0.824 | 0.637 | 0.822 | 0.637
CDI9 | 0.892 | 0.690 | 0.890 | 0.689
CDI10 | 1.064 | 0.823 | 1.062 | 0.823
CDI11 | 0.990 | 0.765 | 1.002 | 0.777
CDI12 | 0.747 | 0.577 | 0.717 | 0.555
CDI13 | 0.689 | 0.532 | 0.685 | 0.531
CDI14 | 0.701 | 0.541 | 0.694 | 0.538
CDI15 | 0.592 | 0.457 | 0.592 | 0.459
CDI16 | 0.833 | 0.644 | 0.832 | 0.645
CDI17 | 0.898 | 0.694 | 0.894 | 0.693
CDI18 | 0.522 | 0.404 | 0.531 | 0.412
CDI19 | 0.516 | 0.399 | 0.500 | 0.388
CDI20 | 1.053 | 0.814 | 1.050 | 0.813
⋮ | ⋮ | ⋮ | ⋮ | ⋮
CDI27 | 0.895 | 0.691 | 0.894 | 0.693

  * Note: The table only lists part of the results because of limited space. For complete results, please see our GitHub.

##  7 Prevalence of the Issue and Practical Suggestions

The issue of estimating factor loading direction in CFA models has often been
overlooked in practice. Yet, its significance cannot be overstated, especially
in comparative and longitudinal studies. Comparative research requires close
attention to factor loading signs, particularly in measurement invariance
tests. As the practical example shows, ignoring the sign of factor loadings
makes it difficult to determine whether the loadings are invariant across
groups or over time. Furthermore, in longitudinal research, which tracks
individual changes over time (Fitzmaurice et al., 2012), inconsistent loading
signs may distort true within-individual changes. Additionally, the issue
extends to cross-sectional studies where mismatched loading signs can make it
difficult to validate the structure of a scale (Lin et al., 2019).

In general, any study using CFA should consider the problem of factor loading
signs in case the results of this study can not be replicated due to loading
signs. Furthermore, this estimation problem is not confined to CFA models but
is also prevalent in other structural equation models, as they employ similar
estimation techniques. It is imperative to address the issue in all SEM models
to avoid a potential replication dilemma.

To address the problem of loading signs, we proposed three solutions to
reflect the true relationship between observed variables and latent factors.
Even if this article only discussed it in CFA models, the solutions can be
applied in all SEM models. The first solution, fixing a positive loading at 1,
can ensure that other loadings in the model are in the expected direction.
Using substantive theory, researchers can identify an observed variable with a
positive correlation with a latent factor and fix the loading on this variable
to 1. However, researchers must identify and fix one positive loading for each
latent factor. The second solution, fixing the factor variance and setting a
bound for loadings, can ensure that all loadings are in the correct direction.
However, to set the right bound for each loading, researchers must determine
all the relationships between observed variables and latent factors. The third
solution, fixing the factor variance and setting the starting values of
loadings at larger values with the same direction as the loadings, leads the
loadings to the desired direction but does not guarantee it. Like the second
solution, it requires a comprehensive understanding of the relationships
between observed variables and latent factors.

##  8 Author contributions

Dandan Tang: Formal Analysis, Investigation, Methodology, Visualization,
Writing-original draft; Steven M. Boker: Methodology, Supervision, Writing-
review & editing; Xin Tong: Supervision, Writing-review & editing.

##  9 Acknowledgements

The authors gratefully thank Dr. Caspar van Lissa for the valuable guidance on
the empirical example.

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###  10.1 Appendix 1

####  10.1.1 Continuous variables

Suppose a one-factor model with three continuous variables. This is the R code
for Solution 1, a positive loading at 1 using lavaan package. If the true
value of the first loading is positive, the first loading can be fixed at 1 by
”auto.fix.first = TRUE”

⬇

# load package

library(lavaan)

# read data

data <\-

read.csv(’data.csv’,header = T)

# CFA model for T1

CFA <- ’F=~x1+x2+x3’

# The true value of the first loading is

# positive, so the first loading is

# fixed at 1 by ”auto.fix.first = TRUE”

result <- cfa(CFA, data = data,

auto.fix.first = TRUE, ordered = FALSE)

# print result

summary(result)

If the first loading is not positive, we can move a positive loading to the
first place. for example, if the loading on item x2 is positive, we can set
the CFA model as follows:

⬇

CFA_reloaction <- ’F=~x2+x1+x3’

This is the R code for Solution 3 using lavaan package. The factor variance
can be fixed by ”std.lv = TRUE”, and the starting values of the loadings can
be set at 1 or -1 by ”start(1)*” or ”start(-1)*”.

⬇

# set start values of the

# loadings at 1 by ”start(1)*”

CFA_start <- ’F=~start(1)*x1+start(1)*x2

+start(1)*x3’

# fix the factor variance by ”std.lv = TRUE”

result <- cfa(CFA_start, data = data,

std.lv = TRUE, ordered = FALSE)

# print results

summary(result)

This is Mplus code for Solution 1, fixing a positive loading at 1. If the true
value of the first loading is positive, the first loading is fixed at 1 by
”@1”.

⬇

! read data

DATA: FILE = data.dat;

! name observed variables

VARIABLE: NAMES = x1-x3;

Model:

! fix the first loading at 1 by ”@1”

F by x1@1 x2 x3;

If the first loading is not positive, we can move a positive loading to the
first place. for example, if the loading on item x2 is positive, we can set
its loadings at 1 by ”@1”.

⬇

DATA: FILE = data.dat;

VARIABLE: NAMES = x1-x3;

Model:

! fix the second loading at 1 by ”@1”

F by x1 x2@1 x3;

This is Mplus code for Solution 3, fixing the factor variance by ”@1” and
setting the starting values of the loadings to 1 or -1 by ”*1” or ”*-1”.

⬇

DATA: FILE = data.dat;

VARIABLE: NAMES = x1-x3;

Model:

! set the starting values of

! the loadings to 1 by ”*1”

F by x1-x3*1;

! fix the factor variance by ”@1”

F@1;

This is the R code for Solution 1, a positive loading at 1 using OpenMx
package. If the true value of the first loading is positive, the first loading
is fixed at 1 by ”mxPath(from=c(”F”), to=c(”x1”), arrows=1, free=FALSE,
values=1)”.

⬇

# load package

library(OpenMx)

# read data

data <- read.csv(”data.csv”)

# name variables and parameters

indicators <- names(data)

latents <- c(”F”)

loadingLabels <\-

paste(”b_”,indicators, sep=””)

uniqueLabels <\-

paste(”U_”,indicators, sep=””)

meanLabels <\-

paste(”M_”, indicators, sep=””)

factorVarLabels <\-

paste(”Var_”, latents, sep=””)

# build model

oneFactorRaw1 <- mxModel(

”Single␣factor␣Model␣with␣Fixed␣Loading”,

type= ”RAM”,

manifestVars=indicators,

latentVars=latents,

mxPath(from=latents, to=indicators,

arrows=1, connect= ”unique.bivariate”,

free=TRUE, values=.2,

labels=loadingLabels),

mxPath(from=c(”F”), to=c( ”x1”),

arrows=1, free=FALSE, values=1),

mxPath(from=indicators,

arrows=2,

free=TRUE, values=.8,

labels=uniqueLabels),

mxPath(from=latents,

arrows=2,

free=TRUE, values=.8,

labels=factorVarLabels),

mxPath(from=”one”, to=indicators,

arrows=1, free=TRUE, values=.1,

labels=meanLabels),

mxData(observed=data, type=”raw”)

)

oneFactorRaw1Out <- mxRun(oneFactorRaw1)

# print results

summary(oneFactorRaw1Out)

If the first loading is not positive, we can move a positive loading to the
first place. for example, if the loading on item x2 is positive, we can set
its loadings at 1 by ”mxPath(from=c(”F”), to=c(”x2”), arrows=1, free=FALSE,
values=1)”.

⬇

# build model

oneFactorRaw1 <- mxModel(

”Single␣factor␣Model␣with␣Fixed␣Loading”,

type= ”RAM”,

manifestVars=indicators,

latentVars=latents,

mxPath(from=latents, to=indicators,

arrows=1, connect= ”unique.bivariate”,

free=TRUE, values=.2,

labels=loadingLabels),

mxPath(from=c(”F”), to=c( ”x2”),

arrows=1, free=FALSE, values=1),

mxPath(from=indicators,

arrows=2,

free=TRUE, values=.8,

labels=uniqueLabels),

mxPath(from=latents,

arrows=2,

free=TRUE, values=.8,

labels=factorVarLabels),

mxPath(from=”one”, to=indicators,

arrows=1, free=TRUE, values=.1,

labels=meanLabels),

mxData(observed=data, type=”raw”)

)

This is the R code for Solution 3 using OpenMx package. The factor variance is
fixed to 1 by setting ”free=FALSE and values=1” and the starting values of the
loadings are set at 1 by ”free=TRUE and values=1”.

