to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{2.07999pt}{0.0pt}\pgfsys@lineto{13.82639pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{2.27998pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02638pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}e$, the products
$e^{*}(gs)$ and $(tg)e^{*}$ both evaluate to $e^{*}ge^{*}$. Conversely, we
have the following:

######  Theorem 6.3 (see [57, Corollary 2.7]).

Any regular $*$-boset is the $*$-boset of a regular $*$-semigroup. ∎

We denote by ${\bf RSBS}$ the category of regular $*$-bosets. Morphisms in
${\bf RSBS}$ are called _regular $*$-bimorphisms_, and are the regular
bimorphisms $\phi:E\to E^{\prime}$ (as above) that respect the involutions,
meaning that

$(e^{*})\phi=(e\phi)^{*}\qquad\text{for all $e\in E$.}$

As above, regular $*$-bosets $E$ and $E^{\prime}$ are isomorphic in ${\bf
RSBS}$ if and only if there is a bisomorphism $E\to E^{\prime}$ that also
respects the involutions.

The assignment $S\mapsto{\bf E}(S)$ is the object part of a (forgetful)
functor

${\bf E}:{\bf RSS}\to{\bf RSBS}.$ (6.4)

A $*$-morphism $\phi:S\to S^{\prime}$ in ${\bf RSS}$ is sent to its
restriction ${\bf E}(\phi)=\phi|_{{\bf E}(S)}:{\bf E}(S)\to{\bf
E}(S^{\prime})$, which is a regular $*$-bimorphism in ${\bf RSBS}$.

###  6.2 From projection algebras to $*$-biordered sets

Theorem 6.19 below establishes a category equivalence ${\bf
PA}\leftrightarrow{\bf RSBS}$. For this we need functors in both directions
between the two categories. We can immediately obtain a functor

${\bf E}:{\bf PA}\to{\bf RSBS}$ (6.5)

by composing the functors

${\bf PA}\to{\bf
RSS}:P\mapsto\mathcal{C}_{P}=\operatorname{\textup{{PG}}}(P)\qquad\text{and}\qquad{\bf
RSS}\to{\bf RSBS}:S\mapsto{\bf E}(S)$

from Theorem 5.1 and (6.4). Note that the functors ${\bf RSS}\to{\bf RSBS}$
and ${\bf PA}\to{\bf RSBS}$ in (6.4) and (6.5) are both denoted by ${\bf E}$.
As these take different kinds of arguments (regular $*$-semigroups or
projection algebras, and their morphisms), it will always be clear from
context which one is meant.

The functor ${\bf E}:{\bf PA}\to{\bf RSBS}$ in (6.5) maps a projection algebra
$P$ to the regular $*$-boset of $\operatorname{\textup{{PG}}}(P)$. Using
(CP5), we have

${\bf E}(P)={\bf
E}(\operatorname{\textup{{PG}}}(P))=\\{[[p,q]]:(p,q)\in{\mathrel{\mathscr{F}}}\\}$
(6.6)

as a set. We will describe the biordered structure of ${\bf E}(P)$ in
Proposition 6.9 below; the involution is of course given by
$[[p,q]]^{*}=[[q,p]]$.

The functor ${\bf E}:{\bf PA}\to{\bf RSBS}$ maps a projection algebra morphism
$\phi:P\to P^{\prime}$ to the regular $*$-bimorphism

${\bf E}(\phi)={\bf E}(\overline{\phi}):{\bf E}(P)\to{\bf E}(P^{\prime}),$

where
$\overline{\phi}:\operatorname{\textup{{PG}}}(P)\to\operatorname{\textup{{PG}}}(P^{\prime})$
is the $*$-morphism from Theorem 5.9. So ${\bf E}(\phi)$ is the restriction of
$\overline{\phi}$ to ${\bf E}(P)={\bf E}(\operatorname{\textup{{PG}}}(P))$;
explicitly, we have $[[p,q]]{\bf E}(\phi)=[[p\phi,q\phi]]$ for
$(p,q)\in{\mathrel{\mathscr{F}}}$.

For the next two proofs, we note that the product of idempotents $e=[[p,q]]$
and $f=[[r,s]]$ in $\operatorname{\textup{{PG}}}(P)$ is given by

$e\bullet
f=[[p^{\prime},q^{\prime},r^{\prime},s^{\prime}]],\qquad\text{where}\qquad
p^{\prime}=r\theta_{q}\theta_{p},\quad q^{\prime}=r\theta_{q},\quad
r^{\prime}=q\theta_{r}\quad\text{and}\quad s^{\prime}=q\theta_{r}\theta_{s}.$
(6.7)

In particular, if $e\bullet f$ happens to be an idempotent, then

$e\bullet
f=[[p^{\prime},s^{\prime}]]=[[r\theta_{q}\theta_{p},q\theta_{r}\theta_{s}]].$
(6.8)

######  Proposition 6.9.

If $P$ is a projection algebra, and if $e=[[p,q]]$ and $f=[[r,s]]$, then

  1. (i)

$e\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{2.07999pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{2.27998pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}f\textup{
(i.e.~{}$e=e\bullet f$)}\ \Leftrightarrow\ {\bf r}(e)\leq{\bf r}(f)\
\Leftrightarrow\ q\leq s$, in which case $f\bullet
e=[[p\theta_{s}\theta_{r},s\theta_{p}\theta_{q}]]$,

  2. (ii)

$e\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{13.82639pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02638pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}f\textup{
(i.e.~{}$e=f\bullet e$)}\ \Leftrightarrow\ {\bf d}(e)\leq{\bf d}(f)\
\Leftrightarrow\ p\leq r$, in which case $e\bullet
f=[[r\theta_{q}\theta_{p},q\theta_{r}\theta_{s}]]$.

###### Proof.

For the first part (the second is dual), it is enough to show that $e=e\bullet
f\ \Leftrightarrow\ q\leq s$, as the expression for $f\bullet e$ will then
follow from (6.8). Write $e\bullet
f=[[p^{\prime},q^{\prime},r^{\prime},s^{\prime}]]$, as in (6.7). If
$e=e\bullet f$, then $q=s^{\prime}=q\theta_{r}\theta_{s}\leq s$. Conversely,
suppose $q\leq s$, so that $q=q\theta_{s}$. Combining this with
$\theta_{s}=\theta_{s}\theta_{r}\theta_{s}$ (which holds by (PA5)), we
calculate

$q=q\theta_{s}=q\theta_{s}\theta_{r}\theta_{s}=q\theta_{r}\theta_{s}=s^{\prime}.$

From $q\leq s\mathrel{\mathscr{F}}r$ it follows from (PA2) that
$q\leq_{\mathrel{\mathscr{F}}}r$, and so $q=r\theta_{q}=q^{\prime}$. But also

$p^{\prime}=r\theta_{q}\theta_{p}=q\theta_{p}=p,$

and so $e\bullet
f=[[p^{\prime},q^{\prime},r^{\prime},s^{\prime}]]=[[p,q,r^{\prime},q]]=[[p,q]]=e$,
as required. ∎

The functor ${\bf E}:{\bf PA}\to{\bf RSBS}$ constructs ${\bf E}(P)$ from a
projection algebra $P$ by first passing through
$\operatorname{\textup{{PG}}}(P)$. Perhaps surprisingly, it turns out that we
could have passed through _any_ regular $*$-semigroup $S$ with projection
algebra $P={\bf P}(S)$, and we would obtain the same $*$-boset up to
isomorphism, ${\bf E}(S)\cong{\bf E}(P)$:

######  Proposition 6.10.

If $S$ is a regular $*$-semigroup with projection algebra $P={\bf P}(S)$, then
the map

${\bf E}(P)\to{\bf E}(S):[[p,q]]\mapsto pq$

is an isomorphism of $*$-bosets.

###### Proof.

As in Remark 5.12, there exists a $*$-morphism
$\Phi(=\overline{\operatorname{id}}_{P}):\operatorname{\textup{{PG}}}(P)\to S$
with $p\Phi=p$ for all $p\in P$. Applying the functor ${\bf E}$ from (6.4), we
obtain the regular $*$-bimorphism

$\phi={\bf E}(\Phi)=\Phi|_{{\bf E}(P)}:{\bf E}(P)\to{\bf E}(S),$

which is the map from the statement. As explained in Subsection 6.1, and since
we already know that $\phi$ is a (regular) $*$-bimorphism, we can complete the
proof that this is an isomorphism by checking that:

  1. (i)

$\phi$ is a bijection, and

  2. (ii)

$\operatorname{\textup{{BP}}}({\bf
E}(P))\phi=\operatorname{\textup{{BP}}}({\bf E}(S))$.

(i). This follows by applying (RS7) in both $S$ and
$\operatorname{\textup{{PG}}}(P)$, and keeping in mind $p\bullet q=[[p,q]]$.

(ii). Since $\phi$ is a bijection, this amounts to showing that

$(e,f)\in\operatorname{\textup{{BP}}}({\bf E}(P))\ \Leftrightarrow\
(e\phi,f\phi)\in\operatorname{\textup{{BP}}}({\bf E}(S))\qquad\text{for all
$e,f\in{\bf E}(P)$.}$

The forward implication is clear, as $\phi$ is a bimorphism. Conversely,
suppose $(e\phi,f\phi)\in\operatorname{\textup{{BP}}}({\bf E}(S))$, where
$e,f\in{\bf E}(P)$, and write $e=[[p,q]]$ and $f=[[r,s]]$. By symmetry we may
assume that ${e\phi\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{2.07999pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{2.27998pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}f\phi}$, i.e.
$e\phi=(e\phi)(f\phi)$; we complete the proof by showing that
$e\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{2.07999pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{2.27998pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}f$, i.e. ${e=e\bullet
f}$. To do so, first write $e\bullet
f=[[p^{\prime},q^{\prime},r^{\prime},s^{\prime}]]$, as in (6.7). By Theorem
3.9, the $*$-morphism $\Phi:\operatorname{\textup{{PG}}}(P)\to S$ is also a
chained projection functor ${{\bf
G}(\operatorname{\textup{{PG}}}(P))=(P,\overline{\mathscr{C}},\nu)\to{\bf
G}(S)=(P,\mathcal{G},\varepsilon)}$. Since $\Phi$ is the identity on
$P=v\overline{\mathscr{C}}=v\mathcal{G}$, it follows that $\Phi$ is a
$v$-functor $\overline{\mathscr{C}}\to\mathcal{G}$, and so

$s^{\prime}={\bf r}(e\bullet f)={\bf r}((e\bullet f)\Phi)={\bf
r}((e\Phi)(f\Phi))={\bf r}((e\phi)(f\phi))={\bf r}(e\phi)={\bf r}(e)=q.$

Consequently, $q=s^{\prime}=q\theta_{r}\theta_{s}\leq s$, and we then obtain
$e\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{2.07999pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{2.27998pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}f$ from Proposition
6.9. ∎

######  Remark 6.11.

Combining Theorem 6.3 and Proposition 6.10, it follows that every regular
$*$-boset is isomorphic to ${\bf E}(P)$ for some projection algebra $P$.

######  Remark 6.12.

Another approach to constructing a boset ${\bf E}(P)$ from a projection
algebra $P$ would be to take the underlying set

${\bf E}(P)={\mathrel{\mathscr{F}}}=\\{(p,q)\in P\times P:p=q\theta_{p}\text{
and }q=p\theta_{q}\\},$

and define the $\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{2.07999pt}{0.0pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{2.27998pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ and
$\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{13.82639pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02638pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ pre-orders, and
basic products, as in Proposition 6.9. One would then need to check that the
boset axioms are satisfied. Taking this approach, Proposition 6.10 would then
state that $(p,q)\mapsto pq$ is an isomorphism ${\bf E}(P)\to{\bf E}(S)$ for
any regular $*$-semigroup $S$ with projection algebra $P={\bf P}(S)$.

###  6.3 From $*$-biordered sets to projection algebras

Now that we have constructed a functor ${\bf E}:{\bf PA}\to{\bf RSBS}$, we
wish to construct a functor ${\bf P}:{\bf RSBS}\to{\bf PA}$ in the reverse
direction. (Again, this functor ${\bf RSBS}\to{\bf PA}$ has the same name as
the forgetful functor ${\bf P}:{\bf RSS}\to{\bf PA}$ considered earlier.) This
is somewhat more involved than the functor ${\bf E}$, which was simply the
composition of two previously existing functors.

Consider a regular $*$-boset $E=(E,{\cdot},{}^{*})$. The underlying set of the
projection algebra ${\bf P}(E)$ is simply the set of fixed points under the
involution,

${\bf P}(E)=\\{p\in E:p=p^{*}\\},$

the elements of which are called the _projections_ of $E$. To define the
operations in ${\bf P}(E)$ we need the following special case of [57,
Proposition 2.5]. We give a simple adaptation of the proof for completeness.

######  Lemma 6.13.

If $E$ is a regular $*$-boset, and if $p,q\in{\bf P}(E)$, then there exists a
unique element $e=e(p,q)$ of the sandwich set $\SS(p,q)$ for which
$pe,eq\in{\bf P}(E)$. Moreover, we have

$e=qp\qquad\text{and}\qquad pe=pqp$

in any regular $*$-semigroup $S$ with $*$-boset $E={\bf E}(S)$.

###### Proof.

Fix an arbitrary regular $*$-semigroup $S$ with $E={\bf E}(S)$. We obtain
$qp\in\SS(p,q)$ from (6.2) and (RS2), and $p(qp),(qp)q\in{\bf P}(E)$ from
(RS5).

