On the structure of lower bounded HNN extensions

Abstract

This paper studies the structure and preservational properties of lower bounded HNN extensions of inverse semigroups, as introduced by Jajcayová. We show that if $S^* = [ S;\; U_1,U_2;\; \phi ]$ is a lower bounded HNN extension then the maximal subgroups of $S^*$ may be described using Bass-Serre theory, as the fundamental groups of certain graphs of groups defined from the $\mathcal{D}$-classes of $S$, $U_1$ and $U_2$. We then obtain a number of results concerning when inverse semigroup properties are preserved under the HNN extension construction. The properties considered are completely semisimpleness, having finite $\mathcal{R}$-classes, residual finiteness, being $E$-unitary, and $0$-$E$-unitary. Examples are given, such as an HNN extension of a polycylic inverse monoid.

1. Introduction

Higman et al. [10] introduced the concept of an HNN extension of a group. In combinatorial group theory, HNN extensions play an important role in algorithmic problems.

Yamamura [20] showed the usefulness of HNN extensions in the variety of inverse semigroups by proving the undecidability of any Markov property and the undecidability of several non-Markov properties. Jajcayová introduced lower bounded HNN extensions in [12], mirroring the definition of lower bounded amalgams of inverse semigroups given in Bennett [3] and [4]. It was proved in Jajcayová [13] that an HNN extension of a free inverse semigroup with finitely generated subsemigroups has decidable word problem.

HNN extensions of a finite inverse semigroup have been considered by Cherubini and Rodaro [6], showing that an HNN extension of a finite inverse semigroup has decidable word problem. More recently, Ayyash and Cherubini [1] and [2] give necessary and sufficient conditions for an HNN extension of a finite inverse semigroup or a lower bounded HNN extension to be completely semisimple. Ayyash [1] also described the maximal subgroups in the finite case.

In the current paper, we use Bass-Serre theory to describe the maximal subgroups of a lower bounded HNN extension $S^*$ containing the idempotents of $S$ (Theorem 4.4). The maximal subgroups are the fundamental groups of graph of groups constructed from $\mathcal{D}$ -classes and maximal subgroups of $S$ . All other maximal subgroups of $S^*$ are isomorphic to subgroups of $S$ (Theorem 4.6). Conditions are given for $S^*$ to have finite $\mathcal{R}$ -classes (Theorem 4.16). Conditions are given for $S^*$ to be $E$ -unitary and $0$ - $E$ -unitary (Theorem 4.19). We show that the HNN extension of a polycyclic inverse monoid can be $0$ - $E$ -unitary, with group of units isomorphic to a free group and all other maximal subgroups are trivial.

2. Preliminaries

A semigroup $S$ is an inverse semigroup if for all $s \in S$ there is a unique element $s^{-1}$ , the inverse of $s$ , such that $s s^{-1} s = s$ and $s^{-1} s s^{-1} = s^{ -1 }$ . The semilattice of idempotents of $S$ is the set $E( S ) = \{ e \in S \;:\; e^2 = e \}$ . The natural partial order $\leq$ of $S$ is defined by $a \leq b$ if and only if $a = eb$ , for some $e \in E( S )$ , for $a, b \in S$ . A subsemigroup $U$ of $S$ is an inverse subsemigroup if $u^{-1} \in U$ , for all $u \in U$ . For inverse semigroups, see Howie [11], Petrich [18], and Lawson [16].

A presentation for an inverse semigroup $S$ is a pair $\langle X \mid R \rangle$ , where $X$ is a non-empty set and $R$ is a binary relation on $( X \cup X^{-1} )^+$ , with $S \cong ( X \cup X^{-1} )^+/ \tau$ , where $\tau$ is the congruence generated by $R$ and the Vagner congruence $\rho$ . We then say $S$ is presented by the generators $X$ and relations $R$ , written $S = Inv \langle X \mid R \rangle$ .

We study $\langle X \mid R \rangle$ by considering the Schützenberger automaton $\mathcal{A} ( X, R, w )$ of $w$ , for $w \in ( X \cup X^{-1} )^+$ . The automaton $\mathcal{A} ( X, R, w )$ has underlying graph $S\Gamma ( X, R, w )$ , with vertices $R_{w \tau }$ , the $\mathcal{R}$ -class of $S$ containing $w \tau$ , and an edge from $s$ to $t$ labeled by $y$ , for $s, t \in R_{w \tau }$ and $y \in X \cup X^{-1}$ where $s \cdot y \tau = t$ in $S$ . The initial state is $ww^{-1} \tau$ and the terminal state is $w \tau$ . We also denote $\langle X \mid R \rangle$ , $S \Gamma ( X, R, w )$ , $\mathcal{A} ( X, R, w )$ by $\langle S \rangle$ , $S \Gamma ( S, w )$ , $\mathcal{A} ( S, w )$ , respectively. For presentations, see Stephen [19].

For any non-empty set $X$ , an inverse word graph $\Gamma$ over $X$ is a connected graph with edges labeled over the set $X \cup X^{-1}$ , such that for any edge from $v_1$ to $v_2$ labeled by $y$ , there is an inverse edge from $v_2$ to $v_1$ labeled by $y^{-1}$ . The inverse word graph $\Gamma$ is deterministic if no two distinct edges have the same initial vertex and label. We denote the vertex and edge by $V( \Gamma )$ and $E( \Gamma )$ , respectively.

A (birooted) inverse automaton over $X$ is a triple $\mathcal{A} = ( \alpha, \Gamma, \beta )$ , where $\Gamma$ is an inverse word graph over $X$ and $\alpha$ , $\beta$ are vertices, called the initial and terminal roots of $\mathcal{A}$ , respectively. The language $L [ \mathcal{A} ]$ of the automaton $\mathcal{A}$ is the set of all words labeling paths from $\alpha$ to $\beta$ . An inverse automaton $\mathcal{A}$ over $X$ is called an approximate automaton of $\mathcal{A} ( X, R, w )$ if $L [ \mathcal{A} ] \subseteq L [ \mathcal{A} ( X, R, w ) ]$ , and there is some word $w_1 \in L [ \mathcal{A} ]$ with $w_1 = w$ in $S = Inv \langle X \mid R \rangle$ , written $\mathcal{A} \leadsto \mathcal{A} ( X, R, w )$ . The notation $\cong$ is used to indicate when two inverse word graphs (automata) are isomorphic.

If $\Gamma$ and $\Gamma _1$ are disjoint inverse word graphs, $v_1, v_2 \in V( \Gamma )$ and $\alpha _1, \beta _1 \in V( \Gamma _1 )$ then we sew on $( \alpha _1, \Gamma _1, \beta _1 )$ from $v_1$ to $v_2$ by taking the quotient of $\Gamma \cup \Gamma _1$ by the $V$ -equivalence generated by $\{ ( v_1, \alpha _1 ), ( v_2, \beta _1 ) \}$ . The linear automaton of $w = z_1z_2 \cdots z_n \in ( X \cup X^{-1} )^+$ , for $z_k \in X \cup X^{-1}$ , is the inverse automaton with vertices $v_0 = \alpha _w$ , $v_1$ , $\ldots$ , $v_{n-1}$ , $v_n = \beta _w$ and edges $v_{k-1} \rightarrow ^{ z_k } v_k$ , $v_k \rightarrow ^{ z^{-1}_k } v_{k-1}$ , for $k = 1, 2, \ldots, n$ . If $( r, s )$ is a relation in $R$ and there is a path $v_1 \rightarrow ^r v_2$ in $\Gamma$ , with no path $v_1 \rightarrow ^s v_2$ , then we perform an elementary expansion, relative to $\langle X \mid R \rangle$ , by sewing on the linear automaton of $s$ from $v_1$ to $v_2$ . A deterministic inverse word graph (automaton) over $X$ is closed relative to $\langle X \mid R \rangle$ if no elementary expansion can be performed.

If $\Gamma$ is an inverse graph over $X$ , then we say there is a path from vertex $v_1$ to vertex $v_2$ labeled by $s \in S$ , written $v_1 \rightarrow ^s v_2$ , if there is a path $v_1 \rightarrow ^w v_2$ , for some $w \in ( X \cup X^{-1} )^+$ such that $w \tau = s$ in $S$ . If $\Gamma$ is closed, relative to $\langle X \mid R \rangle$ , and we have a path $v_1 \rightarrow ^w v_2$ , for some $w \in ( X \cup X^{-1} )^+$ with $w \tau = s$ , then we also have a path $v_1 \rightarrow ^y v_2$ , for any $y \in ( X \cup X^{-1} )^+$ with $y \tau \geq s$ .

3. HNN extensions of inverse semigroups

The theory of lower bounded HNN extensions has been generalized by the authors in [5]. An inverse subsemigroup $U$ is called lower bounded in $S$ if, for any $u \in U$ and $e \in E( S )$ with $u \geq e$ in $S$ , there exists $f \in E( U )$ with $u \geq f \geq e$ in $S$ . The lower bounded inverse subsemigroup condition is illustrated in Fig. 1. We review some definitions and results from [5].

Figure 1. The lower bounded subsemigroup condition.

We consider an HNN extension $S^* = [ S;\; U_1, U_2;\; \phi ]$ of an inverse semigroup $S$ where $U_1$ and $U_2$ are inverse monoids that are lower bounded in $S$ , with respective identities $e_1$ and $e_2$ , and $\phi \;:\; U_1 \rightarrow U_2$ is an isomorphism. If $U_1$ and $U_2$ are only inverse subsemigroups that are lower bounded in $S$ , then we can study the HNN extension $[ S' ;\; U'_{\!\!1}, U'_{\!\!2};\; \phi ' ]$ , where $S' = S \cup \{ 1 \}$ , the element $1$ is disjoint from $S$ and is the identity of $S \cup \{ 1 \}$ , $U'_{\!\!1} = U_1 \cup \{ 1 \}$ and $U'_{\!\!2} = U_2 \cup \{ 1 \}$ are inverse monoids that are lower bounded in $S \cup \{ 1 \}$ and $\phi '$ is the isomorphism $U'_{\!\!1} \rightarrow U'_{\!\!2}$ induced by $\phi \;:\; U_1 \rightarrow U_2$ and $1 \rightarrow 1$ .

