# Using a Penalized Likelihood to Detect Mortality Deceleration

Silvio C. Patricio<EMAIL_ADDRESS>The Interdisciplinary Centre on Population
Dynamics, University of Southern Denmark Trifon I. Missov  The
Interdisciplinary Centre on Population Dynamics, University of Southern
Denmark

###### Abstract

In this paper, we suggest a novel method for detecting mortality deceleration.
We focus on the gamma-Gompertz frailty model and suggest the subtraction of a
penalty in the log-likelihood function as an alternative to traditional
likelihood inference and hypothesis testing. Over existing methods, our method
offers advantages, such as avoiding using a p-value, hypothesis testing, and
asymptotic distributions. We evaluate the performance of our approach by
comparing it with traditional likelihood inference on both simulated and real
mortality data. Results have shown that our approach is more accurate in
detecting mortality deceleration and provides more reliable estimates of the
underlying parameters. The proposed method is a significant contribution to
the literature as it offers a powerful tool for analyzing mortality patterns.
Keywords: Gompertz model; gamma-Gompertz model, mortality deceleration;
penalized likelihood function; maximum a posteriori probability.

##  1 Introduction

Human death-rate patterns are astoundingly log-linear over a wide range of
adult ages. The Gompertz distribution (Gompertz, , 1825) with an exponentially
increasing hazard function captures this accurately. The theory of unobserved
heterogeneity and the associated frailty model (Vaupel et al., , 1979)
predicts a downward deviation at the oldest ages, to which only the most
robust individuals in the population survive. Detecting such a deceleration in
real data is not always successful (Gavrilova and Gavrilov, , 2015; Newman, ,
2018), even though the vast majority of studies indicate that death rates at
older ages increase at lower rates and can even level off (Curtsinger et al.,
, 1992; Fukui et al., , 1993, 1996; Carey et al., , 1995; Khazaeli et al., ,
1998; Gampe, , 2010, 2021; Rootzén and Zholud, , 2017; Alvarez et al., , 2021;
Camarda, , 2022; Belzile et al., , 2022). In a frailty model setting, testing
for mortality deceleration is equivalent to testing whether the non-negative
frailty parameter is strictly positive.

Formally, denote by $X$ a non-negative continuous random variable that
describes individual human lifespans (complete or after a given adult age). If
$X\sim$ Gompertz$(a,b)$, where $a$ is the mortality level at the initial age
and $b$ is the rate of aging, the associated hazard function (force of
mortality) at time $x$

$\mu(x)=\lim_{\varepsilon\downarrow 0}\mathbb{P}(x\leq X<x+\varepsilon|X\geq
x).$

is $\mu(x)=ae^{bx}$ . Vaupel et al.,  (1979) introduce a positive continuous
random variable $Z$, called frailty, that acts multiplicatively on $\mu(x)$
and captures one’s unobserved susceptibility to death. The force of mortality
for an individual with frailty $Z=z$ is

$\mu(x\,|\,Z=z)=z\,\mu(x)\,.$

For a gamma-distributed frailty with $\mathbb{E}(Z)=1$ and
$\mathbb{VAR}(Z)=\sigma^{2}$, the force of mortality of the population, i.e.,
the marginal hazard is

$\bar{\mu}(x)=\frac{ae^{bx}}{1+\sigma^{2}\frac{a}{b}\left(e^{bx}-1\right)}$
(1)

(see Vaupel et al.,  (1979) and Vaupel and Missov,  (2014) for all
technicalities). Note that the variance of $Z$ is often denoted by $\gamma$
(e.g., in Vaupel and Missov, , 2014) because it is also equal to the squared
coefficient of variation of the distribution of frailty among survivors to any
age $x$. If $\sigma^{2}>0$, the force of mortality for the population
$\bar{\mu}(x)$ starts deviating from the exponential pattern with increasing
$x$ and reaches an asymptote $b/\sigma^{2}$. When $\sigma^{2}=0$, i.e., when
there is no unobserved heterogeneity, the model for the population reduces to
the (Gompertz) model for individuals with an exponentially increasing hazard
function $\mu(x)=ae^{bx}$.

Testing for mortality deceleration in this setting reduces to statistical
testing whether $\sigma^{2}=0$ given the alternative $\sigma^{2}>0$. The
frailty parameter $\sigma^{2}$ can take a value on the boundary of the
parameter space ($\sigma^{2}=0$). This violates the standard underlying
assumptions about the asymptotic properties of likelihood-based inference and
statistical hypothesis testing (see, for example, Böhnstedt and Gampe, ,
2019). As a result, the asymptotic distribution of the maximum likelihood
estimator may not be Gaussian.

In this paper, we treat the problem of identifying whether $\sigma^{2}>0$ or
$\sigma^{2}=0$ as a model misspecification problem, i.e., we consider the
gamma-Gompertz model when it is the Gompertz model that actually holds. In
this setting, we suggest subtracting a penalty from the log-likelihood
function. This penalty will be responsible for shrinking $\sigma^{2}$ to zero
when there is no heterogeneity, as well as for adding a small bias to the
Maximum Likelihood Estimator (MLE) when the effect of unobserved heterogeneity
is non-negligible. We carry out Monte Carlo simulation experiments to evaluate
the accuracy and precision of the estimates obtained by maximizing the
likelihood function, on the one hand, and the penalized likelihood function,
on the other.

In Section 2, we formulate the model misspecification problem and introduce
inference methodology taking advantage of the maximum a posteriori probability
(MAP). Then we carry out a Monte Carlo simulation study to compare the
performance of maximizing a standard and a penalized likelihood. In Section 3,
we compare the latter on mortality data for France, Japan and the USA. Section
4 discusses the advantages and drawbacks of applying our method to detect
heterogeneity (deceleration) in mortality patterns.

##  2 Methodology

Suppose $\bm{X}$ is a random sample with a cumulative distribution function
$G(x)$, and we fit the incorrect family of densities
$\\{f(x;\bm{\theta}),\bm{\theta}\in\Theta\\}$ to the data using MLE. The
misspecified log-likelihood is

$\ell(\bm{\theta};\bm{X})=\sum_{i=1}^{n}\log f(X_{i};\bm{\theta}).$

Applying the law of large numbers, we get in the limit what the misspecified
log-likelihood function $\ell(\bm{\theta};\bm{X})$ looks like for each
$\bm{\theta}\in\Theta$ (see the right-hand side below):

$\frac{1}{n}\ell(\bm{\theta};\bm{X})=\frac{1}{n}\sum_{i=1}^{n}\log
f(X_{i};\bm{\theta})\xrightarrow[]{a.s}\mathbb{E}_{g}\left(\log
f(X_{1};\bm{\theta})\right)=\int_{\text{Im}(X_{1})}\log
f(x;\bm{\theta})dG(x)\,.$ (2)

Assume there is no heterogeneity in the data ($\sigma^{2}=0$), and we fit a
gamma-Gompertz model. In other words, we observe an exponential death-rate
increase in the data, but we estimate a model that implies a downward
deviation from the exponential at the oldest ages. As shown in (2), we will
estimate $\sigma^{2}$ close to but never equal to zero.

In this model setting, the standard technique is to estimate both the Gompertz
and the gamma-Gompertz models and compare their goodness of fit. However,
minor changes in the data can result in different models being selected, which
can reduce prediction accuracy and lead to misinterpretations about the
mortality deceleration and the mortality plateau. Böhnstedt and Gampe,  (2019)
derive the asymptotic distribution of the likelihood ratio test statistic to
detect heterogeneity. Here, we would like to suggest an alternative that does
not involve hypothesis testing. Using the latter has been widely discussed and
rethought in the Statistics community (Berk et al., , 2010; Head et al., ,
2015; Vidgen and Yasseri, , 2016; Bruns and Ioannidis, , 2016), especially in
relation to the arbitrary choice of the $\alpha$-level (most often 0.1, 0.05,
or 0.01) and sample size issues.

Maximum likelihood estimators, obtained by maximizing the log-likelihood
function, often have low bias and large variance. Estimation accuracy can
sometimes be improved by shrinking some parameters to zero (Tibshirani, ,
1996). The associated shrinkage estimator improves the overall prediction
accuracy at the expense of introducing a small bias to reduce the variance of
the parameters. This class of estimators is implicit in Bayesian inference and
penalized likelihood inference. Using shrinkage estimators is applied as an
alternative to hypothesis testing. Lasso, Ridge and Stein-type estimators are
the most widely used examples of penalizing methods (see, for example, Hastie
et al., , 2009).

###  2.1 Inference

Let $D_{x}$ be the number of deaths in a given age interval $[x,x+1)$ for
$x=0,\ldots,m$, and $E_{x}$ denote the number of person-years lived in the
same interval (see, for example Brillinger, , 1986; Macdonald et al., , 2018).
Define $\bm{D}=(D_{0},D_{1},\dots,D_{m})^{\top}$ and
$\bm{E}=(E_{0},E_{1},\dots,E_{m})^{\top}$. In addition, let
$\bm{\theta}=(a,b,\sigma^{2})^{\top}\in\Theta$ be the parameter vector that
characterizes the force of mortality at age $x$ of the gamma-Gompertz model
given by (1).

Assume $D_{x}$ are Poisson-distributed with
$\mathbb{E}(D_{x})=\mathbb{VAR}(D_{x})=\mu(x;\bm{\theta})E_{x}$ for
$x=0,\ldots,m$ (Brillinger, , 1986). Under this assumption, the log-likelihood
function for $\bm{\theta}=(a,b,\sigma^{2})^{\top}$ is given by

$\ell(\bm{\theta})=\ell(\bm{\theta}|\bm{D},\bm{E})=\sum_{x=0}^{m}\left[D_{x}\ln\mu(x;\bm{\theta}))-E_{x}\,\mu(x;\bm{\theta})\right].$
(3)

Maximizing $\ell(\bm{\theta})$ with respect to
$\bm{\theta}=(a,b,\sigma^{2})^{\top}$ yields the maximum-likelihood (ML)
estimate $\hat{\bm{\theta}}$.

Let us now define a penalized log-likelihood function as

$\ell_{p}(\bm{\theta})=\ell(\bm{\theta})-p(\sigma^{2})\,,$ (4)

where $\ell(\bm{\theta})$ is the standard log-likelihood (3), while
$p(\sigma^{2})$ is a penalty function. The penalized maximum-likelihood
estimate is obtained by maximizing $\ell_{p}(\bm{\theta})$ with respect to
$\bm{\theta}=(a,b,\sigma^{2})^{\top}$. For the problem addressed in this
paper, the penalty function $p(\sigma^{2})$ must be a non-decreasing monotonic
continuous function and $\displaystyle{\lim_{\sigma^{2}\downarrow
0}p(\sigma^{2})}>p(\sigma^{2})$ for all $\sigma^{2}>0$.

In a Bayesian framework, maximizing (4) is equivalent to maximizing a
posterior distribution in a setting, in which ${\rm
e}^{-p(\sigma^{2})}/C_{p}$, $C_{p}:=\int_{\Theta}{\rm
e}^{-p(\sigma^{2})}\nabla\bm{\theta}<\infty$, is taken as a prior distribution
of $\bm{\theta}$. This procedure yields the maximum a posteriori probability
(MAP) estimator. MAP is the only Bayesian estimator that minimizes the
expected canonical loss (Pereyra, , 2019) and is widely used in image and
video processing (Greig et al., , 1989; Afonso et al., , 2010; Belekos et al.,
, 2010).