For uniqueness, suppose $e\in\SS(p,q)$ is such that $pe,eq\in{\bf P}(E)$. We
first claim that

$pqp=pe\qquad\text{and}\qquad qpq=eq.$ (6.14)

We prove the first, and the second is analogous. Define the projections
$s=pqp$ and $t=pe$. Using (RS2) and (6.2) in the indicated places, we
calculate

$s=\underline{pq}p=\underline{pe}qp=tqp\ \Rightarrow\
s=ts\qquad\text{and}\qquad
t=p\underline{e}=\underline{pq}ep=\underline{pqp}qep=sqep\ \Rightarrow\ t=st.$

Since $s$ and $t$ are projections it then follows that
$s=s^{*}=(ts)^{*}=s^{*}t^{*}=st=t$, as claimed. Combining (RS2), (6.2) and
(6.14), we obtain

$e=\underline{e}e=e\underline{pe}=\underline{ep}qp=\underline{eq}p=\underline{qpqp}=qp.\qed$

From now on we fix the notation $e(p,q)$ from Lemma 6.13. It is important to
note that while the products $qp$ and $pqp$ in this lemma are taken in the
semigroup $S$, the products $pe$ and $eq$ exist in the boset $E$ itself.

######  Lemma 6.15.

If $\phi:E\to E^{\prime}$ is a regular $*$-bimorphism, then
$e(p,q)\phi=e(p\phi,q\phi)$ for all $p,q\in{\bf P}(E)$.

###### Proof.

With $e=e(p,q)$, Lemma 6.13 gives $e\in\SS(p,q)$ and $pe,eq\in{\bf P}(E)$. We
deduce ${e\phi\in\SS(p\phi,q\phi)}$ from regularity of $\phi$, and
$(p\phi)(e\phi),(e\phi)(q\phi)\in{\bf P}(E^{\prime})$ because $\phi$ is a
$*$-bimorphism. It then follows from uniqueness in Lemma 6.13 that
$e\phi=e(p\phi,q\phi)$. ∎

######  Definition 6.16.

For a regular $*$-boset $E$, we define ${\bf P}(E)$ to be the projection
algebra with underlying set ${\bf P}(E)=\\{p\in E:p=p^{*}\\}$, and operations
given by

$q\theta_{p}=pe(p,q)\qquad\text{for $p,q\in{\bf P}(E)$.}$

This is well defined by Lemma 6.13, which also tells us that ${\bf P}(E)={\bf
P}(S)$ is the projection algebra of any regular $*$-semigroup $S$ with
$*$-boset $E={\bf E}(S)$.

For a regular $*$-bimorphism ${\phi:E\to E^{\prime}}$, we define ${\bf
P}(\phi)$ to be the restriction

${\bf P}(\phi)=\phi|_{{\bf P}(E)}:{\bf P}(E)\to{\bf P}(E^{\prime}).$ (6.17)

(Note that $\phi$ maps projections to projections because it is a
$*$-bimorphism.)

######  Proposition 6.18.

${\bf P}$ is a functor ${\bf RSBS}\to{\bf PA}$.

###### Proof.

To show that ${\bf P}(\phi):{\bf P}(E)\to{\bf P}(E^{\prime})$ as in (6.17) is
a projection algebra morphism, let ${p,q\in{\bf P}(E)}$. We then use Lemma
6.15 to calculate

$(q\theta_{p})\phi=(pe(p,q))\phi=(p\phi)(e(p,q)\phi)=(p\phi)e(p\phi,q\phi)=(q\phi)\theta_{p\phi}.$

It is again clear that the laws ${\bf P}(\phi\circ\phi^{\prime})={\bf
P}(\phi)\circ{\bf P}(\phi^{\prime})$ and ${\bf
P}(\operatorname{id}_{E})=\operatorname{id}_{{\bf P}(E)}$ hold. ∎

###  6.4 A category equivalence, and more on freeness

We now establish the promised category equivalence.

######  Theorem 6.19.

We have

${\bf P}\circ{\bf E}=\operatorname{id}_{\bf PA}\qquad\text{and}\qquad{\bf
E}\circ{\bf P}\cong\operatorname{id}_{\bf RSBS}.$

Consequently, the functors ${\bf P}$ and ${\bf E}$ furnish an equivalence of
the categories ${\bf PA}$ and ${\bf RSBS}$.

###### Proof.

To show that ${\bf P}\circ{\bf E}=\operatorname{id}_{\bf PA}$ we need to show
that

  * •

${\bf P}({\bf E}(P))=P$ for any projection algebra $P$, and

  * •

${\bf P}({\bf E}(\phi))=\phi$ for any projection algebra morphism $\phi:P\to
P^{\prime}$.

Since ${\bf E}(\operatorname{\textup{{PG}}}(P))={\bf E}(P)$, it follows from
Definition 6.16 that ${\bf P}({\bf E}(P))={\bf
P}(\operatorname{\textup{{PG}}}(P))=P$. For the statement concerning $\phi$,
we follow the definitions to compute

${\bf P}({\bf E}(\phi))={\bf E}(\phi)|_{{\bf P}({\bf E}(P))}={\bf
E}(\phi)|_{P}=\phi.$

To show that ${\bf E}\circ{\bf P}\cong\operatorname{id}_{\bf RSBS}$, we will
construct a natural isomorphism $\eta:{\bf E}\circ{\bf
P}\to\operatorname{id}_{\bf RSBS}$. Towards this, we claim that for any
regular $*$-boset $E$, the map

$\eta_{E}:{\bf E}({\bf P}(E))\to E:[[p,q]]\mapsto e(q,p)$

is a $*$-bisomorphism. To see this, let $S$ be a regular $*$-semigroup with
$*$-boset $E={\bf E}(S)$, and write $P={\bf P}(E)={\bf P}(S)$. Since
$pq=e(q,p)$ in $S$ by Lemma 6.13, it follows that $\eta_{E}$ is precisely the
isomorphism from Proposition 6.10.

We now show that $\eta=(\eta_{E})$ is a natural isomorphism ${{\bf E}\circ{\bf
P}\to\operatorname{id}_{\bf RSBS}}$. Since each $\eta_{E}$ is an isomorphism,
it remains to check that for any regular $*$-bimorphism $\phi:E\to E^{\prime}$
the following diagram commutes:

${{\bf E}\circ{\bf P}(E)}$ ${{\bf E}\circ{\bf P}(E^{\prime})}$ ${E}$
${E^{\prime}.}$$\scriptstyle{{\bf E}\circ{\bf
P}(\phi)}$$\scriptstyle{\eta_{E}}$$\scriptstyle{\eta_{E^{\prime}}}$$\scriptstyle{\phi}$

But the elements of ${\bf E}\circ{\bf P}(E)$ have the form $[[p,q]]$ for
$\mathrel{\mathscr{F}}$-related projections $p,q\in{\bf P}(E)$, and we use
Lemma 6.15 to calculate

$[[p,q]]\mapstochar\xrightarrow{\ \eta_{E}\ }e(q,p)\mapstochar\xrightarrow{\
\phi\ }e(q\phi,p\phi)\qquad\text{and}\qquad[[p,q]]\mapstochar\xrightarrow{\
{\bf E}\circ{\bf P}(\phi)\ }[[p\phi,q\phi]]\mapstochar\xrightarrow{\
\eta_{E^{\prime}}\ }e(q\phi,p\phi).\qed$

Combining Theorems 5.1 and 6.19, we obtain the following:

######  Theorem 6.20.

The functor ${\bf RSBS}\to{\bf RSS}:E\mapsto\operatorname{\textup{{PG}}}({\bf
P}(E))$ is a left adjoint to the forgetful functor ${\bf RSS}\to{\bf
RSBS}:S\mapsto{\bf E}(S)$, and ${\bf RSBS}$ is coreflective in ${\bf RSS}$. ∎

It follows that the chain semigroups are the free objects in ${\bf RSS}$ with
respect to both forgetful functors

${\bf P}:{\bf RSS}\to{\bf PA}:S\mapsto{\bf P}(S)\qquad\text{and}\qquad{\bf
E}:{\bf RSS}\to{\bf RSBS}:S\mapsto{\bf E}(S).$

The regular $*$-semigroups $\operatorname{\textup{{PG}}}({\bf P}(E))$ are
defined in terms of the regular $*$-boset $E$. Analogously to
$\operatorname{\textup{{IG}}}(E)$ and $\operatorname{\textup{{RIG}}}(E)$, we
denote this by

$\operatorname{\textup{{IG}}}^{*}(E)=\operatorname{\textup{{PG}}}({\bf
P}(E)).$

Note that it is possible for regular $*$-bosets $E$ and $E^{\prime}$ to have
different underlying sets, but have identical projection algebras ${\bf
P}(E)={\bf P}(E^{\prime})$. This would occur if $E$ and $E^{\prime}$ were
isomorphic, with different underlying sets, but with the same set of
projections. In this case we would of course have
$\operatorname{\textup{{IG}}}^{*}(E)=\operatorname{\textup{{IG}}}^{*}(E^{\prime})$.
This means that the assignment $E\mapsto\operatorname{\textup{{IG}}}^{*}(E)$
is not injective.

One could get around this ‘problem’ by instead defining
$\operatorname{\textup{{IG}}}^{*}(E)$ to be a copy of
$\operatorname{\textup{{PG}}}({\bf P}(E))$ in which we identify each $e\in E$
with $[[ee^{*},e^{*}e]]\in\operatorname{\textup{{PG}}}({\bf P}(E))$. This
would then reflect the situation of projection algebras, where
$\operatorname{\textup{{PG}}}(P)=\operatorname{\textup{{PG}}}(P^{\prime})\
\Leftrightarrow\ P=P^{\prime}$, as follows from Theorem 4.22.

##  7 Presentations

In this section we establish a number of presentations for the free
(projection-generated) regular $*$-semigroup
${\operatorname{\textup{{PG}}}(P)=\mathcal{C}_{P}}$ over an arbitrary
projection algebra $P$. The first presentation (Theorem 7.2) involves the
generating set $P$. The second and third (Theorems 7.9 and 7.11) are both in
terms of the generating set $E={\bf E}(P)$, and these highlight the
connections between $\operatorname{\textup{{PG}}}(P)$ and the free
(idempotent-generated) semigroup $\operatorname{\textup{{IG}}}(E)$ and the
free regular (idempotent-generated) semigroup
$\operatorname{\textup{{RIG}}}(E)$.

###  7.1 Preliminaries on presentations

We begin by establishing the notation we will use for presentations; for more
details, see for example [39, Section 1.6]. We also prove a general technical
lemma that will be used in this section and later in the paper.

A _congruence_ on a semigroup $S$ is an equivalence relation $\sigma$ on $S$
that is compatible with multiplication, in the sense that
$a\mathrel{\sigma}a^{\prime}$ and $b\mathrel{\sigma}b^{\prime}$ together imply
$ab\mathrel{\sigma}a^{\prime}b^{\prime}$. The quotient $S/\sigma$ is then a
semigroup under the induced operation on $\sigma$-classes. Given a semigroup
homomorphism $\phi:S\to T$, the _kernel_ $\ker(\phi)=\\{(a,b)\in S\times
S:a\phi=b\phi\\}$ is a congruence on $S$, and the fundamental homomorphism
theorem for semigroups states that
$S/{\ker(\phi)}\cong\operatorname{im}(\phi)$.

For a set $X$, we denote by $X^{+}$ the free semigroup over $X$, which
consists of all non-empty words over $X$, under concatenation. For a set
$R\subseteq X^{+}\times X^{+}$ of pairs of words, we write $R^{\sharp}$ for
the congruence on $X^{+}$ generated by $R$, i.e. the least congruence on
$X^{+}$ containing $R$. We often write $[w]_{R}$ for the $R^{\sharp}$-class of
$w\in X^{+}$. We say a semigroup $S$ has _presentation_ $\langle X:R\rangle$
if $S\cong X^{+}/R^{\sharp}$, i.e. if there is a surjective semigroup
homomorphism $X^{+}\to S$ with kernel $R^{\sharp}$. At times we identify
$\langle X:R\rangle$ with the semigroup $X^{+}/R^{\sharp}$ itself. The
elements of $X$ and $R$ are called generators and (defining) relations,
respectively. A relation $(u,v)\in R$ is typically displayed as an equality:
$u=v$.

######  Lemma 7.1.

Consider semigroups $S=\langle X:R\rangle$ and $T=\langle Y:Q\rangle$, for
which the following hold:

  1. (i)

$X\subseteq Y$,

  2. (ii)

$R\subseteq Q^{\sharp}$,

  3. (iii)

every $y\in Y$ is $Q^{\sharp}$-equivalent to a word over $X$,

  4. (iv)

there is a morphism $\phi:Y^{+}\to S$ with $Q\subseteq\ker(\phi)$, and
$x\phi=[x]_{R}$ for all $x\in X$.

Then $S\cong T$.

