数理解析研究所講究録 1222 巻 2001 年 74-82

74

Van Kampen Diagrams and E-unitary Coextensions
Akihiro Yamamura (山村 明弘)

1*

Introduction

Semidirect products and HNN extensions of groups resemble each other in the sense that bothhave apresentation described by apseudo commutative law, however, no approach to unify
and generalize them has appeared so far. In [12], aunified treatment of semidirect products
and HNN extensions of groups are presented. In this paer, akey lemma in [12] is presented.
First of all, we explain the generalization of semidirect products and HNN extensions givenof ain [12]. Let , $H$ be groups. Each generator of $H$ corresponds to an isomorphism$K$of . Letbe the group presented bysubgroupof onto asubgroup$G$$A_{h}$$h$$G$$B_{h}$$\phi_{h}$$G$

Gp(G, $H|ax=x(a\phi_{x})$ for$\forall a\in A_{x}$,$\forall x\in X$

),

(1.1)

of $G$ into $K$ an isomorphism?
where $H$ is generated by $X$ . Is the natural homomorphismof $H$ into $K$ an isomorphism?
Is the natural homomorphism$G$
It is shown in [12] that is strongly embedded into $K$ under acertain condition (we will
not touch on the embedding problem in this paper). The embedding problem is inherently
related to the algebraic system of partial automorphisms. The key idea is to apply inverse
semigroup theory to group theory. In this paper, we present the key lemma concerning Eunitary coextensions of groups, which is crucial in [12]. Our approach is rather geometrical.
We use van Kampen diagrams, which are very powerful method in group theory.
$\xi c$$\xi_{H}$2Inverse semigroups
Partial one to one mappings of anon-empty set constitute an algebraic system called thesymmetric inverse semigroup. Conversely any abstract inverse semigroup is embedded in thesymmetric inverse semigroup of acertain set (known as Wagner-Preston theorem). Thus, thetheory of inverse semigroups is an ideal tool to formalize and study algebraic structure of asystem of partial one to one morphisms and local properties of mathematical objects. As amatter of fact, inverse semigroup theory was initiated to study local properties of manifolds by
Wagner [10]. Ehresmann [1] also considered inverse semigroups with some extra assumptionsin the context of topology and differential geometry under the name of pseudogroups. Afterthe introduction of the concept, numerous researches had been done mainly on its algebraic
structure. Against its first motivation, only alittle attention had been paid to the theoryof partial automorphisms of aspecific mathematical structure, such as groups, rings, vector
spaces, topological spaces, manifolds, or graphs. The significance of partial automorphisms
in other areas of mathematics has been rediscovered recently. For instance, the study on theCommunications Research Laboratory, 4-2-1, Nukui-Kitamachi, Koganei, Tokyo, 184-8795Japan E-mail:.go.jp$\mathrm{a}\mathrm{k}\mathrm{i}\Phi \mathrm{c}\mathrm{r}\mathrm{l}$75relationship between -algebras, groupoids, and inverse semigroups can be found in [8]. The
reader is referred to [3] for historical account and detailed information on the theory of inverse
semigroups.
An inverse semigroup is aregular semigroup in which every element has the unique inverse(see [3]). The class of inverse semigroups forms avariety, and hence, every inverse semigroup
has apresentation. We should remark that an inverse semigroup presentation is given by asetof generators and aset of relations which is asubset of $(X\cup X^{-1})^{+}\mathrm{x}(X\cup X^{-1})^{+}$ , whereasagroup presentation is usually given by aset of generators and aset of relators which is asubset of $(X\cup X^{-1})^{+}$ . We frequently consider agroup presentation by aset of generators andaset of relations as. Apresentation for agroup generated by $X$ subject to $R$ is denotedby Gp(X $|R$), and apresentation for an inverse semigroup generated by $X$ subject to $R$ isdenoted by Inv(X $|R$) in this paper. The inverse semigroup presented by Inv(X $|R$) is thefreest inverse semigroup subject to the relation $R$ in the following sense.
If agroup $H$ is ahomomorphic image of an inverse semigroup , then we say that is aninverse semigroup coextension of $H$ . We say that an inverse semigroup coextension of $H$ has
the same set of generators as $H$ if both and $H$ are generated by $X$ and, whereandare the natural mappings, and is ahomomorphism of onto $H$ .
An inverse semigroup coextension of $H$ is called $E$-unitary if $\sigma^{-1}(1)=E(S)$ . In suchacase, the group $H$ is the maximal group homomorphic image of . We remark that the
maximal group homomorphic image $H$ of an inverse semigroup has the universal mapping
property: for any homomorphism of into any group $Q$ , there exists ahomomorphismof $H$ into $Q$ such that
. An inverse semigroup is called $E$-unitary if any element
above an idempotent is an idempotent (see [3]). It is easy to see that an inverse semigroupis -unitary if and only if it is an -unitary coextension of its maximal group homomorphic
image.
Agroupoid is small category in which every morphism is invertible. Any inverse semigroupcan be endowed with agroupoid structure ([6, 9]) and we denote it by Gpd(5) in this paper.
The set of objects of Gpd(S) is $E(S)$ . For , $f\in E(S)$ , amorphism of to $f$ is any element$s\in S$ such that $e=ss^{-1}$ and $f=s^{-1}s$ . The inverse of the morphism
is given by. If$e\in E(S)$ , thenis the identity morphism of the object . The trace product of an inverse
semigroupis the groupoid product in Gpd(S), that is, theproduct for si, S2 in
is defined to be
if and only if
in . Thenis amorphism fromtosince
and $(s_{1}s_{2})^{-1}(s_{1}s_{2})=$. Using the terminology in semigroup theory, we define theproduct
if and only if there exists $e\in E(S)$ such that
, where and 72
are Green’s relations (see [3]). We illustrate the morphismin Gpd(S) and the locationof , , ,in $D$ class of in Figure 1. Furthermore, Gpd(5) is endowed with an order
structure, however, we do not touch on it.
$C^{*}$$\mathrm{w}\mathrm{e}\mathrm{u}$