⬇

# build model

oneFactorRaw1 <- mxModel(

”Single␣Factor␣model␣with␣Fixed␣Variance”,

type= ”RAM”,

manifestVars=indicators,

latentVars=latents,

mxPath(from=latents, to=indicators,

arrows=1, connect= ”unique.bivariate”,

# set the starting values of

# the loadings at 1.

free=TRUE, values=1,

labels=loadingLabels),

mxPath(from=indicators,

arrows=2,

free=TRUE, values=.8,

labels=uniqueLabels),

mxPath(from=latents,

arrows=2,

# fixing factor variance to 1.

free=FALSE, values=1,

labels=factorVarLabels),

mxPath(from=”one”, to=indicators,

arrows=1, free=TRUE, values=.1,

labels=meanLabels),

mxData(observed=data, type=”raw”)

)

the R code for Solution 2, a positive loading at 1 using OpenMx package. We
can set the lower bound of 0 by ”lbound=0”, or the upper bound of 0 by
”ubound=0”

⬇

oneFactorRaw1 <- mxModel(

”Single␣Factor␣model␣with␣Fixed␣Variance”,

type= ”RAM”,

manifestVars=indicators,

latentVars=latents,

mxPath(from=latents, to=indicators,

arrows=1, connect= ”unique.bivariate”,

# setting the lower

# bound of the loadings >0.

free=TRUE, lbound=0,

labels=loadingLabels),

mxPath(from=indicators,

arrows=2,

free=TRUE, values=.8,

labels=uniqueLabels),

mxPath(from=latents,

arrows=2,

# fixing factor variance to 1.

free=FALSE, values=1,

labels=factorVarLabels),

mxPath(from=”one”, to=indicators,

arrows=1, free=TRUE, values=.1,

labels=meanLabels),

mxData(observed=data, type=”raw”)

)

####  10.1.2 Categorical variables

Suppose a one-factor model with three categorical variables. If using lavaan,
”ordered = FALSE” should be changed into ”ordered = TRUE” and the rest part
should be kept the same.

This is Mplus code for Solution 1, fixing a positive loading at 1. If the true
value of the first loading is positive, the first loading is fixed at 1 by
”@1”.

⬇

DATA: FILE = data.dat;

VARIABLE: NAMES = x1-x3;

CATEGORICAL ARE x1-x3;

! The estimation method is WLSMV

ANALYSIS: ESTIMATOR = WLSMV;

Model:

!! fix the first loading at 1 by ”@1”

F by x1@1 x2-x3*1;

OUTPUT: TECH1 TECH8;

This is Mplus code for Solution 3, fixing the factor variance by ”@1” and
setting the starting values of the loadings to 1 or -1 by ”*1” or ”*-1”.

⬇

DATA: FILE = data.dat;

VARIABLE: NAMES = x1-x3;

CATEGORICAL ARE x1-x3;

! The estimation method is WLSMV

ANALYSIS: ESTIMATOR = WLSMV;

Model:

!set the starting values of

! the loadings to 1 by ”*1”

F by x1-x3*1;

! fix the factor variance at 1 by ”@1”

F @1;

OUTPUT: TECH1 TECH8;


Alternative $\mit{M_{2}}$-algebras and ${\mit\Gamma}$-algebras

A. Grishkov, I. Shestakov

###### Abstract

Recently V. H. López Solís and I. Shestakov [9] solved an old problem by
N.Jacobson [2] on describing of unital alternative algebras containing the
$2\times 2$ matrix algebra $M_{2}$ as a unital subalgebra. Here we give
another description of $M_{2}$-algebras via the 6-dimensional alternative
superalgebra $B(4,2)$ and an auxiliar $\mathbf{\mit Z}_{2}$-graded algebra
$\mit\Gamma$. It occurs that the category of alternative $M_{2}$-algebras is
isomorphic to the category of ${\mit\Gamma}$-algebras. For any associative and
commutative algebra $A$, we give a construction of a ${\mit\Gamma}$-algebra
${\mit\Gamma}(A)$, which turns to be a Jordan superalgebra; if $A$ is a domain
then ${\mit\Gamma}(A)$ is a prime superalgebra. We describe also the free
${\mit\Gamma}$-algebras and construct their bases.

To the memory of our dear friend Sasha Anan’in

##  1 Introduction.

The classical Wedderburn Coordinatization Theorem says that if a unital
associative algebra $A$ contains a matrix algebra $M_{n}(F)$ over a field $F$
with the same identity element then it is itself a matrix algebra, $A\cong
M_{n}(D)$, “coordinated” by $D$. Generalizations and analogues of this theorem
were proved for various classes of algebras and superalgebras [2, 4, 7, 8, 10,
11, 12, 14, 15]. The common content of all these results is that if an algebra
(or superalgebra) contains a certain subalgebra (matrix algebra, octonions,
Albert algebra) with the same unit then the algebra itself has the same
structure, but not over the basic field rather over a certain algebra that
“coordinatizes” it. The Coordinatization Theorems play important role in
structure theories, especially in classification theorems, and also in the
representation theory, since quite often an algebra $A$ coordinated by $D$ is
Morita equivalent to $D$, though they could belong to different classes (for
instance, Jordan algebras are coordinated by associative and alternative
algebras).

I. Kaplansky [4] proved an analogue of Wedderburn’s theorem for alternative
algebras containing the Cayley algebra. He showed that if $A$ is an
alternative algebra with identity element $1$ which contains a subalgebra
${B}$ isomorphic to a Cayley algebra and if $1$ is contained in ${B}$, then
$A$ is isomorphic to the Kronecker product ${B}\otimes T$, where $T$ is the
center of $A$.

The Wedderburn coordinatization theorem in the case $n\geq 3$ admits a
generalization for alternative algebras, since every alternative algebra $A$
which contains a subalgebra $M_{n}(F)$ $(n\geq 3)$ with the same identity
element is associative (see [15, Corollary $11$, Chapter $2$]). The result is
not true for $n=2$, the split Cayley algebra and its 6-dimensional subalgebra
are counterexamples. The problem of description of alternative algebras
containing $M_{2}(F)$ or, more generally, a generalized quaternion algebra
$\mathbb{H}$ with the same identity element was posed by Jacobson [2].

In [9], this problem was solved for the split case $\mathbb{H}\cong M_{2}(F)$.
The corresponding $M_{2}(F)$-coordinatization in [9] involves two ingredients:
an associative algebra $D$ and a commutative $D$-bimodule $V$ (that is, $V$ is
annihilated by any commutator of elements of $D$), on which a skew-symmetric
mapping is defined with values in the center of $D$, satisfying Plücker
relations. More exactly, $A=M_{2}(D)\oplus V^{2}$, with a properly defined
multiplication.

Here we give another characterization of $M_{2}(F)$-algebras, based on the
6-dimensional simple alternative superalgebra $B=B(4,2)$ [17] and an auxiliar
$\mathbf{\mit Z}_{2}$-graded algebra ${\mit\Gamma}$: a unital alternative
algebra $A$ is an $M_{2}(F)$-algebra if and only if $A$ is a
${\mit\Gamma}$-envelope of the superalgebra $B$: $A={\mit\Gamma}_{0}\otimes
B_{0}+{\mit\Gamma}_{1}\otimes B_{1}$. Moreover, the category of alternative
$M_{2}$-algebras is isomorphic to the category of ${\mit\Gamma}$-algebras. For
any associative and commutative algebra $A$, we give a construction of a
${\mit\Gamma}$-algebra ${\mit\Gamma}(A)$, which turns to be a Jordan
superalgebra; if $A$ is a domain then ${\mit\Gamma}(A)$ is a prime
superalgebra. Finally, we describe free ${\mit\Gamma}$-algebras and construct
their bases. It occurs that these algebras are closely related to coordinate
algebras of grassmannians $Gr(2,n)$.

Throughout this paper the ground field F is of arbitrary characteristic.

##  2 Definitions, examples, and preliminary results

Let $A$ be a composition algebra (see [3, 6, 15, 18]). Recall that $A$ is a
unital alternative algebra, it has an involution $a\mapsto\bar{a}$ such that
the trace $t(a)=a+\bar{a}$ and norm $n(a)=a\bar{a}$ lie in $F$.

An alternative bimodule $V$ over a composition algebra $A$ is called a Cayley
bimodule if it satisfies the relation

$av=v{\bar{a}},$ (1)

where $a\in A$, $v\in V$, and $a\rightarrow{\bar{a}}$ is the canonical
involution in $A$.