###### Proof.

By (i) we have a well defined morphism $\psi:X^{+}\to T:x\mapsto[x]_{Q}$. By
(ii) and (iv), $\psi$ and $\phi$ induce morphisms

$\Psi:S\to T:[x]_{R}\mapsto[x]_{Q}\qquad\text{and}\qquad\Phi:T\to
S:[y]_{Q}\mapsto y\phi.$

By (iii), $T$ is generated by $\\{[x]_{Q}:x\in X\\}$, and of course $S$ is
generated by $\\{[x]_{R}:x\in X\\}$. Since

$[x]_{R}\Psi=[x]_{Q}\qquad\text{and}\qquad[x]_{Q}\Phi=x\phi=[x]_{R}\qquad\text{for
all $x\in X$, by \ref{lempres4},}$

it follows that $\Psi$ and $\Phi$ are mutually inverse isomorphisms of $S$ and
$T$. ∎

###  7.2 Presentation over $P$

Here is the main result of this section:

######  Theorem 7.2.

For any projection algebra $P$, the free regular $*$-semigroup
$\operatorname{\textup{{PG}}}(P)$ has presentation

$\operatorname{\textup{{PG}}}(P)\cong\langle X_{P}:R_{P}\rangle,$

where $X_{P}=\\{x_{p}:p\in P\\}$ is an alphabet in one-one correspondence with
$P$, and where $R_{P}$ is the set of relations

$\displaystyle x_{p}^{2}$ $\displaystyle=x_{p}$ for all $p\in P$, (R1)
$\displaystyle(x_{p}x_{q})^{2}$ $\displaystyle=x_{p}x_{q}$ for all $p,q\in P$,
(R2) $\displaystyle x_{p}x_{q}x_{p}$ $\displaystyle=x_{q\theta_{p}}$ for all
$p,q\in P$. (R3)

######  Remark 7.3.

If $P={\bf P}(S)$ is the projection algebra of a regular $*$-semigroup $S$,
then recall that $q\theta_{p}=pqp$ for $p,q\in P$, where the product $pqp$ is
taken in $S$. Thus, relations of type (R3) have the form
$x_{p}x_{q}x_{p}=x_{pqp}$ in this case.

To prove Theorem 7.2, we require some technical lemmas. But first, it is worth
observing that relations (R1)–(R3) closely resemble projection algebra axioms
(P2), (P4) and (P5), i.e. those that are stated purely in terms of the
$\theta$ maps.

For the rest of this subsection, we fix $P$, $X_{P}$ and $R_{P}$ as in Theorem
7.2. We write ${\sim}=R_{P}^{\sharp}$ for the congruence on $X_{P}^{+}$
generated by relations (R1)–(R3), and use $\sim_{1}$ to indicate equivalence
by one or more applications of (R1), and similarly for $\sim_{2}$ and
$\sim_{3}$.

######  Lemma 7.4.

If $p,q\in P$ are such that $q\leq_{\mathrel{\mathscr{F}}}p$, then
$x_{p}x_{q}\sim x_{p^{\prime}}x_{q}$ for some $p^{\prime}\in P$ with
$q\mathrel{\mathscr{F}}p^{\prime}\leq p$.

###### Proof.

Let $p^{\prime}=q\theta_{p}\leq p$. Combining $q\leq_{\mathrel{\mathscr{F}}}p$
with (PA1), it follows that
$q=p\theta_{q}\mathrel{\mathscr{F}}q\theta_{p}=p^{\prime}$. We also have

$x_{p}x_{q}\sim_{2}x_{p}x_{q}x_{p}x_{q}\sim_{3}x_{q\theta_{p}}x_{q}=x_{p^{\prime}}x_{q}.\qed$

######  Lemma 7.5.

For any $p_{1},\ldots,p_{k}\in P$ we have $x_{p_{1}}\cdots x_{p_{k}}\sim
x_{p_{1}^{\prime}}\cdots x_{p_{k}^{\prime}}$ for some
$p_{1}^{\prime},\ldots,p_{k}^{\prime}\in P$ with
$p_{1}^{\prime}\mathrel{\mathscr{F}}\cdots\mathrel{\mathscr{F}}p_{k}^{\prime}$
and $p_{i}^{\prime}\leq p_{i}$ for all $i$.

###### Proof.

We prove the lemma by induction on $k$. The $k=1$ case being trivial, we
assume $k\geq 2$. With $p_{1}^{\prime\prime}=p_{2}\theta_{p_{1}}\leq p_{1}$
and $p_{2}^{\prime\prime}=p_{1}\theta_{p_{2}}\leq p_{2}$ we have

$x_{p_{1}}x_{p_{2}}\sim_{2}x_{p_{1}}x_{p_{2}}x_{p_{1}}x_{p_{2}}x_{p_{1}}x_{p_{2}}\sim_{3}x_{p_{2}\theta_{p_{1}}}x_{p_{1}\theta_{p_{2}}}=x_{p_{1}^{\prime\prime}}x_{p_{2}^{\prime\prime}},$

and (PA1) gives
$p_{1}^{\prime\prime}\mathrel{\mathscr{F}}p_{2}^{\prime\prime}$. By induction,
we have

$x_{p_{2}^{\prime\prime}}x_{p_{3}}\cdots x_{p_{k}}\sim
x_{p_{2}^{\prime}}x_{p_{3}^{\prime}}\cdots x_{p_{k}^{\prime}}$

for some $p_{2}^{\prime},\ldots,p_{k}^{\prime}\in P$ with
$p_{2}^{\prime}\mathrel{\mathscr{F}}\cdots\mathrel{\mathscr{F}}p_{k}^{\prime}$,
$p_{2}^{\prime}\leq p_{2}^{\prime\prime}$ and $p_{i}^{\prime}\leq p_{i}$ for
$i=3,\ldots,k$. Note that also $p_{2}^{\prime}\leq p_{2}^{\prime\prime}\leq
p_{2}$. Since $p_{2}^{\prime}\leq
p_{2}^{\prime\prime}\mathrel{\mathscr{F}}p_{1}^{\prime\prime}$, (PA2) gives
$p_{2}^{\prime}\leq_{\mathrel{\mathscr{F}}}p_{1}^{\prime\prime}$. It then
follows from Lemma 7.4 that $x_{p_{1}^{\prime\prime}}x_{p_{2}^{\prime}}\sim
x_{p_{1}^{\prime}}x_{p_{2}^{\prime}}$ for some $p_{1}^{\prime}\in P$ with
$p_{2}^{\prime}\mathrel{\mathscr{F}}p_{1}^{\prime}\leq p_{1}^{\prime\prime}$,
and again we observe that $p_{1}^{\prime}\leq p_{1}^{\prime\prime}\leq p_{1}$.
Putting everything together we have

$x_{p_{1}}x_{p_{2}}x_{p_{3}}\cdots x_{p_{k}}\sim
x_{p_{1}^{\prime\prime}}x_{p_{2}^{\prime\prime}}x_{p_{3}}\cdots x_{p_{k}}\sim
x_{p_{1}^{\prime\prime}}x_{p_{2}^{\prime}}x_{p_{3}^{\prime}}\cdots
x_{p_{k}^{\prime}}\sim
x_{p_{1}^{\prime}}x_{p_{2}^{\prime}}x_{p_{3}^{\prime}}\cdots
x_{p_{k}^{\prime}},$

with all conditions met. ∎

Given a $P$-path
$\mathfrak{p}=(p_{1},\ldots,p_{k})\in\mathscr{P}=\mathscr{P}(P)$, we define
the word

$w_{\mathfrak{p}}=x_{p_{1}}\cdots x_{p_{k}}\in X_{P}^{+}.$

It follows from Lemma 7.5 that every word over $X_{P}$ is $\sim$-equivalent to
some $w_{\mathfrak{p}}$. Using (R1), it is easy to see that

$w_{\mathfrak{p}}w_{\mathfrak{q}}\sim
w_{\mathfrak{p}\circ\mathfrak{q}}\qquad\text{for any
$\mathfrak{p},\mathfrak{q}\in\mathscr{P}$ with ${\bf r}(\mathfrak{p})={\bf
d}(\mathfrak{q})$.}$ (7.6)

The next result refers to the congruence ${\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}}=\Xi^{\sharp}$
on $\mathscr{P}$ from Definition 4.7.

######  Lemma 7.7.

For any $\mathfrak{p},\mathfrak{q}\in\mathscr{P}$, we have
$\mathfrak{p}\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}\mathfrak{q}\
\Rightarrow\ w_{\mathfrak{p}}\sim w_{\mathfrak{q}}$.

###### Proof.

It suffices to assume that $\mathfrak{p}$ and $\mathfrak{q}$ differ by a
single application of ($\mathsf{\Omega}$1)–($\mathsf{\Omega}$3), i.e. that

$\mathfrak{p}=\mathfrak{p}^{\prime}\circ\mathfrak{s}\circ\mathfrak{p}^{\prime\prime}\qquad\text{and}\qquad\mathfrak{q}=\mathfrak{p}^{\prime}\circ\mathfrak{t}\circ\mathfrak{p}^{\prime\prime}\qquad\text{for
some $\mathfrak{p}^{\prime},\mathfrak{p}^{\prime\prime}\in\mathscr{P}$ and
$(\mathfrak{s},\mathfrak{t})\in\Xi\cup\Xi^{-1}$.}$

Since $w_{\mathfrak{p}}\sim
w_{\mathfrak{p}^{\prime}}w_{\mathfrak{s}}w_{\mathfrak{p}^{\prime\prime}}$ and
$w_{\mathfrak{q}}\sim
w_{\mathfrak{p}^{\prime}}w_{\mathfrak{t}}w_{\mathfrak{p}^{\prime\prime}}$ by
(7.6), it is in fact enough to prove that

$w_{\mathfrak{s}}\sim w_{\mathfrak{t}}\qquad\text{for all
$(\mathfrak{s},\mathfrak{t})\in\Xi$.}$

We consider the three forms the pair $(\mathfrak{s},\mathfrak{t})\in\Xi$ can
take.

($\mathsf{\Omega}$1). This follows immediately from (R1).

($\mathsf{\Omega}$2). If $\mathfrak{s}=(p,q,p)$ and $\mathfrak{t}=(p)$ for
some $(p,q)\in{\mathrel{\mathscr{F}}}$, then
$w_{\mathfrak{s}}=x_{p}x_{q}x_{p}\sim_{3}x_{q\theta_{p}}=x_{p}=w_{\mathfrak{t}}$.

($\mathsf{\Omega}$3). Finally, suppose
$\mathfrak{s}=\lambda(e,p,f)=(e,e\theta_{p},f)$ and
$\mathfrak{t}=\rho(e,p,f)=(e,f\theta_{p},f)$ for some $p\in P$, and some
$p$-linked pair $(e,f)$. Then

$w_{\mathfrak{s}}=x_{e}x_{e\theta_{p}}x_{f}\sim_{3}x_{e}x_{p}x_{e}x_{p}x_{f}\sim_{2}x_{e}x_{p}x_{f}\sim_{2}x_{e}x_{p}x_{f}x_{p}x_{f}\sim_{3}x_{e}x_{f\theta_{p}}x_{f}=w_{\mathfrak{t}}.\qed$

We can now tie together the loose ends.

###### Proof of Theorem 7.2..

Define the homomorphism

$\Psi:X_{P}^{+}\to\operatorname{\textup{{PG}}}(P)\qquad\text{by}\qquad
x_{p}\Psi=p=[[p]]\qquad\text{for $p\in P$.}$

To see that $\Psi$ is surjective, let
$\mathfrak{c}\in\operatorname{\textup{{PG}}}(P)$, so that
$\mathfrak{c}=[[\mathfrak{p}]]$ for some
$\mathfrak{p}=(p_{1},\ldots,p_{k})\in\mathscr{P}$. We claim that

$[[p_{1},\ldots,p_{i}]]=p_{1}\bullet\cdots\bullet p_{i}\qquad\text{for all
$1\leq i\leq k$.}$

Indeed, the $i=1$ case is clear, and if $2\leq i\leq k$ then

$\displaystyle p_{1}\bullet\cdots\bullet p_{i}$
$\displaystyle=[[p_{1},\ldots,p_{i-1}]]\bullet p_{i}$ by induction
$\displaystyle=[[p_{1},\ldots,p_{i-1}]]{\downharpoonright}_{p_{i-1}^{\prime}}\circ[[p_{i-1}^{\prime},p_{i}^{\prime}]]\circ{}_{p_{i}^{\prime}}{\downharpoonleft}p_{i}$
where $p_{i-1}^{\prime}=p_{i}\theta_{p_{i-1}}$ and
$p_{i}^{\prime}=p_{i-1}\theta_{p_{i}}$
$\displaystyle=[[p_{1},\ldots,p_{i-1}]]\circ[[p_{i-1},p_{i}]]\circ p_{i}$
since $p_{i-1}^{\prime}=p_{i-1}$ and $p_{i}^{\prime}=p_{i}$, as
$p_{i-1}\mathrel{\mathscr{F}}p_{i}$
$\displaystyle=[[p_{1},\ldots,p_{i-1},p_{i}]],$

proving the claim. It then follows that

$\mathfrak{c}=[[p_{1},\ldots,p_{k}]]=p_{1}\bullet\cdots\bullet
p_{k}=(x_{p_{1}}\cdots x_{p_{k}})\Psi=w_{\mathfrak{p}}\Psi,$ (7.8)

completing the proof that $\Psi$ is surjective.

Next, we note that $R_{P}\subseteq\ker(\Psi)$, meaning that $u\Psi=v\Psi$ (in
$\operatorname{\textup{{PG}}}(P)$) for all $(u,v)\in R_{P}$. Indeed, this is
clear when $(u,v)$ has type (R1), and follows from (RS2) or (CP4) for type
(R2) or (R3).