$S$

$S$

$S$

$S$

$S$

$\iota s$$\iota_{H}$$\sigma\circ\iota s$$=\iota H$$S$$\sigma$

$S$

$S$

$S$

$S$

$\rho$$\nu$$S$$\nu\circ\sigma=\rho$$\mathrm{E}$$\mathrm{B}$

$S$

$)$

$e$

$e$

$s^{-1}$

$s$

$e$

$e$

$S$

$s_{1}^{-1}s_{1}=s_{2}s_{2}^{-1}$$s_{1}s_{2}$$s_{1}s_{1}^{-1}$$S$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$S$$s_{1}s_{2}$$(s_{1}s_{2})(s_{1}s_{2})^{-1}=s_{1}s_{2}s_{2}^{-1}s_{1}^{-1}=s_{1}s_{1}^{-1}s_{1}s_{1}^{-1}=s_{1}s_{1}^{-1}$$s_{2}^{-1}s_{2}$$s_{2}^{-1}s_{1}^{-1}s_{1}s_{2}=s_{2}^{-1}s_{2}s_{2}^{-1}s_{2}=s_{2}^{-1}s_{2}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$s_{1}s_{2}$$s_{1}\mathcal{L}e\mathcal{R}s_{2}$$\mathcal{L}$$s_{1}s_{2}$$e$$s_{1}$$s_{2}$$S$$s_{1}s_{2}$$e$$\ldots$$\ldots$$\sim$.
.

.

.

$\cdot$$s1\underline{s_{1}^{-1}s_{1}^{-1}s_{1}--s_{2}s_{2}^{-1}s_{2}^{-1}}s_{2}$$s_{1}$$\cdot$......

$s_{2}$$s_{1}$Morphism$s_{1}s_{2}$in Gpd(S)$D$

class ofFigure 1:
product
be amapping of aset $X$ into an inverse semigroup . Let $w=h_{1}h_{2}\cdots$$s_{1}\iota_{2}$$S$$\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$Let$\tau$$S$$h_{n}$

be aword

76

product in . The word is called acyclic trace product in with respect to if every cyclic
products andproduct in with respect to . The language of all
conjugate of is a$(S,X,\tau)$ ,$(5,
X,\tau)$LOOP
andproducts with respect to are denoted by TRACE
all cyclic
in general, whereas
respectively. Note that LOOP(S, $X,\tau$)mapping of $X$
any
and
group
for any
TRACE(G, $X,\tau$ )into $G$ .
$S$

$S$

$w$

$w$

$S$

$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\tau$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\tau$$\tau$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\subset \mathrm{T}\mathrm{R}\mathrm{A}\mathrm{C}\mathrm{E}(5,X,\tau)\subset(X\cup X^{-1})^{+}$$G$$=\mathrm{L}\mathrm{O}\mathrm{O}\mathrm{P}(G,X,\tau)=(X\cup X^{-1})^{+}$

Lemma 2,1 Suppose$sem\dot{lg}1vup$ $S$$w=h\mathrm{x}h2$ $\ldots h$$\in(X\cup X^{-1})^{+}$.