Typical examples of composition algebras are the algebras of (generalized)
quaternions $\mathbb{H}$ and octonions $\mathbb{O}$ (or a Cayley algebra) with
symplectic involutions. Recall that $\mathbb{O}=\mathbb{H}\oplus v\mathbb{H}$,
with the product defined by

$\displaystyle{{a\cdot b=ab,\ a\cdot vb=v(\bar{a}b),\ vb\cdot a=v(ab),\
va\cdot vb=(b\bar{a})v^{2},}}$ (2)

where $a,b\in\mathbb{H},\ 0\neq v^{2}\in F,\ a\mapsto\bar{a}$ is the
symplectic involution in $\mathbb{H}$.

The subspace $v\mathbb{H}\subset\mathbb{O}$ is invariant under multiplication
by elements of $\mathbb{H}$ and it gives an example of a Cayley bimodule over
$\mathbb{H}$. If $\mathbb{H}$ is a division algebra then $v\mathbb{H}$ is
irreducible, otherwise $\mathbb{H}\cong M_{2}(F)$ and

$v\mathbb{H}=\langle ve_{{22}},-ve_{{12}}\rangle\oplus\langle-
ve_{{21}},ve_{11}\rangle,$

where $M_{2}(F)$-bimodules $\langle ve_{{22}},\,-ve_{{12}}\rangle$ and
$\langle-ve_{{21}},ve_{11}\rangle$ are both isomorphic to the 2-dimensional
Cayley bimodule ${\rm Cay}=F\cdot m_{1}+F\cdot m_{2}$, with the action of
$M_{2}(F)$ given by

$\displaystyle e_{ij}\cdot m_{k}=\delta_{ik}m_{j},\ \ \ \ m\cdot
a={\bar{a}}\cdot m,$ (3)

where $a\in M_{2}(F),\,m\in{\rm Cay},\ i,j,k\in\\{1,2\\}$ and
$a\mapsto{\bar{a}}$ is the symplectic involution in $M_{2}(F)$. In the last
case the algebra $\mathbb{O}=M_{2}(F)\oplus vM_{2}(F)$ is called the split
octonion algebra.

Let us call a unital alternative algebra $A$ an $M_{2}$-algebra if there
exists a homomorphism of unital algebras $\phi:M_{2}\rightarrow A$, where
$M_{2}=M_{2}(F)$ .

Examples of $M_{2}$-algebras:

  1. 1.

$A$ associative, $A\supseteq M_{2}\ni 1_{A}\Rightarrow A=M_{2}(B)\cong
M_{2}\otimes B,\ B$ associative.

  2. 2.

$\mathbb{O}$ split octonion algebra, $\mathbb{O}=M_{2}\oplus(M_{2})v$.

  3. 3.

$S=M_{2}(F)\oplus{\rm Cay}\subseteq\mathbb{O},\ {\rm Cay}^{2}=0$, the split
null extension of $M_{2}$ by bimodule ${\rm Cay}$.

  4. 4.

$G(B(4,2))=G_{0}\otimes M_{2}+G_{1}\otimes{\rm Cay},\ char\,F=3$, the
Grassmann envelope of the simple alternative superalgebra
${B(4,2)=M_{2}\oplus{\rm Cay}}$ (see [17]), with the following multiplication
in ${{\rm Cay\,}}$:

$\displaystyle{m_{1}^{2}=e_{21},\ m_{2}^{2}=-e_{12},\ m_{1}\,m_{2}=-e_{11},\
m_{2}\,m_{1}=e_{22}.}$

###### Remark 1

The odd product in $B(4,2)$ in [17] has different sign; one can get the old
product by the following change of the basis:

$\displaystyle e_{21}\leftrightarrow-e_{21}.\ e_{12}\leftrightarrow-e_{12},\
m_{2}\leftrightarrow-m_{2}.$

Any $M_{2}$-algebra $A$ may be considered as a unital alternative
$M_{2}$-bimodule. The structure of such bimodules is given by the following
result:

###### Theorem 1

[3, 17] Let $V$ be a unital alternative $M_{2}$-bimodule. Then $V$ is
completely reducible, moreover, $V=V_{a}\oplus V_{c}$, where $V_{a}$ is an
associative $M_{2}$-bimodule and is a direct some of regular bimodules ${\rm
Reg\,}M_{2}$, while $V_{c}$ is a Cayley $M_{2}$-bimodule which is a direct sum
of irreducible Cayley bimodules of type ${\rm Cay\,}.$

In particular, any $M_{2}$-algebra $A$ may be written as ${A=A_{a}\oplus
A_{c}}$. It was proved in [9] that

$\displaystyle{{A_{c}A_{a}+A_{a}A_{c}\subseteq A_{c},\ A_{a}A_{a}\subseteq
A_{a},\ A_{c}A_{c}\subseteq A_{a}.}}$

That is, $A$ is a $\mit{Z}_{2}$-graded algebra. We have $A_{a}=({\rm
Reg\,}\,M_{2})^{n},\ A_{c}=({\rm Cay\,})^{m}$. Moreover, the subalgebra
$A_{a}$ is associative [9].

The ${M_{2}(F)}$-coordinatization in [9] involves two ingredients: an
alternative ${M_{2}(F)}$-algebra $A$ is “coordinated” by an associative
algebra $D$ and by a commutative $D$-bimodule ${V}$ (that is, ${V}$ is
annihilated by any commutator of elements of ${D}$), on which a skew-symmetric
form is defined with values in the center of ${D}$, satisfying Plücker
relations. More exactly,

${A=M_{2}(D)\oplus V^{2},}$

with a properly defined multiplication.

In more details, let ${D}$ be an associative unital algebra and ${V}$ be a
left ${D}$-module such that ${[D,D]}$ annihilates ${V}$. Clearly, in this case
${V}$ has a structure of a commutative ${D}$-bimodule with ${v\cdot a=a\cdot
v,\ v\in V,\,a\in D}$. Assume that there exists a ${D}$-bilinear skew-
symmetric mapping ${\langle,\rangle:V^{2}\rightarrow D}$ such that ${\langle
V,V\rangle\subseteq Z(D)}$ and for any ${u,v,w\in V}$

$\displaystyle{\langle u,v\rangle w+\langle v,w\rangle u+\langle w,v\rangle
u=0.}$ (4)

Consider ${A=M_{2}(D)\oplus V^{2}}$. Let $X,Y\in A,$ $X=X_{a}+(x,y),\
Y=Y_{a}+(z,t)$, where $X_{a},\ Y_{a}\in M_{2}(D)$ and $(x,y),(z,t)\in V^{2}$.
Define a product in $A$ by formula:

$\displaystyle{XY=X_{a}Y_{a}+\left(\begin{array}[]{cc}-\langle
x,t\rangle&-\langle y,t\rangle\\\ \langle x,z\rangle&\langle
y,z\rangle\end{array}\right)+(z,t)X_{a}+(x,y)(Y_{a})^{*},}$

where $\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)^{*}=\left(\begin{array}[]{cc}d&-b\\\
-c&a\end{array}\right)$.

###### Theorem 2

[9] The algebra $A$ with the product defined above is an alternative unital
algebra containing $M_{2}(F)$ with the same identity element. Conversely,
every unital alternative algebra that contains the matrix algebra $M_{2}(F)$
with the same identity element has this form.

##  3 The new construction

Consider the vector space direct sum ${{\mit\Gamma}}=D\oplus V$, where $D$ and
$V$ are taken from theorem 2, and define a multiplication on it as follows

$\displaystyle(a+u)(b+v)=(ab+\langle u,v\rangle)+(av+bu).$

Then ${{\mit\Gamma}}$ becomes a ${\mit{Z}}_{2}$-graded algebra with
${{\mit\Gamma}}_{0}=D,\,{{\mit\Gamma}}_{1}=V$, that satisfied the following
conditions:

  * (i)

${{\mit\Gamma}}_{0}$ is a unital associative algebra,
$[{{\mit\Gamma}}_{0},{{\mit\Gamma}}_{1}]=({\mit\Gamma}_{0},{\mit\Gamma},{\mit\Gamma})=0$,

  * (ii)

${{\mit\Gamma}}_{1}^{2}\subseteq Z({{\mit\Gamma}}_{0})$,

  * (iii)

$xy+yx=0,\ x,y\in{{\mit\Gamma}_{1}}$,

  * (iv)

$(xy)z+(yz)x+(zx)y=0,\ x,y,z\in{{\mit\Gamma}_{1}}$.

