11institutetext: State University of New York at Binghamton, Binghamton NY
13902, USA 22institutetext: Slippery Rock University, Slippery Rock PA 16057,
USA 22email<EMAIL_ADDRESS>
33institutetext: Waseda Innovation Lab, Waseda University, Tokyo, Japan
33email<EMAIL_ADDRESS>
# Modeling and Analysis of the Lead-Lag Network of Economic Indicators

Amanda Goodrick
1122 0000-0001-8555-590X Hiroki Sayama 1133 0000-0002-2670-5864

###### Abstract

We propose a method of analyzing multivariate time series data that
investigates lead-lag relationships among economic indicators during the
COVID-19 era with a weighted directed network of lagged variables. The
analysis includes a stock index, average unemployment, and several variables
that are used to calculate inflation. Three complex networks are created with
these variables and several lags of each as the network nodes. Network edges
are weighted based on three relationship metrics: correlation, mutual
information, and transfer entropy. In each network, nodes are merged, and
edges are aggregated to simplify the weighted directed graph. Pagerank is used
to determine the most influential and the most influenced node over the time
period. Results were reasonably robust within each network, but they were
heavily dependent on the choice of metric.

###### Keywords:

Lead-lag networks PageRank time series multivariate economic indicators

##  1 Introduction

Economic activity is relevant to everyone; it impacts daily choices and
determines lifestyles. The economy has been quite dynamic in recent years,
particularly throughout the COVID-19 era. Understanding the dynamics of some
of the underlying economic factors can help us to make choices that will
improve our quality of life. Fluctuations in the stock market have
historically been thought to indicate a future change in the economy, but
those fluctuations are frequent and unpredictable. The unemployment rate tends
to vary within a reasonable range, and it has been considered to indicate past
changes in the economy, as a poor economy leads to fewer jobs. With fewer
jobs, less money is circulating in the system, and the economy is further
impacted by the change in the value of a basket of goods, impacting inflation.
Relationships among these indicators can be investigated by considering the
lead-lag effect in this system. A lead-lag effect in time series is a
relationship between lead variables and lagged variables, where lagged
variables are the variables shifted backward in time. A lead variable is a
variable that happens first and has a relationship with other variables at
some future time, and it provides information about future system behavior. A
lagged variable is a variable that has a relationship with other variables at
some past time, and it provides information about past system behavior. The
stock market has historically been considered as a lead indicator that will
predict future activity [1, 2, 3]. Unemployment rate and inflation have both
been considered as lagged indicators that are the result of the system state
[1, 2, 4]. We typically consider each of these indicators to determine overall
economic health, but how do they each interact? To be better active consumers,
we must understand the lead-lag relationships and the most relevant variables
in this system, particularly as those factors pertain to our everyday basic
needs.

Historical approaches to infer influential relations among time series
indicators involve using Granger causality, which infers an influential
relationship over time using lagged variables. Previous relevant research
indicates that there is a Granger causality from the stock market to the
economy, though the relationship is not reciprocated [3]. That bivariate study
used GDP (Gross Domestic Product) to measure the economy as it relates to the
S&P 500. Another study focused on multivariate Granger causality between
financial development, investment, and economic growth and found a difference
in results in the short-term vs. the long term [5]. Neither of these studies
used a network model. Multivariate Granger causality results are similar to
what we want to investigate, and literature is rich in this topic, but most
studies do not incorporate a network analysis, and Granger causality requires
stationarity to fit a VAR model to linear time series. Granger causality has
been extended to a method to detect nonlinear causality, and it has been found
that transfer entropy can be used in place of Granger causality for Gaussian
processes [6, 8]. Other studies use Granger with sliding windows over time to
detect nonlinear relationships in each window of time [9, 10]. However,
Granger causality indicates the existence of a pairwise relationship between
original variables, but it does not provide an estimation of the strength of
the relationship. As network analysis has become more prevalent, networks with
edge weights based on correlation have commonly been used to represent the
relationship strength between two variables. In much of the literature, when
time series are represented by correlation networks, the goal is to detect
community structure over the entire time period [11]. Recently, network
clustering methods were used to detect lead-lag relationships in stock indices
[12]. Other current methods are based on constructing multilayer temporal
networks with instantaneous correlations as intralayer edges [13]. Almost all
network studies that use correlation as an edge weight are based on only
positive correlations above a certain threshold in order to use only
statistically significant correlations, including one recent study that
investigates lead-lag correlations [14]. However, even low or negative
correlations are a measure of a pairwise relationship. As Information Theory
has developed, mutual information has been used instead of correlation to
quantify nonlinear relations, since correlation is intended for linear
relations. Transfer entropy has recently become more popular as a connection
method, and a multivariate transfer entropy has been proposed with clustering
methods to infer network properties [15]. Transfer entropy has been used for
network connections to rank financial nodes based on Random Forest and
Gradient Boosting [16, 17]. Pagerank (PR) can be used for ranking, as it is an
iterative method intended to detect the most important nodes in a network. It
is based on incoming edges, so it is most appropriate for directed networks
[18]. PR has previously been used to extract features in climate data using
lagged positive correlations with a threshold, and it has been used with
statistically significant correlations of one-minute stock log returns to rank
foreign exchange markets [19, 20].

This model analysis investigates the relationship between economic indicators
during the COVID-19 era using a weighted directed network analysis approach to
investigate the lead-lag effect in this system. PR is used to determine the
most influential and the most influenced node in a directed correlation
network of lagged variables using three different relationship metrics: linear
correlation, mutual information, and transfer entropy. Variables used to
represent the system were several variables that are used in the calculation
of Consumer Price Index (CPI), which is a measure of inflation, average
unemployment rate, and the S&P 500 stock index. We investigate the best
relationship metric choice, the most influential variables, and the most
influenced variables in this system. To our knowledge, there is no study that
investigates this phenomenon with a complex network that retains and
incorporates relationship strength with time difference and a damping
parameter. For each network, the graph is simplified, while retaining all
relational information, and PR is used. This novel method can be applied to
multivariate time series in any field to determine lead-lag relationships,
better understand time series systems, and make more informed decisions.

##  2 Methodology

We used historical monthly data of $13$ economic variables from July 2019 to
December 2022 to investigate the economic system during the COVID-19 era. The
data was from the Northeast Region of USA, as classified by the Bureau of
Labor Statistics [21]. This region includes 9 states and Puerto Rico. The
first $11$ variables were amount spent on: apparel, food at home, food away
from home, alcohol, housing, medical, recreation, new cars, used cars, motor
fuel, and fuels & utilities [21]. Average unemployment rate for the region was
calculated from monthly unemployment rate per state, and monthly closing
prices for the S&P 500 stock index were obtained from Yahoo! Finance [21, 22].
This stock index was chosen because it is often considered to be an accurate
overall market gauge since it is value-related as opposed to price-related
[2].