and$\tau$$\tau$

is a mapping of $X$ into an inverse(1) If $w$ lies in TRACE(S, $X,\tau$) and $w=w_{1}w_{2}$ , where $w_{1},w_{2}\in(X\cup X^{-1})^{+}$ , thenin $S$ under .
(fl) If $w$ lies in LOOP(S, $X,\tau$ ), then so does any cyclic conju gate of .
(3) If $w$ lies in LOOP(S, $X,\tau$ ) and $E(S)$ , then so does any cyclic conjugate of $w$ .
and $w=w_{1}1102$ , where $w_{1},\eta,w_{3}\in(X\cup X^{-1})^{+}$ . Then
(4) Suppose.
if and only
. $Ifw$ lies in TRACE(S, $X,\phi$), then we have(5) Let be a mapping $ofX$ into
for every $:=1,2$ , , $n-1$ . If $w$ lies in LOOP(Tc, $X,\phi$), then Ran(\phi h)=$w_{1}^{-1}w_{1}=$$w_{2}w_{2}^{-1}$$\tau$$w$$103$$w\in \mathrm{T}\mathrm{R}\mathrm{A}\mathrm{C}\mathrm{E}(5,X,\tau)$$\dot{l}fw_{2}\in \mathrm{L}\mathrm{O}\mathrm{O}\mathrm{P}(S,X,\tau)$$w_{1}w_{3}\in \mathrm{T}\mathrm{R}\mathrm{A}\mathrm{C}\mathrm{E}(S,\mathrm{X},\mathrm{r})$$\mathrm{D}\mathrm{o}\mathrm{m}(\phi\iota_{+1}.)$$\mathrm{D}\mathrm{o}\mathrm{m}(\phi_{h_{1}})$$\mathrm{R}\mathrm{a}\mathrm{n}(\phi_{h}:)=$$\mathrm{T}\mathrm{G}$$\phi$$\ldots$.

$\square$

Aword $w$ in $(X\cup X^{-1})^{+}$ is cffied Dycfc if it is the identity in the ffee group
Dyck word on $X$ is an idempotent in any inverse semigroup generated by $X$ .
Lemma 2.2 (1) Ifthen $w$ is the identity morphism of the object$w=h\mathrm{x}h2$$\tau$$\ldots$$\mathrm{F}\mathrm{G}(X)$. Anylies in LOOP(S, $X,\tau$ ) and $E(S)$ under ,$\tau(h_{1})\tau(h_{1})^{-1}=\tau(h_{n})^{-1}\tau(h_{n})$ in Gpd(S) under$h_{\mathfrak{n}}\in(X\cup X^{-1})^{+}$$\tau$.

(1) Let $d=h_{1}h_{2}\cdots h_{n}$ be a Dyck word on X. If belongs to TRACE(S, $X$ , ), then $d\in$
LOOP(S, $X,\tau$ ) and is the identity morphism of the object $\tau(h_{1})\tau(h_{1})^{-1}=\tau(h_{n})^{-1}\tau(h_{n})$ in
Gpd(5) under .
$d$$\tau$$d$$\square$$\tau$

Lemma 2.3 Let be an inverse semigroup. Suppose a word $w\in(X\cup X^{-1})^{+}$ belongs to. $\in E(S)$ for every $:=1,2$ ,, ,$E(S)$ , and $w=w_{0}w_{1}\cdots w_{n}$ , where $w:\in(X\cup X^{-1})^{+}$ . If$e_{n}w_{n}\in
E(S)$
.
and
then
$S$

$\ldots$$\mathrm{q}$$n$$\square$$w0e_{1}w_{1}e_{2}\cdots$$w0e_{1}w_{1}e_{2}\cdots$$e_{\mathfrak{n}}w_{n}\leq w$