###### Theorem 3

The “${{\mit\Gamma}}$-envelope”
${{\mit\Gamma}}(B(4,2))={{\mit\Gamma}}_{0}\otimes
M_{2}+{{\mit\Gamma}}_{1}\otimes Cay$ is isomorphic to the algebra
$M_{2}({{\mit\Gamma}}_{0})\oplus{{\mit\Gamma}}_{1}^{2}$ from theorem 2. In
particular, an algebra $A$ is an alternative $M_{2}$-algebra if and only if
$A={{\mit\Gamma}}(B(4,2))$ for a certain $\mit{Z}_{2}$-graded algebra
${{\mit\Gamma}}$ satisfying the above conditions.

###### Remark 2

Note that the superalgebra $B(4,2)$ is alternative only in $char\,F=3$ case,
but the theorem holds in any characteristic.

Proof of the theorem. It is clear that ${{\mit\Gamma}}_{0}\otimes M_{2}\cong
M_{2}({{\mit\Gamma}}_{0})$. Let us prove that the mapping

$\displaystyle\varphi:X+(x,y)\mapsto X+x\otimes m_{1}+y\otimes m_{2},\ X\in
M_{2}({{\mit\Gamma}}_{0}),\,x,y\in{{\mit\Gamma}}_{1},$

is an isomorphism of $M_{2}({{\mit\Gamma}}_{0})\oplus{{\mit\Gamma}}_{1}^{2}$
and ${{\mit\Gamma}}(B(4,2))$. Let $X=\left(\begin{array}[]{cc}d_{11}&d_{12}\\\
d_{21}&d_{22}\end{array}\right),d_{ij}\in{{\mit\Gamma}}_{1}$. Consider

$\displaystyle\varphi(X\cdot(x,y))$ $\displaystyle=$
$\displaystyle\varphi(xd_{11}+yd_{21},xd_{12}+yd_{22})$ $\displaystyle=$
$\displaystyle(xd_{11}+yd_{21})\otimes m_{1}+(xd_{12}+yd_{22})\otimes m_{2}.$

On the other hand,

$\displaystyle\varphi(X)\varphi(x,y)$ $\displaystyle=$ $\displaystyle
X(x\otimes m_{1}+y\otimes m_{2})=(\sum_{ij}d_{ij}\otimes e_{ij})(x\otimes
m_{1}+y\otimes n)$ $\displaystyle=$ $\displaystyle(d_{11}x+d_{21}y)\otimes
m_{1}+(d_{12}x+d_{22}y)\otimes m_{2}.$

Since $[{{\mit\Gamma}}_{0},{{\mit\Gamma}}_{1}]=0$, we have
$\varphi(X\cdot(x,y))=\varphi(X)\varphi(x,y)$. Similarly, $\varphi((x,y)\cdot
X)=\varphi(x,y)\varphi(X)$. Let now $z,t\in{{\mit\Gamma}}_{1}$, consider

$\displaystyle\varphi((x,y)(z,t))$ $\displaystyle=$
$\displaystyle\varphi\left(\begin{array}[]{cc}-\langle x,t\rangle&-\langle
y,t\rangle\\\ \langle x,z\rangle&\langle
y,z\rangle\end{array}\right)=\left(\begin{array}[]{cc}-\langle
x,t\rangle&-\langle y,t\rangle\\\ \langle x,z\rangle&\langle
y,z\rangle\end{array}\right).$

On the other hand,

$\displaystyle\varphi(x,y)\varphi(z,t)$ $\displaystyle=$
$\displaystyle(x\otimes m_{1}+y\otimes n)(z\otimes m_{1}+t\otimes m_{2})$
$\displaystyle=$ $\displaystyle\langle x,z\rangle\otimes e_{21}-\langle
x,t\rangle\otimes e_{11}+\langle y,z\rangle e_{22}-\langle y,t\rangle\otimes
e_{12}$ $\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}-\langle
x,t\rangle&-\langle y,t\rangle\\\ \langle x,z\rangle&\langle
y,z\rangle\end{array}\right).$

This proves the theorem.

$\Box$

Let us consider the examples of $M_{2}$-algebras from section 2 and determine
the structure of the corresponding algebra ${{\mit\Gamma}}$ in every case.

  1. 1.

$A$ associative, $A\supseteq M_{2}\ni 1_{A}\Rightarrow A=M_{2}(B)\cong
M_{2}\otimes B,\ B$ associative. In this case
${{\mit\Gamma}}={{\mit\Gamma}}_{0}=B$.

  2. 2.

$\mathbb{O}$ split octonion algebra, $\mathbb{O}=M_{2}\oplus(M_{2})v$. Here
${{\mit\Gamma}}=B(1,2)$, the 3-dimensional simple superalgebra from [17] which
is alternative in characteristic 3 case. $B(1,2)_{0}=F,\ B(1,2)_{1}=Fx+Fy,\
x^{2}=y^{2}=0,xy=-yx=1$.

  3. 3.

$S=M_{2}(F)\oplus{\rm Cay}\subseteq\mathbb{O},\ {\rm Cay}^{2}=0$, the split
null extension of $M_{2}$ by bimodule ${\rm Cay}$. In this case
${{\mit\Gamma}}=F+Fx,\,x^{2}=0,$ is the 2-dimensional algebra with
${{\mit\Gamma}}_{0}=F,\,{{\mit\Gamma}}_{1}=Fx$.

  4. 4.

$G(B(4,2))=G_{0}\otimes M_{2}+G_{1}\otimes{\rm Cay\,},\ char\,F=3$, the
Grassmann envelope of the simple alternative superalgebra $B(4,2)$. Here
${{\mit\Gamma}}=G$ since the Grassmann algebra $G$ satisfies the conditions
for ${{\mit\Gamma}}$ in the case of characteristic 3.

##  4 Tensor algebras of bimodules and free ${{\mit\Gamma}}$-algebras

Recall the definition of a tensor algebra of bimodule (see [5]). Let $A$ be an
algebra in a variety $\cal M$ and let $V$ be an $\cal M$-bimodule over $A$.
Consider the free $\cal M$-algebra $F_{\cal M}[A\oplus V]$ and let $I$ be the
ideal of this algebra generated by the set $\\{a*b-ab,\,a*v-a\cdot
v,\,v*a-v\cdot a\,|\,a,b\in A,\,v\in V\\}$, where $*$ denotes the
multiplication in $F_{\cal M}[A\oplus V]$ and $a\cdot v,\,v\cdot a$ denote the
action of $A$ on $V$. Then the quotient algebra $F_{\cal M}[A\oplus V]/I$ is
called the tensor algebra of the $A$-bimodule $V$.

By the standard arguments, one can prove the following universal property of
tensor algebra.

###### Proposition 1

Let $B\in\cal M$ and let $\varphi:A\rightarrow B$ be a homomorphism of
algebras. Then $B$ has a natural structure of an $A$-bimodule. Now, for any
homomorphism of $A$-bimodules $\psi:V\rightarrow B$ there exists a unique
homomorphism of algebras $\tilde{\psi}:F_{\cal M}[A\oplus V]\rightarrow B$
such that $\tilde{\psi}(a)=\varphi(a),\,\tilde{\psi}(v)=\psi(v)$ for any $a\in
A,\,v\in V$.

In particular, the tensor algebra $M_{2}[V]$ of an alternative
$M_{2}$-bimodule $V$ plays a role of a free object in the category of
alternative $M_{2}$-algebras: for any $M_{2}$-algebra $B$, any homomorphism of
$M_{2}$-bimodules $\varphi:V\rightarrow B$ is uniquely extended to an algebra
homomorphism $\tilde{\varphi}:M_{2}[V]\rightarrow B$.

Let us call a $\mit{Z}_{2}$-graded algebra satisfying conditions (i) - (iv) of
section 3 a ${{\mit\Gamma}}$-algebra. We want to prove

###### Theorem 4

The category of alternative $M_{2}$-algebras is isomorphic to the category of
${{\mit\Gamma}}$-algebras.

Proof. There is a natural functor from the category of ${\mit\Gamma}$-algebras
to the category of $M_{2}$-algebras:
$F:{\mit\Gamma}\rightarrow{\mit\Gamma}(B(4,2))$, which sends a morphism of
${\mit\Gamma}$-algebras $\varphi:{\mit\Gamma}\rightarrow{\mit\Gamma}^{\prime}$
to the morphism
$F(\varphi):{\mit\Gamma}(B(4,2))\rightarrow{\mit\Gamma}^{\prime}(B(4,2))$
identical on $B(4,2)$.