It remains to show that $\ker(\Psi)\subseteq R_{P}^{\sharp}$. To do so, fix
some $(u,v)\in\ker(\Psi)$, so that $u,v\in X_{P}^{+}$ and $u\Psi=v\Psi$; we
must show that $u\sim v$ (recall that we write $\sim$ for $R_{P}^{\sharp}$).
By Lemma 7.5, we have $u\sim w_{\mathfrak{p}}$ and $v\sim w_{\mathfrak{q}}$
for some $\mathfrak{p},\mathfrak{q}\in\mathscr{P}$. Using (7.8), and
remembering that ${\sim}\subseteq\ker(\Psi)$, we have

$[[\mathfrak{p}]]=w_{\mathfrak{p}}\Psi=u\Psi=v\Psi=w_{\mathfrak{q}}\Psi=[[\mathfrak{q}]],$

meaning that $\mathfrak{p}\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}\mathfrak{q}$.
But then $w_{\mathfrak{p}}\sim w_{\mathfrak{q}}$ by Lemma 7.7, so $u\sim
w_{\mathfrak{p}}\sim w_{\mathfrak{q}}\sim v$, as required. ∎

###  7.3 Presentation as a quotient of $\operatorname{\textup{{IG}}}(E)$

Consider a boset $E$, and recall that a product $ef$ is defined in $E$
precisely when $(e,f)\in\operatorname{\textup{{BP}}}(E)$ is a basic pair, i.e.
when $\\{ef,fe\\}\cap\\{e,f\\}\not=\varnothing$. The _free (idempotent-
generated) semigroup over $E$_ has presentation

$\operatorname{\textup{{IG}}}(E)=\langle X_{E}:x_{e}x_{f}=x_{ef}\ \text{for
all $(e,f)\in\operatorname{\textup{{BP}}}(E)$}\rangle,$

where here $X_{E}=\\{x_{e}:e\in E\\}$ is an alphabet in one-one correspondence
with $E$. The boset of $\operatorname{\textup{{IG}}}(E)$ is isomorphic to $E$
[26], and consists of all equivalence classes of letters $x_{e}$ ($e\in E$).

Now consider a projection algebra $P$, and let $E={\bf E}(P)={\bf
E}(\operatorname{\textup{{PG}}}(P))$ be the (regular $*$-) boset of
$\operatorname{\textup{{PG}}}(P)$. Since $\operatorname{\textup{{PG}}}(P)$ is
a semigroup with biordered set $E={\bf E}(P)$, general theory [26, Theorem
3.3] tells us that the mapping $x_{e}\mapsto e$ ($e\in E$) induces a
surmorphism
${\operatorname{\textup{{IG}}}(E)\to\operatorname{\textup{{PG}}}(P)}$. Thus,
we know in advance that there exists a presentation for
$\operatorname{\textup{{PG}}}(P)$ extending the above presentation for
$\operatorname{\textup{{IG}}}(E)$ by means of additional relations. Theorem
7.9 below gives an explicit such presentation, with additional relations
$x_{p}x_{q}=x_{pq}$ for projections $p,q\in P(\subseteq E)$. These additional
relations can also be viewed as a generating set for the kernel of the
surmomorphism
$\operatorname{\textup{{IG}}}(E)\to\operatorname{\textup{{PG}}}(P)$. Note that
the product $pq$ might not exist in the boset $E$, but it certainly exists in
the semigroup $\operatorname{\textup{{PG}}}(P)$, and is an idempotent, and
hence a well-defined element of $E$; in fact, $pq$ is the element $e(q,p)$
from Lemma 6.13. For simplicity, we will denote the product in
$\operatorname{\textup{{PG}}}(P)$ by juxtaposition instead of $\bullet$
throughout this subsection.

######  Theorem 7.9.

For any projection algebra $P$, the free regular $*$-semigroup
$\operatorname{\textup{{PG}}}(P)$ has presentation

$\operatorname{\textup{{PG}}}(P)\cong\langle X_{E}:R_{E}\rangle,$

where $X_{E}=\\{x_{e}:e\in E\\}$ is an alphabet in one-one correspondence with
$E={\bf E}(P)$, and where $R_{E}$ is the set of relations

$\displaystyle x_{e}x_{f}$ $\displaystyle=x_{ef}$ for all
$(e,f)\in\operatorname{\textup{{BP}}}(E)$, (R1)′ $\displaystyle x_{p}x_{q}$
$\displaystyle=x_{pq}$ for all $p,q\in P$. (R2)′

###### Proof.

By Theorem 7.2 we have $\operatorname{\textup{{PG}}}(P)\cong\langle
X_{P}:R_{P}\rangle$. Thus, we can prove the current theorem by applying Lemma
7.1 with $S=\langle X_{P}:R_{P}\rangle$ and ${T=\langle X_{E}:R_{E}\rangle}$.
To do so, we must show that:

  1. (i)

$X_{P}\subseteq X_{E}$,

  2. (ii)

$R_{P}\subseteq R_{E}^{\sharp}$,

  3. (iii)

every $x_{e}$ ($e\in E$) is $\sim^{\prime}$-equivalent to a word over $X_{P}$,
where ${\sim^{\prime}}=R_{E}^{\sharp}$, and

  4. (iv)

there is a morphism $\phi:X_{E}^{+}\to\langle X_{P}:R_{P}\rangle$ such that
$R_{E}\subseteq\ker(\phi)$, and $x_{p}\phi=[x_{p}]$ for all $p\in P$, where we
write $[w]$ for the $R_{P}^{\sharp}$-class of $w\in X_{P}^{+}$.

Item (i) is clear. For (ii), we check the relations from $R_{P}$ in turn. We
use $\sim^{\prime}_{1}$ and $\sim^{\prime}_{2}$ to denote equivalence via
(R1)′ or (R2)′.

(R1). This is contained in (R1)′ (and in (R2)′).

(R2). If $p,q\in P$, then
$x_{p}x_{q}\sim^{\prime}_{2}x_{pq}\sim^{\prime}_{1}x_{pq}^{2}\sim^{\prime}_{2}(x_{p}x_{q})^{2}$.

(R3). If $p,q\in P$, then $pq\in E$, and $(p,pq)$ is a basic pair in $E$. It
follows that (R1)′ contains the relation $x_{pq}x_{p}=x_{pqp}$. But then
$x_{p}x_{q}x_{p}\sim^{\prime}_{2}x_{pq}x_{p}\sim^{\prime}_{1}x_{pqp}=x_{q\theta_{p}}$.

For (iii), fix some $e\in E$. Since $e=ee^{*}e^{*}e$, with $ee^{*},e^{*}e\in
P$, we have

$x_{e}=x_{ee^{*}e^{*}e}\sim^{\prime}_{2}x_{ee^{*}}x_{e^{*}e}\in X_{P}^{+}.$

For (iv), we first define a morphism

$\psi:X_{E}^{+}\to\operatorname{\textup{{PG}}}(P):x_{e}\mapsto e.$

To see that $R_{E}\subseteq\ker(\psi)$, fix some $e,f\in E$. If $ef\in E$,
then $\psi$ maps both $x_{e}x_{f}$ and $x_{ef}$ to $ef$, and it follows that
relations (R1)′ and (R2)′ are both preserved. Composing
$\psi:X_{E}^{+}\to\operatorname{\textup{{PG}}}(P)$ with the isomorphism
$\operatorname{\textup{{PG}}}(P)\to\langle X_{P}:R_{P}\rangle:p\mapsto[x_{p}]$
gives $\phi:X_{E}^{+}\to\langle X_{P}:R_{P}\rangle$ with the required
properties. ∎

######  Remark 7.10.

Relations (R2)′ reflect the fact that the product of two projections is always
an idempotent in a regular $*$-semigroup. In fact, any idempotent is the
product of two $\mathrel{\mathscr{F}}$-related projections by (RS3).
Consequently, one might wonder if we could replace (R2)′ by the subset
consisting of the relations $x_{p}x_{q}=x_{pq}$ for
$(p,q)\in{\mathrel{\mathscr{F}}}$, and still define
$\operatorname{\textup{{PG}}}(P)$. It turns out that this is not possible in
general. For example, consider the three element semilattice $S=\\{0,e,f\\}$
with $ef=0$, and note that $P=E=S$ and ${\mathrel{\mathscr{F}}}=\Delta_{P}$ in
this case. As (R2)′ is a copy of the multiplication table of $S$ (which is the
case for any semilattice), it follows that
$\operatorname{\textup{{PG}}}(P)\cong S$. Since
${\mathrel{\mathscr{F}}}=\Delta_{P}$, the relations in (R2)′ arising from
friendly pairs are just $x_{p}^{2}=x_{p}$ for $p\in E$, which are already
contained in (R1)′. So if we reduce the presentation in this way, we actually
arrive at the free idempotent-generated semigroup
$\operatorname{\textup{{IG}}}(E)$. As shown in [12, Example 2],
$\operatorname{\textup{{IG}}}(E)$ is infinite (and non-regular), and so
certainly not isomorphic to $\operatorname{\textup{{PG}}}(P)$. In the next
subsection we will see that if instead of $\operatorname{\textup{{IG}}}(E)$ we
start with a presentation for $\operatorname{\textup{{RIG}}}(E)$, then the
relations from (R2)′ arising from friendly pairs are indeed sufficient to
define $\operatorname{\textup{{PG}}}(P)$.

###  7.4 Presentation as a quotient of $\operatorname{\textup{{RIG}}}(E)$

Consider again a projection algebra $P$, and its associated boset ${E={\bf
E}(P)={\bf E}(\operatorname{\textup{{PG}}}(P))}$. Since $E$ is regular (i.e.
the boset of a regular semigroup, namely $\operatorname{\textup{{PG}}}(P)$),
we also have the _free regular (idempotent-generated) semigroup_
$\operatorname{\textup{{RIG}}}(E)$. This was defined by Nambooripad in [54]
using his groupoid machinery, and in [59] by means of the presentation

$\displaystyle\operatorname{\textup{{RIG}}}(E)=\langle X_{E}:{}$
$\displaystyle x_{e}x_{f}=x_{ef}\ \ \ \hskip 18.49411pt\text{for all
$(e,f)\in\operatorname{\textup{{BP}}}(E)$,}$ $\displaystyle
x_{e}x_{g}x_{f}=x_{e}x_{f}\ \ \ \text{for all $e,f\in E$ and
$g\in\SS(e,f)$}\rangle.$

Here $\SS(e,f)$ is the sandwich set of the idempotents $e,f\in E$, defined in
Subsection 6.1. Note that the characterisation of $\SS(e,f)$ in (6.2) applies
to $S=\operatorname{\textup{{PG}}}(P)$, as $\operatorname{\textup{{PG}}}(P)$
is a regular semigroup with boset $E$. The above presentation shows that
$\operatorname{\textup{{RIG}}}(E)$ is a quotient of
$\operatorname{\textup{{IG}}}(E)$. In turn, our next result shows that
$\operatorname{\textup{{PG}}}(P)$ is a quotient of
$\operatorname{\textup{{RIG}}}(E)$, with the additional relations generating
the kernel of the canonical surmorphism
$\operatorname{\textup{{RIG}}}(E)\to\operatorname{\textup{{PG}}}(P)$.

######  Theorem 7.11.

For any projection algebra $P$, the free regular $*$-semigroup
$\operatorname{\textup{{PG}}}(P)$ has presentation

$\operatorname{\textup{{PG}}}(P)\cong\langle X_{E}:R_{E}^{\prime}\rangle,$

where $X_{E}=\\{x_{e}:e\in E\\}$ is an alphabet in one-one correspondence with
$E={\bf E}(P)$, and where $R_{E}^{\prime}$ is the set of relations

$\displaystyle x_{e}x_{f}$ $\displaystyle=x_{ef}$ for all
$(e,f)\in\operatorname{\textup{{BP}}}(E)$, (R1)′′ $\displaystyle x_{e}x_{f}$
$\displaystyle=x_{e}x_{g}x_{f}$ for all $e,f\in E$ and $g\in S(e,f)$, (R2)′′
$\displaystyle x_{p}x_{q}$ $\displaystyle=x_{pq}$ for all
$(p,q)\in{\mathrel{\mathscr{F}}}$. (R3)′′

###### Proof.

We begin with the presentation $\langle X_{E}:R_{E}\rangle=\langle
X_{E}:\text{\ref{R1'},\ \ref{R2'}}\rangle$ from Theorem 7.9, via the mapping

$\psi:X_{E}^{+}\to\operatorname{\textup{{PG}}}(P):x_{e}\mapsto e.$

Since $\psi$ maps $x_{e}x_{g}x_{f}$ and $x_{e}x_{f}$ both to $egf=ef$ for
$g\in\SS(e,f)$, we can add relations (R2)′′ to the presentation. Noting that
(R1)′ and (R1)′′ are the same sets of relations, the presentation has now
become

$\langle X_{E}:\text{\ref{R1''},\ \ref{R2''},\ \ref{R2'}}\rangle.$

Since (R3)′′ is contained in (R2)′, we can complete the proof by showing that
each relation in (R2)′ is implied by those in $R_{E}^{\prime}$. To do so, let
$p,q\in P$ be arbitrary, and let $p^{\prime}=q\theta_{p}$ and
$q^{\prime}=p\theta_{q}$, so that $p^{\prime}\mathrel{\mathscr{F}}q^{\prime}$
and $p^{\prime}q^{\prime}=pq$. We then calculate (again writing
$\sim_{1}^{\prime\prime}$ for equivalence by (R1)′′, and so on)

$\displaystyle x_{p}x_{q}$
$\displaystyle\sim^{\prime\prime}_{2}x_{p}x_{qp}x_{q}$ as $qp\in\SS(p,q)$
$\displaystyle\sim^{\prime\prime}_{1}x_{p}x_{qp}x_{qp}x_{q}$
$\displaystyle\sim^{\prime\prime}_{1}x_{p\cdot qp}x_{qp\cdot q}$ as $(qp,p)$
and $(q,qp)$ are basic pairs
$\displaystyle=x_{p^{\prime}}x_{q^{\prime}}\sim^{\prime\prime}_{3}x_{p^{\prime}q^{\prime}}=x_{pq}.$

##  8 Adjacency semigroups and bridging path semigroups

We now turn to explicit examples of free projection-generated regular
$*$-semigroups, starting with those arising from adjacency semigroups, as
introduced in Subsection 2.2.