Let $S$ have inverse semigroup presentation $\langle X \mid R \rangle$ . We also denote this presentation for $S$ by $\langle S \rangle$ . Let $t$ be disjoint from $S$ . The free product $S * FIS( t )$ in the variety of inverse semigroups has presentation $\langle X \cup \{ t \} \mid R \rangle$ , where $FIS( t )$ is the free inverse semigroup on $\{ t \}$ . We also denote this presentation for $S * FIS( t )$ by $\langle S \cup \{ t \} \rangle$ . The HNN extension $S^*$ has inverse semigroup presentation $\langle X \cup \{ t \} \mid R \cup R^* \rangle$ , where $R^*$ consists of the relations $tt^{-1} = e_1$ , $t^{-1} t = e_2$ and $t^{-1} u t = ( u ) \phi$ , for $u \in U_1$ . We also denote this presentation for $S^*$ by $\langle S^* \rangle$ . In [20], it was proved that $S$ is embedded into $S^*$ . The HNN extension is illustrated in Fig. 2. For $w \in ( X \cup X^{-1} )^*$ , we let $S\Gamma ( S, w )$ and $\mathcal{A}( S, w )$ denote the Schützenberger graph and automaton of $w$ , respectively, relative to $\langle S \rangle$ . For $w \in ( X \cup X^{-1} \cup \{ t, t^{-1} \} )^*$ , we let $S\Gamma ( S^*, w )$ and $\mathcal{A}( S^*, w )$ denote the Schützenberger graph and automaton of $w$ , respectively, relative to $\langle S^* \rangle$ .

Figure 2. The HNN extensions $S^* = [ S;\; U_1, U_2;\; \phi ]$ .

We briefly describe the algorithm given in [5] for constructing the Schützenberger automata of $S^*$ . Let $\Gamma$ be an inverse word graph over $X \cup \{ t \}$ . An $\langle S \rangle$ -lobe of $\Gamma$ is a maximal connected subgraph with edges labeled over $X \cup X^{-1}$ . A $\langle t \rangle$ -lobe of $\Gamma$ is a maximal connected subgraph with edges labeled over $\{ t, t^{-1} \}$ . The $\langle S \rangle$ -lobe containing $v \in V ( \Gamma )$ is denoted by $\Delta ( v )$ . Any path $v_1 \rightarrow ^t v_2$ is called a $t$ -edge. If $v_1 \rightarrow ^t v_2$ is a $t$ -edge where $v_1$ and $v_2$ belong to distinct $\langle S \rangle$ -lobes $\Delta ( v_1 )$ and $\Delta ( v_2 )$ , respectively, then $\Delta ( v_1 )$ and $\Delta ( v_2 )$ are called adjacent and we say $\Delta ( v_2 )$ is connected to $\Delta ( v_1 )$ by a $t$ -edge.

An $\langle S \rangle$ -lobe path is a finite sequence of $\langle S \rangle$ -lobes $\Delta _1, \Delta _2, \ldots, \Delta _n$ , where $\Delta _k$ is adjacent to $\Delta _{k+1}$ , for $1\leq k \leq n-1$ . The $\langle S \rangle$ -lobe path is reduced if it is not of the form $\Delta _1, \Delta _2, \Delta _1$ and the $\langle S \rangle$ -lobes are distinct, except possibly the first and last. There is a unique reduced $\langle S \rangle$ -lobe path between any two $\langle S \rangle$ -lobes if and only if there are no non-trivial reduced $\langle S \rangle$ -lobe loops. The $\langle S \rangle$ -lobe graph of $\Gamma$ is the graph with vertices consisting of the $\langle S \rangle$ -lobes of $\Gamma$ and edges consisting of all pairs $( \Delta _1, \Delta _2 )$ of adjacent $\langle S \rangle$ -lobes, where there is a $t$ -edge $v_1 \rightarrow ^t v_2$ from a vertex $v_1$ of $\Delta _1$ to a vertex $v_2$ of $\Delta _2$ . The $\langle S \rangle$ -lobe graph of $\Gamma$ is a tree if and only if there are no non-trivial reduced $\langle S \rangle$ -lobe loops. An $\langle S \rangle$ -lobe of $\Gamma$ is a called extremal if it is adjacent to precisely one other $\langle S \rangle$ -lobe.

We say $\Gamma$ is $t$ -cactoid if it has finitely many $\langle S \rangle$ -lobes, every $t$ -edge $v_1 \rightarrow ^t v_2$ connects distinct $\langle S \rangle$ -lobes, for any such $t$ -edge there are loops $v_1 \rightarrow ^{ e_1 } v_1$ and $v_2 \rightarrow ^{ e_2 } v_2$ in $\Gamma$ , where $e_1$ and $e_2$ are the identities of $U_1$ and $U_2$ , respectively, adjacent $\langle S \rangle$ -lobes are connected by precisely one $t$ -edge and the $\langle S \rangle$ -lobe graph of $\Gamma$ is a finite tree. An inverse automaton over $X \cup \{ t \}$ is $t$ -cactoid if its underlying graph is.

Construction 3.1. [5, Construction 3.5] Let $\mathcal{A}$ be a $t$ -cactoid inverse automaton over $X \cup \{ t \}$ that is closed, relative to $\langle X \cup \{ t \} \mid R \rangle$ . Suppose $v_1 \rightarrow ^t v_2$ is a $t$ -edge of $\mathcal{A}$ and we have a loop $v_1 \rightarrow ^f v_1$ in $\Delta ( v_1 )$ , for some $f \in E( U_1 )$ , and no loop $v_2 \rightarrow ^{ ( f ) \phi } v_2$ in $\Delta ( v_2 )$ . Let $\mathcal{A}'$ be the closed form, relative to $\langle X \cup \{ t \} \mid R \rangle$ , of the automaton obtained from $\mathcal{A}$ by sewing on the linear automaton of any word that defines $( f ) \phi$ in $S$ at $v_2$ . The construction is illustrated in Fig. 3, where the circles represent $\langle S \rangle$ -lobes, the dots represent vertices of $\mathcal{A}$ , the arrows represent paths, and the dashed arrow represents the linear automaton of $( f ) \phi$ . We have an analogous construction when we have a loop $v_2 \rightarrow ^{ ( f ) \phi } v_2$ in $\Delta ( v_2 )$ , for some $f \in E( U_1 )$ , and no loop $v_1 \rightarrow ^f v_1$ in $\Delta ( v_1 )$ .

Figure 3. Construction 3.1 illustrated.

Construction 3.2. [5, Construction 3.12] Let $\mathcal{B} = ( \alpha _2, \Gamma _2, \beta _2 )$ be a $t$ -cactoid inverse automaton over $X \cup \{ t \}$ that is closed, relative to $\langle X \cup \{ t \} \mid R \rangle$ . Suppose there are $t$ -edges $v_1 \rightarrow ^t v_2$ , $v_3 \rightarrow ^t v_4$ and paths $v_1 \rightarrow ^u v_3$ , $v_2 \rightarrow ^{ ( u ) \phi } v_5$ in $\Gamma _2$ , for some $u \in U_1$ . The situation is illustrated in Fig. 4.

Figure 4. Construction 3.2 illustrated.

Since the $\langle S \rangle$ -lobe graph of $\Gamma _2$ is a tree, the unique reduced $\langle S \rangle$ -lobe path from an $\langle S \rangle$ -lobe of $\Gamma _2$ to $\Delta ( v_1 )$ either contains $\Delta ( v_2 )$ or does not. Let $\Sigma _1$ be the subgraph of $\Gamma _2$ containing $\Delta ( v_1 )$ and any $\langle S \rangle$ -lobe where the unique reduced $\langle S \rangle$ -lobe path to $\Delta ( v_1 )$ does not contain $\Delta ( v_2 )$ , including all $t$ -edges connecting these $\langle S \rangle$ -lobes. Similarly, let $\Sigma _2$ be the subgraph of $\Gamma _2$ containing $\Delta ( v_2 )$ and any $\langle S \rangle$ -lobe where the unique reduced $\langle S \rangle$ -lobe path to $\Delta ( v_2 )$ does not contain $\Delta ( v_1 )$ , including all $t$ -edges connecting these $\langle S \rangle$ -lobes. Thus $\Sigma _1 \cup \Sigma _2$ is equal to $\Gamma _2$ , minus the $t$ -edge $v_1 \rightarrow ^t v_2$ , and $\Sigma _1 \cap \Sigma _2 = \emptyset$ .

Let $\Sigma ^*_1$ and $\Sigma ^*_2$ denote disjoint copies of $\Sigma _1$ and $\Sigma _2$ , respectively. Let $\alpha ^*$ and $\beta ^*$ denote the unique respective images of $\alpha _2$ and $\beta _2$ in $\Sigma ^*_1 \cup \Sigma ^*_2$ . Then, let $\eta$ denote the $V$ -equivalence on $ \Sigma ^*_1 \cup \Sigma ^*_2$ generated by $\{ ( v_4, v_5 ) \}$ , letting $v_4$ and $v_5$ denote their unique images in $ \Sigma ^*_1 \cup \Sigma ^*_2$ . Put $\mathcal{C} = ( \alpha ^* \eta, ( \Sigma ^*_1 \cup \Sigma ^*_2 )/ \eta, \beta ^* \eta )$ . Let $\mathcal{B}'$ denote the closed form of $\mathcal{C}$ , relative to $\langle X \cup \{ t \} \mid R \rangle$ .

We have an analogous construction if there are $t$ -edges $v_2 \rightarrow ^t v_1$ , $v_4 \rightarrow ^t v_3$ and paths $v_1 \rightarrow ^{ ( u ) \phi } v_3$ , $v_2 \rightarrow ^u v_5$ in $\Gamma _2$ , for some $u \in U_1$ .

Let $\Gamma$ be an inverse word graph over $X \cup \{ t \}$ . The graph $\Gamma$ has the idempotent property if for every loop $v \rightarrow ^s v$ in $\Gamma$ , where $s \in S$ , there is a loop $v \rightarrow ^e v$ , for some $e \in E( S )$ with $s \geq e$ in $S$ . The graph $\Gamma$ has the equality property if, for every $t$ -edge $v_1 \rightarrow ^t v_2$ in $\Gamma$ , connecting two distinct $\langle S \rangle$ -lobes, there is a loop $v_1 \rightarrow ^u v_1$ in $\Delta ( v_1 )$ if and only if there is a loop $v_2 \rightarrow ^{ ( u )\phi } v_2$ in $\Delta ( v_2 )$ , for all $u \in U_1$ .