3Van Kampen diagrams and trace products
product structure on boundary cycles of van Kampen
In this section, we investigate thediagrams and give acondition for an inverse semigroup coextension of agroup to be unitary.
This result is crucial in the embedding proble for groups (see [12]).
consists of two sets Vert(l’) and Edge(Y), together with the
A -oemploe(or graph). We
and : Edge(Y), :Edge(Y)functions : Edge(Y),the elements of Vert(Y) vertices, and the elements of Edge(Y) dges. For
is called the initial $ve\hslash*$ the terminal vertex and the inverse edge, respectively. A 2-complex (or amap) $M$ consists of a1-complex $M’$ , its l-skeleton’ together with aset Face(M) and two functions aand $m$ defined on Face(M). We call the elements of Face(M)is theis the boundary cycle of $F$ , andfaces (or -cells). For each $F$ in Face(Af),. The geometric realization of avertex, an edge
inverse of $F$ satisying$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\mathrm{E}$$\mathrm{Y}$$\mathrm{J}$$\alpha$$arrow \mathrm{V}\mathrm{e}\mathrm{r}\mathrm{t}(\mathrm{Y})$$arrow \mathrm{V}\mathrm{e}\mathrm{r}\mathrm{t}(1 )$$\omega$$arrow \mathrm{E}\mathrm{d}\mathrm{g}\mathrm{e}(\mathrm{Y})$$\eta_{1}$$y\in \mathrm{E}\mathrm{d}\mathrm{g}\mathrm{e}(\mathrm{Y})$$\mathrm{c}\mathrm{a}\mathrm{U}$$\alpha(y),\omega(y),\eta_{1}(y)$$\partial F$$l$$\eta_{1}(\partial F)=\partial(\varphi(F))$$\eta_{2}(F)$77
and aface of aplanar 2-complex is apoint, abounded subset homeomorphic to an open unit
interval and abounded subset homeomorphic to an open unit disk in the Euclidean space ,respectively. We denote the boundary cycle of aconnected and simply connected 2-c0mplex$M$ by. Here the boundary cycle of $M$ is the word that is read along the contour of $M$
counterclockwise starting and ending at the distinguished vertex. The subgraph consisting of
as well. We denote the interior and the closure of athe boundary of $M$ is denoted by,respectively.
by
space
and
topological$H$Let
be agroup presented by Gp(-Y $|R$), where is cyclicffiy reduced. Suppose isafreely reduced word in $(X\cup X^{-1})^{+}$ and $w=1$ in $H$ . Avan Kampen diagram for over
$H$ is afinite planar connected and simply connected 2-complex $M$ such that the boundary
cycle
of any face $F$ of $M$ is acyclic conjugate of arelator in $R$ and the boundary cycle$M$
as words. The
is equal tostarting and ending at the distinguished vertex
ofreader is referred to $[4, 7]$ for van Kampen diagrams and van Kampen’s lemma. We shall call amaximal subcomplex homeomorphic to aclosed disk ageneralized face in this paper. Asta& of, and their initial and,avan Kampen diagram $M$ is asubpath consisting of edges
Af’.
Note
that
is aunion of generalzed
disjoint
every
from
is
that
terminal vertices suchis aunion of stalcs of $M$ . Thus, any van Kampen diagram is aunion
faces of $M$ , andof finitely many generalized faces and stalks.
$\mathrm{E}^{2}$$\partial M$$\partial M$$Z\subset \mathrm{E}^{2}$$\overline{Z}$$Z^{\mathrm{o}}$

$R$

$w$

$w$

$\partial F$$\partial M$$w$$\mathrm{m}$$y_{1},y_{2}$$\ldots$$y_{k}$$\overline{M^{\mathrm{o}}}$$y\dot{.}$$\overline{M\backslash \overline{M^{\mathrm{o}}}}$

Figure 2:
The ramification number of avertex (or ageneralized face $D$ ) of afinite planar connected
and simply connected 2-complex $M$ is the number of connected components of the topological. Then $M_{1}=C\cup\{v\}$). Let $C$ be aconnected component of Af
space
(or
is aconnected and simply connected 2-complex and called abranch of $M$ at . Abranchat ageneralized face $D$ is similarly defined (see Figure 3and 4). Afinite planar connected
and simply connected 2-complex $M$ is caUed chain-shaped if any vertex and any generalized
face has ramification number at most two. In Figure 5, achain-shaped 2-complex with seven
generalized faces, in which the path P. may be empty, is illustrated.
$v$$M\backslash \{v\}$$\backslash \{v\}$$M\backslash D$$v$$\cdot$Figure 3: Several vertices and generalzed faces having ramification number 3$\nearrow_{v}v\downarrow v\backslash$$\nearrow_{vv}\backslash \mathrm{O}_{v}$

–

(a)

(b)

$\backslash \nearrow$(c)

Figure 4: Branches in Figure 3$0$$\backslash \nearrow$

(d)

78

Figure 5: Chain-shaped 2-c0mplex

Lemma 3.1 (Van Kampen [11]) Let $w\in(X\cup X^{-1})^{+}$ be a freely reduced word such that$w=1$ in H. Therea van Kampen diagram for over H.
0$\dot{\varpi}sh$$w$.
Lemma 3.2 $Let\tau$ be a mapping $ofX$ into an inverse semigroup such that
Suppose $M$ is a van Kampen diagram over $H$ for a freely reduced $wod$ $w\in(X\cup X^{-1})^{+}$ representing 1in $H$ .
(1) Let $F$ be a face of M. Then
belongs to LOOP(S, $X,\tau$). If $R\subset E(S)$ under , then$E(S)$
belongs tounder .
(2) Let $D$ be a generalized face of M. Any word in $X\cup X^{-1}$ labeling a path in $D$ belongs tobelongs to LOOP(S, $X,\tau$ ).
TRACE $(5, X,\tau)$ . In particular,(3) If $R\subset E(S)$ under , thenbelongs to $E(S)$ un&r for every generalized face $D$ of$S$$R\subset \mathrm{L}\mathrm{O}\mathrm{O}\mathrm{P}(S, X, \tau)$$\partial F$$\tau$$\partial F$$\tau$$\partial D$$\partial D$$\tau$$M$$\tau$.