It is clear from the proof of theorem 3 that this functor is bijective on
objects: every $M_{2}$-algebra $A=M_{2}(D)\oplus V^{2}$ defines uniquely
${\mit\Gamma}_{0}=D$ and ${\mit\Gamma}_{1}=V$. It remains to show that any
morphism
$\varphi:{\mit\Gamma}(B(4,2))\rightarrow{\mit\Gamma}^{\prime}(B(4,2))$ is
induced by a morphism $\psi:{\mit\Gamma}\rightarrow{\mit\Gamma}^{\prime}$ such
that $\varphi=F(\psi)$. Denote $A={\mit\Gamma}(B(4,2)),\
A^{\prime}={\mit\Gamma}^{\prime}(B(4,2))$. Since $\varphi$ is identical on
$M_{2}$, it is a homomorphism of $M_{2}$-bimodules; in particular,
$\varphi(A_{a})=\varphi(M_{2}({\mit\Gamma}_{0}))\subset
A^{\prime}_{a}=M_{2}({\mit\Gamma}^{\prime}_{0})$ and
$\varphi(A_{c})=\varphi({\mit\Gamma}_{1}\otimes{\rm
Cay})\subset(A^{\prime})_{c}={\mit\Gamma}^{\prime}_{1}\otimes{\rm Cay}$. It is
well known that a homomorphism of matrix algebras is induced by a homomorphism
of their coordinates, hence there exists a homomorphism
$\psi_{0}:{\mit\Gamma}_{0}\rightarrow{\mit\Gamma}^{\prime}_{0}$ which induces
$\varphi|_{A_{a}}$.

Now, fix $0\neq\gamma\in{\mit\Gamma}_{1}$. Let $\varphi(\gamma\otimes
m_{1})=\alpha\otimes m_{1}+\beta\otimes m_{2},\ \varphi(\gamma\otimes
m_{2})=\lambda\otimes m_{1}+\mu\otimes m_{2}$ for some
$\alpha,\beta,\lambda,\mu\in{\mit\Gamma}^{\prime}_{1}$. We have

$\displaystyle\alpha\otimes m_{1}+\beta\otimes m_{2}$ $\displaystyle=$
$\displaystyle\varphi(\gamma\otimes m_{1})=\varphi((1\otimes
e_{11})(\gamma\otimes m_{1}))$ $\displaystyle=$ $\displaystyle(1\otimes
e_{11})\varphi(\gamma\otimes m_{1})=\alpha\otimes m_{1},$

which implies $\beta=0$. Similarly, $\lambda=0$. Futhermore,

$\displaystyle\mu\otimes m_{2}$ $\displaystyle=$
$\displaystyle\varphi(\gamma\otimes m_{2})=\varphi((1\otimes
e_{12})(\gamma\otimes m_{1}))$ $\displaystyle=$ $\displaystyle(1\otimes
e_{12})\varphi(\gamma\otimes m_{1})=\alpha\otimes m_{2},$

which implies $\alpha=\mu$. Therefore, $\varphi(\gamma\otimes{\rm
Cay})=\alpha\otimes{\rm Cay}$, and we put $\psi_{1}(\gamma)=\alpha$. One can
easily check that
$\psi=\psi_{0}+\psi_{1}:{\mit\Gamma}\rightarrow{\mit\Gamma}^{\prime}$ is a
homomorphism of algebras such that $F(\psi)=\varphi$.

$\Box$

###### Corollary 1

Let ${{\mit\Gamma}}[X_{0},X_{1}]$ be the free ${\mit\Gamma}$-algebra on sets
$X_{0}$ and $X_{1}$ of even and odd generators. Then
${{\mit\Gamma}}[X_{0},X_{1}](B(4,2))\cong M_{2}[V]$, where
$V=Reg^{\\#X_{0}}\oplus{\rm Cay}^{\\#X_{1}}.$

In view of the Corollary, it seems important to determine the structure of
free ${\mit\Gamma}$-algebras.

Let $V=F^{n}$ be an $n$-dimensional vector space over $F$ and $F[Gr_{2}(V)]$
be the coordinate algebra of the Grassmannian $Gr(2,n)=Gr(2,V)$. Recall that
$F[Gr(2,V)]\cong F[V^{\wedge 2}]/P$, where $P$ is the ideal generated by the
double Plücker relations

$\displaystyle(u\wedge v)(w\wedge z)+(u\wedge w)(z\wedge v)+(u\wedge
z)(v\wedge w),\ u,v,w,z\in V.$

Furthermore, consider the tensor product $F[Gr(2,V)]\otimes V$ which has a
natural structure of $F[Gr(2,V)]$-module. Denote by $F[Gr(2,V)]_{1}$ the
quotient module $(F[Gr(2,V)]\otimes V)/I$ where $I$ is the
$F[Gr(2,V)]$-submodule generated by the ordirnary Plücker relations

$\displaystyle(u\wedge v)\otimes w+(v\wedge w)\otimes u+(w\wedge u)\otimes v,\
u,v,w\in V.$

Define multiplication in $F[Gr(2,V)]_{1}$ with results in $F[Gr(2,V)]$ by
setting

$\displaystyle(a\otimes u)(b\otimes v)=ab(u\wedge v)\otimes 1,$

where $a,b\in F[Gr(2,V)],\ u,v\in V$. The product is defined correctly, since

$\displaystyle((u\wedge v)\otimes w+(v\wedge w)\otimes u+(w\wedge u)\otimes
v)(a\otimes z)$ $\displaystyle=$ $\displaystyle a((u\wedge v)(w\wedge
z)+(v\wedge w)(u\wedge z)+(w\wedge u)(v\wedge z))\otimes 1$ $\displaystyle=$
$\displaystyle 0.$

###### Theorem 5

The $\mit{Z}_{2}$-graded algebra
${\mit\Gamma}[\emptyset;V]=F[Gr(2,V)]+F[Gr(2,V)]_{1}$ is a free
${\mit\Gamma}$-algebra generated by the space of odd generators $V$.

Proof. In fact, it is easy to check that the unital algebra
${\mit\Gamma}[\emptyset;V]$ satisfies conditions (i) – (iv) defining
${\mit\Gamma}$-algebras, and is generated by the space $V$. Let
${\mit\Gamma}={\mit\Gamma}_{0}+{\mit\Gamma}_{1}$ be a ${\mit\Gamma}$-algebra
and $\varphi:V\rightarrow{\mit\Gamma}_{1}$ a linear mapping. For $v\in V$
denote by $\bar{v}$ its image in ${\mit\Gamma}_{1}$. The mapping $\varphi$ is
extended to a linear mapping $V^{\wedge 2}\rightarrow{\mit\Gamma}_{0},\
u\wedge v\mapsto\bar{u}\bar{v}$ and further to an algebra homomorphism
$F[V^{\wedge 2}]\rightarrow{\mit\Gamma}_{0}$. In view of condition (iv) the
ordinary Plücker relation holds in ${\mit\Gamma}$. Moreover, for any
$u,v,w,z\in V$ we have in ${\mit\Gamma}$

$\displaystyle(\bar{u}\bar{v})(\bar{w}\bar{z})+(\bar{u}\bar{w})(\bar{z}\bar{v})+(\bar{u}\bar{z})(\bar{v}\bar{w})=\bar{u}(\bar{v}\bar{w}\bar{z}+\bar{w}\bar{z}\bar{v})+\bar{z}\bar{v}\bar{w})=0.$

Therefore, the mapping $\varphi$ can be extended to an algebra homomorphism
$\tilde{\varphi}:{\mit\Gamma}[\emptyset;V]\rightarrow{\mit\Gamma}$.

$\Box$

Let now $U$ be another vector space, construct the free ${\mit\Gamma}$-algebra
${\mit\Gamma}[U;V]$ generated by the space $U$ of even generators and the
space $V$ of odd generators. Denote by $F\langle U\rangle$ and by $F[U]$ the
free associative and polynomial algebras over the space $U$. Furthermore, by
$F[Gr(2,V)]^{0}$ denote the augmentation ideal of the algebra $F[Gr(2,V)]$,
that is, the ideal of elements without scalar terms. Consider the
$\mit{Z}_{2}$-graded vector space

$\displaystyle{\mit\Gamma}[U;V]=(F\langle U\rangle\oplus(F[U]\otimes
F[Gr(2,V)]^{0}))\oplus(F[U]\otimes F[Gr(2,V)]_{1}),$

with ${\mit\Gamma}[U;V]_{0}=F\langle U\rangle\oplus(F[U]\otimes
F[Gr(2,V)]^{0})$ and ${\mit\Gamma}[U;V]_{1}=F[U]\otimes F[Gr(2,V)]_{1}$.
Observe that

$\displaystyle I=(F[U]\otimes F[Gr(2,V)]^{0})\oplus(F[U]\otimes
F[Gr(2,V)]_{1})=$ $\displaystyle
F[U]\otimes(F[Gr(2,V)]^{0}+F[Gr(2,V)]_{1})\subseteq
F[U]\otimes{\mit\Gamma}[\emptyset;V]$

Define multiplication on ${\mit\Gamma}[U;V]$ in the following way: the space
$I$ is an ideal of ${\mit\Gamma}(U;V)$ with the product defined as in a
subalgebra of the algebra $F[U]\otimes{\mit\Gamma}[\emptyset;V]$; the algebra
$F\langle U\rangle$ is a subalgebra of ${\mit\Gamma}[U;V]$, and the element
$f\in F\langle U\rangle$ acts on $I$ by

$\displaystyle f\cdot(g\otimes a+h\otimes\ b\otimes v)=\bar{f}g\otimes
a+\bar{f}h\otimes\ b\otimes v,$

where $g,h\in F[U],\,a,b\in F[Gr(2,V)],\,v\in V,$ and $\bar{f}\in F[U]$ is the
image of $f$ under the natural epimorphism $F\langle U\rangle\rightarrow
F[U]$.