For an $t$ -edge $v_1 \rightarrow ^t v_2$ of $\Gamma$ , the set of related pairs of $v_1 \rightarrow ^t v_2$ consists of $( v_1, v_2 )$ and all pairs $( v_3, v_4 )$ of vertices for which we have a path $v_1 \rightarrow ^u v_3$ in $\Delta ( v_1 )$ and a path $v_2 \rightarrow ^{( u ) \phi } v_4$ in $\Delta ( v_2 )$ , for some $u \in U_1$ . If $( v_3, v_4 )$ is a related pair of $v_1 \rightarrow ^t v_2$ , then $v_3$ and $v_4$ are called its first and second coordinates, respectively. The graph $\Gamma$ has the separation property if the related pairs of any two $t$ -edges, connecting different pairs of $\langle S \rangle$ -lobes, share no common first coordinates and no common second coordinates.

We say that a $t$ -edge $v_1 \rightarrow ^t v_2$ of $\Gamma$ has identified related pairs if there is a $t$ -edge $v_3 \rightarrow ^t v_4$ for every related pair $( v_3, v_4 )$ of $v_1 \rightarrow ^t v_2$ . If, in addition, the pair $( v_3, v_4 )$ is a related pair of $v_1 \rightarrow ^t v_2$ , for every $t$ -edge $v_3 \rightarrow ^t v_4$ from $\Delta ( v_1 )$ to $\Delta ( v_2 )$ , then we say $\Delta ( v_1 )$ and $\Delta ( v_2 )$ are $t$ -saturated by $v_1 \rightarrow ^t v_2$ . The graph $\Gamma$ has the $t$ -saturation property if any two adjacent $\langle S \rangle$ -lobes are $t$ -saturated by some $t$ -edge.

If $\Gamma$ has the equality property, then the related pairs of any $t$ -edge $v_1 \rightarrow ^t v_2$ define a partial one-one map between $V( \Delta ( v_1 ) )$ and $V( \Delta ( v_2 ) )$ . If $\Gamma$ has the equality and separation properties and $v_1 \rightarrow ^t v_2$ is the only $t$ -edge from $\Delta ( v_1 )$ to $\Delta ( v_2 )$ , then we can $t$ -saturate $\Delta _1( v )$ and $\Delta _2( v )$ by sewing on a $t$ -edge from $v_3$ to $v_4$ , for every related pair $( v_3, v_4 )$ of $v_1 \rightarrow ^t v_2$ , other than $( v_1, v_2 )$ . If $\Gamma$ has the equality and separation properties and there is precisely one $t$ -edge connecting adjacent $\langle S \rangle$ -lobes, then the $t$ -saturated form of $\Gamma$ is obtained by $t$ -saturating every pair of adjacent $\langle S \rangle$ -lobes.

The graph $\Gamma$ is $t$ -opuntoid if every $t$ -edge connects two distinct $\langle S \rangle$ -lobes, the idempotent, equality and $t$ -saturation properties hold and there are no non-trivial reduced $\langle S \rangle$ -lobe loops. A $t$ -subopuntoid subgraph of a $t$ -opuntoid graph $\Gamma$ is a connected subgraph that is also $t$ -opuntoid and is formed by a collection of the $\langle S \rangle$ -lobes of $\Gamma$ . If $\Gamma$ is $t$ -opuntoid, then a $v \in V( \Gamma )$ is a bud if there is a loop $v \rightarrow ^f v$ in $\Delta ( v )$ , for some $f \in E( U_1 )$ , and no $t$ -edge $v \rightarrow ^t v'$ , or if there a loop $v \rightarrow ^{ ( f ) \phi } v$ in $\Delta ( v )$ , for some $f \in E( U_1 )$ , and no $t$ -edge $v' \rightarrow ^t v$ . Any of the above graph properties holds for an inverse automaton over $X \cup \{ t \}$ if it holds for its underlying graph.

Construction 3.3. [5, Construction 3.17] Let $\mathcal{D}$ be a $t$ -opuntoid automaton that is closed, relative to $\langle X \cup \{ t \} \mid R \rangle$ , and has a bud $v_1$ . If $v_1 \in V( \mathcal{D} )$ and there is a loop $v_1 \rightarrow ^f v_1$ in $\Delta ( v_1 )$ , for some $f \in E( U_1 )$ , and no $t$ -edge $v_1 \rightarrow ^t v_2$ , then we form the automaton $\mathcal{E}$ from $\mathcal{D}$ by sewing on a $t$ -edge $v_1 \rightarrow ^t v_2$ and then sewing the linear automaton of any word that defines $( f ) \phi$ in $S^*$ at $v_2$ , for every $f \in E( U_1 )$ that labels a loop at $v_1$ in $\Delta ( v_1 )$ . In Fig. 5, the dashed arrows represent the automata that are sewed and the dashed circle represents the new $\langle S \rangle$ -lobe created. Let $\mathcal{E}'$ denote the closed form of $\mathcal{E}$ , relative to $\langle S \cup \{ t \} \rangle$ . Let $v'_{\!\!1} \rightarrow ^t v'_{\!\!2}$ denote the image of $v_1 \rightarrow ^t v_2$ in $\mathcal{E}$ . Then, $\mathcal{E}'$ is obtained from $\mathcal{E}$ by closing $\Delta ( v'_{\!\!2} )$ , relative to $\langle S \rangle$ . Let $v''_{\!\!\!1} \rightarrow ^t v''_{\!\!\!2}$ denote the image of $v'_{\!\!1} \rightarrow ^t v'_{\!\!2}$ in $\mathcal{E}'$ . Then, let $\mathcal{D}'$ be the automaton obtained from $\mathcal{E}'$ by sewing on a $t$ -edge from $v_3$ to $v_4$ , for every related pair $( v_3, v_4 )$ of $v''_{\!\!\!1} \rightarrow ^t v''_{\!\!\!2}$ , other than $( v''_{\!\!\!1}, v''_{\!\!\!2} )$ .

Figure 5. Construction 3.3 illustrated.

We have an analogous construction if we have a vertex $v_2 \in V( \mathcal{D} )$ and there is a loop $v_2 \rightarrow ^{ ( f ) \phi } v_2$ in $\Delta ( v_2 )$ , for some $f \in E( U_1 )$ , and no $t$ -edge $v_1 \rightarrow ^t v_2$ .

A $t$ -opuntoid graph $\Gamma$ is complete if it has no buds. A complete $t$ -opuntoid graph is illustrated in Fig. 6, where the circles represent $\langle S \rangle$ -lobes and the arrows represent $t$ -edges.

Figure 6. A complete $t$ -opuntoid graph.

Lemma 3.4. [5, Lemmas 3.18, 3.19] Let $\mathcal{D}$ be a $t$ -opuntoid automaton, and let $\mathcal{D}'$ be obtained from $\mathcal{D}$ by Construction 3.3. Then $\mathcal{D}'$ is a $t$ -opuntoid automaton and $\mathcal{D}$ is a $t$ -subopuntoid subautomaton of $\mathcal{D}'$ . Further, if $\mathcal{D} \leadsto \mathcal{A}( S^*, w )$ then $\mathcal{D}' \leadsto \mathcal{A}( S^*, w )$ . We have a directed system of all automata obtained from $\mathcal{D}$ by a finite number of applications of Construction 3.3. The direct limit $\mathcal{E}$ is a complete $t$ -opuntoid automaton. Thus, if $\mathcal{D} \leadsto \mathcal{A}( S^*, w )$ and we have a loop $v_1 \rightarrow ^{ e_1 } v_1$ for every $t$ -edge $v_1 \rightarrow ^t v_2$ then $\mathcal{E} \cong \mathcal{A}( S^*, w )$ .

Algorithm 3.5. [5, Algorithm 3.20] For $w \in ( X \cup X^{-1} \cup \{ t, t^{-1} \} )^*$ , the Schützenberger automaton of $w$ , relative to $\langle S^* \rangle$ , is constructed as follows:

1. (i) Construct $\mathcal{A} = \mathcal{A}( S \cup \{ t \}, w )$ , using [15]. We can assume $\mathcal{A}$ is $t$ -cactoid and has the idempotent property.

2. (ii) Construct the direct limit $\mathcal{B}$ of the directed system of all automata obtained from $\mathcal{A}$ by a finite number of applications of Construction 3.1. Then, $\mathcal{B}$ is $t$ -cactoid, has the idempotent and equality propertie,s and has at most as many $\langle S \rangle$ -lobes as $\mathcal{A}$ .

3. (iii) If necessary, construct $\mathcal{B}'$ from $\mathcal{B}$ using Construction 3.2. Then , $\mathcal{B}'$ is $t$ -cactoid, has the idempotent and equality properties, and has fewer $\langle S \rangle$ -lobes than $\mathcal{B}$ .

4. (iv) Steps (ii) and (iii) can be applied at most a finite number of times. The resulting automaton $\mathcal{C}$ is $t$ -cactoid and has the idempotent, equality, and separation properties.

5. (v) The $t$ -saturated form $\mathcal{D}$ of $\mathcal{C}$ is $t$ -opuntoid and has finite $\langle S \rangle$ -lobes.

6. (vi) Construct the direct limit $\mathcal{E}$ of the directed system of all automata obtained from $\mathcal{D}$ by a finite number of applications of Construction 3.3. Then, $\mathcal{E}$ is a complete $t$ -opuntoid automaton and $\mathcal{E} \cong \mathcal{A} ( S^*, w )$ .