As $\sigma^{2}$ describes the variance of frailty at the starting age of
analysis, the standard approach would be to specify an inverse gamma prior
distribution for it (Gelman et al., , 1995). The inverse gamma distribution is
heavy-tailed and keeps probability mass further from zero than the gamma
distribution. In addition, while the inverse-gamma mode is always positive,
the gamma mode can also be zero (Llera and Beckmann, , 2016). As we aim to
test whether $\sigma^{2}=0$ or $\sigma^{2}>0$, we will use the log-kernel of
the gamma distribution to define the penalty function as

$p(\sigma^{2})=\lambda\left(\sigma^{2}+\ln\sigma^{2}\right)$ (5)

for some non-negative $\lambda$. When $\lambda<1$, using (5) is equivalent to
specifying a gamma prior distribution for $\sigma^{2}$ with parameters
$\alpha=1-\lambda$ and $\beta=\lambda$.

When $m\to\infty$, the effect of the penalty diminishes regardless of the size
of $\lambda$. For human life table data $m$ is finite, thus $\lambda\geq 0$ is
a constant that controls the relative impact of the penalty function on the
estimates. When $\lambda=0$, the penalty term has no effect, and maximizing
the penalized likelihood will produce the standard maximum likelihood
estimates (MLE). However, as $\lambda\rightarrow\infty$, the impact of the
penalty grows, and the maximum penalized likelihood estimates for $\sigma^{2}$
will approach zero, providing high precision, but low accuracy.

Choosing $\lambda$ is sensible in a wide range of applications (Li et al., ,
2009; Bhattacharya and McNicholas, , 2014). Therefore, in accordance with the
recommendations in Li et al.,  (2009), we carry out a pilot simulation study,
in which we find that choosing $\lambda=\frac{1}{2}$ provides similar
precision to the one by MLE when $\sigma^{2}>0$, but better accuracy and
precision when $\sigma^{2}=0$ (simulation results are presented in the next
subsection). As a result, the final expression for the penalized log-
likelihood we propose is

$\displaystyle\ell_{p}(\bm{\theta})=\sum_{x=0}^{m}\left[D_{x}\ln\mu(x;\bm{\theta})-E_{x}\,\mu(x;\bm{\theta})\right]-\frac{1}{2}\left(\ln\sigma^{2}+\sigma^{2}\right).$
(6)

From a Bayesian perspective, choosing $\lambda=\frac{1}{2}$ provides an
informative prior distribution for $\sigma^{2}$. As for human populations we
are likely to estimate $\sigma^{2}<1$ (Missov, , 2013), the specified prior
will provide for $\sigma^{2}$ a distribution with a mode equal to zero, a
median equal to $0.4549$, and mean equal to $1$. Furthermore, the prior
provides a probability mass of $0.6826$ in the interval $(0,1]$.

Figure 1: Plots of the profile log-likelihood (first row) and penalized log-
likelihood functions (second row) of the parameters. In the first row we used
synthetic data from a gamma-Gompertz model with parameters $a=0.0001$, $b=0.1$
and $\sigma^{2}=0.1$, in the second row we from a Gompertz model with
parameters $a=0.0001$ and $b=0.1$.

Figure 1 shows the log-likelihood and penalized log-likelihood functions for
all parameters when $\sigma^{2}>0$ (first row) and $\sigma^{2}=0$ (second
row). When $\sigma^{2}>0$, the penalty function affects neither the shape of
the log-likelihood nor the location of its maximum. However, when
$\sigma^{2}=0$, adding a penalty yields a higher maximum at 0. Moreover, when
$\sigma^{2}=0$, the first and second derivatives of the penalized log-
likelihood are higher than their respective counterparts of the log-
likelihood. As a result, derivative-based optimization methods may reach the
maximum point faster, and the estimator $\hat{\sigma^{2}}$ may have a smaller
variance.

###  2.2 Monte Carlo simulations

We carry out Monte Carlo simulations to explore the performance of the MAP and
ML methods in estimating the gamma-Gompertz model parameters. We use the R
software (Team et al., , 2022) to maximize the log-likelihood and the
penalized log-likelihood functions via the optim function applying as a pre-
step differential evolution (Storn and Price, , 1997; Ardia et al., , 2011).
The performance of the ML and MAP estimators are evaluated by calculating two
measures: the bias and the standard deviation.

We generate 10,000 random samples from this model for some parameter values
(scenarios with sample sizes of 2,000 and 5,000 were also considered, and are
presented in the appendix). From these samples, we generate life tables and
use them to estimate model parameters via the MAP and MLE methods. This
process was repeated 2,000 times. In the presence of unobserved heterogeneity,
the true parameter values are $a_{1}=0.0001$ and $a_{2}=0.00001$ for $a$,
$b_{1}=0.1$ and $b_{2}=0.15$ for $b$, and $\sigma^{2}_{1}=0.2$ and
$\sigma^{2}_{2}=0.8$ for $\sigma^{2}$. When there is no heterogeneity
($\sigma^{2}=0$), the true parameter values are $a_{1}=0.0001$, $a_{2}=0.0003$
and $a_{3}=0.0005$ for $a$, and $b_{1}=0.09$, $b_{2}=0.10$ and $b_{3}=0.11$
for $b$.

Table 1: Simulation results: gamma-Gompertz model and sample size 10,000.

| There is heterogeneity
---|---
| MLE estimator | MAP estimator
| Bias | Standard deviation | Bias | Standard deviation
Parameter | $a$ | $b$ | $\sigma^{2}$ | $a$ | $b$ | $\sigma^{2}$ | $a$ | $b$ | $\sigma^{2}$ | $a$ | $b$ | $\sigma^{2}$
$(a_{1},b_{1},\sigma^{2}_{1})$ | 0.000053 | -0.000051 | -0.000223 | 0.000051 | 0.001499 | 0.020721 | 0.000055 | -0.000134 | -0.001626 | 0.000052 | 0.001502 | 0.020791
$(a_{1},b_{1},\sigma^{2}_{2})$ | 0.000060 | -0.000292 | -0.007787 | 0.000056 | 0.001784 | 0.035822 | 0.000061 | -0.000354 | -0.009229 | 0.000056 | 0.001783 | 0.035795
$(a_{1},b_{2},\sigma^{2}_{1})$ | 0.000077 | 0.000131 | 0.004431 | 0.000056 | 0.002181 | 0.020569 | 0.000080 | 0.000015 | 0.003096 | 0.000057 | 0.002186 | 0.020631
$(a_{1},b_{2},\sigma^{2}_{2})$ | 0.000085 | -0.000262 | -0.003547 | 0.000061 | 0.002557 | 0.034714 | 0.000087 | -0.000348 | -0.004920 | 0.000061 | 0.002556 | 0.034687
$(a_{2},b_{1},\sigma^{2}_{1})$ | 0.000007 | -0.000349 | -0.003349 | 0.000008 | 0.001315 | 0.019464 | 0.000008 | -0.000417 | -0.004597 | 0.000008 | 0.001318 | 0.019523
$(a_{2},b_{1},\sigma^{2}_{2})$ | 0.000009 | -0.000635 | -0.013592 | 0.000008 | 0.001515 | 0.032551 | 0.000009 | -0.000683 | -0.014810 | 0.000008 | 0.001514 | 0.032528
$(a_{2},b_{2},\sigma^{2}_{1})$ | 0.000009 | -0.000170 | 0.002295 | 0.000008 | 0.001963 | 0.019377 | 0.000009 | -0.000268 | 0.001078 | 0.000008 | 0.001966 | 0.019430
$(a_{2},b_{2},\sigma^{2}_{2})$ | 0.000011 | -0.000650 | -0.007809 | 0.000009 | 0.002273 | 0.032534 | 0.000011 | -0.000721 | -0.009014 | 0.000009 | 0.002272 | 0.032510
| There is no heterogeneity
| MLE estimator | MAP estimator
| Bias | Standard deviation | Bias | Standard deviation
Parameter | $a$ | $b$ | $\sigma^{2}$ | $a$ | $b$ | $\sigma^{2}$ | $a$ | $b$ | $\sigma^{2}(10^{-16})$ | $a$ | $b$ | $\sigma^{2}(10^{-15})$
$(a_{1},b_{1},\sigma^{2})$ | 0.000005 | -0.000055 | 0.000181 | 0.000006 | 0.000875 | 0.008460 | 0.000007 | -0.000239 | 0.125942 | 0.000006 | 0.000791 | 0.6308937
$(a_{1},b_{2},\sigma^{2})$ | 0.000006 | -0.000050 | 0.000222 | 0.000006 | 0.000968 | 0.008907 | 0.000007 | -0.000407 | 0.060433 | 0.000006 | 0.000870 | 0.1460452
$(a_{1},b_{3},\sigma^{2})$ | 0.000006 | -0.000057 | 0.000428 | 0.000006 | 0.001082 | 0.009726 | 0.000008 | -0.000305 | 0.089035 | 0.000006 | 0.000978 | 8.212895
$(a_{2},b_{1},\sigma^{2})$ | 0.000011 | 0.000138 | 0.001864 | 0.000016 | 0.000913 | 0.009010 | 0.000016 | -0.000198 | 0.004470 | 0.000015 | 0.000811 | 0.294978
$(a_{2},b_{2},\sigma^{2})$ | 0.000014 | 0.000117 | 0.001094 | 0.000016 | 0.001042 | 0.009873 | 0.000019 | -0.000264 | 0.003661 | 0.000015 | 0.000859 | 4.922260
$(a_{2},b_{3},\sigma^{2})$ | 0.000015 | 0.000164 | 0.002204 | 0.000016 | 0.001150 | 0.010369 | 0.000022 | -0.000216 | 0.007167 | 0.000016 | 0.001017 | 3.551245
$(a_{3},b_{1},\sigma^{2})$ | 0.000018 | 0.000143 | 0.001062 | 0.000025 | 0.000979 | 0.009712 | 0.000027 | -0.000138 | 0.001124 | 0.000025 | 0.000876 | 1.898573
$(a_{3},b_{2},\sigma^{2})$ | 0.000024 | 0.000076 | 0.000505 | 0.000025 | 0.001057 | 0.009721 | 0.000030 | -0.000112 | 0.001177 | 0.000025 | 0.000942 | 8.114498
$(a_{3},b_{3},\sigma^{2})$ | 0.000025 | 0.000117 | 0.001427 | 0.000025 | 0.001178 | 0.009999 | 0.000033 | -0.000171 | 0.001415 | 0.000024 | 0.000990 | 0.000025

The simulation results are presented in Table 1. In the presence of unobserved
heterogeneity, both methods underestimate $b$ and $\sigma^{2}$. They also
introduce a small positive bias to $a$, the one provided by ML estimator being
slightly smaller. However, in general the ML and MAP estimators perform
equally well, with a similar bias and standard deviation.