of any face $F$ of $M$ is acyclic conjugate of arelator in,
by Lemma 2.1 (2). If $R\subset E(S)$ , then $\partial F\in E(S)$ by Lemma 2.1 (3).
,, are edges in $D$ entering and each(2) Let be avertex in $D$ . Supposeis$h_{l}\inX\cupX^{-1}$
product in Gpd(S),. It suffices to show thatis alabeled byin for all $1\leq:,j\leq n$ . Since $D$ is aplanar diagram,
equivalently,
we may assume , , , are enumerated counterclockwise around in this order. Since$D$ is homeomorphic
to aclosed disc, there are two possible cases: (Case 1) lies in the interior$D$of , and (Case 2) lies on the boundary of $D$ (see Figure 6).

Proof (1) Since the boundary cycle$R$$\partial F$$\partial F\in \mathrm{L}\mathrm{O}\mathrm{O}\mathrm{P}(S,X,\tau)$$v$$y_{1},y_{2}$$\ldots$$v$$y_{n}$$\tau(k.)\tau(h_{\mathrm{j}})^{-1}$$\tau(h_{\dot{1}})^{-1}\tau(k.)=\tau(h_{j})^{-1}\tau(h_{j})$$y_{1}$$y_{2}$$\ldots$$y_{\dot{1}}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$

$S$

$v$

$y_{n}$$v$$v$(Case 1)Figure 6: Vertex$v$and edges entering it) form asubpath(Case 1) The edges $y_{+1}$ and} with $1\leq:\leq n-1$ (and and
$D$
$F$
(1),
belongs to LOOP(S, $X,\tau$ ), andin . Byof the boundary cycle of acertain face.
hence, $\tau(h_{i+1})\tau(|4.)^{-1}$ belongs to TRACE(S, $X,\tau$ ) for every as
is asubword$\tau(\iota_{+1}.)^{-1}\tau(h+1)=\tau(h\iota)^{-1}\tau(|4.)$
every
for
It
follows
thatHence,.
ofandfor aU:, . (Case 2) We may assume without loss of generalty thatare on the boundary of $D$ . For each consecutive pair $y$ and $\mathfrak{R}+1(1\leq:\leq n-1)$ , there exists
aface in which $y$ and $\Re+1$ form asubpath of the boundary cycle, and hence, we can provefor all:, as in (Case 1). It follows that any word labelng apath$D$$X,\tau$). Since any cyclic conjugate ofbelongs
is apath in $D$ ,in belongs to TRACE(S,$\eta_{1}(y.\cdot$$\eta_{1}(y_{||})$$y_{1}$$\partial F.\cdot$

$|$

:.

$\partial F_{}$$\tau(h_{\mathrm{j}})^{-1}\tau(h_{j})$$\tau(l4.+1)\tau(|\mathrm{h}.)^{-1}$$\tau(l\mathrm{h}.)^{-1}\tau(k.)=$$j$$y_{1}$$\tau(h_{i})^{-1}\tau(l4.)=\tau(h_{j})^{-1}\tau(h_{\mathrm{j}})$$y_{n}$$j$$\partial D$$\partial D$

to LOOP $(5,X,\tau)$ .
(3) Suppose $R\subset E(S)$ under . Let $D$ be ageneralized face of $M$ . We shall show that aboundary cycle of any connected and simply connected subcomplex of $D$ belongs to $E(S)$under using induction on the number of faces in . First we note that aconnected andis aDyck
simply connected subcomplex with no faces is atree. Hence, the boundary cycle$E(S)$. Second, we suppose has only one face . Then is obtained
word and belongs to
by attaching finitely many trees to . Note that aboundary cycle of any tree is
ffom$\tau$