The original data had $42$ monthly values per variable. These variables were
transformed to monthly rate of change to create stationarity and remove
autocorrelation, resulting in $41$ monthly rates of change. The first $12$
lags for each variable were created, which increased the number of variables
to $169$, since there were $12$ new variables for each of the existing $13$
variables. Each lagged variable had a sample size one less than the previous
lag, so the variable of the greatest lag had 29 monthly rate of change values.
We let $X_{ik}$ denote the lag $k$ of variable $X_{i}$ for $i=1,2,...,13$ and
$k=0,1,...,12$, where $k=0$ indicates the original variables. A directed graph
was created with the $169$ variables as nodes with a directed edge from node
$X_{ik}$ to node $X_{jm}$ if $k>m$, for $i\neq j$ and $k\neq m$. Then each
existing edge was directed toward the pairwise lead variable, and
instantaneous correlations and any lingering autocorrelations were neglected.
Edge weights were based on: pairwise absolute simple linear correlations,

$|r_{ik,jm}|=\left|\frac{COV(X_{ik},X_{jm})}{\sqrt{VAR(X_{ik})VAR(X_{jm})}}\right|,\hskip
3.61371pt\mathrm{for}\hskip 3.61371pti\neq j\hskip 3.61371pt\mathrm{and}\hskip
3.61371ptk\neq m,$ (1)

pairwise lag difference,

$D_{km}=|k-m|,\hskip 3.61371pt\mathrm{for}\hskip 3.61371ptk\neq m,$ (2)

and a parameter $a$ where $0\leq a\leq 1$. This parameter represents the
proportion of correlation remaining over time $D_{km}$. Then each edge weight
is given by

$w_{ik,jm}=|r_{ik,jm}|a^{D_{km}}.$ (3)

This $169$-node network was simplified by aggregating nodes of each variable
over all lag values. For each node $i$, $X_{ik}$ was merged into a single node
for all $k$, resulting in a $13$ node graph. For each directed edge from node
$i$ to node $j$, edge weights were summed so that each node pair had exactly
two directed edges, each with a single weight,

$w_{i,j}=\sum_{k}\sum_{m}w_{ik,jm}.$ (4)

This simplified graph had $13$ nodes and $156$ edges. Weighted edges were
directed toward the lead variable, so the node with the greatest PR value was
considered to be the most influential node over the time investigated. Edge
weights remained the same, and the direction of each edge was reversed and
directed toward the lagged variable of the pair. PR was used to determine the
most influenced node during the COVID-19 era.

The PR algorithm is an iterative process originally used by Google to
determine the most important websites. The PR of a node is based on the
incoming edges of that node divided by the number of total edges in the graph.
It is given by

$PR_{i}=\frac{1-d}{N}+d\sum{\frac{PR_{j}}{c_{j}}}$ (5)

where $PR_{i}$ is the pagerank of node $i$, $d$ is the damping factor,
$PR_{j}$ is the PR value for all other nodes in the network where $j\neq i$,
and $c_{j}$ is the number of outgoing edges from node $j$ for $j\neq i$ [18].
For a weighted graph, the pagerank algorithm in the networkx package in python
uses edge weights rather than counts [24]. The damping factor is based on the
idea that a web surfer will eventually stop clicking links on the internet.
The default value in the pagerank algorithm was $d=0.85$, which we used for
this project because the overall economy dynamics do not dampen over time, and
the time factor was already accounted for in the edge weights. Results were
not impacted by other reasonable choices for $d$, as the variable
relationships remained intact regardless of this parameter value.

Since Eq. 5 calculates a measure of importance based on incoming links, when
edges were directed backward in time toward the pairwise lead variable, the
node with the greatest PR was considered to be the most influential. When
edges were directed toward the pairwise lag variable, the node with the
greatest PR value was considered as the most influenced variable. It is
important to note that influential and influenced in no way imply causation.
An influential observation in regression is one that influences the regression
equation, and the regression equation will change if the influential
observation is removed. This terminology is analogous to that situation.

This process was replicated with a network that had edge weights based on
mutual information, $I(X;Y)$, instead of linear correlation, where
$I(X;Y)=I(Y;X)$ and

$I(X;Y)=H(X)+H(Y)-H(X,Y)=\int\int{f(xy)log\frac{f(xy)}{f(x)f(y)}}.$ (6)

In Eq. 6, $H(X)$ is defined as the entropy of a variable, $H(X,Y)$ is the
joint entropy, $f(x)$ is a probability density function, and $I(X;Y)$ is the
amount of information obtained about a variable $X$ by knowing about a
variable $Y$ [25]. The logarithmic function is typically base $2$ and mutual
information returned is in bits of information. When variables are discrete,
the integral is replaced with a summation. For independent variables,
$I(X;Y)=0$, and for dependent variables, $I(X;Y)>0$, so all of the mutual
information values used as edge weights were positive.

While correlation is based on the variable outcomes, mutual information is
determined by the probability that each outcome will occur, and the outcome
itself is irrelevant. Correlation is based on linear relationships, but mutual
information detects all relations, so it is more appropriate to use this
metric when nonlinear relationships exist. Since the variables were
continuous, to avoid variations based on binning, pairwise mutual information
was calculated using the mutual_info_regression function in python from the
sklearn.feature_library [23]. This method estimates a probability density
function for each continuous variable based on the distance between each
outcome and it’s k-nearest neighbors [27]. The algorithm calculates the
distance from the observation to the farthest neighbor, and then it counts the
number of unique observations within that distance to estimate a frequency for
calculating the necessary probability density functions [27, 28]. The
necessary marginal probabilities hold for all values of $k$, but the joint
probabilities are more sensitive to choices of $k$. In order to avoid
numerical variations based on the number of nearest-neighbors between the
marginal and joint probabilities, this method starts by applying the chosen
$k$-value to the joint probability estimate and works backward to fit the
marginal probabilities based on the nearest neighbors used for the joint
probabilities [26, 27]. The default value in this function is for $k=3$
nearest neighbors, and this value was used to calculate pairwise mutual
information because values of $k$ greater than $3$ may result in an estimation
with less variation but perhaps more bias [27, 28]. We tried several other
values to create the network, but results were much more consistent for values
of the parameter $a$ when $k=3$. Then Eq. 6 was substituted into Eq. 3 in
place of Eq. 1 to obtain edge weights, and PR was used to determine the most
influential node and the most influenced node of the mutual information
network.