In the absence of unobserved heterogeneity, the ML estimator provides again a
smaller bias for $a$ and $b$ than the MAP estimator. However, in this case,
the MAP method estimates more precisely the frailty parameter $\sigma^{2}$,
with a bias and a standard deviation close to zero ($\propto 10^{-15}$). The
MAP estimator also provides a slight reduction in the standard deviation of
parameter $b$.

By the Monte Carlo simulation we also calculate the proportion of trials in
which MAP estimates $\sigma^{2}>0$ when the true values is $\sigma^{2}=0$
(error type I), as well as the proportion of trials in which MAP estimates
$\sigma^{2}=0$ when the true values is $\sigma^{2}>0$ (error type II). Based
on our simulations, the type I errro equals $0.001502$, while the type II
error is $0.001126$.

The Monte Carlo simulations show that using a penalizing likelihood function
(6) is an alternative to hypothesis testing, the latter being dependent on the
asymptotic distribution of the ML estimator, sample size and the arbitrary
choice of the $\alpha$-level (Böhnstedt and Gampe, , 2019).

##  3 Performance of MAP and ML estimators on HMD data

In this section, we estimate the gamma-Gompertz model via ML and MAP using
mortality data from the Human Mortality Database (HMD, , 2022). We take
exposures and raw death counts for the female population of France, Japan and
the USA in the years 1960, 1980, 2000, and 2020, after age 70. We apply again
R (Team et al., , 2022) to compute the ML and MAP estimates of
$\bm{\theta}=(a,b,\sigma^{2})^{\prime}$ by using differential evolution. We
use the mean squared error given by

$MSE=\frac{1}{n}\sum_{x=0}^{m}\left(\ln
m_{x}-\ln\bar{\mu}(x;\hat{\bm{\theta}})\right)^{2},$

to assess the goodness of fit.

Table 2: Life expectancy: gamma Gompertz–Makeham model and ML estimates.

|  | ML Estimates | MAP Estimates
---|---|---|---
Country | Year | $a$ | $b$ | $\sigma^{2}$ | MSE | $a$ | $b$ | $\sigma^{2}$ | MSE
France | 1960 | 0.003582 | 0.107599 | 0.016726 | 0.100588 | 0.003593 | 0.107483 | 0.015902 | 0.100540
| 1980 | 0.002210 | 0.112032 | 0.003393 | 0.045776 | 0.002220 | 0.111729 | 0.003117 | 0.046747
| 2000 | 0.001247 | 0.117749 | 0.000001 | 0.073648 | 0.001250 | 0.117689 | 0 | 0.073262
| 2020 | 0.000957 | 0.119494 | 0.000002 | 0.094243 | 0.000960 | 0.119416 | 0 | 0.093697
Japan | 1960 | 0.004782 | 0.105858 | 0.039989 | 0.011366 | 0.004782 | 0.105845 | 0.039593 | 0.011493
| 1980 | 0.002009 | 0.117886 | 0.015251 | 0.053944 | 0.002009 | 0.117941 | 0.015942 | 0.053328
| 2000 | 0.001140 | 0.115268 | 0.000015 | 0.064604 | 0.001142 | 0.115118 | 0 | 0.063728
| 2020 | 0.000575 | 0.125814 | 0.000233 | 0.104944 | 0.000574 | 0.125870 | 0.000225 | 0.105597
USA | 1960 | 0.004701 | 0.095797 | 0.032146 | 0.111711 | 0.004699 | 0.095814 | 0.032802 | 0.110740
| 1980 | 0.003612 | 0.093688 | 0.000001 | 0.054483 | 0.003609 | 0.093720 | 0 | 0.054652
| 2000 | 0.002712 | 0.100566 | 0.000003 | 0.030967 | 0.002714 | 0.100540 | 0 | 0.030873
| 2020 | 0.002473 | 0.101652 | 0.000001 | 0.022735 | 0.002476 | 0.101612 | 0 | 0.022618

Table 2 shows the results of applying ML and MAP methods to the datasets
described above. The MAP estimator provides lower MSEs in 8 of the 12
datasets. When the standard ML method estimates $\sigma^{2}<10^{-4}$, our
novel method estimates $\sigma^{2}=0$ and provides a smaller MSE. This
suggests that the MAP provides a slightly better fit to the data. Overall, MAP
performs better than ML when unobserved heterogeneity is not detected, and
while for estimates of $\hat{\sigma}^{2}>0$ ML has a slight advantage.

The results from the real-data application back up the results from the Monte
Carlo simulations in Section 2. In the presence of unobserved heterogeneity,
the MLE method provides the most precise and accurate estimates. The MAP
method, though, has just slightly lower precision. On the other hand, in the
absence of unobserved heterogeneity, the MAP provides a smaller bias and
variance in its estimates compared to MLE.

###  3.1 Examples when MAP and ML estimators yield different outcomes

Using MAP and ML estimators does not always lead to the same statistical
inference. One of them can detect heterogeneity in cases when the other does
not. We will illustrate this on HMD data for the Japanese female population in
2009 and the French female population born in 1848, ages 70+. To assess the
goodness of fit, we will use MSE again.

For Japanese females in 2009, ML yields estimates
$\hat{\bm{\theta}}_{MLE}=(0.006359,0.133805,0.070513)^{\prime}$ with standard
errors $SE(a)=0.000188$, $SE(b)=0.002263$ and $SE(\sigma^{2})=0.021156$. The
95% confidence interval for $\sigma^{2}$ is $(0.029047,0.111978)$ indicating
stasitically significant unobserved heterogeneity, i.e., the existence of
mortality deceleration. On the other hand, the MAP method estimates
$\hat{\bm{\theta}}_{MAP}=(0.006966,0.125440,0)^{\prime}$, indicating the
absence of unobserved heterogeneity. Comparing the goodness of fit of both
methods speaks in favor of the MAP outcome: MAP’s MSE is by 37% lower than
ML’s LSE (0.018691 for MAP vs 0.029958 for ML). It indicates that unobserved
heterogeneity is negligible and that the gamma-Gompertz model is misspecified.

Figure 2: MAP and MLE estimates of the force of mortality for the Japanese
population in 2009 and the Swedish population born in 1881, after age 70

The left panel of Figure 2 shows that both methods estimate a similar
logarithmic force of mortality at most ages. However, after age 100, the MLE
deviates downward from the observed logarithmic death rates.

The MAP also provides a better fit and different conclusion for the cohort of
French females born in 1848. While ML estimates
$\hat{\bm{\theta}}_{MLE}=(0.053748,0.090552,0.008604)^{\prime}$ with
$SE(a)=0.000317$, $SE(b)=0.001273$, $SE(\sigma^{2})=0.007562$ and provides an
MSE equal 0.046222, MAP estimates
$\hat{\bm{\theta}}_{MAP}=(0.053113,0.094921,0.036466)^{\prime}$ and provides
$MSE=0.034226$, i.e., MAP’s MSE is by 26% smaller than ML’s MSE.

Furthermore, while the MAP estimate of $\sigma^{2}$ suggests that there is
non-negligible unobserved heterogeneity, the ML estimate and standard error
for $\sigma^{2}$ indicates the opposite: the amount of unobserved
heterogeneity is not statistically significant. The right panel of Figure 2
shows the difference between these estimates. MAP’s estimate shows a leveling-
off in the force of mortality, while the MLE shows a log-linear increase in
the hazard function.

###  3.2 Comparison between MAP and ML estimators for different populations

To evaluate and compare empirically the performance of the ML and MAP methods
we are going to use them to estimate the force of mortality for the male and
female populations of France, Denmark, Sweden, Italy, Japan, Czechia, and the
United States of America from 1950 to 2019, resulting in 980 populations. To
access the goodness of fit we are using the MSE.

Figure 3 presents the method that provides a better fit (which has the
smallest MSE). Overall both methods provide similar goodness of fit, with MAP
providing a slightly lower MSE (on average 0.5% smaller than the ML method).
Over the 980 populations, the MAP provides a better fit for 502 of them. For
Czechia, Sweden, and France, the MAP also provided a slightly better fit than
the ML method; however, the MSE difference is small within the country.

Figure 3: Method providing the best goodness of fit.

We also estimate the standard error for the ML estimates of $\sigma^{2}$ to
test if the parameter is not statistically significantly different from zero.
The results were compared with the MAP estimate. Figure 4 presents for which
populations $\sigma^{2}$ is zero through the ML and MAP method. From both
methods, it is clear that the non-estimation of unobserved heterogeneity is
related to the male populations - especially for the Danes. It might be caused
by the small male population surviving after age 105.

In about 96% of the populations, the methods agree on the significance or non-
significance of $\sigma^{2}$. However, when the MAP estimates $\sigma^{2}=0$
the ML provided positive confidence intervals for about 5.1948% of the cases,
which corresponds to the chosen $\alpha$-level (probability of Type I error)
of 5% for those intervals.

Figure 4: Statistical non-significance of $\sigma^{2}$

##  4 Concluding remarks

Böhnstedt and Gampe,  (2019) introduced a formal procedure to identify whether
$\sigma^{2}>0$ or $\sigma^{2}=0$ in a hypothesis testing setting: they studied
the asymptotic properties of the maximum likelihood estimator and the
likelihood ratio test (LRT) for $H_{0}:\sigma^{2}=0$ vs. $H_{1}:\sigma^{2}=0$
for the gamma-Gompertz model. However, LRTs are based on the asymptotic
distribution of the maximum likelihood estimator, hence its convergence
depends on the sample size. Moreover, conclusions drawn from hypothesis tests
are dependent on the arbitrary choice of the significance level or $p$-value.

We suggest an alternative method by considering the problem as model
misspecification. We add a penalty function to the likelihood so that we make
sure that $\hat{\sigma}^{2}=0$ when there is no heterogeneity. We also present
a Bayesian interpretation (MAP) to our method. We take advantage of robust
Monte Carlo simulations to measure the bias and standard deviation of the ML
and MAP methods in scenarios with and without unobserved heterogeneity. We
also compare the performance of both methods for estimating the gamma-Gompertz
model parameters using actual mortality data from the Human Mortality
Database. The two methods work almost equally well, the ML having a slight
advantage, in the presence of unobserved heterogeneity. However, in the
absence of the latter, the MAP method provides an estimate closer to 0
($\hat{\sigma}^{2}\approx 10^{-20}$) and a better fit to the model in
comparison to ML. As a result, the method we propose here can be used as an
alternative to likelihood ratio testing for the gamma-Gompertz model with
$H_{0}:\sigma^{2}=0$ vs. $H_{1}:\sigma^{2}>0$. On the one hand, the MAP method
does not depend on any asymptomatic distribution, its performance is not
strongly affected by sample size, and it also does not depend on the arbitrary
choice of the significance level. On the other hand, MAP provides similar
estimates to the ones by ML when $\sigma^{2}>0$ and more accurate estimates
when $\sigma^{2}=0$.

## Acknowledgments

The research leading to this publication is a part of a project that has
received funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (Grant agreement No.
884328 – Unequal Lifespans). Silvio C. Patricio gratefully acknowledges the
support provided from AXA Research Fund, through the funding for the “AXA
Chair in Longevity Research”.