Let $\Gamma=(P,E)$ be a symmetric, reflexive digraph, and let $A_{\Gamma}$ be
its adjacency semigroup. We keep the notation of Subsection 2.2, including the
projection algebra $P_{0}=P\cup\\{0\\}$, whose operations were given in (2.2).
We now consider the structure of the free regular $*$-semigroup
$\operatorname{\textup{{PG}}}(P_{0})=\mathcal{C}_{P_{0}}$.

The path category $\mathscr{P}=\mathscr{P}(P_{0})$ consists of tuples of the
form $(0,\ldots,0)$, and $(p_{1},\ldots,p_{k})$ where each $(p_{i},p_{i+1})\in
E(\subseteq{\mathrel{\mathscr{F}}})$; the latter are simply the paths in
$\Gamma$ in the usual graph-theoretical sense, with repeated vertices allowed
as in Subsection 3.3.

As in Remark 3.6, any non-zero path $\mathfrak{p}\in\mathscr{P}$ is
$\approx$-equivalent to a unique _reduced_ path
$\overline{\mathfrak{p}}=(p_{1},\ldots,p_{k})$, where each $p_{i}$ is distinct
from $p_{i+1}$ (if $i\leq k-1$) and from $p_{i+2}$ (if $i\leq k-2$). By
identifying a non-zero chain ($\approx$-class of a path) with the unique
reduced path it contains, we may identify the chain groupoid
$\mathscr{C}=\mathscr{C}(P_{0})$ with the set of reduced paths in $\Gamma$,
along with $0=[0]$.

Using (2.2), we see that $(e,f)$ is $p$-linked if and only if $e=f=0$ or
$(e,p)$ and $(p,f)$ are both edges of $\Gamma$. In these respective cases, we
have

$\lambda(e,p,f)=\rho(e,p,f)=(0,0,0)\qquad\text{or}\qquad\lambda(e,p,f)=\rho(e,p,f)=(e,p,f).$

It follows that the congruences $\approx$ and $\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}$ are
equal, and so $\overline{\mathscr{C}}=\mathscr{C}$, and
$[[\mathfrak{p}]]=[\mathfrak{p}]$ for all $\mathfrak{p}\in\mathscr{P}$.

The free regular $*$-semigroup
$\operatorname{\textup{{PG}}}(P_{0})=\mathcal{C}_{P_{0}}$ can therefore be
viewed as follows:

  * •

The elements are $0$ and the reduced paths in $\Gamma$.

  * •

The product of reduced paths $\mathfrak{p}=(p_{1},\ldots,p_{k})$ and
$\mathfrak{q}=(q_{1},\ldots,q_{l})$ is given by

$\mathfrak{p}\bullet\mathfrak{q}=\begin{cases}\overline{\mathfrak{p}\oplus\mathfrak{q}}&\text{if
$(p_{k},q_{1})\in E$ is an edge of $\Gamma$}\\\
0&\text{otherwise.}\end{cases}$

Here $\oplus$ denotes concatenation, so
$\mathfrak{p}\oplus\mathfrak{q}=(p_{1},\ldots,p_{k},q_{1},\ldots,q_{l})$.

  * •

The involution is given by reversal of paths.

  * •

The non-zero projections are the empty paths $p=(p)$, which are in one-one
correspondence with the vertices of $\Gamma$.

  * •

The remaining non-zero idempotents are the non-loop edges $p\bullet q=(p,q)$
of $\Gamma$.

The authors have not seen these specific semigroups in the literature, but we
note that they are closely related to the _graph inverse semigroups_
introduced in [4], in which $\mathfrak{p}\bullet\mathfrak{q}$ is only non-zero
when $p_{k}=q_{1}$. In our case $\mathfrak{p}$ and $\mathfrak{q}$ can also be
composed if there is an edge $p_{k}\to q_{1}$, which ‘bridges’ the end of
$\mathfrak{p}$ and the start of $\mathfrak{q}$. We therefore call
$\operatorname{\textup{{PG}}}({\bf P}(A_{\Gamma}))$ the _bridging path
semigroup of $\Gamma$_, and denote it by $B_{\Gamma}$. Note that bridging path
semigroups can be defined starting from an arbitrary digraph, i.e. without
assuming symmetry and reflexivity a priori. The properties of such a semigroup
would then depend on the properties of the graph $\Gamma$ and this may be an
interesting direction for study. In particular, one can verify that
$B_{\Gamma}$ is a regular $*$-semigroup with projections $P_{0}$ and the
involution given by reversal of paths if and only if $\Gamma$ is symmetric and
reflexive.

In the special case that $\Gamma$ is the complete digraph, the adjacency
semigroup $A_{\Gamma}$ is simply the square band $B_{P}=P\times P$ with a zero
adjoined. This band $B_{P}$ is a regular $*$-semigroup, with operations
$(p,q)(r,s)=(p,s)$ and $(p,q)^{*}=(q,p)$. Every element of $B_{P}$ is an
idempotent; the projections are $p=(p,p)$; and the projection algebra ${\bf
P}(B_{P})=P$ has a trivial structure, in the sense that the $\theta_{p}$
operations are all constant maps. The above analysis shows that the semigroup
$\operatorname{\textup{{PG}}}(P)$ consists of all reduced tuples, with product
$\mathfrak{p}\bullet\mathfrak{q}=\overline{\mathfrak{p}\oplus\mathfrak{q}}$.

######  Example 8.1.

If $|P|\leq 2$ then $\operatorname{\textup{{PG}}}(P)=B_{P}$ is finite. By
contrast, the corresponding free idempotent-generated semigroup
$\operatorname{\textup{{IG}}}({\bf E}(B_{P}))$ is infinite when $|P|=2$; this
is folklore and can be deduced from [35, Theorem 5].

######  Example 8.2.

If $|P|\geq 3$ then $\operatorname{\textup{{PG}}}(P)$ is infinite, and in
particular $\operatorname{\textup{{PG}}}(P)\not=B_{P}$. It is instructive to
consider the case where $P=\\{p,q,r\\}$ has size $3$. For $s,t\in P$, let
$H_{s,t}$ be the set of all reduced paths from $s$ to $t$, so that
$\operatorname{\textup{{PG}}}(P)=\bigsqcup_{s,t\in P}H_{s,t}$. Note for
example that

$H_{p,p}=\\{p\\}\cup\\{(p,q,r,p)^{k},(p,r,q,p)^{k}:k=1,2,\ldots\\}.$

It is easy to check that $H_{p,p}$ is isomorphic to the infinite cyclic group
$H=\langle a\rangle$. The identity of $H_{p,p}$ is $p$, and $(p,q,r,p)^{k}$
and $(p,r,q,p)^{k}$ are inverses of each other. An analogous argument shows
that each $H_{s,t}\cong H$. It follows from general structure theory (see [39,
Chapter 3]) that $\operatorname{\textup{{PG}}}(P)$ is a $3\times 3$ Rees
matrix semigroup over $H$, with respect to the sandwich matrix
$\biggl{(}\begin{smallmatrix}1&1&1\\\ 1&1&a\\\
1&a^{-1}&1\end{smallmatrix}\biggr{)}$. By way of comparison,
$\operatorname{\textup{{IG}}}({\bf
E}(B_{P}))=\operatorname{\textup{{RIG}}}({\bf E}(B_{P}))$ is a $3\times 3$
Rees matrix semigroup over the free group of rank $4$, again following from
[35, Theorem 5].

The structure of an arbitrary bridging path semigroup
$B_{\Gamma}=\operatorname{\textup{{PG}}}({\bf P}(A_{\Gamma}))$ can be
similarly described in terms of Rees $0$-matrix semigroups over free groups.
The description requires an analysis of the maximal subgroups of free regular
$*$-semigroups, which is the subject of the forthcoming paper [31].

Our final example in this section is an extension $P^{\prime}$ of the
projection algebra $P$ of the $3\times 3$ rectangular band from Example 8.2 by
a single extra projection. It was introduced in [28] (where it had an identity
adjoined), and was originally discovered by Michael Kinyon. We note that
Kinyon’s projection algebra is not the projection algebra of an adjacency
semigroup. It illustrates the fact that it is possible for a projection
algebra $P$ with $\operatorname{\textup{{PG}}}(P)$ infinite to be contained in
a projection algebra $P^{\prime}$ with
$\operatorname{\textup{{PG}}}(P^{\prime})$ finite.

######  Example 8.3.

Let $P^{\prime}=\\{p,q,r,e\\}$ be the projection algebra with operations

$\theta_{p}=\left(\begin{smallmatrix}p&q&r&e\\\
p&p&p&p\end{smallmatrix}\right),\qquad\theta_{q}=\left(\begin{smallmatrix}p&q&r&e\\\
q&q&q&q\end{smallmatrix}\right),\qquad\theta_{r}=\left(\begin{smallmatrix}p&q&r&e\\\
r&r&r&r\end{smallmatrix}\right)\qquad\text{and}\qquad\theta_{e}=\left(\begin{smallmatrix}p&q&r&e\\\
p&q&q&e\end{smallmatrix}\right).$

Note that $p,q,r$ are all $\mathrel{\mathscr{F}}$-related, but $e$ is
$\mathrel{\mathscr{F}}$-related only to itself. As in Example 8.2, it follows
that

$\mathscr{C}=\mathscr{C}(P^{\prime})=\\{e\\}\cup\bigsqcup_{s,t\in\\{p,q,r\\}}H_{s,t}.$

However, $\overline{\mathscr{C}}$ is a proper quotient of $\mathscr{C}$ here,
as there is a non-trivial linked pair. Specifically, $(p,r)$ is $e$-linked,
and we have

$\lambda(p,e,r)=(p,p\theta_{e},r)=(p,p,r)\qquad\text{and}\qquad\rho(p,e,r)=(p,r\theta_{e},r)=(p,q,r).$

Consequently, $[[p,q,r]]=[[p,p,r]]=[[p,r]]$ in
$\operatorname{\textup{{PG}}}(P^{\prime})$. It follows from this that
$[[s,t,u]]=[[s,u]]$ for distinct $s,t,u\in\\{p,q,r\\}$. For example,

$[[p,r,q]]=[[p,q,r,q]]=[[p,q]]\qquad\text{and}\qquad[[q,p,r]]=[[q,p,q,r]]=[[q,r]].$

The other three cases are obtained by inverting the three already considered.
This all shows that $\operatorname{\textup{{PG}}}(P^{\prime})$ contains
exactly ten elements:

  * •

the projections $e,p,q,r$, and

  * •

the remaining idempotents $s\bullet t=[[s,t]]$, for distinct
$s,t\in\\{p,q,r\\}$.

The entire multiplication table of $\operatorname{\textup{{PG}}}(P^{\prime})$
can be obtained from the fact that
$\operatorname{\textup{{PG}}}(P^{\prime})\setminus\\{e\\}=\langle
p,q,r\rangle$ is a $3\times 3$ rectangular band, together with the rules

$p\bullet e=p,\qquad q\bullet e=q\qquad\text{and}\qquad r\bullet e=[[r,q]].$

##  9 Temperley–Lieb monoids

In Section 8 we saw examples of naturally occurring (albeit very small)
regular $*$-semigroups $S$ that were isomorphic to their associated free
regular $*$-semigroup $\operatorname{\textup{{PG}}}({\bf P}(S))$. In this
section, we present a decidedly non-trivial family of semigroups displaying
this phenomenon, namely the Temperley–Lieb monoids, introduced in Subsection
2.3.

######  Theorem 9.1.

Let $P={\bf P}(\mathcal{T}\\!\mathcal{L}_{n})$ be the projection algebra of a
finite Temperley–Lieb monoid $\mathcal{T}\\!\mathcal{L}_{n}$. Then
$\operatorname{\textup{{PG}}}(P)\cong\mathcal{T}\\!\mathcal{L}_{n}$.