The method was repeated one more time using transfer entropy to quantify the
pairwise relationship between variables. Transfer entropy, $T_{X->Y}$, is the
amount of information gain in future values of $Y$ by knowing past values of
$X$ and past values of $Y$. Transfer entropy is given by

$T_{X\to
Y}=I(Y_{t}^{(l)}:X_{t-1}^{(l)}|Y_{t-1}^{(k)})=H(Y_{t}|Y_{t-1}^{(k)})-H(Y_{t}|Y_{t-1}^{(k)}X_{t-1}^{(l)})$
(7)

where $l$ is the variable lags from $1$ through fixed maximum lag $l$, and
$H(X|Y)$ and $I(X;Y)$ are conditional entropy and mutual information,
respectively [26]. Transfer entropy is similar to mutual information in that
it can be used for dynamic nonlinear relations. However, it is not a symmetric
relation, $T_{X\to Y}\neq T_{Y\to X}$, so the direction of the relation
matters, and $T_{X\to Y}\geq 0$, so all edge weights were positive. For this
network, relations are asymmetric, and edges only exist for relations directed
to a future time where $k>m$, so any relations where $k<m$ were disregarded.
Pairwise transfer entropy was calculated using the te_compute function from
the python implementation PyIF [29]. This algorithm calculates transfer
entropy as the sum of two mutual information values, and it is based on the
same density estimates that we used for calculating mutual information [27,
29]. Using this function requires choosing a value of $k$ for nearest
neighbors in calculating the mutual information required to calculate transfer
entropy, so we again used $k=3$ to more easily compare results and to minimize
bias in the estimates [27, 29]. The fixed maximum length of $l$ was chosen as
$1$ due to the dynamics of each variable and the volatility of those with much
larger standard deviations relative to the others. It was also important to
not overestimate the transfer entropy in the network when lagged variables
were already included. As with Eq. 6, Eq. 7 was substituted into Eq. 3 in
place of Eq. 1. In each case, the simplified graph had $13$ nodes and $156$
edges.

##  3 Results

The box plots in Fig. 1 illustrate some of the basic descriptive statistics of
each variable in the COVID-19 era. These descriptives are based on the monthly
rate of change in percentages. Basic descriptive statistics, in percents, are
provided in Table 1. The mean monthly rate of change is reasonably close, less
than $1\%$, for each variable except apparel, whose rate of change of $0.04\%$
is relatively low, and unemployment rate, whose rate of change of $218\%$ is
relatively high. There is much variation over the time period, particularly in
average unemployment rate ($32.22\%$), motor fuel ($5.82\%$), S&P 500
($5.76\%$), and used cars ($3.18\%$).

Figure 1: Rate of change (y-axis) per variable from July 2019 through December 2022. The y-axis is truncated for visibility, with an off-the-chart value of 197% increase in unemployment from March to April 2020. Table 1: Basic Descriptives per Variable (In Percent) Variable | Mean | Std. Dev. | Min | Max
---|---|---|---|---
Apparel | 0.04 | 2.95 | -5.83 | 6.38
Home Food | 0.47 | 0.72 | -0.62 | 3.25
Away Food | 0.45 | 0.41 | -0.45 | 1.35
Alcohol | 0.28 | 0.40 | -0.51 | 1.13
Housing | 0.32 | 0.36 | -0.41 | 1.03
Fuels & Utils | 0.80 | 2.07 | -2.95 | 6.86
Medical | 0.24 | 0.42 | -0.65 | 0.94
Recreation | 0.20 | 0.89 | -2.77 | 3.12
New Cars | 0.41 | 0.81 | -1.31 | 2.09
Used Cars | 0.76 | 3.18 | -4.00 | 11.00
Motor Fuel | 0.69 | 5.82 | -11.25 | 18.10
Unemployment | 2.18 | 32.22 | -17.57 | 197.20
S&P500 | 0.91 | 5.76 | -12.51 | 12.68

The initial networks were created and are visualized in Figs. 2, 3, and 4,
with edge weights based on correlation, mutual information, and transfer
entropy, respectively. These graphs each had edges directed toward the lead
variable and edge weights with parameter $a=1$. The graph with edge weights
based on correlation had $12168$ edges, while edge weights based on mutual
information resulted in a graph with $6258$ edges, and the graph with edge
weights based on transfer entropy had $6610$ edges. The correlation and mutual
information networks had edges based on all pairwise relations where Eq. 2 was
positive. However, while several correlation values were quite low, there were
no zero correlations, so the correlation network included all possible
relations. The mutual information network had fewer relations because the
number of possible relations was the same as that of the correlation network,
but there were pairwise relations with zero mutual information. The transfer
entropy network had fewer edges than the correlation network and slightly more
edges than the mutual information network.

Figure 2: Correlation network. This directed network has edge weights based on
Eq. 3 with correlation and parameter $a=1$. This graph has $169$ nodes and
$12168$ edges. Figure 3: Mutual information network. This directed network has
edge weights based on Eq. 3 with mutual information Eq. 6 and parameter $a=1$.
This graph has $169$ nodes and $6258$ edges. Figure 4: Transfer entropy
network. This directed network has edge weights based on Eq. 3 with transfer
entropy Eq. 7 and parameter $a=1$. This graph has $169$ nodes and $6610$
edges. Figure 5: PR values (y-axis) per node (x-axis) for networks with edge
weights based on correlation and determined by select values of $a$ in the
legend. The nodes on the x-axis are sorted by average PR. The most influential
node is shown on the left and has greatest PR with edges directed toward the
lead variable. The most influenced node is shown on the right and has greatest
PR with edges directed toward the lagged variable.

Once all nodes were merged and edge weights were summed by Eq. 4, PR was
calculated and plotted to investigate behavior. The PR values based on the
correlation network are shown in Fig. 5, with the most influential nodes shown
on the left, and the most influenced nodes shown on the right. Variables on
the x-axis are sorted by average PR, least to greatest. When edge weights are
based on correlation, for most values of $a$, the most influential nodes are
change in apparel, S&P500, and new cars. With edges directed toward the lagged
variable to determine the most influenced node, PR values are shown on the
right in Fig. 5. The most influenced nodes are change in food at home,
medical, and new cars. The PR values are based on incoming weighted edges, and
outgoing links are not considered, so any node that appears high in both lists
is a node that has heavily weighted incoming links and heavily weighted
outgoing links when edges are directed only toward the lead variable or only
toward the lagged variable, and influence does not mean causality. Consider
the variable New Cars, which is third most influential and third most
influenced. Then when edges are pointed toward the lead variable, New Cars has
the third highest weight of incoming edges, so it is the third most important
when considering correlation with future values of other variables. New Cars
also has the third highest weight of outgoing edges when edges are directed
toward the lead variable, indicating that this variable is the third most
important when considering correlation with past values of other variables.
Correlations with past values of other variables is determined by PR when
edges are directed toward the lag variable, since PR only considers incoming
edges. Variables that switch from the top of one list to the bottom of the
other, or vice versa, have a heavier weight of incoming than outgoing edges,
or vice versa. For instance, Apparel has greater relations with future values
of other variables but few relations with past values of other variables.
Based on $a$, PR values fluctuate the most for those variables with larger
standard deviation, while those with smaller standard deviation were quite
insensitive to changes in $a$.