## References

  * Afonso et al., (2010) Afonso, M. V., Bioucas-Dias, J. M., and Figueiredo, M. A. (2010).  An augmented lagrangian approach to the constrained optimization formulation of imaging inverse problems.  IEEE transactions on image processing, 20(3):681–695.
  * Alvarez et al., (2021) Alvarez, J.-A., Villavicencio, F., Strozza, C., and Camarda, C. G. (2021).  Regularities in human mortality after age 105.  PloS one, 16(7):e0253940.
  * Ardia et al., (2011) Ardia, D., Boudt, K., Carl, P., Mullen, K., and Peterson, B. G. (2011).  Differential evolution with deoptim: an application to non-convex portfolio optimization.  The R Journal, 3(1):27–34.
  * Belekos et al., (2010) Belekos, S. P., Galatsanos, N. P., and Katsaggelos, A. K. (2010).  Maximum a posteriori video super-resolution using a new multichannel image prior.  IEEE Transactions on Image Processing, 19(6):1451–1464.
  * Belzile et al., (2022) Belzile, L., Davison, A., Gampe, J., Rootzen, H., and Zholud, D. (2022).  Is there a cap on longevity? a statistical review.  Annual Review of Statistics and Its Application, 9:21–45.
  * Berk et al., (2010) Berk, R., Brown, L., and Zhao, L. (2010).  Statistical inference after model selection.  Journal of Quantitative Criminology, 26(2):217–236.
  * Bhattacharya and McNicholas, (2014) Bhattacharya, S. and McNicholas, P. D. (2014).  A lasso-penalized bic for mixture model selection.  Advances in Data Analysis and Classification, 8(1):45–61.
  * Böhnstedt and Gampe, (2019) Böhnstedt, M. and Gampe, J. (2019).  Detecting mortality deceleration: Likelihood inference and model selection in the gamma-gompertz model.  Statistics & Probability Letters, 150:68–73.
  * Brillinger, (1986) Brillinger, D. R. (1986).  A biometrics invited paper with discussion: the natural variability of vital rates and associated statistics.  Biometrics, pages 693–734.
  * Bruns and Ioannidis, (2016) Bruns, S. B. and Ioannidis, J. P. (2016).  P-curve and p-hacking in observational research.  PloS one, 11(2):e0149144.
  * Camarda, (2022) Camarda, C. G. (2022).  The curse of the plateau. measuring confidence in human mortality estimates at extreme ages.  Theoretical Population Biology, 144:24–36.
  * Carey et al., (1995) Carey, J., Liedo, P., and Vaupel, J. (1995).  Mortality dynamics of density in the mediterranean fruit fly.  Experimental Gerontology, 30(6):605–629.
  * Curtsinger et al., (1992) Curtsinger, J., Fukui, H., Townsend, D., and Vaupel, J. (1992).  Demography of genotypes: Failure of the limited life-span paradigm in drosophila melanogaster.  Science, 258:461–463.
  * Fukui et al., (1996) Fukui, H., Ackert, L., and Curtsinger, J. (1996).  Deceleration of age-specific mortality rates in chromosomal homozygotes and heterozygotes of drosophila melanogaster.  Experimental Gerontology, 36(4):517–531.
  * Fukui et al., (1993) Fukui, H., Curtsinger, J., and Xiu, L. (1993).  Slowing of age-specific mortality rates in drosophila melanogaster.  Experimental Gerontology, 28:585–599.
  * Gampe, (2010) Gampe, J. (2010).  Human mortality beyond age 110.  In Supercentenarians, pages 219–230. Springer.
  * Gampe, (2021) Gampe, J. (2021).  Mortality of supercentenarians: Estimates from the updated idl.  In Exceptional Lifespans, pages 29–35. Springer, Cham.
  * Gavrilova and Gavrilov, (2015) Gavrilova, N. S. and Gavrilov, L. A. (2015).  Biodemography of old-age mortality in humans and rodents.  Journals of Gerontology Series A: Biomedical Sciences and Medical Sciences, 70(1):1–9.
  * Gelman et al., (1995) Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1995).  Bayesian data analysis.  Chapman and Hall/CRC.
  * Gompertz, (1825) Gompertz, B. (1825).  Xxiv. on the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. in a letter to francis baily, esq. frs &c.  Philosophical transactions of the Royal Society of London, (115):513–583.
  * Greig et al., (1989) Greig, D. M., Porteous, B. T., and Seheult, A. H. (1989).  Exact maximum a posteriori estimation for binary images.  Journal of the Royal Statistical Society: Series B (Methodological), 51(2):271–279.
  * Hastie et al., (2009) Hastie, T., Tibshirani, R., Friedman, J. H., and Friedman, J. H. (2009).  The elements of statistical learning: data mining, inference, and prediction, volume 2.  Springer.
  * Head et al., (2015) Head, M. L., Holman, L., Lanfear, R., Kahn, A. T., and Jennions, M. D. (2015).  The extent and consequences of p-hacking in science.  PLoS biology, 13(3):e1002106.
  * HMD, (2022) HMD (2022).  The human mortality database.  http://www.mortality.org/.
  * Khazaeli et al., (1998) Khazaeli, A., Pletcher, S., and Curtsinger, J. (1998).  The fractionation experiment: reducing heterogeneity to investigate age-specific mortality in drosophila.  Mechanisms of Aging and Development, 105:301–317.
  * Li et al., (2009) Li, P., Chen, J., and Marriott, P. (2009).  Non-finite fisher information and homogeneity: an em approach.  Biometrika, 96(2):411–426.
  * Llera and Beckmann, (2016) Llera, A. and Beckmann, C. (2016).  Estimating an inverse gamma distribution.  arXiv preprint arXiv:1605.01019.
  * Macdonald et al., (2018) Macdonald, A. S., Richards, S. J., and Currie, I. D. (2018).  Modelling mortality with actuarial applications.  Cambridge University Press.
  * Missov, (2013) Missov, T. I. (2013).  Gamma-gompertz life expectancy at birth.  Demographic Research, 28:259–270.
  * Newman, (2018) Newman, S. J. (2018).  Errors as a primary cause of late-life mortality deceleration and plateaus.  PLoS biology, 16(12):e2006776.
  * Pereyra, (2019) Pereyra, M. (2019).  Revisiting maximum-a-posteriori estimation in log-concave models.  SIAM Journal on Imaging Sciences, 12(1):650–670.
  * Rootzén and Zholud, (2017) Rootzén, H. and Zholud, D. (2017).  Human life is unlimited–but short.  Extremes, 20(4):713–728.
  * Storn and Price, (1997) Storn, R. and Price, K. (1997).  Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces.  Journal of Global Optimization, 11:341–359.
  * Team et al., (2022) Team, R. C. et al. (2022).  R: A language and environment for statistical computing.
  * Tibshirani, (1996) Tibshirani, R. (1996).  Regression shrinkage and selection via the lasso.  Journal of the Royal Statistical Society: Series B (Methodological), 58(1):267–288.
  * Vaupel and Missov, (2014) Vaupel, J. and Missov, T. (2014).  Unobserved population heterogeneity: A review of formal relationships.  Demographic Research, 31(22):659–686.
  * Vaupel et al., (1979) Vaupel, J. W., Manton, K. G., and Stallard, E. (1979).  The impact of heterogeneity in individual frailty on the dynamics of mortality.  Demography, 16(3):439–454.
  * Vidgen and Yasseri, (2016) Vidgen, B. and Yasseri, T. (2016).  P-values: misunderstood and misused.  Frontiers in Physics, 4:6.

Table 3: Simulation results: gamma-Gompertz model and sample size 2,000.

There is heterogeneity
---
| MLE estimator | MAP estimator
| Bias | Standard deviation | Bias | Standard deviation
Parameter | $a$ | $b$ | $\sigma^{2}$ | $a$ | $b$ | $\sigma^{2}$ | $a$ | $b$ | $\sigma^{2}$ | $a$ | $b$ | $\sigma^{2}$
$(a_{1},b_{1},\sigma^{2}_{1})$ | 0.000089 | -0.001172 | -0.020361 | 0.000111 | 0.003286 | 0.047684 | 0.000104 | -0.001709 | -0.029261 | 0.000117 | 0.003465 | 0.051082
$(a_{1},b_{1},\sigma^{2}_{2})$ | 0.000110 | -0.002075 | -0.048634 | 0.000128 | 0.003972 | 0.078621 | 0.000118 | -0.002397 | -0.055984 | 0.000129 | 0.003970 | 0.078467
$(a_{1},b_{2},\sigma^{2}_{1})$ | 0.000104 | -0.000906 | -0.008883 | 0.000124 | 0.004826 | 0.047186 | 0.000120 | -0.001603 | -0.016820 | 0.000130 | 0.005032 | 0.049798
$(a_{1},b_{2},\sigma^{2}_{2})$ | 0.000124 | -0.001978 | -0.029154 | 0.000139 | 0.005738 | 0.077069 | 0.000133 | -0.002420 | -0.036104 | 0.000140 | 0.005733 | 0.076901
$(a_{2},b_{1},\sigma^{2}_{1})$ | 0.000030 | -0.003810 | -0.046694 | 0.000021 | 0.003024 | 0.045092 | 0.000034 | -0.004407 | -0.057147 | 0.000024 | 0.003388 | 0.052275
$(a_{2},b_{1},\sigma^{2}_{2})$ | 0.000038 | -0.005056 | -0.097144 | 0.000023 | 0.003338 | 0.070146 | 0.000039 | -0.005316 | -0.103429 | 0.000024 | 0.003333 | 0.069973
$(a_{2},b_{2},\sigma^{2}_{1})$ | 0.000027 | -0.003998 | -0.030152 | 0.000021 | 0.004259 | 0.043531 | 0.000030 | -0.004673 | -0.038167 | 0.000022 | 0.004481 | 0.046813
$(a_{2},b_{2},\sigma^{2}_{2})$ | 0.000032 | -0.005191 | -0.065676 | 0.000025 | 0.005158 | 0.075375 | 0.000034 | -0.005565 | -0.071836 | 0.000025 | 0.005146 | 0.075141
There is no heterogeneity
| MLE estimator | MAP estimator
| Bias | Standard deviation | Bias | Standard deviation
Parameter | $a$ | $b$ | $\sigma^{2}$ | $a$ | $b$ | $\sigma^{2}$ | $a$ | $b$ | $\sigma^{2}(10^{-12})$ | $a$ | $b$ | $\sigma^{2}(10^{-12})$
$(a_{1},b_{1},\sigma^{2})$ | 0.000015 | -0.001225 | 0.000002 | 0.000013 | 0.001615 | 0.006546 | 0.000016 | -0.001341 | 0.021800 | 0.000014 | 0.001693 | 0.000287
$(a_{1},b_{2},\sigma^{2})$ | 0.000014 | -0.001224 | 0.000002 | 0.000013 | 0.001833 | 0.007389 | 0.000016 | -0.001312 | 0.018541 | 0.000014 | 0.001858 | 0.004010
$(a_{1},b_{3},\sigma^{2})$ | 0.000014 | -0.001323 | 0.000002 | 0.000013 | 0.001998 | 0.008857 | 0.000015 | -0.001324 | 0.024195 | 0.000015 | 0.002098 | 0.001130
$(a_{2},b_{1},\sigma^{2})$ | 0.000028 | -0.000591 | 0.000003 | 0.000031 | 0.001703 | 0.009242 | 0.000031 | -0.000983 | 0.000923 | 0.000033 | 0.001666 | 0.000003
$(a_{2},b_{2},\sigma^{2})$ | 0.000029 | -0.000837 | 0.000003 | 0.000031 | 0.001857 | 0.010440 | 0.000031 | -0.000842 | 0.000731 | 0.000031 | 0.001804 | 0.000009
$(a_{2},b_{3},\sigma^{2})$ | 0.000030 | -0.000810 | 0.000004 | 0.000031 | 0.002067 | 0.011604 | 0.000035 | -0.001196 | 0.000701 | 0.000034 | 0.002086 | 0.000008
$(a_{3},b_{1},\sigma^{2})$ | 0.000034 | -0.000398 | 0.000003 | 0.000046 | 0.001739 | 0.011618 | 0.000037 | -0.000606 | 0.000220 | 0.000048 | 0.001681 | 0.000001
$(a_{3},b_{2},\sigma^{2})$ | 0.000036 | -0.000503 | 0.000005 | 0.000049 | 0.002062 | 0.013431 | 0.000040 | -0.000670 | 0.000151 | 0.000048 | 0.001901 | 0.000002
$(a_{3},b_{3},\sigma^{2})$ | 0.000040 | -0.000238 | 0.000008 | 0.000050 | 0.002247 | 0.014168 | 0.000045 | -0.000722 | 0.000108 | 0.000051 | 0.002061 | 0.000008

Table 4: Simulation results: gamma-Gompertz model and sample size 5,000.