###### Theorem 6

The algebra ${\mit\Gamma}[U;V]$ with the multiplication defined above is a
free ${\mit\Gamma}$-algebra generated by the spaces $U$ and $V$ of even and
odd generators.

Proof. First of all, one can easily check that the algebra ${\mit\Gamma}[U;V]$
satisfies conditions (i) – (iv). Furthermore, let
${\mit\Gamma}={\mit\Gamma}_{0}+{\mit\Gamma}_{1}$ be a ${\mit\Gamma}$-algebra
and $\varphi:U\rightarrow{\mit\Gamma}_{0},\ \psi:V\rightarrow{\mit\Gamma}_{1}$
be linear mappings. By above, $\psi$ can be extended to an algebra
homomorphism $\tilde{\psi}:{\mit\Gamma}[\emptyset;V]\rightarrow{\mit\Gamma}$.
By the property of free algebras, there exists also an algebra homomorphism
$\tilde{\varphi}:F\langle U\rangle\rightarrow{\mit\Gamma}_{0}$ extending
$\varphi$. Now the mapping

$\displaystyle(\tilde{\varphi}+\tilde{\psi}):f+(h\otimes a+g\otimes b\otimes
v)\mapsto\tilde{\varphi}(f)+\tilde{\varphi}(h)\tilde{\psi}(a)+\tilde{\varphi}(g)\tilde{\psi}(b)\psi(v)$

for $f\in F\langle U\rangle,\ g,h\in F[U],\,a,b\in F[Gr(2,V)]$ and $v\in V$ is
an algebra homomorphism of ${\mit\Gamma}[U;V]$ to ${\mit\Gamma}$ extending
$\varphi+\psi$.

$\Box$

##  5 ${\mit\Gamma}$-algebras and Jordan superalgebras

Observe that if ${\mit\Gamma}$ is a ${\mit\Gamma}$-algebra with commutative
even part ${\mit\Gamma}_{0}$ then ${\mit\Gamma}$ is a commutative
superalgebra. Moreover, in this case it is a Jordan superalgebra.

###### Proposition 2

Let ${\mit\Gamma}$ is a ${\mit\Gamma}$-algebra with commutative even part
${\mit\Gamma}_{0}$. Then ${\mit\Gamma}$ is a Jordan superalgebra. Moreover, if
$char\,F=3$ then ${\mit\Gamma}$ is an alternative superalgebra.

Proof. Recall that a commutative superalgebra is called a Jordan superalgebra
if it satisfies the super-identity

$\displaystyle(xy,z,t)+(-1)^{\bar{y}\bar{z}+\bar{y}\bar{t}+\bar{z}\bar{t}}(xt,z,y)+(-1)^{\bar{x}(\bar{y}+\bar{z}+\bar{t})+\bar{z}\bar{t}}(yt,z,x)=0,$
(8)

where $\bar{x}$ for $x\in{\mit\Gamma}_{0}\cup{\mit\Gamma}_{1}$ denotes the
parity of element $x$: $\bar{x}=i\Leftrightarrow x\in{\mit\Gamma}_{i}$. Note
that $[{\mit\Gamma}_{0},{\mit\Gamma}]=0$, hence ${\mit\Gamma}_{0}$ is
contained in the center of ${\mit\Gamma}$. Therefore, if at least 2 elements
of $x,y,z,t$ lie in ${\mit\Gamma}_{0}$ or $z\in{\mit\Gamma}_{0}$, all the
associators in identity (8) vanish. If all the elements $x,y,z,t$ are in
${\mit\Gamma}_{1}$ then $xy,\,xt,\,yt\in{\mit\Gamma}_{0}$, and again (8)
holds. Therefore, it suffices to consider the case when
$x,y,z\in{\mit\Gamma}_{1},\,t=a\in{\mit\Gamma}_{0}$. We have

$\displaystyle(xy,z,a)-(xa,z,y)+(ya,z,x)=-(xa,z,y)+(ya,z,x)$ $\displaystyle=$
$\displaystyle a(-(x,z,y)+(y,z,x))=a(-xz\cdot y+x\cdot zy+yz\cdot x-y\cdot
zx)$ $\displaystyle=$ $\displaystyle a(-(xz+zx)a+x(zy+yz))=0.$

Furthermore, let $char\,F=3$. Since ${\mit\Gamma}_{0}\subseteq
Z({\mit\Gamma})$, in order to check alternatively we have to consider only
associators on odd generators. Let $x,y,z\in{\mit\Gamma}_{1}$, then we have

$\displaystyle(x,y,z)-(x,z,y)$ $\displaystyle=$ $\displaystyle xy\cdot
z-x\cdot yz-xz\cdot y+x\cdot zy=xy\cdot z-2yz\cdot x+zx\cdot y$
$\displaystyle=$ $\displaystyle xy\cdot z+yz\cdot x+zx\cdot y=0,$

and similarly $(x,y,z)-(y,x,z)=0$, hence the superalgebra ${\mit\Gamma}$ is
alternative.

$\Box$

An important example of supercommutative ${\mit\Gamma}$-algebras can be
obtained as follows. Let $A$ be a commutative associative algebra, consider
${\mit\Gamma}(A)=A\oplus A^{2}$ with the grading
${\mit\Gamma}(A)_{0}=A,\,{\mit\Gamma}(A)_{1}=A^{2}$ and the following
multiplication:

$\displaystyle a\cdot b=ab,\,a\cdot(b,c)=(b,c)\cdot
a=(ab,ac),\,(a,b)\cdot(c,d)=ad-bc;\ a,b,c,d\in A.$

We have only to check condition (iv) in the definition of
${\mit\Gamma}$-algebra. Consider

$\displaystyle(a,b)(c,d)\cdot(e,f)+(c,d)(e,f)\cdot(a,b)+(e,f)(a,b)\cdot(c,d)$
$\displaystyle=(ad-bc)(e,f)+(cf-de)(a,b)+(eb-fa)(c,d)=(0,0).$

Therefore, ${\mit\Gamma}(A)$ is a ${\mit\Gamma}$-algebra. Since $A$ is
commutative, ${\mit\Gamma}(A)$ is a Jordan superalgebra.

###### Proposition 3

Let $A$ be a domain, then ${\mit\Gamma}(A)$ is a central order in the simple
Jordan superalgebra of type $B(1,2)$ (the superalgebra of a skew-symmetric
bilinear form on a 2-dimensional vector space). In particular, in this case
${\mit\Gamma}(A)$ is prime and special.

Proof. In fact, let $K$ be the quotient field of $A$, then we have an
inclusion ${\mit\Gamma}(A)\subseteq{\mit\Gamma}(K)$; moreover, since
$K=A^{-1}A$ and $A=Z({\mit\Gamma}(A))$, we have
$A^{-1}{\mit\Gamma}(A)={\mit\Gamma}(K)$. It is clear that
$\dim_{K}{\mit\Gamma}(K)=3$ and ${\mit\Gamma}(K)=B(1,2)$ as a
$K$-superalgebra.

It is well known that $B(1,2)$ is a special superalgebra, hence so is
${\mit\Gamma}(A)$. Finally, a central order in a simple (super)algebra is
evidently prime.

$\Box$

##  6 Bases of free ${\mit\Gamma}$-algebras

In this section we will construct bases of free ${\mit\Gamma}$-algebras
defined in terms of free generators.

Let $A_{n}=F[x_{1},\ldots,x_{n};y_{1},\ldots,y_{n}],$ consider the
${\mit\Gamma}$-algebra ${\mit\Gamma}(A_{n})$. We want to prove that the
subalgebra of ${\mit\Gamma}(A_{n})$ generated by the odd elements
$v_{i}=(x_{i},y_{i}),\,i=1,\ldots,n,$ is a free ${\mit\Gamma}$-algebra on
these set of generators.