###### Proof.

We begin with the presentations $\mathcal{T}\\!\mathcal{L}_{n}\cong\langle
X_{T}:R_{T}\rangle$ and $\operatorname{\textup{{PG}}}(P)\cong\langle
X_{P}:R_{P}\rangle$ from Theorems 2.3 and 7.2, and apply Lemma 7.1 to show
that they are equivalent. For this, we first need to convert the monoid
presentation $\langle X_{T}:R_{T}\rangle$ into a semigroup presentation
$\langle X_{T}^{\prime}:R_{T}^{\prime}\rangle$ with $X_{T}^{\prime}\subseteq
X_{P}$, which we do as follows. First, we identify each generator $t_{i}\in
X_{T}$ with $x_{\tau_{i}}\in X_{P}$, noting that they both represent the
element $\tau_{i}$ of $\mathcal{T}\\!\mathcal{L}_{n}$; see (2.4). Then we
define $X_{T}^{\prime}=\\{e\\}\cup X_{T}$, where here $e=x_{1}$ represents the
identity $1$ of $\mathcal{T}\\!\mathcal{L}_{n}$. Finally we add new relations
that ensure $e$ acts as the identity. The resulting set $R_{T}^{\prime}$ of
relations is:

$\displaystyle t_{i}^{2}$ $\displaystyle=t_{i}$ for all $i$ (T1)
$\displaystyle t_{i}t_{j}$ $\displaystyle=t_{j}t_{i}$ if $|i-j|>1$ (T2)
$\displaystyle t_{i}t_{j}t_{i}$ $\displaystyle=t_{i}$ if $|i-j|=1$ (T3)
$\displaystyle e^{2}$ $\displaystyle=e$ (T4) $\displaystyle et_{i}=t_{i}e$
$\displaystyle=t_{i}$ for all $i$. (T5)

It is clear that $\mathcal{T}\\!\mathcal{L}_{n}\cong\langle
X_{T}^{\prime}:R_{T}^{\prime}\rangle$. We now work towards applying Lemma 7.1,
with ${S=\langle X_{T}^{\prime}:R_{T}^{\prime}\rangle}$ and $T=\langle
X_{P}:R_{P}\rangle$. For this we need to show that:

  1. (i)

$X_{T}^{\prime}\subseteq X_{P}$,

  2. (ii)

$R_{T}^{\prime}\subseteq R_{P}^{\sharp}$,

  3. (iii)

every $x_{p}$ ($p\in P$) is $\sim$-equivalent to a word over $X_{T}^{\prime}$,
where again ${\sim}=R_{P}^{\sharp}$, and

  4. (iv)

there is a morphism $\phi:X_{P}^{+}\to\langle
X_{T}^{\prime}:R_{T}^{\prime}\rangle$ such that $R_{P}\subseteq\ker(\phi)$,
and $x\phi=[x]_{R_{T}^{\prime}}$ for all $x\in X_{T}^{\prime}$.

We have already noted that (i) holds. For (ii), we consider the relations from
$R$ in turn. Again we write $\sim_{1}$ to denote equivalence via (R1), and so
on.

(T1) and (T4). These are contained in (R1).

(T2). Fix $i,j$ with $|i-j|>1$, and define the projection
$p=\tau_{i}\tau_{j}\tau_{i}=\tau_{j}\tau_{i}\tau_{j}(=\tau_{i}\tau_{j})$. Then

$t_{i}t_{j}=x_{\tau_{i}}x_{\tau_{j}}\sim_{2}x_{\tau_{i}}x_{\tau_{j}}x_{\tau_{i}}x_{\tau_{j}}x_{\tau_{i}}x_{\tau_{j}}\sim_{3}x_{\tau_{i}\tau_{j}\tau_{i}}x_{\tau_{j}\tau_{i}\tau_{j}}=x_{p}x_{p}\sim_{1}x_{p}\quad\text{and
similarly}\quad t_{j}t_{i}\sim x_{p}.$

(T3). We have
$t_{i}t_{j}t_{i}=x_{\tau_{i}}x_{\tau_{j}}x_{\tau_{i}}\sim_{3}x_{\tau_{i}\tau_{j}\tau_{i}}=x_{\tau_{i}}=t_{i}$.

(T5). Writing $p=\tau_{i}$ for simplicity, we have

$et_{i}=x_{1}x_{p}\sim_{2}x_{1}x_{p}x_{1}x_{p}\sim_{3}x_{1p1}x_{p}=x_{p}x_{p}\sim_{1}x_{p}=t_{i}\qquad\text{and
similarly}\qquad t_{i}e\sim t_{i}.$

For (iii) we first note that $x_{1}=e\in X_{T}^{\prime}$. For $p\not=1$ we
have $p=\tau_{i_{1}}\cdots\tau_{i_{k}}$ (in $\mathcal{T}\\!\mathcal{L}_{n}$)
for some $i_{1},\ldots,i_{k}$, and then

$p=p^{*}p=\tau_{i_{k}}\cdots\tau_{i_{1}}\tau_{i_{1}}\cdots\tau_{i_{k}}=\tau_{i_{k}}\cdots\tau_{i_{2}}\tau_{i_{1}}\tau_{i_{2}}\cdots\tau_{i_{k}}=\tau_{i_{1}}\theta_{\tau_{i_{2}}}\cdots\theta_{\tau_{i_{k}}}.$

It follows that
$x_{p}=x_{\tau_{i_{1}}\theta_{\tau_{i_{2}}}\cdots\theta_{\tau_{i_{k}}}}\sim_{3}x_{\tau_{i_{k}}}\cdots
x_{\tau_{i_{2}}}x_{\tau_{i_{1}}}x_{\tau_{i_{2}}}\cdots
x_{\tau_{i_{k}}}=t_{i_{k}}\cdots t_{i_{2}}t_{i_{1}}t_{i_{2}}\cdots t_{i_{k}}$.

Finally, for (iv) we begin by defining a morphism

$\psi:X_{P}^{+}\to\mathcal{T}\\!\mathcal{L}_{n}:x_{p}\mapsto p.$

We then have $R_{P}\subseteq\ker(\psi)$ since $P={\bf
P}(\mathcal{T}\\!\mathcal{L}_{n})$. We then obtain the desired
$\phi:X_{P}^{+}\to\langle X_{T}^{\prime}:R_{T}^{\prime}\rangle$ by composing
$\psi:X_{P}^{+}\to\mathcal{T}\\!\mathcal{L}_{n}$ with the canonical
isomorphism $\mathcal{T}\\!\mathcal{L}_{n}\to\langle
X_{T}^{\prime}:R_{T}^{\prime}\rangle$. ∎

It is natural to wonder about the relationships between other diagram monoids
and their associated free projection-generated regular $*$-semigroups. These
relationships tend to be more complicated than the case of
$\mathcal{T}\\!\mathcal{L}_{n}$, as glimpsed in Example 10.17 below, and
treated in detail for the partition monoid $\mathcal{P}_{n}$ in the
forthcoming paper [31].

##  10 Topological interpretation

In this section we provide an alternative, topological interpretation of the
groupoids $\mathscr{C}=\mathscr{C}(P)$ and
$\overline{\mathscr{C}}=\overline{\mathscr{C}}(P)$ associated to a projection
algebra $P$. Specifically, we can view $\mathscr{C}$ as the fundamental
groupoid of a natural graph $G_{P}$ built from the
$\mathrel{\mathscr{F}}$-relation of $P$, and $\overline{\mathscr{C}}$ as the
fundamental groupoid of either of two $2$-complexes $K_{P}$ and
$K_{P}^{\prime}$. These complexes both have $G_{P}$ as their $1$-skeleton, and
their $2$-cells are induced by linked pairs of projections. The complex
$K_{P}$ is defined in terms of all linked pairs, and $K_{P}^{\prime}$ in terms
of a special subset that generates the others; in particular $K_{P}^{\prime}$
is simplicial. Maximal subgroups of the semigroup
$\operatorname{\textup{{PG}}}(P)$ coincide up to isomorphism with fundamental
groups of either of these two complexes.

###  10.1 Special and degenerate linked pairs

Consider a $p$-linked pair $(e,f)$ in a projection algebra $P$. The
projections $e,f,e\theta_{p},f\theta_{p}$ (which are used to define the paths
$\lambda(e,p,f)$ and $\rho(e,p,f)$) need not all be distinct, and an important
special case occurs when $e=e\theta_{p}$ or $f=f\theta_{p}$ (i.e. when $e\leq
p$ or $f\leq p$). In this case, we say $(e,f)$ is a _special $p$-linked pair_.

Let $\Xi^{\prime}$ be the subset of $\Xi$ consisting of all pairs
$(\mathfrak{s},\mathfrak{t})\in\mathscr{P}\times\mathscr{P}$ of the form
($\mathsf{\Omega}$1) and ($\mathsf{\Omega}$2), as well as:

  1. ($\mathsf{\Omega}$3)′

$\mathfrak{s}=\lambda(e,p,f)$ and $\mathfrak{t}=\rho(e,p,f)$, for some $p\in
P$, and some special $p$-linked pair $(e,f)$,

and let ${\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}^{\prime}}=\Xi^{\prime\sharp}$
be the congruence on $\mathscr{P}$ generated by $\Xi^{\prime}$.

######  Proposition 10.1.

We have ${\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}^{\prime}}={\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}}$.

###### Proof.

We just need to show that $\mathfrak{s}\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}^{\prime}\mathfrak{t}$
whenever $(\mathfrak{s},\mathfrak{t})$ is a pair of type ($\mathsf{\Omega}$3).
So suppose $\mathfrak{s}=\lambda(e,p,f)$ and $\mathfrak{t}=\rho(e,p,f)$ for
some $p$-linked pair $(e,f)$, and let $e^{\prime}=e\theta_{p}$ and
$f^{\prime}=f\theta_{p}$. Then one can show that $(e,f^{\prime})$ and
$(e^{\prime},f)$ are special $p$-linked pairs (with $f^{\prime}\leq p$ in the
first, and $e^{\prime}\leq p$ in the second), and we have

$\displaystyle\lambda(e,p,f^{\prime})$
$\displaystyle=(e,e^{\prime},f^{\prime}),$
$\displaystyle\lambda(e^{\prime},p,f)$
$\displaystyle=(e^{\prime},e^{\prime},f)\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}^{\prime}(e^{\prime},f),$
(10.4) $\displaystyle\rho(e,p,f^{\prime})$
$\displaystyle=(e,f^{\prime},f^{\prime})\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}^{\prime}(e,f^{\prime}),$
$\displaystyle\rho(e^{\prime},p,f)$ $\displaystyle=(e^{\prime},f^{\prime},f).$
(10.7)

Since $\Xi^{\prime}$ contains the pairs
$(\lambda(e,p,f^{\prime}),\rho(e,p,f^{\prime}))$ and
$(\lambda(e^{\prime},p,f),\rho(e^{\prime},p,f))$, it follows that
$(e,e^{\prime},f^{\prime})\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}^{\prime}(e,f^{\prime})$
and $(e^{\prime},f)\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}^{\prime}(e^{\prime},f^{\prime},f)$.
But then

$\mathfrak{s}=\lambda(e,p,f)=(e,e^{\prime},f)\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}^{\prime}(e,e^{\prime},f^{\prime},f)\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}^{\prime}(e,f^{\prime},f)=\rho(e,p,f)=\mathfrak{t}.\qed$

Another case in which we do not need to include a pair
$(\mathfrak{s},\mathfrak{t})$ of type ($\mathsf{\Omega}$3) is when we already
have $\mathfrak{s}\approx\mathfrak{t}$.

######  Definition 10.8.

We say a $p$-linked pair $(e,f)$ is _degenerate_ if
$\lambda(e,p,f)\approx\rho(e,p,f)$.

The next result characterises such degenerate pairs at the level of the
projection algebra structure.

######  Proposition 10.9.

A $p$-linked pair $(e,f)$ is degenerate if and only if
$e\theta_{p}=f\theta_{p}$ or $e,f\leq p$.

###### Proof.

Throughout the proof we write $e^{\prime}=e\theta_{p}$ and
$f^{\prime}=f\theta_{p}$, and also $\lambda=\lambda(e,p,f)=(e,e^{\prime},f)$
and $\rho=\rho(e,p,f)=(e,f^{\prime},f)$.

($\Leftarrow$). If $e^{\prime}=f^{\prime}$, then $\lambda=\rho$. If $e,f\leq
p$, then $e=e^{\prime}$ and $f=f^{\prime}$, so that

$\lambda=(e,e,f)\approx(e,f)\approx(e,f,f)=\rho.$

($\Rightarrow$). Suppose $\lambda\approx\rho$. If $\lambda$ and $\rho$ are
reduced (in the sense of Remark 3.6), then we must have $\lambda=\rho$, which
implies $e^{\prime}=f^{\prime}$. So now suppose $\lambda$ and $\rho$ are not
reduced. In particular, $e\not=f$; cf. ($\mathsf{\Omega}$2). For $\lambda$ not
to be reduced, it must then be the case that $e^{\prime}\in\\{e,f\\}$, and
similarly $f^{\prime}\in\\{e,f\\}$. If $e^{\prime}=f^{\prime}$ then we are
done. Otherwise,
$\\{e,f\\}=\\{e^{\prime},f^{\prime}\\}=\\{e\theta_{p},f\theta_{p}\\}$, and it
follows that $e,f\leq p$. ∎

It follows from Propositions 10.1 and 10.9 that the congruence
$\mathrel{\raise
1.07639pt\hbox{\oalign{$\approx$\crcr\kern-3.61664pt\cr$\approx$}}}$ on
$\mathscr{P}$ is generated by the set $\Xi^{\prime\prime}$ of all pairs
$(\mathfrak{s},\mathfrak{t})\in\mathscr{P}\times\mathscr{P}$ of the form
($\mathsf{\Omega}$1) and ($\mathsf{\Omega}$2), as well as:

  1. ($\mathsf{\Omega}$3)′′

$\mathfrak{s}=\lambda(e,p,f)$ and $\mathfrak{t}=\rho(e,p,f)$, for some $p\in
P$, and some non-degenerate special $p$-linked pair $(e,f)$.