There is heterogeneity
---
| MLE estimator | MAP estimator
| Bias | Standard deviation | Bias | Standard deviation
Parameter | $a$ | $b$ | $\sigma^{2}$ | $a$ | $b$ | $\sigma^{2}$ | $a$ | $b$ | $\sigma^{2}$ | $a$ | $b$ | $\sigma^{2}$
$(a_{1},b_{1},\sigma^{2}_{1})$ | 0.000067 | -0.000379 | -0.004470 | 0.000071 | 0.002078 | 0.029046 | 0.000071 | -0.000552 | -0.007370 | 0.000072 | 0.002090 | 0.029281
$(a_{1},b_{1},\sigma^{2}_{2})$ | 0.000070 | -0.000582 | -0.015178 | 0.000077 | 0.002506 | 0.049888 | 0.000074 | -0.000708 | -0.018078 | 0.000077 | 0.002504 | 0.049808
$(a_{1},b_{2},\sigma^{2}_{1})$ | 0.000090 | -0.000273 | 0.001254 | 0.000077 | 0.002969 | 0.028061 | 0.000095 | -0.000509 | -0.001483 | 0.000078 | 0.002982 | 0.028253
$(a_{1},b_{2},\sigma^{2}_{2})$ | 0.000094 | -0.000514 | -0.007569 | 0.000083 | 0.003583 | 0.048230 | 0.000097 | -0.000687 | -0.010324 | 0.000084 | 0.003579 | 0.048152
$(a_{2},b_{1},\sigma^{2}_{1})$ | 0.000014 | -0.001320 | -0.014568 | 0.000011 | 0.001786 | 0.026808 | 0.000014 | -0.001465 | -0.017208 | 0.000011 | 0.001795 | 0.027016
$(a_{2},b_{1},\sigma^{2}_{2})$ | 0.000015 | -0.001650 | -0.033574 | 0.000013 | 0.002344 | 0.049087 | 0.000016 | -0.001748 | -0.036030 | 0.000013 | 0.002342 | 0.049016
$(a_{2},b_{2},\sigma^{2}_{1})$ | 0.000014 | -0.001299 | -0.006177 | 0.000012 | 0.002706 | 0.026681 | 0.000015 | -0.001505 | -0.008713 | 0.000012 | 0.002717 | 0.026863
$(a_{2},b_{2},\sigma^{2}_{2})$ | 0.000015 | -0.001571 | -0.019479 | 0.000014 | 0.003352 | 0.046735 | 0.000016 | -0.001714 | -0.021904 | 0.000014 | 0.003349 | 0.046666
There is no heterogeneity
| MLE estimator | MAP estimator
| Bias | Standard deviation | Bias | Standard deviation
Parameter | $a$ | $b$ | $\sigma^{2}$ | $a$ | $b$ | $\sigma^{2}$ | $a$ | $b$ | $\sigma^{2}(10^{-12})$ | $a$ | $b$ | $\sigma^{2}(10^{-12})$
$(a_{1},b_{1},\sigma^{2})$ | 0.000008 | -0.000312 | 0.000009 | 0.000008 | 0.001139 | 0.006564 | 0.000008 | -0.000441 | 0.024410 | 0.000009 | 0.001142 | 0.000058
$(a_{1},b_{2},\sigma^{2})$ | 0.000008 | -0.000339 | 0.000011 | 0.000009 | 0.001290 | 0.007315 | 0.000009 | -0.000497 | 0.024494 | 0.000009 | 0.001259 | 0.000179
$(a_{1},b_{3},\sigma^{2})$ | 0.000008 | -0.000237 | 0.000011 | 0.000009 | 0.001398 | 0.007896 | 0.000009 | -0.000483 | 0.037635 | 0.000009 | 0.001417 | 0.000985
$(a_{2},b_{1},\sigma^{2})$ | 0.000015 | 0.000055 | 0.000823 | 0.000021 | 0.001212 | 0.008106 | 0.000016 | -0.000102 | 0.001137 | 0.000022 | 0.001184 | 0.000012
$(a_{2},b_{2},\sigma^{2})$ | 0.000017 | 0.000028 | 0.000732 | 0.000022 | 0.001363 | 0.009211 | 0.000020 | -0.000228 | 0.001954 | 0.000023 | 0.001319 | 0.001225
$(a_{2},b_{3},\sigma^{2})$ | 0.000019 | 0.000122 | 0.000182 | 0.000022 | 0.001473 | 0.009371 | 0.000021 | -0.000024 | 0.002634 | 0.000022 | 0.001387 | 0.000014
$(a_{3},b_{1},\sigma^{2})$ | 0.000023 | 0.000212 | 0.001357 | 0.000033 | 0.001265 | 0.009088 | 0.000023 | -0.000068 | 0.000266 | 0.000033 | 0.001200 | 0.000001
$(a_{3},b_{2},\sigma^{2})$ | 0.000025 | 0.000198 | 0.000845 | 0.000034 | 0.001391 | 0.009982 | 0.000030 | -0.000154 | 0.000558 | 0.000035 | 0.001332 | 0.000048
$(a_{3},b_{3},\sigma^{2})$ | 0.000026 | 0.000127 | 0.000722 | 0.000035 | 0.001590 | 0.011029 | 0.000034 | -0.000151 | 0.001303 | 0.000035 | 0.001464 | 0.003727


# On semi-Peano algebras

C. Cardó

###### Abstract.

A semi-Peano algebra is an algebra for which each operation is injective, and
the images of the operations are pairwise disjoint. The most straightforward
non-trivial kind of finitely presented semi-Peano algebra are algebras with a
single unary operation. There are two possible directions of generalization:
algebras with a single operation of any arity, and algebras with several unary
operations. The first can be solved easily by adapting results on
equidecomposable groupoids from [2]. However, the second way is somewhat
different. We will show that a finitely presented multi-unary semi-Peano
algebra is the free product of cyclic semi-Peano algebras and that a unique
relation defines such cyclic algebras. In addition, we will characterize each
cyclic algebra up to isomorphism.

MSC(2010): Primary: 08A60, 08B20; Secondary: 08B25.
Keywords: Peano algebra, generalized Peano algebra, semi-Peano algebra, unary
algebra.
$*$Corresponding author

##  1\. Introduction

Peano axioms of arithmetic of natural numbers can be understood as a cyclic
algebra $\mathcal{N}=(N;s)$, such that the successor operation
$s:N\longrightarrow N$ is injective, and such that the generator, that is, the
zero, cannot be decomposed. _Generalized Peano algebras_ , or simply here
_Peano algebras_ , are a generalization of $\mathcal{N}$ for several
operations and several generators:

######  Definition 1.1.

An element of an algebra is said to be _decomposable_ if it belongs to the
image of some operation. A _Peano algebra_ is an algebra $\mathcal{A}=(A;F)$
satisfying the two following conditions:

  1. (i)

for any $f,g\in F$, if $f(x_{1},\ldots,x_{n})=g(y_{1},\ldots,y_{m})$, then
$f=g$, and $x_{i}=y_{i}$ for all $i=1,\ldots,n=m$.

  2. (ii)

$\mathcal{A}$ is generated by its indecomposable elements.

When only (i) is required, the algebra is said to be a _semi-Peano algebra_.
The class of those algebras is denoted by $\mathbf{SP}$.

The class of Peano algebras corresponds precisely to the class of absolutely
free algebras; see [3], Proposition 3, for the statement, and [13] for a
proof. Similarly to natural numbers, Peano algebras provide an _algebraic
induction_. If a property $P$ is satisfied for every generator, and for each
operation $f$ we have that:

$\displaystyle x_{1},\ldots,x_{n}\mbox{ satisfy }P\implies
f(x_{1},\ldots,x_{n})\mbox{ satifies }P,$

then each element of the algebra satisfies the property $P$. Peano algebras
were introduced early to construct formal languages for algebraic logics with
the possibility of infinitary connectives, see Diener [3, 4, 5, 6]. More
recently, the format of free algebras as Peano algebras is used in computer
science, in the area of knowledge representation through the concept of a
labeled stratified graph, [10, 11, 13].

Let us fix some notation. $\mathcal{A}=(A;\\{f_{i}\\}_{i\in I})$ denotes an
algebra with operations indexed by $I$. When the index is not necessary, we
will write $\mathcal{A}=(A;F)$, where $F=\\{f_{i}\\}_{i\in I}$. From now on, a
_unary_ algebra will mean a multi-unary algebra. A _mono-unary algebra_ is an
algebra with only one operation. Absolutely free algebras over the generating
set $X$ are denoted by $\mathscr{F}(X)$. For cyclic free algebras with
generator $x$ we write $\mathscr{F}(x)$ or simply $\mathscr{F}$. $X\sqcup Y$
means $X\cup Y$, but with the condition that the sets are disjoint. For other
common concepts and notations on universal algebra, we follow [1].

Our primary interest is finitely presented semi-Peano algebras. Before
starting, let us comment on some elementary properties and examples of such
algebras. It is routine to prove that the class $\mathbf{SP}$ is closed under
subalgebras and arbitrary direct products but not under quotients. Finite non-
nullary semi-Peano algebras have at most one operation. Consider two
operations $f:A^{n}\longrightarrow A$ and $g:A^{m}\longrightarrow A$, such
that $n\geq 1$ or $m\geq 1$. If $n\geq 1$, $f$ is injective. Since $A$ is
finite, $f$ is bijective, and then
$\operatorname{im}f\cap\operatorname{im}g\not=\emptyset$, which is absurd. If
$m\geq 1$, the argument is similar. In particular, only the class of algebras
with a single operation possesses trivial algebras. If an operation
$f:A^{n}\longrightarrow A$ is injective and $A$ is finite, then
$\lvert{A}\rvert^{n}\leq\lvert{A}\rvert$, which only is possible if
$\lvert{A}\rvert=1$ or $0\leq n\leq 1$. Consequently, a non-trivial semi-Peano
algebra is finite if and only if it is nullary or mono-unary.