Figure 6: PR values (y-axis) per node (x-axis) for networks with edge weights
based on mutual information and determined by select values of $a$ in the
legend. The nodes on the x-axis are sorted by average PR. The most influential
node is shown on the left and has greatest PR with edges directed toward the
lead variable. The most influenced node is shown on the right and has greatest
PR with edges directed toward the lagged variable.

When edge weights are based on mutual information, with variables sorted by
average PR, the most influential nodes are shown on the left in Fig. 6, and
they are rate of change in housing, motor fuel, and S&P500. These most
influential nodes each have greater relations with future values of other
variables. The most influenced nodes, shown on the right in Fig. 6, are change
in food at home, fuels & utilities, and medical, and these variables each have
greater relations with past values of other variables. Food at home is the
least influential and the most influenced, indicating that change in amount
spent on food at home is most related to change in variables over time in
previous months, though it has the lowest relational weight to other variables
at a future time. Comparing the results from the mutual information network
with those of the correlation network, S&P500 is in the top three of most
influential, and food at home and medical are in the top three of the most
influenced. Considering the nonlinearity of the system, mutual information is
likely the better metric choice.

Figure 7: PR values (y-axis) per node (x-axis) for networks with edge weights
based on transfer entropy and determined by select values of $a$ in the
legend. The nodes on the x-axis are sorted by average PR. The most influential
node is shown on the left and has greatest PR with edges directed toward the
lead variable. The most influenced node is shown on the right and has greatest
PR with edges directed toward the lagged variable.

When edge weights are based on transfer entropy, with variables sorted by
average PR, the most influential variables are shown on the left in Fig. 7.
The most influential nodes are change in motor fuel, apparel, and
unemployment, so these variables have greater relations with future values of
other variables. The most influenced nodes based on transfer entropy are shown
on the right in Fig. 7, and they are change in recreation, fuels & utilities,
and apparel, so these variables have a greater relation with past values of
other variables. PR results vary less for edge weights based on transfer
entropy than for edge weights based on both correlation and mutual
information. There is a slight variation in PR remaining for variables with
larger standard deviation, though results are most robust for the network with
node relations based on transfer entropy.

##  4 Conclusion & Limitations

Many methods to investigate time series are concerned with prediction, and
they work only with univariate data. This method is a useful way to analyze
and understand multivariate time series data while removing the specific time
dependence, and it can be applied to analyze other multivariate time series
data in many fields. For each method used to quantify the relationships over
time, the overall results are not sensitive to variations in the value of the
input parameter $a$, for $0\leq a\leq 1$, verifying the robustness of the
method within each metric. However, the results are extremely sensitive to the
choice of metric. Results are most insensitive when transfer entropy is used
to determine the most influenced node, and transfer entropy was also the
metric that had the greatest range of PR, making the ranking of variable
influence easier to distinguish than those from the correlation network and
the mutual information network. Transfer entropy was also the least sensitive
to variables with more variation. While correlation and mutual information
provided useful information, transfer entropy is definitely the best measure
to quantify the relationships among economic indicators over time during the
COVID-19 era.

This method can be applied only to historical data; results are not
predictive. Each relationship measure quantifies only pairwise relations with
equal sample sizes. While mutual information and transfer entropy both capture
nonlinear relations, correlation quantifies only linear relations. Future work
includes considering a different initial transformation to achieve better
stationarity and test the difference in results, using Granger causality to
determine relevant relations, and using a higher order network to extend the
analysis beyond pairwise relations.

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The set $T_{E}$ is an inverse subsemigroup of $\mathcal{I}(E)$ whose
semilattice of idempotents is isomorphic to E.

The semigroup $T_{E}$ is called the Munn semigroup of the semilattice $E$. We
can now construct inverse semigroups having specific semilattices of
idempotents.

######  Theorem 7.6 (Munn representation theorem).

Let $S$ be an inverse semigroup. Then there is an idempotent-separating
homomorphism $\delta\colon S\rightarrow T_{\mathsf{E}(S)}$ whose image is a
wide inverse subsemigroup of $T_{\mathsf{E}(S)}$. The kernel of $\delta$ is
$\mu$.

###### Proof.

For each $s\in S$ define the function

$\delta_{s}\colon\mathbf{d}(s)^{\downarrow}\rightarrow\mathbf{r}(s)^{\downarrow}$

by $\delta_{s}(e)=ses^{-1}$. We first show that $\delta_{s}$ is well-defined.
Let $e\leq s^{-1}s$. Then $ss^{-1}\delta_{s}(e)=\delta_{s}(e)$, and so
$\delta_{s}(e)\leq ss^{-1}$. To show that $\delta_{s}$ is isotone, let $e\leq
f\in(s^{-1}s)^{\downarrow}$. Then

$\delta_{s}(e)\delta_{s}(f)=ses^{-1}sfs^{-1}=sefs^{-1}=\delta_{s}(e).$

Thus $\delta_{s}(e)\leq\delta_{s}(f)$. Consider now the function
$\delta_{s^{-1}}\colon(ss^{-1})^{\downarrow}\to(s^{-1}s)^{\downarrow}$. This
is isotone by the argument above. For each $e\in(s^{-1}s)^{\downarrow}$, we
have that

$\delta_{s^{-1}}(\delta_{s}(e))=\delta_{s^{-1}}(ses^{-1})=s^{-1}ses^{-1}s=e.$

Similarly, $\delta_{s}(\delta_{s^{-1}}(f))=f$ for each
$f\in(ss^{-1})^{\downarrow}$. Thus $\delta_{s}$ and $\delta_{s^{-1}}$ are
mutually inverse, and so $\delta_{s}$ is an order isomorphism.