Let $\Gamma$ be a $t$ -opuntoid graph. Let $\Delta _1$ and $\Delta _2$ be adjacent $\langle S \rangle$ -lobes of $\Gamma$ . Then, $\Delta _2$ feeds off $\Delta _1$ if there is a $t$ -edge $v_1 \rightarrow ^t v_2$ of $\Gamma$ from $\Delta _1$ to $\Delta _2$ such that, for any loop $v_2 \rightarrow ^y v_2$ in $\Delta _2$ , there is a loop $v_2 \rightarrow ^g v_2$ in $\Delta _2$ , for some $g \in E( U_2 )$ with $y \geq g$ in $S$ . We also say that $\Delta _2$ feeds off $\Delta _1$ if there is a $t$ -edge $v_2 \rightarrow ^t v_1$ of $\Gamma$ from $\Delta _2$ to $\Delta _1$ such that, for any loop $v_2 \rightarrow ^y v_2$ in $\Delta _2$ , there is a loop $v_2 \rightarrow ^f v_2$ in $\Delta _2$ , for some $f \in E( U_1 )$ with $y \geq f$ in $S$ . For non-adjacent $\langle S \rangle$ -lobes $\Delta _1$ and $\Delta _n$ of $\Gamma$ , we say $\Delta _n$ feeds off $\Delta _1$ if there is a sequence of $\langle S \rangle$ -lobes $\Delta _1, \Delta _2, \ldots, \Delta _n$ , where $\Delta _{k+1}$ is adjacent to $\Delta _k$ and $\Delta _{k+1}$ feeds off $\Delta _k$ , for $1 \leq k \leq n-1$ ,

Let $\Gamma '$ be a $t$ -subopuntoid subgraph of $\Gamma$ . An $\langle S \rangle$ -lobe of $\Gamma$ that does not belong to $\Gamma '$ is called external to $\Gamma '$ . An extremal $\langle S \rangle$ -lobe of $\Gamma '$ is called a parasite if it feeds off the unique $\langle S \rangle$ -lobe of $\Gamma '$ to which it is adjacent. The subgraph $\Gamma '$ is parasite-free if it has no parasites. The subgraph $\Gamma '$ is a host of $\Gamma$ if it has finitely many $\langle S \rangle$ -lobes, is parasite-free, and every $\langle S \rangle$ -lobe of $\Gamma$ that is external to $\Gamma '$ feeds off some $\langle S \rangle$ -lobe of $\Gamma '$ .

Theorem 3.6. [5, Theorem 3.26] Let $S^* = [ S;\; U_1, U_2;\; \phi ]$ be an HNN extension of an inverse semigroup $S$ , where $U_1$ and $U_2$ are inverse monoids that are lower bounded in $S$ . Then, the Schützenberger automata of $S^*$ are complete $t$ -opuntoid automata with a host.

Lemma 3.7. [5, Lemma 3.23] Let $\Gamma$ be a $t$ -opuntoid graph. Then, a host of $\Gamma$ is a maximal parasite-free $t$ -subopuntoid subgraph. If $\Gamma$ has more than one host, then every host is an $\langle S \rangle$ -lobe of $\Gamma$ . The unique reduced $\langle S \rangle$ -lobe path between any two hosts, in the $\langle S \rangle$ -lobe tree of $\Gamma$ , consists entirely of $\langle S \rangle$ -lobes that are hosts.

Thus, we can associate a number with a $t$ -opuntoid graph $\Gamma$ that has a host, by defining $n( \Gamma )$ to be the number of $\langle S \rangle$ -lobes in any host. Either $\Gamma$ has one host, in which case $n( \Gamma ) \geq 1$ , or every host of $\Gamma$ is an $\langle S \rangle$ -lobe, in which case $n( \Gamma ) = 1$ .

Lemma 3.8. [5, Lemma 3.24] Let $\mathcal{D}$ be a $t$ -opuntoid automaton with finitely many $\langle S \rangle$ -lobes and a host $\Sigma$ . If $\mathcal{D}'$ is obtained from $\mathcal{D}$ by Construction 3.3, then $\Sigma$ is also a host of $\mathcal{D}'$ .

Lemma 3.9. [5, Corollary 3.29] Let $\Gamma$ and $\Gamma '$ be complete $t$ -opuntoid graphs that have hosts and let $\Sigma$ be any host of $\Gamma$ . Then, every isomorphism from $\Sigma$ onto some host of $\Gamma '$ extends (uniquely) to an isomorphism of $\Gamma$ onto $\Gamma '$ .

Lemma 3.10. [5, Lemma 3.31] If $\Gamma$ is a $t$ -opuntoid graph with finitely many $\langle S \rangle$ -lobes, then the automorphism group of $\Gamma$ is embedded into the automorphism group of some $\langle S \rangle$ -lobe of $\Gamma$ .

Lemma 3.11. [5, Lemma 3.32] Let $\Gamma$ be a complete $t$ -opuntoid graph that has a host. Let $\Gamma '$ be the subgraph that consists of the $\langle S \rangle$ -lobes of every host of $\Gamma$ and the $t$ -edges connecting them. Then, $\Gamma '$ is a $t$ -subopuntoid subgraph of $\Gamma$ and the automorphism group of $\Gamma$ is isomorphic to the automorphism group of $ \Gamma '$ .

4. Lower bounded HNN extensions

In this section, let $U_1$ and $U_2$ denote inverse monoids of an inverse semigroup $S$ , with respective identities $e_1$ and $e_2$ , let $\phi \;:\; U_1 \rightarrow U_2$ be an isomorphism and let $S^* = [ S;\; U_1, U_2;\; \phi ]$ be a lower bounded HNN extension, as defined in [12]. That is, for each $e \in E( S )$ and $i \in \{ 1, 2 \}$ , the set $\{ u \in U_i \;:\; u \geq e \}$ is either empty or has a minimal element, denoted by $f_i ( e )$ , and there does not exist an infinite sequence $\{ u_k \}$ , where $u_k \in E( U_i )$ and $u_k > f_i ( e u_k ) > u_{ k+1 }$ , for all $k$ . The monoids $U_1$ and $U_2$ are also lower bounded in $S$ , as defined in Section 3, thus we can use the results of Section 3.

Theorem 4.1. Let $S^* = [ S;\; U_1, U_2;\; \phi ]$ be a lower bounded HNN extension. Then, the Schützenberger automata of $S^*$ are, up to isomorphism, the complete $t$ -opuntoid automata that have a host, a loop $v_1 \rightarrow ^{ e_1 } v_1$ for every $t$ -edge $v_1 \rightarrow ^t v_2$ and $\langle S \rangle$ -lobes isomorphic to Schützenberger graphs of $\langle S \rangle$ .

Proof. The result is a restatement of Jajcayová [14, Theorem 4.1] using the definitions of Section 3.

The Bass-Serre theory can be used to describe the maximal subgroups of $S^*$ , as follows; see Cohen [7] and Dicks and Dunwoody [9, Chapters 1, 2, 3] for notation and definitions.

Notation 4.2. Let $\Gamma$ be a complete $t$ -opuntoid graph that has a host, and let $\Gamma '$ denote the $t$ -subopuntoid subgraph of $\Gamma$ that consists of the $\langle S \rangle$ -lobes of every host and all $t$ -edges connecting these hosts. Let $T( \Gamma ' )$ denote the $\langle S \rangle$ -lobe tree of $\Gamma '$ , and let $G$ denote the automorphism group of $\Gamma '$ . For each $\alpha \in G$ and $\langle S \rangle$ -lobe $\Delta$ of $\Gamma '$ , we define the action of $\alpha$ on $\Delta$ , written $\alpha \cdot \Delta$ , to be $( \Delta ) \alpha$ , the image of $\Delta$ under $\alpha$ , which is also an $\langle S \rangle$ -lobe of $\Gamma '$ . We extend this action to an action of $T( \Gamma ' )$ by defining the action of $\alpha$ on the edge $( \Delta _1, \Delta _2 )$ to be equal to $( ( \Delta _1 ) \alpha, ( \Delta _2 ) \alpha )$ .

The quotient graph $G \backslash T( \Gamma ' )$ is the graph of orbits of the action of $G$ on $T( \Gamma ' )$ and is connected. There exist subsets $T_0 \subseteq T \subseteq T( \Gamma ' )$ such that $T_0$ is a subtree of $T( \Gamma ' )$ , $T_0$ and $T$ have the same vertices, the initial vertex of every edge of $T$ is also a vertex of $T$ and $T$ , is a $G$ -transversal in $T( \Gamma ' )$ ; that is, $T$ meets each $G$ -orbit exactly once, and thus, the map $T \rightarrow G \backslash T( \Gamma ' ) \;:\; y \rightarrow G \cdot y$ , defined for all edges and vertices $y$ , is bijective.

We can make $T$ into a graph by specifying the initial vertex of each edge $( \Delta _1,\Delta _2 )$ to be $\Delta _1$ and specifying the terminal vertex to be the unique vertex $\Delta '_{\!\!2}$ of $T$ which lies in the same $G$ -orbit as $\Delta _2$ . It then follows that the graph $T$ is isomorphic to $G \backslash T( \Gamma ' )$ under the above map $y \rightarrow G \cdot y$ , and $T_0$ is a maximal subtree of $T$ , as well as a subtree of $T( \Gamma ' )$ .

For any edge $y = ( \Delta _1, \Delta _2 )$ of $T$ , the $\langle S \rangle$ -lobes $\Delta _2$ and $\Delta '_{\!\!2}$ lie in the same $G$ -orbit, and thus, we can choose an element $\alpha _y \in G$ such that $\alpha _y \cdot \Delta '_{\!\!2} = \Delta _2$ . If $y \in E( T_0 )$ , then $\Delta _2 \in V( T_0 ) = V( T )$ and so $\Delta '_{\!\!2} = \Delta _2$ , in which case we take $\alpha _y$ as the identity of $G$ . Next, let $G( y )$ denote the stabilizer group of $y$ under the action of $G$ ; that is, the group $G( y )$ is the subgroup of $G$ consisting of all automorphisms of $\Gamma '$ which map $\Delta _1$ onto itself and $\Delta _2$ onto itself. Similarly, for each vertex, we let $G( \Delta )$ denote the stabilizer group of $\Delta$ under the action of $G$ . For any edge $y = ( \Delta _1, \Delta _2 )$ of $T$ , we have $G( y ) \subseteq G( \Delta _1 )$ and the map $t_y \;:\; G( y ) \rightarrow G( \Delta '_{\!\!2} ) \;:\; \alpha \rightarrow \alpha _y \circ \alpha \circ \alpha ^{-1}_y$ defines a group monomorphism.