Denote $\alpha_{ij}=v_{i}v_{j}=x_{i}y_{j}-x_{j}y_{i}\,(1\leq i<j\leq n),\
S_{n}=F[\alpha_{12},\ldots,\alpha_{(n-1)n}]\subset A_{n},\ \
V_{i}=Fv_{i},\,V=\sum_{i=1}^{n}V_{i}$. We have the relations

$\displaystyle\alpha_{ij}v_{k}+\alpha_{jk}v_{i}+\alpha_{ki}v_{j}$
$\displaystyle=$ $\displaystyle 0,\ $ (9) $\displaystyle\alpha_{ij}+a_{ji}$
$\displaystyle=$ $\displaystyle 0,$ (10)
$\displaystyle\alpha_{ij}\alpha_{kl}+\alpha_{ik}\alpha_{lj}+\alpha_{il}\alpha_{jk}$
$\displaystyle=$ $\displaystyle 0.$ (11)

The following lemma is well known (see, for instance, [13]).

###### Lemma 1

The algebra $S_{n}$ is the free algebra modulo relations (10),(11). Moreover,
it has the following base over $F$:

$\displaystyle
B_{n}=\\{\alpha_{i_{1}j_{1}}\alpha_{i_{2}j_{2}}\cdots\alpha_{i_{r}j_{r}}\,|\,i_{1}\leq
i_{2}\leq\cdot\leq i_{r},\ j_{1}\leq j_{2}\leq\cdot\leq
j_{r};\,i_{s}<j_{s}\\}.$ (12)

In fact, the algebra $S_{n}=F[Gr(2,n)]$ is the coordinate algebra of
grassmanian $Gr(2,n)$ (see, for example, [16, vol.1, p.42]).

Let $S_{n,m}=F[\alpha_{ij}\,|\,j\geq m]$, then we have

$\displaystyle S_{n,n}\subseteq S_{n,n-1}\subseteq\ldots\subseteq
S_{n,2}=S_{n,1}=S_{n}.$

Denote by $I_{m}$ be the ideal of $S_{n}$ generated by the set
$\\{\alpha_{ij}\,|\,i<j<m\\},\,3\leq m\leq n$; then $0=I_{2}\subseteq
I_{3}\subseteq I_{4}\subseteq\cdots\subseteq I_{n}$. Let also
$\bar{S}_{n,m}=(S_{n,m}+I_{m})/I_{m}$.

###### Lemma 2

The image in $\bar{S}_{n,m}$ of the following set forms a base of the algebra
$\bar{S}_{n,m}$ over $F$:

$\displaystyle
B_{n,m}=\\{\alpha_{i_{1}j_{1}}\alpha_{i_{2}j_{2}}\cdots\alpha_{i_{r}j_{r}}\in
B_{n}\,|\,j_{1}\geq m\\}.$

Proof. It is easy to prove using identities (11) that the set $B_{n,m}$ spans
$S_{n,m}$ modulo $I_{m}$. Let us prove that it is linearly independent modulo
$I_{m}$. Consider the algebra
$E=E_{n,m}=F[x_{m},\ldots,x_{n};t;y_{1},\ldots,y_{n}]$ and the homomorpism

$\displaystyle\phi:A_{n}\rightarrow E,\ y_{i}\mapsto
y_{i},\,i=1,\ldots,n;\,x_{j}\mapsto x_{j},\,j\geq m;x_{k}\mapsto
ty_{k},\,k<m.$

Let $D_{n,m}=\phi(S_{n})$. Note that $\phi(\alpha_{ij})=0$ if $j<m$, hence
$I_{m}\subseteq\ker\phi$. Introduce the $\deg{\rm lex}$ order in $E$ by
setting

$\displaystyle x_{n}>x_{n-1}>\cdots>x_{m}>y_{1}>y_{2}>\cdots>y_{n}>t,$

and let $\bar{f}$ denotes the leading term of polynomial $f$. Then for
$i<m\leq j$ we have
$\overline{\phi(\alpha_{ij})}=\overline{ty_{i}y_{j}-x_{j}y_{i}}=-x_{j}y_{i}$,
hence

$\displaystyle\overline{\phi(\alpha_{i_{1}j_{1}}\alpha_{i_{2}j_{2}}\cdots\alpha_{i_{s}j_{s}})}=(-1)^{s}x_{j_{1}}\cdots
x_{j_{s}}y_{i_{1}}\cdots y_{i_{s}}.$

Therefore, if $u$ and $v$ are monomials in $\phi(\alpha_{ij})$ then $u=v$ if
and only if $\bar{u}=\bar{v}$. This easily implies that the set
$\phi(B_{n,m})$ is linearly independent over $F$, and thus the set $B_{n,m}$
is linearly independent modulo $I_{m}$.

$\Box$

Consider the elements of the base $B_{n}$ with more details. For any $u\in
B_{n}$ of form (12) there exist uniquely defined numbers $l<p$ such that

$\displaystyle
u=\alpha_{i_{1}j_{1}}\cdots\alpha_{i_{l}j_{l}}\cdots\alpha_{i_{p}j_{p}}\cdots\alpha_{i_{r}j_{r}},$
(13)

where $j_{l}<m,j_{l+1}\geq m;\ i_{p}<m,i_{p+1}\geq m$.

###### Lemma 3

The intersection $I_{m}\cap S_{n,m}$ has a base formed by elements (13) with
$r-p\geq l\geq 1$.

Proof. Let us first prove that every element $u$ of form (13) with $r-p\geq
l\geq 1$ belongs to $I_{m}\cap S_{n,m}$. Since $l>1,\ u\in I_{m}$. In order to
prove that $u\in S_{n,m}$, it suffices to show that

$\displaystyle(\alpha_{i_{1}j_{1}}\cdots\alpha_{i_{l}j_{l}})(\alpha_{i_{p+1},j_{p+1}}\cdots\alpha_{i_{r}j_{r}})\in
S_{n,m}.$

Since $r-p\geq l$, it suffices to prove that every product
$\alpha_{ij}\alpha_{kl}$ with $j<m,\,k>m$ belongs to $S_{n,m}$. But this
follows easily from relation (11).

In order to prove the inverse inclusion, we associate with any element $u\in
B_{n}$ the set of its indices ${\rm
ind}(u)=\\{i_{1},j_{1},\ldots,i_{r},j_{r}\\}$. Note that relation (11) does
not change the set of indices, hence the algebras $S_{n}$ and $S_{n,m}$ are
homogeneous with respect to the sets of indices, that is, they may be
represented as direct sums of subspaces with the same sets of indices.
Moreover, so is the ideal $I_{m}$. Since the elements of $S_{n,m}$ are
polynomials in $\alpha_{ij}$ with $j\geq m$, it is clear that for any
homogeneous element $u\in S_{n,m}$ with ${\rm
ind}(u)=\\{i_{1},j_{1},\ldots,i_{r},j_{r}\\}$ we should have at least $r$
indices that are greater or equal to $m$. Assume now that
$\sum\lambda_{i}u_{i}\in I_{m}\cap S_{n,m}$ for some $u_{i}$ of form (13),
then we have ${\rm ind}(u_{i})={\rm
ind}(u_{j})=\\{i_{1},j_{1},\ldots,i_{l},j_{l},\ldots,i_{p},j_{p},\ldots,i_{r},j_{r}\\}$
with $l\geq 1$ for all the summonds $u_{i},u_{j}$. The set ${\rm ind}(u_{i})$
has $l+p$ indices which are smaller than $m$, hence the sum lies in $S_{n,m}$
only if $l+p\leq r$ or $r-p\geq l$.