Define $\delta\colon S\rightarrow T_{\mathsf{E}(S)}$ by
$\delta(s)=\delta_{s}$. To show that $\delta$ is a homomorphism, we begin by
calculating $\mbox{dom}(\delta_{s}\delta_{t})$ for any $s,t\in S$. We have
that

$\mbox{dom}(\delta_{s}\delta_{t})=\delta^{-1}_{t}((s^{-1}s)^{\downarrow}~{}\cap~{}(tt^{-1})^{\downarrow})=\delta^{-1}_{t}((s^{-1}stt^{-1})^{\downarrow}).$

But $\delta^{-1}_{t}=\delta_{t^{-1}}$ and so

$\mbox{dom}(\delta_{s}\delta_{t})=((st)^{-1}st)^{\downarrow}=\mbox{dom}(\delta_{st}).$

If $e\in\mbox{dom}\,\delta_{st}$ then

$\delta_{st}(e)=(st)e(st)^{-1}=s(tet^{-1})s^{-1}=\delta_{s}(\delta_{t}(e)).$

Hence $\delta_{s}\delta_{t}=\delta_{st}$.

To show that $\delta$ is idempotent-separating, suppose that
$\delta(e)=\delta(f)$ where $e$ and $f$ are idempotents of $S$. Then
$\mbox{dom}\,\delta(e)=\mbox{dom}\,\delta(f)$. Thus $e=f$.

The image of $\delta$ is a wide inverse subsemigroup of $T_{\mathsf{E}(S)}$
because every idempotent in $T_{\mathsf{E}(S)}$ is of the form $1_{[e]}$ for
some $e\in E(S)$, and $\delta_{e}=1_{[e]}$.

Suppose that $\delta(s)=\delta(t)$. Then $(s,t)\in\mathscr{H}$. It is now
immediate from the definition that, precisely, $(s,t)\in\mu$. ∎

The Munn representation should be contrasted with the Wagner-Preston
representation: the Wagner-Preston was injective whereas the Munn
representation has a non-trivial kernel. Fundamental inverse semigroups arise
in the following way.

######  Theorem 7.7.

Let $S$ be an inverse semigroup. Then $S$ is fundamental if and only if $S$ is
isomorphic to a wide inverse subsemigroup of the Munn semigroup
$T_{\mathsf{E}(S)}$.

###### Proof.

Let $S$ be a fundamental inverse semigroup. By Theorem 7.6, there is a
homomorphism $\delta\colon S\rightarrow T_{\mathsf{E}(S)}$ such that
$\mbox{ker}(\delta)=\mu$. By assumption, $\mu$ is the equality congruence by
Lemma 7.3, and so $\delta$ is an injective homomorphism. Thus $S$ is
isomorphic to its image in $T_{\mathsf{E}(S)}$, which is a wide inverse
subsemigroup.

Conversely, let $S$ be a wide inverse subsemigroup of a Munn semigroup
$T_{E}$. Clearly, we can assume that $E=\mathsf{E}(S)$. We calculate the
maximum idempotent-separating congruence of $S$. Let $\alpha,\beta\in S$ and
suppose that $(\alpha,\beta)\in\mu$ in $S$. Then
$\mbox{dom}(\alpha)=\mbox{dom}(\beta)$. Let $e\in\mbox{dom}(\alpha)$. Then
$1_{[e]}\in S$, since $S$ is a wide inverse subsemigroup of
$T_{\mathsf{E}(S)}$. By assumption $\alpha 1_{[e]}\alpha^{-1}=\beta
1_{[e]}\beta^{-1}$. It is easy to check that $1_{[\alpha(e)]}=\alpha
1_{[e]}\alpha^{-1}$ and $1_{[\beta(e)]}=\beta 1_{[e]}\beta^{-1}$. Thus
$\alpha(e)=\beta(e)$. Hence $\alpha=\beta$, and so $S$ is fundamental. ∎

The following is a special case of an argument due to Wagner.

######  Example 7.8.

Let $(X,\tau)$ be a $T_{0}$-space. We prove that $\mathcal{I}(X,\tau)$ is
fundamental. Let $f\in\mathcal{I}(X,\tau)$ be a non-idempotent. Then there is
an element $x\in X$ such that $f(x)\neq x$. Since $X$ is $T_{0}$ there is an
open set $U$ such that either $f(x)\in U$ and $x\notin U$ or $f(x)\notin U$
and $x\in U$. In either event, the elements $f1_{U}$ and $1_{U}f$ are not
equal, where $1_{u}$ is an idempotent. It follows that $\mathcal{I}(X,\tau)$
is fundamental.

As an example of fundamental inverse semigroups, we can easily construct the
fundamental finite Boolean inverse monoids.

######  Proposition 7.9.

A finite Boolean inverse monoid is fundamental if and only its groupoid of
atoms is principal.

###### Proof.

Let $S$ be fundamental. We shall prove that the groupoid of atoms is principal
by proving that the local groups are trivial. Suppose that
$e\stackrel{{\scriptstyle a}}{{\longrightarrow}}e$ where $e$ is an atom and an
idempotent. We shall prove that $a$ is an idempotent. Then $a$ is an atom. Let
$f$ be any idempotent. Then $fa\leq a$. It follows that $fa=0$ or $fa=a$.
Suppose that $fa=0$. Then $fe=0$ and so $af=0$. Thus $fa=af$. Suppose now that
$fa=a$. Then $fe=e=fe$ and so $af=a$. It follows again that $fa=af$. We have
therefore proved that $a$ commutes with every idempotent, but under our
assumption that the inverse semigroup is fundamental, we deduce that $a=e$.

Conversely, suppose that $e\stackrel{{\scriptstyle a}}{{\longrightarrow}}e$,
where $e$ is an atomic idempotent, implies that $a=e$. We shall prove that our
semigroup is fundamental. Let $a$ commute with all idempotents. We prove that
$a$ is an idempotent. We can write $a=\bigvee_{i=1}^{m}a_{i}$ where the
$a_{i}$ are atoms. We prove that $\mathbf{d}(a_{i})=\mathbf{r}(a_{i})$ for all
$i$ from which the result follows. Since
$\mathbf{r}(a_{j})a=a\mathbf{r}(a_{j})$, by assumption, we have that
$a_{j}=\bigvee_{i=1}^{m}a_{i}\mathbf{r}(a_{j})$. But
$a_{i}\mathbf{r}(a_{j})\leq a_{j}$. Thus either $a_{i}\mathbf{r}(a_{j})=0$ or
$a_{i}\mathbf{r}(a_{j})=a_{j}$ since $a_{j}$ is an atom. But
$a_{i}\mathbf{r}(a_{j})\leq a_{i}$. It follows that
$a_{i}\mathbf{r}(a_{j})=a_{i}$ and so $a_{i}=a_{j}$. Thus
$a_{j}\mathbf{r}(a_{j})=a_{j}$. Hence $\mathbf{d}(a_{j})\leq\mathbf{r}(a_{j})$
and so $\mathbf{d}(a_{j})=\mathbf{r}(a_{j})$, since both $\mathbf{d}(a_{j})$
and $\mathbf{r}(a_{j})$ are atoms. By assumption $a_{j}$ is an idempotent. It
follows that $a$ is an idempotent. ∎

The above result can be used to obtain a more explicit description of the
finite fundamental Boolean inverse monoids.