We have a graph of groups $( G(\!-\!), T )$ . Since $T( \Gamma ' )$ is a tree, the fundamental group $\Pi ( G(\!-\!), T, T_0 )$ of the graph of groups $( G(\!-\!), T )$ is then isomorphic to $G$ . By Lemma 3.11, the automorphism group of $\Gamma$ is isomorphic to $G$ . Hence, the automorphism group of $\Gamma$ is isomorphic to $\Pi ( G(\!-\!), T, T_0 )$ . The group $\Pi ( G(\!-\!), T, T_0 )$ is generated by the disjoint union of $E( T )$ and the vertex groups of $( G(\!-\!), T )$ , subject to the relation $y^{-1} \cdot \alpha \cdot y = ( \alpha ) t_y$ , for all $y \in E( T )$ and all $\alpha \in G( y )$ , and the relation $y = 1$ , for all $y \in E( T_0 )$ .

Notation 4.3. We define a graph of groups $( H(\!-\!), Y )$ for the HNN extension $S^* = [ S;\; U_1, U_2;\; \phi ]$ , as follows. The graph $Y$ has vertices $V( Y )$ the $\mathcal{D}$ -classes of $S$ . The graph $Y$ has edges $E( Y )$ the set of all triples $( D_1, D, D_2 )$ , where $D$ is a $\mathcal{D}$ -class of $U_1$ , $D_1$ is the $\mathcal{D}$ -class of $S$ containing $D$ , and $D_2$ is the $\mathcal{D}$ -class of $S$ containing $( D ) \phi$ .

We specify an $\mathcal{H}$ -class group within each $\mathcal{D}$ -class of $S$ and specify an $\mathcal{H}$ -class group within each $\mathcal{D}$ -class of $U_1$ . Let $y = ( D_1, D, D_2 )$ be an edge of $Y$ and let $H_g$ , $H_f$ and $H_h$ be the specified $\mathcal{H}$ -class groups of $D_1$ , $D$ , and $D_2$ , containing idempotents $g$ , $f$ , and $h$ , respectively. Fix $d_1 \in D_1$ such that $f \mathcal{R} d_1 \mathcal{L} g$ in $S$ and fix $d_2 \in D_2$ such that $( f ) \phi \mathcal{R} d_2 \mathcal{L} h$ in $S$ . The maps $H_f \rightarrow H_g \;:\; s \rightarrow d^{-1}_1 s d_1$ and $H_{ ( f ) \phi } \rightarrow H_g \;:\; s \rightarrow d^{-1}_2 s d_2$ are group monomorphisms. Then, $H( y ) = d_1^{-1} H_f d_1$ is a subgroup of $H_g$ and the map $t_y \;:\; H( y ) \rightarrow H_h \;:\; d^{-1}_1 s d_1 \rightarrow d^{-1}_2 \cdot ( s ) \phi \cdot d_2$ is a group monomorphism.

The construction of the graph of groups $( H(\!-\!), Y )$ is completed by defining the vertex group $H( D )$ , of each vertex $D$ , to be the specified $\mathcal{H}$ -class group of $D$ and defining the edge group and monomorphism of each edge $y$ to be the group $H( y )$ and the monomorphism $t_y$ , respectively, as indicated above.

For $e \in E( S )$ , let $Y_e$ denote the connected component of $Y$ containing, as a vertex, the $\mathcal{D}$ -class of $e$ in $T$ . If $e \in E( U_1 )$ , then the $\mathcal{D}$ -class of $e$ in $S$ and the $\mathcal{D}$ -class of $( e ) \phi$ in $S$ are connected by an edge in $Y$ and are thus in the same connected component. Let $( H_e (\!-\!), Y_e )$ denote the restriction of $( H (\!-\!), Y )$ to $Y_e$ .

The following result generalizes Yamamura [21, Theorem 5.2], on locally full HNN extensions, and overlaps with Ayyash [1, Theorem $5.4.1$ ], on HNN extensions of finite inverse semigroups.

Theorem 4.4. Let $S^* = [ S;\; U_1, U_2;\; \phi ]$ be a lower bounded HNN extension and let $e$ be an idempotent of $S$ . Then, the maximal subgroup of $S^*$ containing $e$ is isomorphic to the fundamental group of the graph of groups $( H_e(\!-\!), Y_e )$ .

Proof. Let $\Gamma = S \Gamma ( S^*, e )$ . By Theorem 4.1, the graph $\Gamma$ is a complete $t$ -opuntoid graph with a host such that the $\langle S \rangle$ -lobes are isomorphic to Schützenberger graphs of $\langle S \rangle$ . From Notation 4.2, the fundamental group $\Pi ( G(\!-\!), T, T_0 )$ of the graph of groups $( G(\!-\!), T )$ is isomorphic to the automorphism group of $\Gamma$ and thus is also isomorphic to the maximal subgroup of $S^*$ containing $e$ . Hence, the theorem is completed by showing that the graphs of groups $( G(\!-\!), T )$ and $( H_e(\!-\!), Y_e )$ are conjugate isomorphic. We need to define a graph isomorphism between $T$ and $Y_e$ and define isomorphisms between the corresponding vertex and edge groups, such that the group isomorphisms commute with the corresponding edge monomorphisms.

From Algorithm 3.5, the Schützenberger graph $S \Gamma ( S, e )$ is embedded onto a host of $\Gamma$ . Thus $n ( \Gamma ) = 1$ , as defined in Section 3, and every host is an $\langle S \rangle$ -lobe, by Lemma 3.7. We define a graph map $\psi \;:\; T \rightarrow Y$ as follows.

Let $\Delta \in V( T )$ . If $v_1, v_2 \in V( \Delta )$ , then $( v_i, \Delta, v_i ) \cong \mathcal{A}( S, e( v_i ) )$ , for $i = 1, 2$ , and $e( v_1 ) \mathcal{D} e( v_2 )$ in $S$ . Thus, we can define $( \Delta ) \psi$ to be equal to the $\mathcal{D}$ -class of $S$ containing $e( v )$ , for any vertex $v$ of $\Delta$ .

Let $( \Delta _1, \Delta _2 )$ be an edge of $T$ . Then, $\Delta _1$ and $\Delta _2$ are hosts of $\Gamma$ and thus feed off each other, and there is a $t$ -edge from a vertex of $\Delta _1$ to a vertex of $\Delta _2$ . If $v_1 \rightarrow ^t v_2$ and $v'_{\!\!1} \rightarrow ^t v'_{\!\!2}$ are $t$ -edges, where $v_1$ , $v'_{\!\!1}$ are vertices of $\Delta _1$ and $v_2$ , $v'_{\!\!2}$ are vertices of $\Delta _2$ , then we have $e( v_1 ), e( v'_{\!\!1} ) \in E( U_1 )$ , $e( v_2 ) = ( e( v_1 ) ) \phi$ , $e( v'_{\!\!2} ) = ( e( v'_{\!\!1} ) ) \phi$ , with $e( v_1 ) \mathcal{D} e( v'_{\!\!1} )$ in $U_1$ and $e( v_2 ) \mathcal{D} e( v'_{\!\!2} )$ in $U_2$ . Thus, we can define $( \Delta _1, \Delta _2 ) \psi$ to be the edge $( D_1, D, D_2 )$ , where $D$ is the $\mathcal{D}$ -class of $U_1$ containing $e( v_1 )$ , $D_1$ is the $\mathcal{D}$ -class of $S$ containing $e( v_1 )$ , and $D_2$ is the $\mathcal{D}$ -class of $S$ containing $( e( v_1 ) ) \phi$ , for any $t$ -edge $v_1 \rightarrow ^t v_2$ , from a vertex $v_1$ of $\Delta _1$ to a vertex $v_2$ to $\Delta _2$ .

Let $\Gamma '$ be the $t$ -subopuntoid subgraph of $\Gamma$ consisting of every host and let $G = AUT( \Gamma ' )$ . As described in Notation 4.2, the initial vertex of the edge $( \Delta _1, \Delta _2 )$ is $\Delta _1$ and the terminal vertex of the edge $( \Delta _1, \Delta _2 )$ is the unique vertex $\Delta '_{\!\!2} \in V( T )$ that lies in the same $G$ -orbit as $\Delta _2$ . Let $v_1 \rightarrow ^t v_2$ be a $t$ -edge from a vertex $v_1$ of $\Delta _1$ to a vertex $v_2$ of $\Delta _2$ . Let $e( v_1 ) \in D_1$ and $e( v_2 ) \in D_2$ . Then, $( \Delta _1 ) \psi = D_1$ and $( \Delta '_{\!\!2} ) \psi = D_2$ , since $\Delta '_{\!\!2} \cong \Delta _2$ . Thus, the map $\psi \;:\; T \rightarrow Y$ defines a graph homomorphism.

We now show that $\psi$ defines a monomorphism. Suppose $\Delta$ and $\Delta '$ are vertices of $T$ with $( \Delta ) \psi = ( \Delta ' ) \psi$ . Let $v$ denote a vertex of $\Delta$ and let $v'$ denote a vertex of $\Delta '$ . Then $( \Delta ) \psi = ( \Delta ' ) \psi$ implies that $e( v ) \mathcal{D} e( v' )$ in $S$ , in which case we have $\Delta \cong \Delta '$ . Since $\Delta$ and $\Delta '$ are hosts, the isomorphism between them extends to an automorphism of $\Gamma$ , by Lemma 3.9. Thus $\Delta$ and $\Delta '$ are in the same $G$ -orbit and so $\Delta = \Delta '$ , since $T$ is a transversal. We have shown that $\psi$ is one-one on vertices. Since $T$ is a tree, the map $\psi$ must also be one-one on the edges. Since $\Delta$ has a host that is isomorphic to $S \Gamma ( S, e )$ , we have that $( T ) \psi$ is a connected subgraph of $Y_e$ .