Semi-Peano algebras are related to Jónsson-Tarski algebras, as shown in the
following examples.

######  Example 1.2.

A _Jónsson-Tarski algebra_ [8, 9] is an algebra
$\mathcal{J}=(J;+,\alpha,\beta)$, where $+$ is a binary operation and
$\alpha,\beta$ are unary, satisfying the Cantor identities:

$\displaystyle\alpha(x)+\beta(x)=x,\quad\alpha(x+y)=x,\quad\beta(x+y)=y.$

Since the operation $+:J^{2}\longrightarrow J$ is necessarily bijective,
reducts of Jónsson-Tarski algebras $(J;+)$ are semi-Peano groupoids. This
inclusion of classes is proper. The free product of the trivial groupoid
$\mathcal{I}$ by the cyclic free groupoid $\mathscr{F}$ yields a semi-Peano
groupoid which is not a free groupoid nor a reduct of a Jónsson-Tarski
algebra: $\mathcal{I}*\mathscr{F}\cong\langle a,b\mid a\approx a+a\rangle$.

######  Example 1.3.

Jónsson-Tarski algebras can be generalized in the following obvious way, cf.
[7]. Let $f$ be an $n$-ary operation and let $p_{1}$, …, $p_{n}$ be unary
operations, satisfying the generalized Cantor identities:

$p_{i}(f(x_{1},\ldots,x_{n}))=x_{i},\forall i=1,\ldots,n,\quad
f(p_{1}(x),\ldots,p_{n}(x))=x.$

Given an algebra with a single operation of arity $n>1$, $\mathcal{A}=(A;f)$,
by the _unarisation of $\mathcal{A}$ on the $k$-th component_, denoted by
$\mathfrak{U}_{k}(\mathcal{A})$, we mean the algebra:

$\displaystyle\mathfrak{U}_{k}(\mathcal{A})=(A;\\{f_{\vec{a}_{k}}\\}_{\vec{a}_{k}\in
A^{n-1}})$

where $\vec{a}_{k}=(a_{1},\ldots,a_{k-1},a_{k+1},\ldots,a_{n})$ and

$\displaystyle
f_{\vec{a}_{k}}(x)=f(a_{1},\ldots,a_{k-1},x,a_{k+1},\ldots,a_{n}).$

Then, the unarisation of the reduct on $f$ of a generalized Jónsson-Tarski
algebra is a unary semi-Peano algebra, with possibly infinite operations.

The most straightforward non-trivial kind of finitely presented semi-Peano
algebra are algebras with a single unary operation. The graph of those
algebras consists of a finite set of directed cycles and ray graphs. An
example is the algebra of natural numbers $\mathcal{N}$ whose graph is a ray.
There are two direct generalizations of those algebras: (I) algebras with a
single operation of any arity; and (II) algebras with several unary
operations.

Regarding the way (I), the case of arity two was solved in [2]. An
_equidecomposable groupoid_ $\mathcal{M}=(M;+)$ is groupoid such that
$x+y=x^{\prime}+y^{\prime}$ implies that $x=x^{\prime},y=y^{\prime}$, or in
other terms, the operation $+$ is injective. Therefore, equidecomposable
groupoids and semi-Peano groupoids are the same things. Finitely presented
semi-Peano groupoids can be expressed in a normal form that characterizes them
isomorphically. More specifically, if $X$ is a minimal generating set of a
semi-Peano groupoid $\mathcal{M}$, there is a unique injective mapping
$\Psi:X\longrightarrow\mathscr{F}(X)$, such that for every $x\in X$, $x$
appears at least once in the term $\Psi(x)$, and such that $\mathcal{M}$ is
the quotient of $\mathscr{F}(X)$ by the congruence generated by the relations
$(x,\Psi(x))$. In short, for each semi-Peano groupoid, there is a unique
presentation without redundant relations; see [2] for the details. A
generalization of that result for an operation of arity greater than two does
not present a significant difficulty. It is only necessary to rewrite theorems
from [2] but with a more complicated notation. Thus, the way (I) does not
offer much more mathematical interest.

Here we are concerned with the second case (II). We will show that any
finitely presented unary semi-Peano algebra is the free product of cyclic
unary semi-Peano algebras and that the congruences generated by a single
relation define those cyclic algebras. This relation is given by the action of
a word of the free monoid on the set of operations. Proving the decomposition
is almost an elementary fact. So the contribution of this paper consists of
the study of cyclic unary semi-Peano algebras.

##  2\. Some preliminary facts on semi-Peano algebras and unary algebras

We will need a characterization of semi-Peano algebras in terms of
congruences.

######  Definition 2.1.

A congruence $\theta$ of an algebra $\mathcal{U}=(U;F)$ is said to be:

  1. (i)

_closed_ if for each $f\in F$:

$\displaystyle(f(x_{1},\ldots,x_{n}),f(y_{1},\ldots,y_{n}))\in\theta\implies(x_{1},y_{1}),\ldots,(x_{n},y_{n})\in\theta.$

  2. (ii)

_balanced_ if for each $f,g\in F$:

$\displaystyle(f(x_{1},\ldots,x_{n}),g(y_{1},\ldots,y_{m}))\in\theta\implies
f=g.$

######  Lemma 2.2.

Given an homomorphism of algebras $\Phi$, $\operatorname{im}\Phi$ is a semi-
Peano algebra if and only if $\ker\Phi$ is a closed and balanced congruence.

###### Proof.

Let $\Phi:\mathcal{U}\longrightarrow\mathcal{V}$ be a homomorphism of algebras
and $\mathcal{U}=(U;F)$. We have the following equivalences, where each
statement must be read with the prefix “for all
$x_{1},\ldots,x_{n},y_{1},\ldots,y_{m}\in U$”, and $n$ and $m$ are the
corresponding arities:

$\displaystyle\operatorname{im}\Phi\mbox{ is semi-Peano}$ $\displaystyle\iff$
$\displaystyle\mbox{ if
}f(\Phi(x_{1}),\ldots,\Phi(x_{n}))=g((\Phi(y_{1}),\ldots,\Phi(y_{m}))\mbox{
with }f,g\in F,$ $\displaystyle\mbox{ then }f=g,n=m\mbox{ and
}\Phi(x_{1})=\Phi(y_{1}),\ldots,\Phi(x_{n})=\Phi(y_{n})$ $\displaystyle\iff$
$\displaystyle\mbox{ if
}\Phi(f(x_{1},\ldots,x_{n}))=\Phi(g(y_{1},\ldots,y_{m}))\mbox{ with }f,g\in
F,$ $\displaystyle\mbox{ then }f=g,n=m\mbox{ and
}\Phi(x_{1})=\Phi(y_{1}),\ldots,\Phi(x_{n})=\Phi(y_{n})$ $\displaystyle\iff$
$\displaystyle\mbox{ if
}(f(x_{1},\ldots,x_{n}),g(y_{1},\ldots,y_{m}))\in\ker\Phi\mbox{ with }f,g\in
F,$ $\displaystyle\mbox{ then }f=g,n=m\mbox{ and
}(x_{1},y_{1}),\ldots,(x_{n},y_{n})\in\ker\Phi$ $\displaystyle\iff$
$\displaystyle\ker\Phi\mbox{ is closed and balanced}.$

∎

$\mathbf{SP}$ possesses $\mathbf{SP}$-free algebras. Every absolutely free
algebra is a Peano algebra, which, in its turn, is a semi-Peano algebra. Since
absolutely free algebras satisfy the universal mapping property, the class of
$\mathbf{SP}$-free algebras is the class of absolutely free algebras. Thus,
each semi-Peano algebra $\mathcal{U}$ is of the form $\mathscr{F}(X)/\theta$,
where $\theta$ is a closed and balanced congruence, and $X$ is a generator set
of $\mathcal{U}$. The arbitrary intersection of closed and balanced
congruences is closed and balanced. Therefore, the following definition makes
sense.

######  Definition 2.3.

Given an absolutely free algebra $\mathscr{F}(X)$ and a subset
$R\subseteq\mathscr{F}(X)^{2}$, the least closed and balanced congruence on
$\mathscr{F}(X)$ containing $R$ is denoted by $\boxminus(R)$. Congruences
generated by only one relation $(a,b)$ are abbreviated
$\boxminus(a,b)=\boxminus(\\{(a,b)\\})$. A semi-Peano algebra $\mathcal{U}$ is
_finitely presented_ if there are finite sets $X$ and $R$ such that
$\mathcal{U}\cong\mathscr{F}(X)/\boxminus(R)$.

In addition, we will need some elementary facts on unary algebras. Given a
unary algebra $\mathcal{U}=(U;F)$, a term function is a mapping of the form
$\alpha_{1}\circ\cdots\circ\alpha_{n}:U\longrightarrow U$, for some operations
$\alpha_{1},\ldots,\alpha_{n}\in F$. Consider the free monoid $F^{*}$ of words
over the alphabet $F$. There is a natural action of the monoid $F^{*}$ over
the set $U$ defined by
$\alpha_{1}\cdots\alpha_{n}(x)=(\alpha_{1}\circ\cdots\circ\alpha_{n})(x)$, and
if $1$ denotes the empty word, then $1(x)=x$. The length of a word $\varphi\in
F^{*}$ is denoted by $\lvert{\varphi}\rvert$. Given two words $\varphi$ and
$\psi$ such that $\varphi$ is a prefix of $\psi$, $\varphi{\setminus}\psi$
denotes the unique word such that $\varphi(\varphi{\setminus}\psi)=\psi$. When
$\psi$ is a suffix of $\varphi$, $\varphi/\psi$ denotes the unique word such
that $(\varphi/\psi)\psi=\varphi$. Two words $\omega$ and $\omega^{\prime}$
are _conjugated_ if there is a word $\varphi$ such that
$\omega\varphi=\varphi\omega^{\prime}$. This is equivalent to saying that
there are words $\alpha,\beta$ such that $\omega=\alpha\beta$,
$\omega^{\prime}=\beta\alpha$.

The _free product_ of two unary algebras $\mathcal{U}=(U;\\{f_{i}\\}_{i\in
I})$ and $\mathcal{V}=(V;\\{g_{i}\\}_{i\in I})$ is defined by
$\mathcal{U}\amalg\mathcal{V}=(U\sqcup V;\\{f_{i}\sqcup g_{i}\\}_{i\in I})$
where:

$\displaystyle f_{i}\sqcup g_{i}(x)=\begin{cases}f_{i}(x)&\mbox{if }x\in U,\\\
g_{i}(x)&\mbox{if }x\in V.\end{cases}$

The product can be extended for an arbitrary family of algebras $\coprod_{j\in
J}\mathcal{U}_{j}$. An algebra $\mathcal{W}$ is said to be
$\amalg$-irreducible when $\mathcal{W}=\mathcal{U}\amalg\mathcal{V}$ implies
that $\mathcal{U}=\emptyset$ or $\mathcal{V}=\emptyset$. Consider the
following relation in a unary algebra: $x\sim y$ if and only if there is a
word $\varphi$ such that $\varphi(x)=y$ or $\varphi(y)=x$. $\sim$ is an
equivalence relation and defines a partitition of the algebra into
$\amalg$-irreducible subalgebras. Cyclic unary algebras are always
$\amalg$-irreducible. The reciprocal is not in general true, except for semi-
Peano algebras, as it is shown in the Proposition 3.2.