######  Theorem 7.10.

Let $S$ be a finite Boolean inverse monoid. Then it is fundamental if and only
if $S$ is isomorphic to a finite direct products
$\mathscr{I}_{n_{1}}\times\ldots\times\mathscr{I}_{n_{r}}$.

###### Proof.

It can be checked that the product of two fundamental inverse semigroups is
itself a fundamental inverse semigroup, and that the product of two Boolean
inverse monoids is again a Boolean inverse monoid. So, one direction is easy
to prove. Let, now, $S$ be a fundamental finite Boolean inverse monoid. Then
by Proposition 7.9, $S$ is isomorphic to a Boolean inverse monoid of the form
$\mathsf{K}(G)$ where $G$ is a principal groupoid. Now,
$G=\bigcup_{i=1}^{i=m}H_{i}$ is a finite disjoint union of connected principal
groupoids. It can be checked that
$\mathsf{K}(G)\cong\mathsf{K}(H_{1})\times\ldots\times\mathsf{K}(H_{m})$. Let
$H$ be any finite connected principal groupoid with set of identities $X$.
Then $\mathsf{K}(H)\cong\mathcal{I}(X)$. The result now follows. ∎

The above result can be specialized to characterize the finite symmetric
inverse monoids. Let $S$ be a Boolean inverse monoid. A semigroup ideal
$I\subseteq S$ is said to be an additive ideal if $a,b\in I$ and $a\sim b$
implies that $a\vee b\in I$. Clearly, both $\\{0\\}$ and $S$ itself are
additive ideals. If these are the only ones we say that $S$ is
$0$-simplifying.

We now have the following theorem which can be derived from Theorem 7.10 (or
see [60]).

######  Theorem 7.11.

The finite fundamental $0$-simplifying Boolean inverse monoids are precisely
the finite symmetric inverse monoids.

The above theorem suggests that the groups of units of fundamental
$0$-simplifying Boolean inverse monoids should be regarded as generalizations
of finite symmetric inverse monoids.

##  8\. Congruence-free inverse semigroups with zero

An inverse semigroup is said to be congruence-free if its only congruences are
equality and the universal congruence. In this section, we shall characterize
those inverse semigroups with zero which are congruence-free.161616Douglas
Munn once remarked to me that this was one of the few instances where the
theory for inverse semigroups with zero was easier than it was for the one
without. We begin by ruling out the existence of various kinds of congruence.
An inverse semigroup with zero $S$ is said to be $0$-simple if the only ideals
are $\\{0\\}$ and $S$. The following characterization uses Lemma 6.7 where we
describe the $\mathscr{J}$-relation in terms of the $\mathscr{D}$-relation and
the natural partial order.

######  Lemma 8.1.

Let $S$ be an inverse semigroup with zero. Then $S$ is $0$-simple if and only
if for any two non-zero idempotents $e$ and $f$ in $S$ there exists an
idempotent $i$ such that $e\,\mathcal{D}\,i\leq f$ and an idempotent $j$ such
that $f\,\mathcal{D}\,j\leq e$.

###### Proof.

Suppose first that $S$ is $0$-simple. Let $e$ and $f$ be any two non-zero
idempotents. Observe that $e\in SeS$ and $f\in SfS$. So, both $SeS$ and $SfS$
are not equal to $\\{0\\}$. It follows that $S=SeS=SfS$. Thus
$e\,\mathscr{J}\,f$. We now use Lemma 6.7 to deduce the result where we have
used the fact that the idempotents form an order-ideal in an inverse
semigroup. We now prove the converse. Let $I\neq\\{0\\}$ be any non-zero ideal
of $S$. Suppose that $I\neq S$. Let $a\in S\setminus I$. Observe that
$\mathbf{d}(a)\in S\setminus I$ because if $\mathbf{d}(a)\in I$ then
$a=a\mathbf{d}(a)\in I$, since $I$ is an ideal. Let $b\in I$ be any non-zero
element. Then $\mathbf{d}(b)\in I$ since $I$ is an ideal. By assumption,
$\mathbf{d}(a)$ and $\mathbf{d}(b)$ are nonzero idempotents. It follows by the
assumption and Lemma 6.7, that $S\mathbf{d}(a)S=S\mathbf{d}(b)S$. This implies
that $\mathbf{d}(a)\in I$ and so $a\in I$, which is a contradiction. It
follows that, in fact, $I=S$. ∎

Let $S$ be any inverse semigroup with zero. Define

$(s,t)\in\xi_{S}\Leftrightarrow(\forall a,b\in S)(asb=0\Leftrightarrow
atb=0).$

It is left to the reader to check that this really is a congruence. We shall
denote it by $\xi$ when the semigroup it is defined on is clear. In the case
where $S$ is a meet-semilattice, the above definition simplifies somewhat. Let
$E$ be a meet-semilattice with zero, the operation of which is denoted by
concatenation. Then $(e,f)\in\xi$ if and only ($\forall g\in E$)($eg=0$ if and
only if $fg=0$). A congruence $\rho$ is said to be $0$-restricted if the
$\rho$-class containing $0$ is just $0$. We now have the following
characterization of the congruence $\xi$.

######  Lemma 8.2.

Let $S$ be an inverse semigroup with zero. The congruence $\xi$ is the maximum
$0$-restricted congruence on $S$.

###### Proof.

Observe, first, that $\xi$ is $0$-restricted. Suppose that $(s,0)\in\xi$. By
definition, for all $a,b\in S$ we have that $asb=0$ if and only of $a0b=0$.
However, if we put $a=ss^{-1}$ and $b=s^{-1}s$ then we deduce that $s=0$. Now,
let $\rho$ be any $0$-restricted congruence on $S$ and let $s\,\rho\,t$.
Suppose that $asb=0$. Then $asb\,\rho\,atb$. Since $\rho$ is $0$-restricted,
we have that $atb=0$. Thus $asb=0$ implies that $atb=0$. By symmetry, we
deduce that $a\,\xi\,b$. Thus $\rho\subseteq\xi$, as required. ∎

Our next result shows, amongst other things, the relationship between $\mu$
and $\xi$.