We now show that $( T ) \psi = Y_e$ . Let $( D_1, D, D_2 )$ be an edge of $Y_e$ , where $D_1 = ( \Delta _1 ) \psi$ , for some vertex $\Delta _1$ of $T$ . Let $f$ be an idempotent of $U_1$ that is in $D$ . There exists a vertex $v_1$ of $\Delta _1$ such that $e( v_1 ) = f$ . Then, there must be a $t$ -edge $v_1 \rightarrow ^t v_2$ , where $v_2$ is a vertex of an $\langle S \rangle$ -lobe $\Delta _2$ . Since $\Delta _1$ is a host of $\Gamma$ and $e( v_2 ) = ( f ) \phi$ , it follows that $\Delta _2$ is also a host of $\Gamma$ . Since $T$ meets each $G$ -orbit of $T( \Gamma ' )$ exactly once, there exists an edge $( \Delta '_{\!\!1}, \Delta '_{\!\!2} )$ of $T$ that lies in the same $G$ -orbit as $( \Delta _1, \Delta _2 )$ . Since $\Delta _1 \in V( T )$ , we must have $\Delta _1 = \Delta '_{\!\!1}$ . Then, there exists a $t$ -edge $v'_{\!\!1} \rightarrow ^t v'_{\!\!2}$ from a vertex $v'_{\!\!1}$ of $\Delta _1$ to a vertex $v'_{\!\!2}$ of $\Delta '_{\!\!2}$ , such that $e( v'_{\!\!1} ) = f$ and $e( v'_{\!\!2} ) = ( f ) \phi$ . We then have $( \Delta _1. \Delta '_{\!\!2} ) \psi = ( D_1, D, D_2 )$ . A similar proof shows that if $( D_1, D, D_2 )$ is an edge of $Y_e$ , where $D_2 = ( \Delta _2 ) \psi$ , for some vertex $\Delta _2$ of $T$ , then $( \Delta '_{\!\!1}. \Delta _2 ) \psi = ( D_1, D, D_2 )$ , for some edge $( \Delta '_{\!\!1}. \Delta _2 )$ of $T$ . It now follows that $( T ) \psi$ is a maximal connected subgraph of $Y_e$ . We have shown that $\psi \;:\; T \rightarrow Y$ defines a graph monomorphism onto $Y_e$ .

We now define the vertex group isomorphisms. Let $\Delta$ denote a vertex of $T$ . Let $H( ( \Delta ) \psi ) = H_g$ , the $\mathcal{H}$ -class group of $S$ with identity $g$ . If $v$ is a vertex of $\Delta$ then $e( v ) \mathcal{D} g$ in $S$ . Thus, we have an isomorphism $\pi \;:\; \Delta \rightarrow S \Gamma ( S, g )$ . The group $G( \Delta )$ is the stabilizer group of $\Delta$ , under the action of $G$ . Since $\Delta$ is a host of $\Gamma$ , any automorphism of $\Delta$ extends (uniquely) to an automorphism of $\Gamma$ , by Lemma 3.9. Thus, we have an isomorphism $G( \Delta ) \rightarrow AUT ( \Delta )$ , under the mapping $\alpha \rightarrow \alpha _\Delta$ , where $\alpha _\Delta$ denotes the restriction of $\alpha$ to $\Delta$ . We then have an isomorphism $AUT( \Delta ) \rightarrow AUT( S \Gamma ( S, g ) )$ , defined by $\alpha \rightarrow \pi ^{-1} \circ \alpha _\Delta \circ \pi$ . We have an isomorphism $AUT( S \Gamma ( S, g ) ) \rightarrow H_g$ , under the mapping $\beta \rightarrow ( g ) \beta$ ; the set of vertices of $S \Gamma ( S, g )$ is the $\mathcal{R}$ -class of $S$ containing $g$ . Hence we have an isomorphism $\psi \;:\; G( \Delta ) \rightarrow H( ( \Delta ) \psi )$ , defined by $\alpha \rightarrow ( g ) \pi ^{-1} \circ \alpha _\Delta \circ \pi$ . The map $\psi$ may be expressed by saying that $( \alpha ) \psi = s$ , where $s \in S$ such that $\mathcal{A}( S, s ) \cong ( v, \Delta, ( v ) \alpha )$ , for any vertex $v$ of $\Delta$ with $e( v ) = g$ .

We now define the edge group isomorphisms. Let $y = ( \Delta _1, \Delta _2 )$ be an edge in $T$ and let $( y ) \psi = ( D_1, D, D_2 )$ . Let $H_g$ and $H_f$ denote the specified $\mathcal{H}$ -class groups of $D_1$ and $D$ , containing the identities $g$ and $f$ , respectively. Thus, $H( D_1 ) = H_g$ and $H( ( y ) \psi ) = d^{-1}_1 H_f d_1$ , where $d_1$ is the fixed element of $D_1$ such that $f \mathcal{R} d_1 \mathcal{L} g$ in $S$ . Let $v_1 \rightarrow ^t v_2$ be a $t$ -edge from a vertex $v_1$ of $\Delta _1$ to a vertex $v_2$ of $\Delta _2$ . Then, $e( v_1 ) \mathcal{R} a \mathcal{L} f$ , for some $a \in U_1$ . Thus, we have a path $v_1 \rightarrow ^a v_3$ , where $e( v_3) = f$ , and a path $v_3 \rightarrow ^{ d_1 } v_4$ , where $e( v_4 ) = g$ .

Let $\alpha \in G( y )$ . Then, $\alpha$ stabilizes $\Delta _1$ and $\Delta _2$ and so $( v_1 ) \alpha \rightarrow ^t ( v_2 ) \alpha$ is a $t$ -edge from $\Delta _1$ to $\Delta _2$ . Since $\Delta _1$ and $\Delta _2$ are $t$ -saturated, there is a path $v_1 \rightarrow ^b ( v_1 ) \alpha$ , for some $b \in U_1$ . Since, we have a path $v_1 \rightarrow ^{ a d_1 } v_4$ in $\Delta _1$ , we have a path $( v_1 ) \alpha \rightarrow ^{ a d_1 } ( v_4 ) \alpha$ in $\Delta$ . Thus, $( v_4, \Delta _1, ( v_4 ) \alpha ) \cong \mathcal{A}( S, s )$ , where $s = d^{-1}_1 ( f a^{-1}b a ) d_1$ and $f a^{-1}b a \in H_f$ . Hence, $\psi \;:\; G( \Delta _1 ) \rightarrow H( ( \Delta _1 ) \psi )$ maps $G( y )$ into $H( ( y ) \psi )$ .

Conversely, let $c \in H_f$ . Since $\psi \;:\; G( \Delta _1 ) \rightarrow H( ( \Delta _1 ) \psi )$ is an isomorphism, there exists $\alpha \in G( \Delta _1 )$ such that $( v_4, \Delta, ( v_4 ) \alpha ) \cong \mathcal{A}( S, d^{-1}_1 c d_1 )$ . Then we have $( v_3, \Delta _1, ( v_3 ) \alpha ) \cong \mathcal{A}( S, c )$ and $( v_1, \Delta _1, ( v_1 ) \alpha ) \cong \mathcal{A}( S, a f c a^{-1} )$ . Thus the $t$ -edge $( v_1 ) \alpha \rightarrow ^t ( v_2 ) \alpha$ must also be a $t$ -edge from $\Delta _1$ to $\Delta _2$ . This implies $\alpha \in G( y )$ . Thus the isomorphism $\psi \;:\; G( \Delta _1 ) \rightarrow H( ( \Delta _1 ) \psi )$ maps $G( y )$ onto $H( ( y ) \psi )$ .

Finally, we show that the isomorphisms between the vertex and edge groups of $( G(\!-\!), T )$ and $( H(\!-\!), Y_e )$ commute with the edge monomorphisms. Let $y = ( \Delta _1, \Delta _2 )$ be an edge of $T$ , and let $( y ) \psi$ be equal to $( D_1, D, D_2 )$ . Let $H_g$ , $H_f$ and $H_h$ denote the specified $\mathcal{H}$ -class groups of $D_1$ , $D$ and $D_2$ , containing idempotents $g$ , $f$ and $h$ , respectively. Let $d_1$ and $d_2$ be the fixed elements of $D_1$ and $D_2$ , respectively, such that $f \mathcal{R} d_1 \mathcal{L} g$ and $( f ) \phi \mathcal{R} d_2 \mathcal{L} h$ in $S$ . The map $t_{ ( y ) \psi } \;:\; H( ( y ) \psi ) \rightarrow H_h$ defined by $d^{-1}_1 s d_1 \rightarrow d^{-1}_2 \cdot ( s ) \phi \cdot d_2$ , for $s \in H( ( y ) \psi )$ , is the edge monomorphism for $( y ) \psi$ . Let $\Delta '_{\!\!2}$ be the unique vertex of $T$ that belongs in the same $G$ -orbit as $\Delta _2$ , and let $\alpha _y \in G$ such that $\alpha _y \cdot \Delta '_{\!\!2} = \Delta _2$ . The edge monomorphism $t_y \;:\; G( y ) \rightarrow G( \Delta '_{\!\!2} )$ for $y$ is given by $\alpha \rightarrow \alpha _y \circ \alpha \circ \alpha ^{-1}_y$ .

The composition of the edge map $t_y$ with $\psi \;:\; G( \Delta '_{\!\!2} ) \rightarrow H( ( \Delta '_{\!\!2} )\psi )$ is the map $t_y \circ \psi \;:\; G( y ) \rightarrow H( ( \Delta '_{\!\!2} ) \psi ) \;:\; \alpha \rightarrow s$ , with $\mathcal{A}( S, s ) \cong ( v, \Delta '_{\!\!2}, ( v ) \alpha _y \circ \alpha \circ \alpha ^{-1}_y )$ , for any vertex $v$ of $\Delta '_{\!\!2}$ such that $e( v ) = h$ . Since $\alpha _y$ maps $\Delta '_{\!\!2}$ isomorphically onto $\Delta _2$ , we can redefine this map by saying $( \alpha ) t_y \circ \psi = s$ , where $s \in S$ such that $\mathcal{A}( S, s ) \cong ( v, \Delta _2, ( v ) \alpha )$ , for some vertex $v$ of $\Delta _2$ with $e( v ) = h$ .

The composition of $\psi \;:\; G( y ) \rightarrow H( ( y ) \psi )$ with the edge map $t_{ ( y ) \psi }$ is given by $\psi \circ t_{ ( y ) \psi } \;:\; G( y ) \rightarrow H( ( \Delta '_{\!\!2} ) \psi ) :$ $\alpha \rightarrow d^{-1}_2 \cdot ( r ) \phi \cdot d_2$ , where $r \in H_f$ such that $\mathcal{A}( S, r ) \cong ( v_1, \Delta _1, ( v_1) \alpha )$ , for some $t$ -edge $v_1 \rightarrow ^t v_2$ from a vertex $v_1$ of $\Delta _1$ to a vertex $v_2$ of $\Delta _2$ , with $e( v_1 ) = f$ .