##  3\. Classification of unary semi-Peano algebras

######  Lemma 3.1.

If $\varphi(x)=\psi(y)$ in a unary semi-Peano algebra and
$\lvert{\varphi}\rvert\leq\lvert{\psi}\rvert$, then $\varphi$ is a prefix of
$\psi$ and $x=\varphi{\setminus}\psi(y)$.

###### Proof.

Apply Definition 1.1(i) $n$ times, where $n=\lvert{\varphi}\rvert$. ∎

######  Proposition 3.2.

Let $\mathcal{U}=(U;\\{f_{i}\\}_{i\in I})$ be a unary semi-Peano algebra with
a minimal generating set $A$. Then:

$\displaystyle\mathcal{U}=\coprod_{a\in A}\mathcal{U}_{a},$

where $\mathcal{U}_{a}$ is the cyclic subalgebra $\mathcal{U}_{a}=(\langle
a\rangle;\\{f_{a,i}\\}_{i\in I})$ and $f_{a,i}$ is the operation $f_{i}$
restricted to $\langle a\rangle$.

###### Proof.

First we see that $\langle A\rangle=\bigcup_{a\in A}\langle a\rangle$. The
direction $\supseteq$ is trivial. For the direction $\subseteq$, we notice
that any combination of the operations can only be applied over only one
element, since they are unary. Thus, for every $x\in\langle A\rangle$ there is
a word $\varphi$ and an element $a\in A$, such that $\varphi(a)=x$.

Next we check that the unions are disjoint. Let $a,b\in A$ be two different
generators and supose that $\langle a\rangle\cap\langle
b\rangle\not=\emptyset$. This means that there are two words, $\varphi$ and
$\psi$, such that $\varphi(a)=\psi(b)$. Without lost of generality, we assume
that $\lvert{\varphi}\rvert\leq\lvert{\psi}\rvert$. By Lemma 3.1,
$a=\varphi{\setminus}\psi(b)$, which is a contradiction with the minimality of
the generating set $A$, because the generator $a$ can be removed. It is
routine to prove that each operation $f_{i}$ of $\mathcal{U}$ can be written
as $f_{i}=\bigsqcup_{a\in A}f_{a,i}$. ∎

By Proposition 3.2, a unary semi-Peano algebra is $\amalg$-irreducible if and
only if it is cyclic. In addition, all the minimal generating sets are
equinumerous, whereby it makes sense to define the _rank_ of a unary semi-
Peano algebra as the cardinal of a minimal generating set.

We will abbreviate
$\mathscr{P}/\omega=\mathscr{F}(a_{0})/\boxminus(a_{0},\omega(a_{0}))$, where
$\omega\in F^{*}$ and $F$ is the set of operations of $\mathscr{F}(a_{0})$. We
call the _canonical generator_ of $\mathscr{P}/\omega$ the equivalence class
containing $a_{0}$.

######  Lemma 3.3.

Any cyclic and finitely presented unary semi-Peano algebra is of the form
$\mathscr{P}/\omega$ for some word $\omega$.

###### Proof.

A cyclic and finitely presented unary semi-Peano algebra $\mathcal{U}$ is of
the form:

$\displaystyle\mathcal{U}\cong\faktor{\mathscr{F}(c)}{\boxminus(\\{(a_{i},b_{i})\mid
i\in I\\})}$

for some finite set $I$. Let $a\approx b$ denote two elements
$a,b\in\mathscr{F}(c)$ such that evaluates the same in $\mathcal{U}$. Consider
the congruence generated by a single relation $\boxminus(a,b)$. Since
$\mathcal{U}$ is cyclic, there are words $\varphi,\psi$ such that
$\varphi(c)\approx a$ and $\psi(c)\approx b$. By Lemma 3.1,
$c\approx\varphi{\setminus}\psi(c)$ or $c\approx\psi{\setminus}\varphi(c)$.
That is, $c\approx\phi(c)$ for some word $\phi$. Thus:

$\displaystyle\boxminus(a,b)=\boxminus(\varphi(c),\psi(c))=\boxminus(c,\phi(c)).$

This means that $\mathcal{U}$ we can be rewritten as:

$\displaystyle\mathcal{U}\cong\faktor{\mathscr{F}(c)}{\boxminus(\\{(c,\phi_{i}(c))\mid
i\in J\\})},$

where $J=\\{i\in I\mid a_{i}\not\approx b_{i}\\}$. In the case that
$J=\emptyset$, $\mathcal{U}=\mathscr{F}(c)=\mathscr{P}/1$ and we are done. So,
we assume that $\lvert{J}\rvert>1$. Given a set of relations
$R\subseteq\\{(c,\varphi(c))\mid\varphi\not=1\\}$, we introduce the _weight_
of $R$ as:

$\displaystyle W(R)=\sum_{(c,\varphi(c))\in R}\lvert{\varphi}\rvert.$

Suppose that there are two relations $(c,\varphi(c)),(c,\psi(c))\in R$ such
that $\varphi$ and $\psi$ have different length, say,
$\lvert{\varphi}\rvert<\lvert{\psi}\rvert$. Then we define:

$\displaystyle
R^{\prime}=R\setminus\\{(c,\psi(c))\\}\cup\\{(c,\varphi{\setminus}\psi(c))\\}.$

Notice that $\boxminus(R)=\boxminus(R^{\prime})$, but

$\displaystyle
W(R^{\prime})=W(R)-\lvert{\psi}\rvert+(\lvert{\psi}\rvert-\lvert{\varphi}\rvert)=W(R)-\lvert{\varphi}\rvert<W(R),$

because $\varphi\not=1$. We denote by a superscript $k)$ the successive
application of that transformation: $R,R^{1)},R^{2)},R^{3)},\ldots$. Since
$R=\\{(c,\phi_{i}(c))\mid i\in J\\}$ is finite, $W(R)$ is finite, which means
that we can only apply the transformation finite many times, say $k\geq 0$. In
that point, $\boxminus(R^{k)})=\boxminus(R)$, and for each
$(c,\varphi(c)),(c,\psi(c))\in R^{k)}$,
$\lvert{\varphi}\rvert=\lvert{\psi}\rvert$. But then, by Lemma 3.1,
$\varphi(c)\approx\psi(c)$ implies that $\varphi$ is a prefix of $\psi$ and
vice versa. Therefore, $\varphi=\psi$, and $R^{k)}$ only contains one
relation, say $(c,\omega(c))$, and:

$\displaystyle\mathcal{U}\cong\faktor{\mathscr{F}(c)}{\boxminus(c,\omega(c))}=\faktor{\mathscr{P}}{\omega},$

for some word $\omega$. ∎

######  Lemma 3.4.

Let $\mathscr{P}/\omega$ be a cyclic semi-Peano algebra, and $a$ be the
canonical generator.

  1. (i)

If $\varphi(a)=a$ for some word $\varphi$, then $\varphi=\omega^{n}$ for some
$n\geq 0$.

  2. (ii)

If $\varphi(a)=\psi(a)$ for some words $\varphi,\psi$, with
$\lvert{\varphi}\rvert\leq\lvert{\psi}\rvert$, then $\psi=\varphi\omega^{n}$
for some $n\geq 0$.

  3. (iii)

If $\varphi(b)=b$ for some $b\in\mathscr{P}/\omega$ and some word $\varphi$,
then $\varphi$ and $\omega^{n}$ are conjugated for some $n\geq 0$.

###### Proof.

(i) Let
$\mathscr{P}/\omega=\mathscr{F}(a_{0})/\boxminus(a_{0},\omega(a_{0}))$. The
equivalence class $a=\\{a_{0},\omega(a_{0}),\omega^{2}(a_{0}),$
$\omega^{3}(a_{0}),\ldots\\}$ is the canonical generator of
$\mathscr{P}/\omega$. Suppose that $\varphi(a)=a$, for some word $\varphi$.
Then, $\varphi(a_{0})\in a$, which means that
$\varphi(a_{0})=\omega^{n}(a_{0})$. Since $\mathscr{F}(a_{0})$ is absolutely
free, $\varphi=\omega^{n}$, for some $n\geq 0$.

(ii) If $\varphi(a)=\psi(a)$, then by Lemma 3.1,
$\varphi{\setminus}\psi(a)=a$. By (i), $\varphi{\setminus}\psi=\omega^{n}$,
that is, $\psi=\varphi\omega^{n}$.

(iii) Since $a$ is a generator, $b=\alpha(a)$, for some word $\alpha$. Then,
$\alpha(a)=\varphi(\alpha(a))$, which means that
$a=\alpha{\setminus}\varphi\alpha(a)$. By (i),
$\alpha{\setminus}\varphi\alpha=\omega^{n}$ for some $n\geq 0$, whereby
$\varphi\alpha=\alpha\omega^{n}$. ∎

######  Proposition 3.5.

$\mathscr{P}/\omega\cong\mathscr{P}/\omega^{\prime}$ if and only if $\omega$
and $\omega^{\prime}$ are conjugated words.

###### Proof.

$(\Rightarrow)$ Let
$\Phi:\mathscr{P}/\omega\longrightarrow\mathscr{P}/\omega^{\prime}$ be an
isomorphism, $a$, the generator of $\mathscr{P}/\omega$, and $b=\Phi(a)$.
Since $\omega(a)=a$, applying the homomorphism we get
$\Phi(\omega(a))=\omega(\Phi(a))=\Phi(a)$, that is, we have the identity
$\omega(b)=b$ in $\mathscr{P}/\omega^{\prime}$. By Lemma 3.4(iii), $\omega$
and $\omega^{\prime n}$ are conjugated for some $n\geq 0$. Now, by considering
the inverse isomorphism
$\Phi^{-1}:\mathscr{P}/\omega^{\prime}\longrightarrow\mathscr{P}/\omega$, and
applying the same reasoning, $\omega^{\prime}$ and $\omega^{m}$ are conjugated
for some $m\geq 0$.

If $\omega=\omega^{\prime}=1$, we are done. So, let us suppose that at least
one of the words, say $\omega^{\prime}$, is different to $1$. Join both
conjugations: $\omega\alpha=\alpha\omega^{\prime n}$ and
$\omega^{\prime}\beta=\beta\omega^{m}$. From the first equality,
$\omega=\alpha\omega^{\prime n}/\alpha$. Substituying in the second equality
and taking lengths:

$\displaystyle\omega^{\prime}\beta=\beta(\alpha\omega^{\prime n}/\alpha)^{m}$
$\displaystyle\implies$
$\displaystyle\lvert{\omega^{\prime}}\rvert+\lvert{\beta}\rvert=\lvert{\beta}\rvert+m(\lvert{\alpha}\rvert+n\lvert{\omega^{\prime}}\rvert-\lvert{\alpha}\rvert)$
$\displaystyle\implies$
$\displaystyle\lvert{\omega^{\prime}}\rvert=mn\lvert{\omega^{\prime}}\rvert\implies
mn=1\implies m=n=1,$

where we have used that $\omega^{\prime}\not=1$, that is,
$\lvert{\omega^{\prime}}\rvert\not=0$. Therefore, $\omega$ and
$\omega^{\prime}$ are conjugated.