$L$

$L$

$l$

$\tau$$L$$\partial L$$L$$F_{\phi}$$F_{0}$$F_{0}$$L$

79
aDyck word. Therefore, the boundary cycle of is obtained ffom
by inserting finitely
many Dyck words, and hence, written as, where,,,andare (maybe empty) trees. By Lemma 2.3,.
$L$
Now we suppose the claim is true for any positive integer less than and asubcomplex of$D$
has faces. Suppose has at least two generalized faces. There exists avertex
havingramification number at least two and there exist at least two branches at such that each ofthem contains at least one generalized face. Then the boundary cycle of starting and endingat can be written as
for some subcomplexes
andof having at least one
generalized face, whereandaxe the boundary cycles of and starting and ending
at , respectively (see Figure 7).
$L$$\partial F_{0}$$w_{1}z_{1}(\partial T_{1})z_{2}(T_{2})\cdots$ $z_{n}(\partial T_{n})w_{2}$$\partial F_{0}=\mathrm{Z}1\mathrm{Z}2$$\ldots z_{n}$$T_{0},T_{1}$$w_{1}w_{2}=\partial T_{0}$$T_{n}$$\ldots$$\partial L\in \mathrm{E}(\mathrm{S})$

$l$

$L$

$l$

$v_{1}$$v_{1}$$L$$(\partial L_{1})(\partial L_{2})$$v_{1}$$\partial L_{1}$$L_{1}$$\partial L_{2}$$L_{2}$$L$$L_{2}$$L_{1}$$v_{1}$

Figure 7:$L$

has at least two generalzed faces

By the inductive hypothesis,
belong to $E(S)$ . Thus, the boundary cycle ofandstarting and ending at
is written as. By Lemma 2.1 (3), $\partial L\in E(S)$ . Next we
suppose has only one generalized face with no stalcs. Then we show thatlies in $E(S)$ .
There are two cases. (Case 1) At least two faces of have edges on the boundary of . In
such acase, is aunion of connected and simply connected subcomplexes
andwithmore than one generalized faces that have edges on the boundary
andis empty.
(Case 2) Only one face(see Figure 8).
has the edges in$\partial L_{1}$$L$$\partial L_{2}$$(\partial L_{1})(\partial L_{2})$$v_{1}$$L$$\partial L$

$L$

$L$

$L$

$D_{1}$$\partial L$$D_{2}$$D_{1}^{\mathrm{o}}\cap D_{2}^{\mathrm{o}}$$\partial L$$F_{1}$(Case 1)(Case 2)
Figure 8: Decomposition of $L$(Case 1) We can write,,. Sinceis apath in $D$ ,is aDyck word belonging to TRACE $(5, X,\tau)$ by the part (2). Hence,is theidentity morphism in Gpd(5) by Lemma 2.2 (2). Then we have SIS3
under
. On the other hand,under by the inductive hypothesis. It follows$=s_{1}s_{2}s_{2}^{-1}s_{3}=(\partial D_{1})(\partial D_{2})\in E(S)$ .
that(Case 2) We can write $\partial L=s_{1}$ ,,, whereis aconnected and simply connected subcomplex of$D$ .
Note thathas $l-1$ faces. By the inductive hypothesis,
belongs to $E(S)$ under .
$\partial L=\mathrm{S}1\mathrm{S}3$$\partial D_{1}=\mathrm{S}1\mathrm{S}2$$\partial D_{2}=s_{2}^{-1}s_{3}$$s_{2}s_{2}^{-1}$$s_{2}s_{2}^{-1}$$s_{2}s_{2}^{-1}$$=s_{1}s_{2}s_{2}^{-1}s_{3}$$\partial D_{1},\partial D_{2}\in E(S)$$\tau$$\tau$$\partial L=\mathrm{S}1\mathrm{S}3$$\partial F_{1}=s_{1}ws_{2}^{-1}w^{-1}$$\partial D_{2}=s_{2}$$D_{2}$$D_{2}$$\partial D_{2}$$\tau$

Moreover,. Sinceasis aDyckword and lies in TRACE $(5, X,\tau)$ by the part (2), it is the identity morphism in Gpd(S) by
Lemma 2.2 (2). It follows that $s_{1}=s_{1}ws_{2}^{-1}w^{-1}(ws_{2}^{-1}w^{-1})^{-1}=(\partial F_{1})(\partial D_{2})\in E(S)$ andis
an idempotent in . Hence, if has no stalks, thenlies in $E(S)$ . We now suppose consistsof ageneralized face $D$ and several stalcs. By the argument above, we have $\partial D\in E(S)$ . If
, them$enwnz2$ , where each . is the boundary cycle ofastalk and ZIZ2 is aboundary cycle of astalc. Since astalk is atree, its boundary cycle isaDyck word. By Lemma 2.3, $\partial L\in E(S)$ under . This completes the induction, and hence,$\partial F_{1}$belongs to $E(S)$ under$\tau$$R\subset E(S)$$ws_{2}^{-1}w^{-1}(ws_{2}^{-1}w^{-1})^{-1}$$\partial L$$S$$\partial D=w\mathit{0}w_{1}\cdots w_{n}$

$L$

$L$

$\partial L$$\partial L=\mathrm{z}\mathrm{i}\mathrm{w}\mathrm{o}\mathrm{e}\mathrm{i}\mathrm{w}\mathrm{i}\mathrm{e}2\cdots$$\mathrm{q}$$\tau$$\partial D\in E(S)$.