Since $e( v_2 ) = ( f ) \phi$ and $( f ) \phi \mathcal{R} d_2$ in $S$ , there exists a vertex $v'_{\!\!2}$ of $\Delta _2$ such that $( v_2, \Delta _2, v'_{\!\!2} ) \cong \mathcal{A}( S, d_2 )$ . Since we have a path $v_1 \rightarrow ^r ( v_1 ) \alpha$ in $\Delta _1$ , we have a path $v_2 \rightarrow ^{ ( r ) \phi } ( v_2 ) \alpha$ in $\Delta _2$ . Then, $( v_2, \Delta _2, ( v_2 ) \alpha ) \cong$ $\mathcal{A}( S, ( r ) \phi )$ . Now $( ( v_2 ) \alpha, \Delta _2, ( v'_{\!\!2} ) \alpha ) \cong$ $\mathcal{A}( S, d_2 )$ and so $( v'_{\!\!2}, \Delta _2, ( v'_{\!\!2} ) \alpha ) \cong$ $\mathcal{A}( S, d^{-1}_2 \cdot ( r ) \phi \cdot d_2 )$ . We have $e( v'_{\!\!2} ) = h$ and so $( \alpha ) t_y \circ \psi = d^{-1}_2 \cdot ( r ) \phi \cdot d_2 =$ $\psi \circ t_{ ( y ) \psi }$ , as required, and the proof of the theorem is complete.

Notation 4.5. We define an equivalence $\sim _i$ on $S$ by $s_1 \sim _i s_2$ if and only if $s_1 = s_2$ or $s_1 \mathcal{R} s_2$ with $s_1 = s_2 u$ , for some $u \in U_i$ , for $s_1, s_2 \in S$ , for $i = 1, 2$ .

Theorem 4.6. Let $S^* = [ S;\; U_1, U_2;\; \phi ]$ be a lower bounded HNN extension, and let $e$ be an idempotent of $S^*$ that is not $\mathcal{D}$ -related to any element of $S$ . Then, the maximal subgroup of $S^*$ containing $e$ is isomorphic to a subgroup $H$ of $S$ , whose quotient $H/ \sim _i$ is finite, for some $i \in \{ 1, 2 \}$ .

Proof. Let $\Gamma = S \Gamma ( S^*, e )$ . Then, the maximal subgroup of $S^*$ containing $e$ is isomorphic to the automorphism group of $\Gamma$ . If $n ( \Gamma ) = 1$ , as defined in Section 3, then there exists an $\langle S \rangle$ -lobe $\Delta$ that is a host of $\Gamma$ . Since $\Delta$ is isomorphic to a Schützenberger graph of $S$ , we then have $e \mathcal{D} g$ in $S^*$ , for some $g \in E( S )$ , a contradiction. Thus, $n( \Gamma ) > 1$ and $\Gamma$ has precisely one host $\Sigma$ , consisting of at least two $\langle S \rangle$ -lobes.

The automorphism group of $\Gamma$ is isomorphic to the automorphism group of $\Sigma$ , by Lemma 3.11. From Lemma 3.10, the automorphism group of $\Sigma$ is embedded into the automorphism group of some $\langle S \rangle$ -lobe $\Delta$ of $\Sigma$ , under the embedding $\alpha \rightarrow \alpha _\Delta$ , where $\alpha _\Delta$ denotes the restriction of $\alpha$ to $\Delta$ . Let $v$ be a vertex of $\Delta$ . We have $\Delta \cong S \Gamma ( S, g )$ , where $g = e( v )$ . Then, the map $\psi \;:\; AUT( \Gamma ) \rightarrow H_g$ defined by $\alpha \rightarrow s$ , where $( v, \Delta, ( v ) \alpha ) \cong \mathcal{A}( S, s )$ and $H_g$ is the $\mathcal{H}$ -class of $S$ containing $g$ , defines a group monomorphism.

Let $H$ denote the image of $AUT ( \Gamma )$ under $\psi$ . Let $v_1 \rightarrow ^t v_2$ be a $t$ -edge of $\Sigma$ , where one of the vertices $v_1$ , $v_2$ belongs to $\Delta$ . Suppose $v_1 \in V( \Delta )$ . We can assume $v = v_1$ . Now let $\alpha _1, \alpha _2 \in AUT( \Gamma )$ , $( \alpha _1 ) \psi = s_1$ and $( \alpha _2 ) \psi = s_2$ . If $( v_1 ) \alpha _1 \rightarrow ^t ( v_2 ) \alpha _1$ and $( v_1 ) \alpha _2 \rightarrow ( v_2 ) \alpha _2$ are $t$ -edges from $\Delta$ to an $\langle S \rangle$ -lobe $\Delta '$ of $\Sigma$ , then we have $s_2 = s_1 u$ , for some $u \in U_1$ , and so $s_1 \sim _1 s_2$ . Thus, the number of $\sim _1$ -classes in $H$ is at most the number of $\langle S \rangle$ -lobes in $\Sigma$ that are adjacent to $\Delta$ . Since $\Sigma$ has finitely many $\langle S \rangle$ -lobes, the group $H$ has finitely many $\sim _1$ -classes. If $v_2 \in V( \Delta )$ , then a similar proof shows that the group $H$ has finitely many $\sim _2$ -classes.

Theorems 4.4 and 4.6 tell us that every maximal subgroup of $S^*$ is either isomorphic to the fundamental group of some graph of groups $( H_e (\!-\!), Y_e )$ , where the vertex and edge groups are subgroups of $S$ , or is isomorphic to a subgroup of $S$ .

Corollary 4.7. Let $S^* = [ S;\; U_1, U_2;\; \phi ]$ be a lower bounded HNN extension.

1. (i) If $S$ is combinatorial then every maximal subgroup of $S^*$ is a free group.

2. (ii) $S^*$ is combinatorial if and only if $S$ is combinatorial and $Y$ is a forest.

Proof. The results are immediate from Theorem 4.4.

Corollary 4.8. Let $S^* = [ S;\; U_1, U_2;\; \phi ]$ be a lower bounded HNN extension. Let $f \in E( U_1 )$ and $H_f$ , $H_{ ( f )\phi }$ , $G_f$ denote the maximal subgroups containing $f$ , $( f )\phi$ , $f$ in $S$ , $S$ , $U_1$ , respectively. Assuming $f \mathcal{D} g$ in $S$ implies $f \mathcal{D} g$ in $U_1$ , for $g \in E( U_1 )$ :

1. (ii) If $( f ) \phi \mathcal{R} d \mathcal{L} f$ in $S$ , for $d \in S$ , then the maximal subgroup of $S^*$ containing $f$ is isomorphic to the group HNN extension $[ H_f;\; G_f, d^{-1} \cdot ( G_f ) \phi \cdot d ]$ .

2. (iii) If $( f ) \phi {\mathcal{D}} f$ in $S$ , then the maximal subgroup of $S^*$ containing $f$ is isomorphic to the amalgamated free product of the group amalgam $[ H_f, H_{ ( f )\phi };\; G_f ]$ .

Proof. Suppose $( f ) \phi \mathcal{R} d \mathcal{L} f$ in $S$ , for $d \in S$ . Assuming $f \mathcal{D} g$ in $S$ implies $f \mathcal{D} g$ in $U_1$ , for $g \in E( U_1 )$ , the component $Y_f$ consists of one vertex $D_1$ and one edge $y = ( D_1, D, D_1 )$ . We may assume that the vertex group $H( y )$ is $G_f$ and the vertex group $H( D_1 )$ is $H_f$ . The group monomorphism $t_y \;:\; H( y ) \rightarrow H( D_1 )$ is given by $s \rightarrow d^{-1} \cdot ( s ) \phi \cdot d$ . By Theorem 4.4, the maximal subgroup of $S^*$ containing $e$ is isomorphic to the HNN extension of groups $[ H( D_1 );\; H( y ), ( H( y ) ) t_y;\; t_y ]$

Suppose $( f ) \phi {\mathcal{D}} f$ in $S$ . Assuming $f \mathcal{D} g$ in $S$ implies $f \mathcal{D} g$ in $U_1$ , for $g \in E( U_1 )$ , the component $Y_f$ consists of two vertices $D_1$ and $D_2$ connected by a single edge $y = ( D_1, D, D_2 )$ . We may assume that the vertex group $H( y )$ is $G_f$ , the vertex group $H( D_1 )$ is $H_f$ , and the vertex group $H( D_2 )$ is $H_{ ( f ) \phi }$ . The group monomorphism $t_y \;:\; H( y ) \rightarrow H( D_2 )$ is given by $s \rightarrow ( s ) \phi$ . Then, by Theorem 4.4, the maximal subgroup of $S^*$ containing $e$ is isomorphic to the amalgamated free product of the group amalgam $[ H( D_1 ), H ( D_2 );\; H( y ) \cong H( y ) t_y ]$ .

Notation 4.9. Similar to Ayyash and Cherubini [2], we define a binary relation $\prec _S$ on $E( U_1 ) \cup E( U_2 )$ . For $f, g \in E( U_1 ) \cup E( U_2 )$ , we write $f \prec _S g$ if $f \mathcal{D} h \leq g$ in $S$ , for some $h \in E( S )$ . We then let $\prec$ denote the transitive closure of $\prec _S$ and the set $\{ ( f, ( f ) \phi ), ( ( f ) \phi, f ) \;:\; f \in E( U_1 ) \}$ . As the next result shows, we are interested in when the intersection of $\prec$ and $\succ _S$ is contained in $\prec _S$ .

An inverse semigroup is completely semisimple if two distinct idempotents in any $\mathcal{D}$ -class are not comparable, under the natural partial order. Equivalently, from [3, Lemma 10], an inverse semigroup is completely semisimple if and only if the endomorphism monoid and the automorphism group coincide for every Schützenberger graph. We have the following result for lower bounded HNN extensions, which has been generalized in [5, Theorem 3.30].