$(\Leftarrow)$ Suppose that $\omega=\beta\alpha$ and
$\omega^{\prime}=\alpha\beta$. Consider the mapping
$\Phi:\mathscr{P}/\omega\longrightarrow\mathscr{P}/\omega^{\prime}$ defined by
$\Phi(x)=\varphi\beta(b)$, where $\varphi(a)=x$, and where $a$ and $b$ are the
canonical generators of $\mathscr{P}/\omega$ and
$\mathscr{P}/\omega^{\prime}$, respectively. First we check that the mapping
is well-defined. Suppose that $x=\varphi(a)=\psi(a)$, with
$\lvert{\varphi}\rvert\leq\lvert{\psi}\rvert$. By Lemma 3.4 (ii),
$\psi=\varphi\omega^{n}$, for some $n\geq 0$. Therefore,

$\displaystyle\Phi(\psi(a))$
$\displaystyle=\Phi(\varphi\omega^{n}(a))=\varphi\omega^{n}\beta(b)=\varphi(\beta\alpha)^{n}\beta(b)=\varphi\beta(\alpha\beta)^{n}(b)=\varphi\beta(b)$
$\displaystyle=\Phi(\varphi(a)).$

Clearly $\Phi$ is an homomorphism. If $x=\varphi(a)$, then for each $f\in F$,
$\Phi(f(x))=\Phi(f(\varphi(a)))=\Phi(f\varphi(a))=f\varphi\beta(b)=f(\varphi\beta(b))=f(\Phi(a))=f(\Phi(x))$.
Next, we see that $\Phi$ is an epimorphism. Since
$\alpha(\beta(b))\in\langle\beta(b)\rangle$, we have:

$\displaystyle\langle\alpha(\beta(b))\rangle\subseteq\langle\beta(b)\rangle\subseteq\langle
b\rangle$ $\displaystyle\implies\langle\alpha\beta(b)\rangle=\langle
b\rangle\subseteq\langle\beta(b)\rangle\subseteq\langle b\rangle$
$\displaystyle\implies\langle
b\rangle=\langle\beta(b)\rangle=\langle\Phi(a)\rangle.$

Finally we check that $\Phi$ is a monomorphism. Suppose $x=\varphi(a)$ and
$y=\psi(a)$ with $\lvert{\varphi}\rvert\leq\lvert{\psi}\rvert$. Then:

$\displaystyle\Phi(x)=\Phi(y)$
$\displaystyle\implies\Phi(\varphi(a))=\Phi(\psi(a))\implies\varphi\beta(a)=\psi\beta(a)$
$\displaystyle\implies(\varphi\beta){\setminus}(\psi\beta)=\omega^{\prime n}$
$\displaystyle\implies\psi\beta=\varphi\beta\omega^{\prime
n}=\varphi\beta(\alpha\beta)^{n}=\varphi(\beta\alpha)^{n}\beta,$

for some $n\geq 0$. Since we are in the free monoid $F^{*}$, we can cancel
$\beta$’s:

$\displaystyle\psi\beta=\varphi(\beta\alpha)^{n}\beta\implies\psi=\varphi(\beta\alpha)^{n},$

and then
$\psi(a)=\varphi(\beta\alpha)^{n}(a)=\varphi(\omega^{n}(a))=\varphi(a)$. That
is, $y=x$. ∎

Now we can classify all the finitely presented unary semi-Peano algebras.

######  Theorem 3.6.

For any finitely presented unary semi-Peano algebra $\mathcal{U}$ with rank
$n$ there are $n$ words $\omega_{1},\ldots,\omega_{n}$ such that:

$\displaystyle\mathcal{U}\cong\faktor{\mathscr{P}}{\omega_{1}}\amalg\cdots\amalg\faktor{\mathscr{P}}{\omega_{n}},$

and the decomposition is unique, up to the order of the factors and
conjugation of the words.

###### Proof.

Let $A=\\{a_{1},\ldots,a_{n}\\}$ be a minimal generating set of $\mathcal{U}$.
By Proposition 3.2,
$\mathcal{U}=\mathcal{U}_{a_{1}}\amalg\cdots\amalg\mathcal{U}_{a_{n}}$, where
$\mathcal{U}_{a_{1}},\ldots,\mathcal{U}_{a_{n}}$ are cyclic. By Lemma 3.3,
each $\mathcal{U}_{a_{i}}$ is isomorphic to $\mathscr{P}/\omega_{i}$ for some
word $\omega_{i}$. The uniqueness of the decomposition follows from
Propositions 3.2 and 3.5. ∎

######  Example 3.7.

Consider the bijection
$\diamond:\mathbb{N}\times\mathbb{N}\longrightarrow\mathbb{N}$:

$\displaystyle x\diamond y=\frac{1}{2}(x^{2}+y^{2}+2xy-x-3y+2).$

The groupoid $\mathcal{G}=(\mathbb{N};\diamond)$ is the reduct of a Jónsson-
Tarski algebra, and it has only two non-trivial relations:

$\displaystyle 1\diamond 1=1\quad\mbox{ and }\quad 1\diamond 2=2,$

The unarisation on the second component $\mathfrak{U}_{2}(\mathcal{G})$ gives
a unary semi-Peano algebra, recall Example 1.3, with infinite countable unary
operations $\diamond_{a}(x)=a\diamond x$, for each $a\in\mathbb{N}$. The above
relations in $\mathcal{G}$ become the unary relations in
$\mathfrak{U}_{2}(\mathcal{G})$:

$\displaystyle\diamond_{1}(1)=1\quad\mbox{ and }\quad\diamond_{1}(2)=2.$

It can be proved that $1$ and $2$ are the generators of
$\mathfrak{U}_{2}(\mathcal{G})$:

$\displaystyle\mathbb{N}=\langle 1\rangle\sqcup\langle
2\rangle=\\{1,3,4,6,8,10,13,16,\ldots\\}\sqcup\\{2,5,9,11,14,17,\ldots\\},$

and therefore:

$\displaystyle\mathfrak{U}_{2}(\mathcal{G})\cong\faktor{\mathscr{P}}{\diamond_{1}}\amalg\faktor{\mathscr{P}}{\diamond_{1}}.$

If we take the unarisation on the first component, the operations are of the
form $\widetilde{\diamond}_{a}=x\diamond a$. Only one generator is needed,
$1$, and then:

$\displaystyle\mathfrak{U}_{1}(\mathcal{G})\cong\faktor{\mathscr{P}}{\widetilde{\diamond}_{1}}.$

\begin{overpic}[scale={0.15}]{Fig1} \put(35.0,43.0){$f$} \put(61.0,42.0){$f$}
\put(56.0,57.0){$g$} \put(42.0,57.0){$g$} \put(48.0,31.0){$f$} \end{overpic}
Figure 1. A portion of the graph of $\mathscr{P}/fg^{2}f^{2}$.
\begin{overpic}[scale={0.15}]{Fig2} \put(50.0,35.0){$f$} \put(63.0,43.0){$g$}
\put(62.0,55.0){$h$} \put(50.0,63.0){$f$} \put(36.0,55.0){$g$}
\put(36.0,43.0){$h$} \end{overpic} Figure 2. A portion of the graph of
$\mathscr{P}/(fgh)^{2}$. \begin{overpic}[scale={0.11}]{Fig3}
\par\put(59.0,47.0){$f$} \put(39.0,47.0){$g$} \put(48.0,33.0){$h$}
\end{overpic} Figure 3. A portion of the graph of $\mathscr{P}/fgh$.

The graph of a unary algebra gives us nice representations of semi-Peano
algebras. If $\omega\not=1$, the graph of $\mathscr{P}/\omega$ contains a
unique cycle defined by the word $\omega$, see Figure 1. Every vertex of that
cycle is an alternative generator of $\mathscr{P}/\omega$. The cycle
determines the rest of the graph, which can be completed by adding trees. We
can check many properties visually. For instance, all the proper subalgebras
of a cyclic unary semi-Peano algebra are absolutely free. Or, for instance, If
$q$ divides $p$, there is an epimorphism
$\mathscr{P}/\omega^{p}\longrightarrow\mathscr{P}/\omega^{q}$. Thus,
$\mathscr{P}/fgh$ is a quotient of $\mathscr{P}/(fgh)^{2}$; see Figure 2 and
3. The algebra $\mathscr{P}/fg^{2}f^{2}$ has no proper quotients, Figure 1.

As we have seen, when we consider semi-Peano algebras with several operations
(II), the presentation is no longer unique, as in the case of algebras with
one operation (I). That suggests that the general case could be much more
complex than a simple combination of the ideas of (I) and (II).

## References

  * [1] Burris, S., Sankappanavar, H.P.: A course in Universal Algebra, The Millenium Edition, 2012. http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html.
  * [2] C. Cardó, Equidecomposable magmas, Algebra Universalis 81 (2020), 57–83.
  * [3] K.H. Diener, Order in absolutely free and related algebras, Colloquium Mathematicum 14 (1966), no. 1, 63–72.
  * [4] K.H. Diener and G. Gratzer, A note on absolutely free algebras, Proceedings of the American Mathematical Society (1967), 551–553.
  * [5] K.H. Diener, On the predecessor relation in abstract algebras, Mathematical Logic Quarterly 39 (1993), no. 1, 492–514.
  * [6] K.H. Diener, On the subalgebra lattice of Peano algebras with finitary and infinitary operations, Algebra Universalis 33 (1995). no. 2, 169–185.
  * [7] Dudek, J. and Marczak, A. W., On Cantor identities, Algebra Universalis 68 (2012) no. 3, 237–247
  * [8] R. Freese and J.B. Nation, Free Jónsson-Tarski algebras, http://www.math.hawaii.edu/~jb/Jonsson-Tarski_algebras.pdf (2020).
  * [9] B. Jónsson and A. Tarski, On two properties of free algebras, Mathematica Scandinavica 9 (1961), 95–101.
  * [10] N. Ţăndăreanu, Knowledge bases with output, Knowledge and Information Systems 2 (2000), 2, 4, 438–460.
  * [11] N. Ţăndăreanu, Distinguished representatives for equivalent labelled stratified graphs and applications, Discrete Applied Mathematics 144 (2004) no. 2, 183–208.
  * [12] N. Ţăndăreanu, Lecture Notes on Universal Algebra, Basic Concepts of Peano Algebras and Lattices, Research Report in Artifcial Intelligence, Research Center for Artifcial Intelligence, University of Craiova, Universitaria Publishing House, 301 (2006)
  * [13] N. Ţăndăreanu and I. Tudor, Properties of the Closure Operator in Peano Algebras, Annals of the University of Craiova, Mathematics and Computer Science Series 39 (2012), no. 12, 299–308.

First Author Departament de Ciències Bàsiques, Universitat Internacional de
Catalunya, c/ Josep Trueta s/n, Sant Cugat del Vallès, 08195, Spain. Email:
<EMAIL_ADDRESS>