######  Remark 10.10.

Consider a non-degenerate $p$-linked pair $(e,f)$, and write
$e^{\prime}=e\theta_{p}$ and $f^{\prime}=f\theta_{p}$, and also
$\lambda=\lambda(e,p,f)=(e,e^{\prime},f)$ and
$\rho=\rho(e,p,f)=(e,f^{\prime},f)$. By Proposition 10.9 there are three
possibilities:

  1. (i)

$\\{e,f,e^{\prime},f^{\prime}\\}$ has size $4$,

  2. (ii)

$\\{e,f,e^{\prime},f^{\prime}\\}$ has size $3$ and $e\leq p$ (i.e.
$e=e^{\prime}$),

  3. (iii)

$\\{e,f,e^{\prime},f^{\prime}\\}$ has size $3$ and $f\leq p$ (i.e.
$f=f^{\prime}$),

with $(e,f)$ being special in the second and third. In case (i),
identification of the paths $\lambda$ and $\rho$ (cf. ($\mathsf{\Omega}$3))
amounts to commutativity of the diamond in Figure 4(a), in the groupoid
$\overline{\mathscr{C}}$. In case (ii), we have $\lambda\approx(e,f)$, so
equating $\lambda$ and $\rho$ amounts to commutativity of the triangle in
Figure 4(b). Case (iii) corresponds to Figure 4(c). Considering again case
(i), the proof of Proposition 10.1 showed that $(e,f^{\prime})$ and
$(e^{\prime},f)$ are special $p$-linked pairs; since
$e^{\prime},f^{\prime}\leq p$, these are of types (iii) and (ii),
respectively. This means that the two triangles commute in Figure 4(d);
commutativity of these triangles of course implies commutativity of the outer
diamond in the same diagram, as per the final line of the proof of Proposition
10.1.

On the other hand, a degenerate $p$-linked pair $(e,f)$ leads to one of
diagrams (e) or (f) in Figure 4, in the cases $e^{\prime}=f^{\prime}$ and
$e,f\leq p$, respectively. These diagrams already commute in $\mathscr{C}$,
and we note that (e) and (f) picture the generic case in which all projections
displayed are distinct (it is possible to have even more collapse).

$e$$\phantom{{}^{\prime}}e^{\prime}$$f^{\prime}$$f$(a)$e=e^{\prime}$$f^{\prime}$$f$(b)$e$$e^{\prime}$$f=f^{\prime}$(c)$e$$\phantom{{}^{\prime}}e^{\prime}$$f^{\prime}$$f$(d)

$e$$e^{\prime}$$f$

=

$f^{\prime}$(e)$e$

=

$\phantom{{}^{\prime}}e^{\prime}$$f$

=

$\phantom{{}^{\prime}}f^{\prime}$(f)

Figure 4: Diamonds, triangles and degeneracy of linked pairs of projections;
see Remark 10.10.

###  10.2 Graphs and complexes

We are now in a position to give the promised topological interpretation of
the chain and reduced chain groupoids $\mathscr{C}=\mathscr{C}(P)$ and
$\overline{\mathscr{C}}=\overline{\mathscr{C}}(P)$. We begin by defining a
graph $G_{P}$ with:

  * •

vertex set $P$ (a given projection algebra), and

  * •

an (undirected) edge $\\{p,q\\}$ for every pair of distinct
$\mathrel{\mathscr{F}}$-related projections
$(p,q)\in{\mathrel{\mathscr{F}}}\setminus\Delta_{P}$.

We then define two $2$-complexes, $K_{P}$ and $K_{P}^{\prime}$. These both
have $G_{P}$ as their $1$-skeleton.

  * •

The complex $K_{P}$ has a $2$-cell with boundary
$(e,e^{\prime},f,f^{\prime},e)$ for every $p$-linked pair $(e,f)$, where as
usual we write $e^{\prime}=e\theta_{p}$ and $f^{\prime}=f\theta_{p}$.

  * •

The complex $K_{P}^{\prime}$ is a sub-complex of $K_{P}$, and contains only
the $2$-cells corresponding to non-degenerate special $p$-linked pairs
$(e,f)$. Such a cell has boundary $(e,f,f^{\prime},e)$ or $(e,e^{\prime},f,e)$
when the pair is of type (ii) or (iii), as enumerated in Remark 10.10.

In particular, all cells in $K_{P}^{\prime}$ are triangles.

The following result is essentially the observation that our categories
$\mathscr{C}=\mathscr{P}/\Omega^{\sharp}$ and
${\overline{\mathscr{C}}=\mathscr{P}/\Xi^{\sharp}=\mathscr{P}/\Xi^{\prime\prime\sharp}}$
are constructed in precisely the same way as the fundamental groupoids of the
above graph and complexes. See for example [38, Section 6].

######  Theorem 10.11.

For any projection algebra $P$ we have

$\mathscr{C}(P)\cong\pi_{1}(G_{P})\qquad\text{and}\qquad\overline{\mathscr{C}}\cong\pi_{1}(K_{P})\cong\pi_{1}(K_{P}^{\prime}),$

as (unordered) groupoids. ∎

This has an immediate important consequence concerning the subgroups of the
free regular $*$-semigroups $\operatorname{\textup{{PG}}}(P)$. It follows from
general semigroup theory that in an arbitrary semigroup $S$, maximal subgroups
are precisely the $\mathrel{\mathscr{H}}$-classes of idempotents; see [16,
Exercise 2.3.1]. The relation $\mathrel{\mathscr{H}}$ is one of the five
Green’s equivalences (see [39, Section 2.1]), but we do not require its actual
definition, only that the $\mathrel{\mathscr{H}}$-class of an idempotent $e$
has the form

$H_{e}=\\{s\in S:se=es=s\text{ and }st=ts=e\text{ for some }t\in S\\}.$

If $e$ and $f$ are $\mathrel{\mathscr{R}}$-equivalent idempotents, which here
can be simply taken to mean $e\mathrel{\leavevmode\hbox to14.63pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-2.59998pt\hbox
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}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{0.2pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02638pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}f$ using the notation
introduced in Subsection 6.1, then $H_{e}\cong H_{f}$ [39, Proposition 2.3.6].
Specialising to the case where $S$ is a regular $*$-semigroup, in which every
idempotent $e$ is $\mathrel{\mathscr{R}}$-related to the projection $ee^{*}$
[58, Theorem 2.2], we see that every maximal subgroup of $S$ is isomorphic to
one of the form $H_{p}$ with $p\in{\bf P}(S)$, and that $H_{p}=\\{s\in S\colon
ss^{*}=s^{*}s=p\\}$. Finally, in the special case that
$S=\operatorname{\textup{{PG}}}(P)$ for a projection algebra $P$, we see from
(4.20) that the multiplication in $S$ restricted to $H_{p}$ is precisely the
same as the composition in the reduced chain groupoid
$\overline{\mathscr{C}}=\overline{\mathscr{C}}(P)$ restricted to
$\overline{\mathscr{C}}(p,p)$. Combining this with Theorem 10.11, we obtain
the following as a corollary:

######  Theorem 10.12.

Let $P$ be a projection algebra. Every maximal subgroup of
$\operatorname{\textup{{PG}}}(P)$ is isomorphic to a fundamental group of
$K_{P}$, or (equivalently) of $K_{P}^{\prime}$. Specifically, the
$\mathrel{\mathscr{H}}$-class of any projection $p\in P$ is isomorphic to
$\pi_{1}(K_{P},p)\cong\pi_{1}(K_{P}^{\prime},p)$. ∎

We conclude this section and the paper by recasting the examples from Sections
8 and 9 in the terminology of this section, and adding two new ones. As
always, the reader should have at the back of their mind the category
isomorphism between regular $*$-semigroups and chained projection groupoids
(Subsection 3.5), and that under this isomorphism the free regular
$*$-semigroup $\operatorname{\textup{{PG}}}(P)$ and the reduced chain groupoid
$\overline{\mathscr{C}}(P)$ correspond to each other (Section 4). The
exposition will freely move between these two structures.

######  Example 10.13.

Let $\Gamma=(P,E)$ be a symmetric, reflexive digraph, $A_{\Gamma}$ the
adjacency semigroup, $P_{0}$ its projection algebra, and
$B_{\Gamma}=\operatorname{\textup{{PG}}}(P_{0})$ the bridging path semigroup,
as in Subsection 2.2 and Section 8. The graph $G_{P_{0}}$ is the simple,
undirected reduct of $\Gamma$ with $0$ added as a new, isolated vertex. As we
saw in Section 8, all linked pairs in $P_{0}$ are degenerate, and so
$K_{P_{0}}=K_{P_{0}}^{\prime}=G_{P_{0}}$. In particular,
$\mathscr{C}=\overline{\mathscr{C}}$ can be realised as the fundamental
groupoid of the graph $G_{P_{0}}$, and Theorem 10.12 implies that all maximal
subgroups of $B_{\Gamma}=\operatorname{\textup{{PG}}}(P_{0})$ are free. This
gives another way to see why $\operatorname{\textup{{PG}}}(P)$ is finite in
Example 8.1 and infinite in Example 8.2.

######  Example 10.14.

Let $P^{\prime}=\\{p,q,r,e\\}$ be the projection algebra from Example 8.3. The
graph $G_{P^{\prime}}$ consists of a triangle with vertices $p,q,r$, and an
isolated vertex $e$. The complex $K_{P^{\prime}}=K_{P^{\prime}}^{\prime}$ has
a $2$-cell attached to the triangle in $G_{P^{\prime}}$, coming from the
$e$-linked pair $(p,r)$.

######  Example 10.15.

In Theorem 9.1, we showed that when $P={\bf P}(\mathcal{T}\\!\mathcal{L}_{n})$
is the projection algebra of the Temperley–Lieb monoid
$\mathcal{T}\\!\mathcal{L}_{n}$, the free regular $*$-semigroup
$\operatorname{\textup{{PG}}}(P)$ is isomorphic to
$\mathcal{T}\\!\mathcal{L}_{n}$. This has a topological consequence. The
monoid $\mathcal{T}\\!\mathcal{L}_{n}$ is known to have no non-trivial
subgroups; equivalently, it is $\mathrel{\mathscr{H}}$-trivial [61]. Hence,
all the fundamental groups of the complexes $K_{P}$ and $K_{P}^{\prime}$ are
trivial, i.e. their connected components are simply connected. It does not
seem obvious, a priori, that this ought to be the case.

In the final two examples we use the topological viewpoint to gain some
insight into the free regular $*$-semigroups
$\operatorname{\textup{{PG}}}({\bf P}(\mathcal{M}_{n}))$ associated to the
Motzkin monoids $\mathcal{M}_{n}$, as defined in Subsection 2.3, and the
relationship to the idempotent-generated subsemigroups $\langle
E(\mathcal{M}_{n})\rangle$, which were studied in [22].