######  Lemma 8.3.

Let $S$ be an inverse semigroup with zero.

  1. (1)

$\mu\subseteq\xi$.

  2. (2)

The congruence $\xi$ restricted to $\mathsf{E}(S)$ is precisely
$\xi_{\mathsf{E}(S)}$.

###### Proof.

(1) Let $s\,\mu\,t$. We prove that $s\,\xi\,t$. Suppose that $asb=0$ for
$a,b\in S$. We shall prove that $atb=0$. However, $asb\,\mu\,atb$. By
definition, for all idempotents $e$, we have that
$(asb)e(asb)^{-1}=(atb)e(atb)^{-1}$. Choose $e=b^{-1}b$. It follows that
$\mathbf{r}(asb)=\mathbf{r}(atb)$. We are given that $asb=0$ and so
$\mathbf{r}(asb)=0$. It follows that $\mathbf{r}(atb)=0$ and so $atb=0$. By
symmetry, this shows that $s\,\xi\,t$.

(2) Let $e$ and $f$ be idempotents. To say that $(e,f)\in\xi$ means that for
all $a,b\in S$ we have that $aeb=0$ if and only if $afb=0$. It is clear that
$(e,f)\in\xi_{\mathsf{E}(S)}$. Suppose, now, that
$(e,f)\in\xi_{\mathsf{E}(S)}$. We prove that $(e,f)\in\xi$ in $S$. Let
$aeb=0$. Then $a^{-1}aebb^{-1}=0$. Thus $a^{-1}abb^{-1}e=0$ and so, by
assumption, $a^{-1}abb^{-1}f=0$. Hence $a^{-1}afbb^{-1}=0$ and so $afb=0$. The
reverse direction is proved similarly. ∎

We now link $\xi$ being the equality relation on the meet-semilattice $E$ with
the property of $E$ being $0$-disjunctive which we introduced just before
Lemma 3.15.

######  Lemma 8.4.

Let $E$ be a meet semilattice with zero. Then $\xi$ is the equality relation
on $E$ if and only if $E$ is $0$-disjunctive.

###### Proof.

Suppose first that $\xi$ is the equality relation on $E$. We prove that $E$ is
$0$-disjunctive. Suppose that $0<f<e$ and assume that there does not exist any
$0\neq g\leq e$ such that $fg=0$. If $ei=0$ then clearly $fi=0$. Suppose that
$fi=0$. Then $fi=fei=0$ and so $f(ei)=0$. Clearly, $ei\leq e$ and so by our
assumption above, we must have $ei=0$. It follows that $(e,f)\in\xi$. But this
implies that $e=f$, which is a contradiction.

We assume that $E$ is $0$-disjunctive and prove that $\xi$ is the equality
relation. Suppose that $(e,f)\in\xi$, where $e$ and $f$ are both non-zero.
Then $e\,\xi\,ef$ and so $ef\neq 0$ since $\xi$ is $0$-restricted. Suppose
that $ef\neq e$. Then $0<ef<e$. Then, by assumption, there exists $0\neq g\leq
e$ such that $(ef)g=0$. But, clearly, $g=eg\neq 0$. However, $e\,\xi\,ef$ and
so we have a contradiction. It follows that $ef=e$. Similarly, $ef=f$ and so
$e=f$, as required. ∎

We are nearly at our goal.

######  Lemma 8.5.

Let $S$ be an inverse semigroup with zero. Then $\xi$ is the equality relation
if and only if $\mathsf{E}(S)$ is $0$-disjunctive and $S$ is fundamental.

###### Proof.

Suppose first that $\xi$ is the equality relation. Then $\mathcal{E}(S)$ is
$0$-disjunctive by Lemma 8.3 and Lemma 8.4, and it is fundamental by Lemma
8.3. To prove the converse, suppose that $\mathsf{E}(S)$ is $0$-disjunctive
and $S$ is fundamental. Then by Lemma 8.3 and Lemma 8.4, $\xi$ restricted to
$\mathsf{E}(S)$ is the equality relation and so $\xi$ is idempotent-
separating. It follows by Lemma 7.2 that $\xi\subseteq\mu$. But $S$ is
fundamental and so $\mu$ is the equality relation thus $\xi$ is the equality
relation. ∎

We may now state the characterization of congruence-free inverse semigroups
with zero.

######  Theorem 8.6 (Congruence-free inverse semigroups with zero).

An inverse semigroup with zero $S$ is congruence-free if and only if $S$ is
fundamental, $0$-simple and $\mathsf{E}(S)$ is $0$-disjunctive.

###### Proof.

Suppose that $S$ is congruence-free. Then $\mu$ is equality, there are no non-
trivial ideals and $\xi$ is equality. Thus $S$ is fundamental, $0$-simple and
$\mathsf{E}(S)$ is $0$-disjunctive by Lemma 8.4

To prove the converse, suppose that $S$ is fundamental, $0$-simple and
$\mathsf{E}(S)$ is $0$-disjunctive. Let $\rho$ be a congruence on $S$ which is
not the universal relation. Then $\rho(0)$ is an ideal which is not $S$. Thus
this ideal must be equal to $\\{0\\}$. It follows that $\rho$ is a
$0$-restricted congruence and so $\rho\subseteq\xi$ by Lemma 8.2. Now we use
the fact that the semigroup is fundamental together with Lemma 8.5, to deduce
that $\xi$ is the equality congruence and so $\rho$ is the equality
congruence. ∎

##  9\. Further reading and examples

My interest in inverse semigroups developed as a result of John Fountain’s
undergraduate course in semigroups at the University of York, UK. It was
deepened by attending the seminar at Oxford organized by Peter Cameron,
Wilfrid Hodges, Angus Macintyre, and Peter Neumann, in which partial
automorphisms and the work of Fraïssé played a central role. The connections
between Fraïssé’s work and inverse semigroups was clarified in [114] though
based on the work of Benda to be found [8]. Partial automorphisms also play a
significant role in Fraïssé’s approach to the study of relational structures
[25]. The work of Ehresmann on pseudogroups [21] convinced me that inverse
semigroups really were worth studying. I was, however, bugged by the question
of whether there were ‘natural’ examples of inverse semigroups. But what makes
an area of mathematics ‘natural’? I can think of at least two answers to this
question: the existence of deep problems and the proliferation of good
examples. When I embarked on writing my book [51], I was primarily motivated
by finding good examples. The work of Kellendonk [44, 45, 46] and Girard [29],
as mediated by Peter Hines [34], provided such good examples. Since that time,
it has become clear that partial bijections, and therefore inverse semigroups,
arise naturally in many different parts of mathematics. There are also deep
questions such as whether every finite inverse monoid has a finite $F$-inverse
cover [5] or what can be said about finite semigroups with commuting
idempotents [3]. In this section, I shall concentrate on sources of good
examples, since these look like promising growth points for the theory.