$\Box$

###### Lemma 4

$(S_{n,m}v_{m})\cap(\sum_{j<m}S_{n,j}v_{j})=(S_{n,m}\cap I_{m})v_{m}.$

Proof. Let us first prove that $(S_{n,m}\cap
I_{m})v_{m}\subseteq\sum_{j<m}S_{n,j}v_{j}$. Let $u$ be an element of form
(13) with $r-p\geq l\geq 1$, then $u=\alpha_{i_{1}j_{1}}u^{\prime}$, where
$i_{1}<j_{1}<m$ and $u^{\prime}$ is an element of form (13) with
$r^{\prime}=r-1,\,p^{\prime}=p-1,\,l^{\prime}=l-1$. In particular, we have
$r^{\prime}-p^{\prime}=r-p\geq l>l^{\prime}$, therefore as in the proof of
lemma 3 we have $u^{\prime}\in S_{n,m}$. Now by (9)

$\displaystyle
uv_{m}=u^{\prime}(\alpha_{i_{1}j_{1}}v_{m})=-u^{\prime}(\alpha_{i_{1}m}v_{j_{1}}-\alpha_{j_{1}m}v_{i_{1}})\in
S_{n,m}v_{j_{1}}+S_{n,m}v_{i_{1}}\subseteq\sum_{j<m}S_{n,j}v_{j}.$

Note that lemma 2 implies that $S_{n,m}=(S_{n,m}\cap I_{m})\oplus F\cdot
B_{n,m}$. Let us prove that $B_{n,m}V_{m}\cap(\sum_{j<m}S_{n,j}v_{j})=0$.
Assume that $w_{1}=w_{2}\neq 0$, where

$\displaystyle w_{1}=\sum_{a_{i}\in B_{n,m}}\lambda_{i}a_{i}v_{m},\ \ \
w_{2}=\sum_{j<m,\,b_{j}\in S_{n,j}}\mu_{j}b_{j}v_{j};\ \
\lambda_{i},\mu_{j}\in F.$

In particular, we have
$\sum_{i}\lambda_{i}a_{i}x_{m}=\sum_{j}\mu_{j}b_{j}x_{j}\neq 0$. Consider the
leading terms of both parts with respect to the $\deg{\rm lex}$ order in
$A_{n}$ when

$\displaystyle x_{n}>x_{n-1}>\cdots>x_{1}>y_{1}>y_{2}>\cdots>y_{n}.$

We have

$\displaystyle\overline{\sum_{i}\lambda_{i}a_{i}x_{m}}$ $\displaystyle=$
$\displaystyle\overline{\alpha_{i_{1}j_{1}}\ldots\alpha_{i_{s}j_{s}}x_{m}}=f(y)x_{m}x_{j_{1}}\cdots
x_{j_{s}},\ m\leq j_{1}\leq j_{2}\leq\cdots\leq j_{s},$
$\displaystyle\overline{\sum_{j}\mu_{j}b_{j}x_{j}}$ $\displaystyle=$
$\displaystyle\overline{\alpha_{p_{1}q_{1}}\ldots\alpha_{p_{s}q_{s}}x_{j}}=g(y)x_{j}x_{q_{1}}\cdots
x_{q_{s}},\ j<m,j\leq q_{1}\leq q_{2}\leq\cdots\leq q_{s}.$

Since
$\overline{\sum_{i}\lambda_{i}a_{i}x_{m}}\neq\overline{\sum_{j}\mu_{j}b_{j}x_{j}}$,
we conclude that $w_{1}=w_{2}=0$.

Now, since $(S_{n,m}\cap I_{m})v_{m}\subseteq\sum_{j<m}S_{n,j}v_{j}$, we have

$\displaystyle(S_{n,m}v_{m})\cap(\sum_{j<m}S_{n,j}v_{j})$ $\displaystyle=$
$\displaystyle((S_{n,m}\cap I_{m})\oplus F\cdot
B_{n,m})v_{m}\cap(\sum_{j<m}S_{n,j}v_{j})$ $\displaystyle=$
$\displaystyle((S_{n,m}\cap
I_{m})v_{m}+B_{n,m}V_{m})\cap(\sum_{j<m}S_{n,j}v_{j})$ $\displaystyle=$
$\displaystyle(S_{n,m}\cap
I_{m})v_{m}+B_{n,m}V_{m}\cap(\sum_{j<m}S_{n,j}v_{j})$ $\displaystyle=$
$\displaystyle(S_{n,m}\cap I_{m})v_{m}.$

$\Box$

###### Lemma 5

The subspace $S_{n}V\subseteq A_{n}^{2}$ is decomposed into a vector space
direct sum

$\displaystyle S_{n}V=\oplus_{i=1}^{n}B_{n,i}V_{i}.$

Proof. First of all, note that due to (9) we have
$S_{n}V=\sum_{i=1}^{n}S_{n,i}v_{i}$. Denote
$U_{m}=\sum_{i=1}^{m}S_{n,i}v_{i}$, then $U_{1}\subseteq
U_{2}\subseteq\cdots\subseteq U_{n}=S_{n}V$. Furthermore, let
$W_{i}=U_{i}/U_{i-1}$, then we have a vector space isomorphism
$S_{n}V\cong\oplus_{i=1}^{n}W_{i}$. Finally, for any $m\leq n$ we have

$\displaystyle W_{m}$ $\displaystyle=$
$\displaystyle(\sum_{i=1}^{m}S_{n,i}v_{i})/(\sum_{i=1}^{m-1}S_{n,i}v_{i})\cong
S_{n,m}v_{m}/(S_{n,m}v_{m}\cap(\sum_{i=1}^{m-1}S_{n,i}v_{i}))$
$\displaystyle=$ $\displaystyle\hbox{ (by lemma \ref{lem4})
}=S_{n,m}v_{m}/(S_{n,m}\cap I_{m})v_{m}\cong(S_{n,m}/(S_{n,m}\cap
I_{m}))v_{m}$ $\displaystyle\cong$ $\displaystyle\hbox{ (by lemma \ref{lem2})
}\cong B_{n,m}V_{m}.$

$\Box$

###### Theorem 7

The space $S_{n}+S_{n}V$ is a subalgebra of algebra ${\mit\Gamma}(A_{n})$
which is isomorphic to the free ${\mit\Gamma}$-algebra
${\mit\Gamma}[\emptyset;V]$. It has a base
$B_{n}\cup(\cup_{j=1}^{n}B_{n,j}v_{j})$.

Proof. Consider the epimorphism $\varphi:{\mit\Gamma}[\emptyset;V]\rightarrow
S_{n}+S_{n}V$ defined by the conditions $v_{i}\mapsto(x_{i},y_{i})$. Relations
(9) – (11) hold in the algebra $G[\emptyset;V]$ as well, and using these
relations it is easy to see that it is spanned by the set
$B_{n}\cup(\cup_{j=1}^{n}B_{n,j}v_{j})$. Since its image is linearly
independent in ${\mit\Gamma}(A_{n})$, it forms a base of $G[\emptyset;V]$, and
$\varphi$ is an isomorphism.

$\Box$

###### Theorem 8

The free ${\mit\Gamma}$-algebra
${\mit\Gamma}[t_{1},\ldots,t_{m};v_{1},\ldots,v_{n}]$ on even generators
$t_{1},\ldots,t_{m}$ and odd generators $v_{1},\ldots,v_{n}$ has the following
structure:

$\displaystyle{\mit\Gamma}_{0}$ $\displaystyle=$ $\displaystyle F\langle
t_{1},\ldots,t_{m}\rangle+S_{n}^{\prime}\otimes
F[\bar{t}_{1},\ldots,\bar{t}_{m}],$ $\displaystyle{\mit\Gamma}_{1}$
$\displaystyle=$ $\displaystyle
F[\bar{t}_{1},\ldots,\bar{t}_{m}]\otimes(\oplus_{j=1}^{n}B_{n,j}V_{j}),$

where $F\langle t_{1},\ldots,t_{m}\rangle$ and
$F[\bar{t}_{1},\ldots,\bar{t}_{m}]$ are the free associative and the
polynomial algebras on $m$ variables, $S_{n}^{\prime}$ stands for the
augmentation ideal of the algebra $S_{n}$, $v_{i}\cdot v_{j}=\alpha_{ij}\in
S_{n},\ V_{i}=Fv_{i}$, and for any $f=f(t_{1},\ldots,t_{m})\in F\langle
t_{1},\ldots,t_{m}\rangle$ and $v\in{\mit\Gamma}_{1}$, $f\cdot
v=f(\bar{t}_{1},\ldots,\bar{t}_{m})\otimes v$.

##  7 Acknowledgements

The paper was written during I. Shestakov’s visite to the Shenzhen
International Mathematical Center of Southern University of Science and
Technology (SUSTech). He thanks Professor Efim Zelmanov for the invitation and
the Mathematical department of SUSTech for the support and hospitality. He was
also supported by the Brazilian grants FAPESP 2018/23690-6 and CNPq
304313/2019-0. A.Grishkov acknowledges financial support from FAPESP, grant
2018/23690-6 (Brazil)(1-3) and from RSF, grant 22-21-00745 (Russia)(4-5).

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A. Grishkov, University of São Paulo (São Paulo, Brazil) and Dostoevsky Omsk
State University (Omsk, Russia).
e-mail<EMAIL_ADDRESS>
I. Shestakov, University of São Paulo (São Paulo, Brazil), Sobolev Institute
of Mathematics (Novosibirsk, Russia), and SICM, Southern University of Science
and Technology (Shenzhen, China).
e-mail<EMAIL_ADDRESS>