######  Example 10.16.

Consider the Motzkin monoid $\mathcal{M}_{3}$, and let $P={\bf
P}(\mathcal{M}_{3})$ be the projection algebra of this monoid. The elements of
$\mathcal{M}_{3}$ are shown in Figure 5 in a so-called _egg-box diagram_ (see
[16] for more details), in which the projections are shaded dark grey, and the
non-projection idempotents light grey. The complex $K_{P}^{\prime}$ is shown
in Figure 6. The connected component of $K_{P}^{\prime}$ at the bottom of the
figure contains three triangular $2$-cells, which are indicated by shading.
(The outer triangle of this component is not the boundary of a $2$-cell.) To
see for example that the ‘upper’ triangle is a $2$-cell, denote its vertices
by $e={\lower 4.30554pt\hbox{ \leavevmode\hbox to15.31pt{\vbox
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}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{6.544pt}{0.0pt}\pgfsys@moveto{7.65646pt}{0.0pt}\pgfsys@curveto{7.65646pt}{0.6144pt}{7.1584pt}{1.11246pt}{6.544pt}{1.11246pt}\pgfsys@curveto{5.92961pt}{1.11246pt}{5.43155pt}{0.6144pt}{5.43155pt}{0.0pt}\pgfsys@curveto{5.43155pt}{-0.6144pt}{5.92961pt}{-1.11246pt}{6.544pt}{-1.11246pt}\pgfsys@curveto{7.1584pt}{-1.11246pt}{7.65646pt}{-0.6144pt}{7.65646pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{6.544pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{13.08801pt}{0.0pt}\pgfsys@moveto{14.20047pt}{0.0pt}\pgfsys@curveto{14.20047pt}{0.6144pt}{13.70241pt}{1.11246pt}{13.08801pt}{1.11246pt}\pgfsys@curveto{12.47362pt}{1.11246pt}{11.97556pt}{0.6144pt}{11.97556pt}{0.0pt}\pgfsys@curveto{11.97556pt}{-0.6144pt}{12.47362pt}{-1.11246pt}{13.08801pt}{-1.11246pt}\pgfsys@curveto{13.70241pt}{-1.11246pt}{14.20047pt}{-0.6144pt}{14.20047pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{13.08801pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{19.63203pt}{0.0pt}\pgfsys@moveto{20.74449pt}{0.0pt}\pgfsys@curveto{20.74449pt}{0.6144pt}{20.24643pt}{1.11246pt}{19.63203pt}{1.11246pt}\pgfsys@curveto{19.01764pt}{1.11246pt}{18.51958pt}{0.6144pt}{18.51958pt}{0.0pt}\pgfsys@curveto{18.51958pt}{-0.6144pt}{19.01764pt}{-1.11246pt}{19.63203pt}{-1.11246pt}\pgfsys@curveto{20.24643pt}{-1.11246pt}{20.74449pt}{-0.6144pt}{20.74449pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{19.63203pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} {}{{}}{}{{}{}{}{{}}{{{}{}{}{}}}}{} {} {} {} {}{}{}{{}}{}
{}{}{}{{}}{}{{}{}{}{{}}{{{}{}{}{}}}}{} {} {} {}
{}{}{}\pgfsys@moveto{13.08801pt}{13.08801pt}\pgfsys@curveto{13.08801pt}{11.64236pt}{14.25992pt}{10.47046pt}{15.70557pt}{10.47046pt}\pgfsys@moveto{15.70558pt}{10.47046pt}\pgfsys@lineto{17.01447pt}{10.47046pt}\pgfsys@moveto{17.01447pt}{10.47046pt}\pgfsys@curveto{18.46011pt}{10.47046pt}{19.63202pt}{11.64236pt}{19.63202pt}{13.08801pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}{}{}{{}}{{{}{}{}{}}}}{} {} {} {} {}{}{}{{}}{}
{}{}{}{{}}{}{{}{}{}{{}}{{{}{}{}{}}}}{} {} {} {}
{}{}{}\pgfsys@moveto{13.08801pt}{0.0pt}\pgfsys@curveto{13.08801pt}{1.44565pt}{14.25992pt}{2.61755pt}{15.70557pt}{2.61755pt}\pgfsys@moveto{15.70558pt}{2.61755pt}\pgfsys@lineto{17.01447pt}{2.61755pt}\pgfsys@moveto{17.01447pt}{2.61755pt}\pgfsys@curveto{18.46011pt}{2.61755pt}{19.63202pt}{1.44565pt}{19.63202pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }}$ and $g={\lower
4.30554pt\hbox{ \leavevmode\hbox to15.31pt{\vbox
to15.31pt{\pgfpicture\makeatletter\hbox{\hskip-5.43155pt\lower-1.11246pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{6.544pt}{13.08801pt}\pgfsys@moveto{7.65646pt}{13.08801pt}\pgfsys@curveto{7.65646pt}{13.70241pt}{7.1584pt}{14.20047pt}{6.544pt}{14.20047pt}\pgfsys@curveto{5.92961pt}{14.20047pt}{5.43155pt}{13.70241pt}{5.43155pt}{13.08801pt}\pgfsys@curveto{5.43155pt}{12.47362pt}{5.92961pt}{11.97556pt}{6.544pt}{11.97556pt}\pgfsys@curveto{7.1584pt}{11.97556pt}{7.65646pt}{12.47362pt}{7.65646pt}{13.08801pt}\pgfsys@closepath\pgfsys@moveto{6.544pt}{13.08801pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{13.08801pt}{13.08801pt}\pgfsys@moveto{14.20047pt}{13.08801pt}\pgfsys@curveto{14.20047pt}{13.70241pt}{13.70241pt}{14.20047pt}{13.08801pt}{14.20047pt}\pgfsys@curveto{12.47362pt}{14.20047pt}{11.97556pt}{13.70241pt}{11.97556pt}{13.08801pt}\pgfsys@curveto{11.97556pt}{12.47362pt}{12.47362pt}{11.97556pt}{13.08801pt}{11.97556pt}\pgfsys@curveto{13.70241pt}{11.97556pt}{14.20047pt}{12.47362pt}{14.20047pt}{13.08801pt}\pgfsys@closepath\pgfsys@moveto{13.08801pt}{13.08801pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{19.63203pt}{13.08801pt}\pgfsys@moveto{20.74449pt}{13.08801pt}\pgfsys@curveto{20.74449pt}{13.70241pt}{20.24643pt}{14.20047pt}{19.63203pt}{14.20047pt}\pgfsys@curveto{19.01764pt}{14.20047pt}{18.51958pt}{13.70241pt}{18.51958pt}{13.08801pt}\pgfsys@curveto{18.51958pt}{12.47362pt}{19.01764pt}{11.97556pt}{19.63203pt}{11.97556pt}\pgfsys@curveto{20.24643pt}{11.97556pt}{20.74449pt}{12.47362pt}{20.74449pt}{13.08801pt}\pgfsys@closepath\pgfsys@moveto{19.63203pt}{13.08801pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{6.544pt}{0.0pt}\pgfsys@moveto{7.65646pt}{0.0pt}\pgfsys@curveto{7.65646pt}{0.6144pt}{7.1584pt}{1.11246pt}{6.544pt}{1.11246pt}\pgfsys@curveto{5.92961pt}{1.11246pt}{5.43155pt}{0.6144pt}{5.43155pt}{0.0pt}\pgfsys@curveto{5.43155pt}{-0.6144pt}{5.92961pt}{-1.11246pt}{6.544pt}{-1.11246pt}\pgfsys@curveto{7.1584pt}{-1.11246pt}{7.65646pt}{-0.6144pt}{7.65646pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{6.544pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{13.08801pt}{0.0pt}\pgfsys@moveto{14.20047pt}{0.0pt}\pgfsys@curveto{14.20047pt}{0.6144pt}{13.70241pt}{1.11246pt}{13.08801pt}{1.11246pt}\pgfsys@curveto{12.47362pt}{1.11246pt}{11.97556pt}{0.6144pt}{11.97556pt}{0.0pt}\pgfsys@curveto{11.97556pt}{-0.6144pt}{12.47362pt}{-1.11246pt}{13.08801pt}{-1.11246pt}\pgfsys@curveto{13.70241pt}{-1.11246pt}{14.20047pt}{-0.6144pt}{14.20047pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{13.08801pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{19.63203pt}{0.0pt}\pgfsys@moveto{20.74449pt}{0.0pt}\pgfsys@curveto{20.74449pt}{0.6144pt}{20.24643pt}{1.11246pt}{19.63203pt}{1.11246pt}\pgfsys@curveto{19.01764pt}{1.11246pt}{18.51958pt}{0.6144pt}{18.51958pt}{0.0pt}\pgfsys@curveto{18.51958pt}{-0.6144pt}{19.01764pt}{-1.11246pt}{19.63203pt}{-1.11246pt}\pgfsys@curveto{20.24643pt}{-1.11246pt}{20.74449pt}{-0.6144pt}{20.74449pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{19.63203pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }}$. Then with
$p={\lower 4.30554pt\hbox{ \leavevmode\hbox to15.31pt{\vbox
to15.31pt{\pgfpicture\makeatletter\hbox{\hskip-5.43155pt\lower-1.11246pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{6.544pt}{13.08801pt}\pgfsys@moveto{7.65646pt}{13.08801pt}\pgfsys@curveto{7.65646pt}{13.70241pt}{7.1584pt}{14.20047pt}{6.544pt}{14.20047pt}\pgfsys@curveto{5.92961pt}{14.20047pt}{5.43155pt}{13.70241pt}{5.43155pt}{13.08801pt}\pgfsys@curveto{5.43155pt}{12.47362pt}{5.92961pt}{11.97556pt}{6.544pt}{11.97556pt}\pgfsys@curveto{7.1584pt}{11.97556pt}{7.65646pt}{12.47362pt}{7.65646pt}{13.08801pt}\pgfsys@closepath\pgfsys@moveto{6.544pt}{13.08801pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{13.08801pt}{13.08801pt}\pgfsys@moveto{14.20047pt}{13.08801pt}\pgfsys@curveto{14.20047pt}{13.70241pt}{13.70241pt}{14.20047pt}{13.08801pt}{14.20047pt}\pgfsys@curveto{12.47362pt}{14.20047pt}{11.97556pt}{13.70241pt}{11.97556pt}{13.08801pt}\pgfsys@curveto{11.97556pt}{12.47362pt}{12.47362pt}{11.97556pt}{13.08801pt}{11.97556pt}\pgfsys@curveto{13.70241pt}{11.97556pt}{14.20047pt}{12.47362pt}{14.20047pt}{13.08801pt}\pgfsys@closepath\pgfsys@moveto{13.08801pt}{13.08801pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{19.63203pt}{13.08801pt}\pgfsys@moveto{20.74449pt}{13.08801pt}\pgfsys@curveto{20.74449pt}{13.70241pt}{20.24643pt}{14.20047pt}{19.63203pt}{14.20047pt}\pgfsys@curveto{19.01764pt}{14.20047pt}{18.51958pt}{13.70241pt}{18.51958pt}{13.08801pt}\pgfsys@curveto{18.51958pt}{12.47362pt}{19.01764pt}{11.97556pt}{19.63203pt}{11.97556pt}\pgfsys@curveto{20.24643pt}{11.97556pt}{20.74449pt}{12.47362pt}{20.74449pt}{13.08801pt}\pgfsys@closepath\pgfsys@moveto{19.63203pt}{13.08801pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{6.544pt}{0.0pt}\pgfsys@moveto{7.65646pt}{0.0pt}\pgfsys@curveto{7.65646pt}{0.6144pt}{7.1584pt}{1.11246pt}{6.544pt}{1.11246pt}\pgfsys@curveto{5.92961pt}{1.11246pt}{5.43155pt}{0.6144pt}{5.43155pt}{0.0pt}\pgfsys@curveto{5.43155pt}{-0.6144pt}{5.92961pt}{-1.11246pt}{6.544pt}{-1.11246pt}\pgfsys@curveto{7.1584pt}{-1.11246pt}{7.65646pt}{-0.6144pt}{7.65646pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{6.544pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{13.08801pt}{0.0pt}\pgfsys@moveto{14.20047pt}{0.0pt}\pgfsys@curveto{14.20047pt}{0.6144pt}{13.70241pt}{1.11246pt}{13.08801pt}{1.11246pt}\pgfsys@curveto{12.47362pt}{1.11246pt}{11.97556pt}{0.6144pt}{11.97556pt}{0.0pt}\pgfsys@curveto{11.97556pt}{-0.6144pt}{12.47362pt}{-1.11246pt}{13.08801pt}{-1.11246pt}\pgfsys@curveto{13.70241pt}{-1.11246pt}{14.20047pt}{-0.6144pt}{14.20047pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{13.08801pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{19.63203pt}{0.0pt}\pgfsys@moveto{20.74449pt}{0.0pt}\pgfsys@curveto{20.74449pt}{0.6144pt}{20.24643pt}{1.11246pt}{19.63203pt}{1.11246pt}\pgfsys@curveto{19.01764pt}{1.11246pt}{18.51958pt}{0.6144pt}{18.51958pt}{0.0pt}\pgfsys@curveto{18.51958pt}{-0.6144pt}{19.01764pt}{-1.11246pt}{19.63203pt}{-1.11246pt}\pgfsys@curveto{20.24643pt}{-1.11246pt}{20.74449pt}{-0.6144pt}{20.74449pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{19.63203pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{6.544pt}{13.08801pt}\pgfsys@lineto{6.544pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{13.08801pt}{13.08801pt}\pgfsys@lineto{13.08801pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }}$, one can check
that $(e,f)$ is a special $p$-linked pair, with $e\theta_{p}=e$ and
$f\theta_{p}=g$.

It follows from this that the connected components of $K_{P}^{\prime}$ are
simply connected, and hence that $\operatorname{\textup{{PG}}}(P)$ is
$\mathrel{\mathscr{H}}$-trivial. It follows that
$\operatorname{\textup{{PG}}}(P)$ is isomorphic to its image under the natural
morphism
$\overline{\operatorname{id}}_{P}:\operatorname{\textup{{PG}}}(P)\to\mathcal{M}_{3}$
from Theorem 5.8. This image is the idempotent-generated subsemigroup
$\langle{\bf E}(\mathcal{M}_{3})\rangle$.

Figure 5: The elements of the Motzkin monoid $\mathcal{M}_{3}$; see Example
10.16. Figure 6: The complex $K_{P}^{\prime}$, where $P={\bf
P}(\mathcal{M}_{3})$ is the projection algebra of the Motzkin monoid
$\mathcal{M}_{3}$; see Example 10.16.

One may wonder whether the same holds for larger Motzkin monoids, but this is
not the case, as our final example shows.

######  Example 10.17.

Let $P={\bf P}(\mathcal{M}_{4})$ be the projection algebra of the Motzkin
monoid $\mathcal{M}_{4}$. The complex $K_{P}^{\prime}$ has $35$ vertices, and
eleven connected components, one of which is shown in Figure 7; its six
$2$-cells are shaded. It is apparent that the fundamental group(oid) of this
component is infinite; specifically, a loop around the central square has
infinite order. Consequently, $\operatorname{\textup{{PG}}}(P)$ is infinite,
and hence not isomorphic to $\langle{\bf E}(\mathcal{M}_{4})\rangle$.

Figure 7: A connected component of the complex $K_{P}^{\prime}$, where $P={\bf
P}(\mathcal{M}_{4})$ is the projection algebra of the Motzkin monoid
$\mathcal{M}_{4}$; see Example 10.17.

Maximal subgroups of free projection-generated regular $*$-semigroups will be
the main topic of our paper [31]. This will include a general theory of
presentations for maximal subgroups of arbitrary
$\operatorname{\textup{{PG}}}(P)$, and detailed computations for
$\operatorname{\textup{{PG}}}({\bf P}(\mathcal{P}_{n}))$, the free regular
$*$-semigroups arising from the partition monoids. These results will be
compared and contrasted with known presentations [35] for maximal subgroups of
the free (regular) idempotent-generated semigroups
$\operatorname{\textup{{IG}}}(E)$ and $\operatorname{\textup{{RIG}}}(E)$. This
will again include explicit results for partition monoids, which will
highlight the significant difference between
$\operatorname{\textup{{IG}}}({\bf E}(\mathcal{P}_{n}))$ and
$\operatorname{\textup{{PG}}}({\bf P}(\mathcal{P}_{n}))$, and involve
unexpected connections with _twisted_ partition monoids [32, 33, 46].

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