If there is one area of mathematics which makes essential use of inverse
semigroups, it is in the theory of $C^{\ast}$-algebras. There are so many
books and papers that are relevant here that I will only mention a few to get
you started. The connection between inverse semigroups and $C^{\ast}$-algebras
was first explored in [48] and [101]. A rather more recent book on this topic
is [94] which will repay reading. There are many, many papers on this topic. A
good place to start is [22] and [24] but it is worth checking out [99, page
45] which is on the special case of Cantor minimal systems. Let me, here, give
one example of the theory of $C^{\ast}$-algebras leading to new semigroup
theory. A monoid is said to be finitely aligned if the intersection of any two
principal right ideals is a finitely generated right ideal. This concept was
first defined within the theory of $C^{\ast}$-algebras in [100] and made
extensive use of in [107, 108]. However, it seems to have been first defined
within semigroup theory in [31], quite early on, but was not picked up by
workers in inverse semigroup theory; this is but one illustration of the
importance of mathematicians in different fields talking to one another. See
also the more recent [12]. It would be most naturally applied in [13].

The connection between inverse semigroups and $C^{\ast}$-algebras has come to
be mediated by what are now termed Steinberg algebras; see [110, 112]. Even
where no direct connection between inverse semigroups and $C^{\ast}$-algebras
appears to exist, the theory of $C^{\ast}$-algebras has sometimes led to new
developments in the theory of inverse semigroups. A good example is
Steinberg’s notion of strong Morita equivalence of inverse semigroups
motivated by the corresponding theory for $C^{\ast}$-algebras [109].

The connections between inverse semigroups and quasi-crystals were first spelt
out by Kellendonk [44, 45, 46]. The inverse semigroup perspective on the
topological groupoids that Kellendonk constructs were first clarified by Lenz
[76] and then developed in [67]. Aspects of this connection were described in
[66]. This led to what we might term ‘non-commutative Stone duality’ which
connects, in particular, Boolean inverse monoids with a class of étale
groupoids. For a survey of non-commutative Stone duality, see [65].

I once asked the late Pieter Hofstra, an expert in topos theory, why I kept
seeing inverse semigroups everyone. I don’t know whether Pieter was being
polite, but he explained that it was because étendues were everywhere, where
an étendue is a particular kind of topos. This seems to me a particularly
fruitful area of research. We refer the reader to [27, 28].

If the theory of inverse semigroups is to be more fully developed, then there
will be a need to study invariants associated with inverse semigroups.
Motivated by the Banach-Tarski Paradox [15] and the theory of $K_{0}$-groups
[30], the type monoid was introduced in [47, 116] and its theory developed by
Wehrung [117]. Type monoids are commutative refinement monoids and provide the
first example of invariants of inverse semigroups. Another source of examples
of invariants of inverse semigroups are cohomology theories. These have barely
been developed; see the work of Lausch [74] and, particularly, Loganathan
[77]. An application of the extension theory of inverse semigroups in the case
of Von Neumann algebras can be found in [19]. MV-algebras form a natural class
of invariants for the class of factorizable Boolean inverse monoids $S$ in
which $S/\mathscr{J}$ is a lattice. MV-algebras are another generalization of
Boolean algebras but it was proved in [70] that all countable MV-algebras
arise in this way and this was generalized to all MV-algebras by Wehrung [117]
using very different methods. Thus every MV-algebra arises from a particular
kind of Boolean inverse monoid. The theory of MV-algebras goes back to
Tarski’s book [113], which is also noteworthy in that partial automorphisms of
structures are singled out in [113, Theorem 11.6]. MV-algebras are studied in
their own right in [14, 87, 88] but their applications to the study of certain
kinds of Boolean inverse monoids has so far not been developed.

I have indicated that certain kinds of groups can be constructed from inverse
semigroups. See [81, 82] for how groups arise as so-called ‘topological full
groups’ of certain kinds of groupoids; under non-commutative Stone duality,
this means that the groups arise as the groups of units of certain kinds of
Boolean inverse monoids. The Thompson groups form an important class of such
groups. They, too, can be constructed from inverse monoids [41, 71, 73]
generalizing some initial work in [9]. Groups with a different flavour arise
from what Dehornoy calls ‘geometry monoids’. This is explained in his book
[17] but the explicit connection with inverse monoids is made in [54].

If I were to single out one class of inverse semigroups that justifies the
field it would be the polycyclic inverse monoids introduced by Nivat and
Perrot [92]. They were generalized by Perrot [95, 96]. As I pointed out in
[57], Perrot’s thesis contains the first definition of self-similar group
actions [91] and was notivated entirely by inverse semigroup theory. My paper
was subsequently generalized by Alistair Wallis [116]; see also [68, 69]. The
polycyclic inverse monoids are intimately related to the classical Thompson
groups [55, 56, 64]. The representation theory (by means of partial
bijections) of the polycyclic inverse monoids is the subject of [39, 40, 58]
and was motivated by [10] and by [43], which on the face of it have nothing to
do with the polycyclic inverse monoids. The polycyclic inverse monoids arise
from the free monoid. Analogous inverse monoids arise from free categories [4,
41]. There is a natural connection between such inverse semigroups and Leavitt
path algebras [1]. This is part of the general topic of studying algebras of
various kinds associated with inverse semigroups. This goes back to pioneering
work of Douglas Munn [89] and it reaches its apotheosis in the applications of
inverse semigroups to the theory of $C^{\ast}$-algebras mentioned above. This
is a subject in its own right, but we mention in passing [18] and [72], which
are relevant.

To conclude, I have not said anything about the classical theory of inverse
semigroups. Here, I would simply highlight the papers [2, 83, 86] with the
suggestion that there is more to do here. I have also said nothing about free
inverse semigroups. These are introduced in my book [51, Chapter 6] but a much
more recent paper on their structure is [33]. In passing, I would like to
mention that free inverse monoids can also be constructed using the doubly
pointed pattern classes of Kellendonk applied to the graph of the free group
regadred as a ‘dendritic tiling’ [45]. In fact, it was this example which
convinced me that Kellendonk was on to something. The study of free inverse
monoids leads naturally to the study of presentations of inverse semigroups. A
recent paper on this topic is [32].

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