$\square$

We remark that Lemma 3.2 (2) implies that TRACE $(5, X,\tau)$ includes the language ac

80
cepted by the 1-skeleton $D’$ of any generalized face $D$ considered as an automaton in which ev$X\cup X^{-1}$ery vertex of $D$ is both an initial state and aterminal state. Furthermore, any word inlabeling aclosed path in $D$ belongs to $E(S)$ under . We briefly sketch the proof. Choose an, is aarbitrary maximal subtree (spanning tree) of the 1-skeleton $D’$ . Suppose , ,$w=h_{1}h_{2}\ldots$.
$D$ starting and ending at the vertex
the
word
by
labeledandclosed path in.
There exists aunique geodesic from to any vertex in . We denote the geodesic by
by$D$
denoted,is,byencompassed
Take an edge in . The subcomplex. By the argument above, the boundaryis the inverse path ofTri(y), where$E(S)$
is. Eachinlies
atendingof Tri(y) starting andcycle
hence,) and belongs to TRACE $(5, X,\tau)$ by Lemma 3.2 (2), andDyck word (asis equal to
Gpd(5)by Lemma 2.2 (2) under . Then,
it is an identity morphism in,and hence, it belongs to $E(S)$ . Therefore, $E(S)$ includes the language accepted by the auin which the initial state and the terminal state is the same fixed vertex of $D$ .
tomatonAfinite planar connected and simply connected 2-complex $M$ is said to be good with respect
and $\partial D\in E(S)$to a mapping of $X$ into an inverse semigroup if$D$ of $M$ . In such acase, any cyclic conjugate ofbelongs
under for every generalized face
for adiagram
Kampenanyvanthat
claims
3.2
(3).
Lemma
$E(S)$
under by Lemma 2.1
to
$H$
and
freely reduced word representing 1in is good with respect to if$R\subset E(S)$ under.
$\tau$$T$$y_{2}$$\mathrm{y}_{\mathrm{i}}$$\ldots$$y_{n}$$h_{n}$

$v0$

$v$

$v0$

$T$

$\mathrm{P}(v)$$\mathrm{P}(\alpha(y))$

$y$

$y$

$\eta_{1}(\mathrm{P}(\omega(y)))$$\mathrm{P}(\omega(y))$$\eta_{1}(\mathrm{P}(\omega(y)))$$\eta 1(\mathrm{P}(\omega(y_{\dot{1}})))\mathrm{P}(\alpha(y:+1))$$\partial’\mathrm{I}\mathrm{h}.(y)$$\mathrm{a}$$v\mathit{0}$$\omega(y_{\dot{1}})=\alpha(y_{+1}.\cdot)$$h_{1}h_{2}\ldots$ $h_{n}$$\tau$$h_{1}\eta_{1}(\mathrm{P}(\omega(y_{1})))\mathrm{P}(\alpha(y_{2}))h_{2}\cdots\eta_{1}(\mathrm{P}(\omega(y_{n-1})))\mathrm{P}(\alpha(y_{n}))h_{n}=(\theta \mathrm{I}\mathrm{h}.(y_{1}))(\partial’\mathrm{I}\mathrm{h}.(y_{2}))\cdots(\theta \mathrm{I}\mathrm{k}\mathrm{i}(y_{n}))$$\theta$$S$$\tau$$\partial D\in \mathrm{L}\mathrm{O}\mathrm{O}\mathrm{P}(S,X,\tau)$$\partial D$$\tau$$\tau$$\tau$$R\subset \mathrm{L}\mathrm{O}\mathrm{O}\mathrm{P}(S,X,\tau)$$\tau$

Theorem 3.3 Let $H$ be a group presented by Gp(X $|R$ ), where $R$ is cyclically reduced. Suppose$X$of generators as H. If
is an inverse semigroup coextension of $H$ having the same set$R\subset
E(S)$E-unitary.
,then is
and
$S$

$S$

$R\subset \mathrm{L}\mathrm{O}\mathrm{O}\mathrm{P}(S,X, \iota_{S})$in $H$ for $w\in(X\cup X^{-1})^{+}$ , where
if and only if
for representing
is the normal form for in FG(\"A). If we can prove
1in $H$ , then $to\in E(S)$ by Lemma 2.3. Therefore, we have only to show that $u\in E(S)$ for
every freely reduced word representing 1in $H$ . This is equivalent to show that the boundary
cycle of avan Kampen diagram $M$ for afreely reduced word over $H$ , where $u=1$ in $H$ , is an$(S,X,\iota s)$ and $R\subset E(S)$ , any van Kampen
idempotent in by Lemma 3.1. Since$XarrowS$
is$H$
by Lemma 3.2. We shall show that:
diagram over is good with respect to2-c0mplex
an idempotent in for every good (finite planar connected and simply connected)$M$ using induction on the number of generalized faces of $M$ .
Suppose $M$ consists of only one generalzed face $D$ and finitely many trees attached. There(see Figure 9).
is not inand (Case 2)
are two possible cases: (Case 1) is in

Proof. We first note that $w=1$ in$H$$\mathrm{N}\mathrm{F}(w)=1$$\mathrm{N}\mathrm{F}(w)\in E(S)$$w$$\mathrm{N}\mathrm{F}(w)$

$w$

$u$

$u$

$S$

$R\subset \mathrm{L}\mathrm{O}\mathrm{O}\mathrm{P}$$\partial M$$\iota s$$S$$v_{0}$$\partial D$$v0$$\partial D$

81
. By
be the subcomplex consisting ofmeets $D$ at the vertex . Letlies in $E(S)$ . Then we can write
starting and ending at(Case 1), the boundary cycle. Since
is aDyck word, $\partial T_{1}\in E(S)$ . Hence,, where$\partial M\in E(S)$ by Lemma 2.3.
Therefore, the claim is true for any 2-complex with only one
generalized face. We now suppose the claim is true for any positive integer less than andis in
the number of generalized faces of $M$ is $n\geq 2$ . There are two possible cases: (Case 1)$M$$M$(see Figureand (Case 2) $v0$ is not in acertain generalized face of
all generalized faces of$T_{1}$$(M\backslash T_{1})\cup\{v_{1}\}$$M_{1}$$v_{1}$$\partial M_{1}$$v_{1}$$\partial T_{1}=w_{1}w_{2}$$\partial M=w_{1}(\partial M_{1})w_{2}$$\partial T_{1}$$n$$\mathrm{m}$10).

(Case 1)Figure 10: $M$ has more than one generalized faces(Case 1) Suppose
is in every generalized face of $M$ . An generalized faces are adjacent each,
are the branches at $v0$ and enumerated counterclockwiseother at . Suppose $M_{1},M_{2}$ ,. Note that each M. is aconnectedin this order andaroundhas
and simply connected 2-complex. Since $n\geq 2$ and every generalized face has , eachevery:,
whereby the inductive hypothesis forat most one generalized faces. Thenis the boundary cycle of M. starting and ending at . Therefore, $\partial M\in E(S)$ . (Case. Since the number of generalzed faces of $M$ is2) Suppose
is not in ageneralized face$n\geq 2$ , there exists at least one branch at. Suppose $M_{1},M_{2}$ , ,Ma are branches atcontains . Letinthisorder. We may assume
aroundenumerated counterdodcwise
is adjacent to . We note that each consecutive, ,, be the vertices at which each
, wheremay be equal. Thenvertices :and
$2\leq:\leqs$,forstarting and ending at
is the boundary cycle of$\in E(S)$ for every $1\leq i\leq s$ and. By the inductive hypothesis,
and0. Therefore, $\partial M\in E(S)$ by Lemma 2.3. Consequently is -unitary.
$v_{0}$$M_{s}$$\ldots$$\mathrm{V}\mathrm{q}$$\partial M=(\partial M_{1})(\partial M_{2})\cdots$ $(\partial M_{s})$$v_{0}$$\cdot$$M_{}$$v_{0}$$\partial M.\cdot\in E(S)$$\partial M.\cdot$$v0$$\cdot$$D_{1}$$v_{0}$$D_{1}$$M_{1}$$D_{1}$$v_{1}$$v_{2}$$\ldots$$\partial M=w_{1}z_{1}(\partial M_{2})z_{2}(\partial M_{3})z\mathrm{a}\cdots$ $z_{s-1}(\partial M_{s})z_{s}w_{2}$$v.\cdot+1$$M_{}$$\partial M\dot{.}$$\partial D_{1}=z_{1}z_{2}\cdots z_{\epsilon}$$\partial D_{1}\in E(S)$$\mathrm{v}\mathrm{o}$$D_{1}$$M_{\dot{1}}$$v_{s}$$v$$D_{1}$$\ldots$$\partial M_{1}=w_{1}w_{2}$$v.\cdot$$\partial \mathit{1}\mathit{4}$$S$$\mathrm{E}$

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