Theorem 4.10 (2, Ayyash and Cherubini, Theorem 5.3). Let $S^* = [ S;\; U_1, U_2;\; \phi ]$ be a lower bounded HNN extension. Then, $S^*$ is completely semisimple if and only if $S$ is completely semisimple and $\prec \cap \succ _S \subseteq \prec _S$ .

Corollary 4.11. Let $S^* = [ S;\; U_1, U_2;\; \phi ]$ be a lower bounded HNN extension. Suppose $S$ is completely semisimple and $f \mathcal{D} ( f ) \phi$ , for all $f \in E( U_1 )$ . Then, $\prec \cap \succ _S \subseteq \prec _S$ and so $S^*$ is completely semisimple.

Proof. Let $f_1, g_1, f_2, g_2, \ldots f_n, g_n \in E( U_1 ) \cup E( U_2 )$ , for $n \geq 1$ , where at least one of $f_k = g_k$ , $( f_k ) \phi = g_k$ and $( f_k ) \phi ^{-1} = g_k$ holds, for $1 \leq k \leq n$ , and $g_k \prec _S f_{ k+1 }$ , for $1 \leq k \leq n-1$ . Thus, we have $f_1 \prec g_n$ . Assuming $f \mathcal{D} ( f ) \phi$ , for all $f \in E( U_1 )$ , we have $f_k \prec _S g_k$ , for $1 \leq k \leq n$ . It then follows that $f_1 \prec _S g_n$ , as $\prec _S$ is transitive. Thus, we have $\prec = \prec _S$ . Hence $S^*$ is completely semisimple, by Theorem 4.10.

Corollary 4.12. Let $S^* = [ S;\; U_1, U_2;\; \phi ]$ be a lower bounded HNN extension. Suppose $S$ is completely semisimple and the following hold, for all $f, g \in E( U_1 )$ :

1. (i) We do not have $f \prec _S ( g ) \phi$ in $S$ and so $E( U_1 ) \cap E( U_2 ) = \emptyset$ .

2. (ii) $f \prec _S g$ implies $f \mathcal{R} u \mathcal{L} u_1^{-1} u_1 \leq g$ , for some $u \in U_1$ .

3. (iii) $( f ) \phi \prec _S ( g ) \phi$ implies $( f ) \phi \mathcal{R} ( u ) \phi \mathcal{L} ( u^{-1} u ) \phi \leq ( g ) \phi$ , for some $u \in U_1$ .

Then $\prec \cap \succ _S \subseteq \prec _S$ and so $S^*$ is completely semisimple.

Proof. Let $f_1, g_1, f_2, g_2, \ldots f_n, g_n \in E( U_1 ) \cup E( U_2 )$ , for $n \geq 1$ , where at least one of $f_k = g_k$ , $( f_k ) \phi = g_k$ and $( f_k ) \phi ^{-1} = g_k$ holds, for $1 \leq k \leq n$ , and $g_k \prec _S f_{ k+1 }$ , for $1 \leq k \leq n-1$ . If $f_k = g_k$ , then $g_{ k-1 } \prec _S f_k = g_k \prec _S f_{ k+1 }$ , and we can shorten the sequence. Thus we can assume $f_k \neq g_k$ .

Suppose $f_1 \in E( U_1 )$ and $( f_1 ) \phi = g_1 \in E( U_2 )$ . From condition (i) and $g_1 \prec _S f_2$ , we have $f_2 \notin E( U_1 )$ and so $( f_2 ) \phi ^{-1} = g_2 \in E( U_1 )$ . From condition (iii) and $g_1 \prec _S f_2$ , we have $g_1 \mathcal{R} ( u_1 ) \phi \mathcal{L} ( u^{-1}_1 u_1 ) \phi \leq f_2$ , for some $u_1 \in U_1$ . Then applying $\phi ^{-1}$ , we have $f_1 \mathcal{R} u_1 \mathcal{L} u^{-1}_1 u_1 \leq g_2$ .

From condition (i) and $g_2 \prec _S f_3$ , we have $f_3 \notin E( U_2 )$ and so $( f_3 ) \phi = g_3 \in E( U_ 2 )$ . From condition (ii) and $g_2 \prec _S f_3$ , we have $g_2 \mathcal{R} u_2 \mathcal{L} u^{-1}_2 u_2 \leq f_3$ , for some $u_2 \in U_1$ . Thus, $f_1 \mathcal{R} u_1u_2 \mathcal{L} u^{-1}_2 u^{-1}_1 u_1 u_2 \leq f_3$ , where $u_1 u_2 \in U_1$ .

Continuing in this manner, we have $f_1 \mathcal{R} u \mathcal{L} u^{-1} u \leq f_{ 2k+1 }$ , for some $u \in U_1$ , for $k \geq 1$ . Thus, if we also have $f_1 \succ _S f_{ 2k+1 }$ then $f_1 \mathcal{D} f_{ 2k+1 }$ in $U_1$ , since $S$ is completely semisimple. Similarly, if $f_1 \in E( U_2 )$ and $f_1 \succ _S f_{ 2k+1 }$ then $f_1 \mathcal{D} f_{ 2k+1 }$ in $U_2$ . Hence, $\prec \cap \succ _S \subseteq \prec _S$ and so $S^*$ is completely semisimple, by Theorem 4.10.

We now establish a result that provides sufficient conditions for the HNN extension $S^*$ to have finite $\mathcal{R}$ -classes. For $S^*$ to have finite $\mathcal{R}$ -classes it is necessary for $S$ to have finite $\mathcal{R}$ -classes. Since the bicyclic inverse semigroup has infinite $\mathcal{R}$ -classes, an inverse semigroup with finite $\mathcal{R}$ -classes cannot contain a copy of the bicyclic inverse semigroup and so must be completely semisimple.

Definition 4.13. The relation $\prec$ is reflexive and transitive on $E( U_1 ) \cup E( U_2 )$ . It follows that $\prec \cap \succ$ defines an equivalence on $E( U_1 ) \cup E( U_2 )$ . The $\prec \cap \succ$ equivalence classes are partially ordered by $[ f ] \leq [ g ]$ if and only if $f \prec g$ , where $[ f ]$ and $[ g ]$ denote the $\prec \cap \succ$ -classes of $f, g \in E( U_1 )$ , respectively. We say that $E( U_1 ) \cup E( U_2 )$ is finite $\prec \cap \succ$ -above if every strictly ascending chain of $\prec \cap \succ$ -classes is finite.

Lemma 4.14. Let $S^* = [ S;\; U_1, U_2;\; \phi ]$ be any HNN extension where $S$ is completely semisimple and $\prec \cap \succ _S \subseteq \prec _S$ . If $f \prec \cap \succ g$ , where $f, g \in E( U_1 ) \cup E( U_2 )$ , then $f$ and $g$ are related by the equivalence on $E( U_1 ) \cup E( U_2 )$ generated by the $\mathcal{D}$ -relation on $S$ and the mapping $\phi$ .

Proof. For $f \prec \cap \succ g$ , where $f, g \in E( U )$ , we have:

1. (i) As $f \prec g$ , there exists $f_1, g_1, f_2, g_2, \ldots f_n, g_n \in E( U_1 ) \cup E( U_2 )$ , $n \geq 1$ , where $f = f_1$ , $g = g_n$ , at least one of $f_k = g_k$ , $( f_k ) \phi = g_k$ and $( f_k ) \phi ^{-1} = g_k$ holds, for $1 \leq k \leq n$ , and $g_k \prec _S f_{ k+1 }$ , for $1 \leq k \leq n-1$ .

2. (ii) Since $f \succ g$ , there exists $h_1, j_1, h_2, j_2, \ldots h_n, j_m \in E( U_1 ) \cup E( U_2 )$ , $m \geq 1$ , where $g = h_1$ , $f = j_m$ , at least one of $h_k = j_k$ , $( h_k ) \phi = j_k$ and $( h_k ) \phi ^{-1} = j_k$ holds, for $1 \leq k \leq m$ , and $j_k \prec _S h_{ k+1 }$ , for $1 \leq k \leq m-1$ .

3. (iii) Since $\prec \cap \succ _S \subseteq \prec _S$ and $g_k \prec _S f_{ k+1 } \prec g \prec f = f_1 \prec g_k$ , for $1 \leq k \leq n-1$ , we then have $g_k \succ _S f_{ k+1 }$ , for $1 \leq k \leq n-1$ ,

4. (iv) As $\prec \cap \succ _S \subseteq \prec _S$ and $j_k \prec _S h_{ k+1 } \prec f \prec g = h_1 \prec j_k$ , for $1 \leq k \leq m-1$ , we then have $j_k \succ _S h_{ k+1 }$ , for $1 \leq k \leq m-1$ .

5. (v) Since $S$ is completely semisimple. the relation $\prec _S \cap \succ _S$ is the $\mathcal{D}$ -relation on $S$ . Thus $g_k \mathcal{D} f_{ k+1 }$ , for $1 \leq k \leq n-1$ , and $j_k \mathcal{D} h_{ k+1 }$ , for $1 \leq k \leq m-1$ .

Hence $f$ and $g$ are related by the equivalence on $E( U_1 ) \cup E( U_2 )$ generated by the $\mathcal{D}$ -relation on $S$ and the mapping $\phi$ .

The fundamental group of a graph of groups whose underlying graph is a finite tree is obtained inductively by a process of repeating amalgamated free products of groups or HNN extensions of groups, one for each edge. The fundamental group is then referred to as a finite tree product of the vertex groups.

Lemma 4.15. Let $S^* = [ S;\; U_1, U_2;\; \phi ]$ be a lower bounded HNN extension. If $S^*$ has finite $\mathcal{R}$ -classes, then every component of $Y$ is a finite tree and the resulting tree product is not proper.

Proof. Suppose $S^*$ has finite $\mathcal{R}$ -classes. Let $e \in E( S )$ . Then, from Theorem 4.4, the fundamental group $\Pi ( H_e(\!-\!), Y_e )$ is isomorphic to the $\mathcal{H}$ -class of $S^*$ containing $e$ and so $\Pi ( H_e(\!-\!), Y_e )$ finite. If $Y_e$ is not a tree then $\Pi ( H_e(\!-\!), Y_e )$ is necessarily infinite, since it contains a free group. Thus $Y_e$ is a tree.

Suppose $Y_e$ is an infinite tree. From Theorem 4.4, the graph $Y_e$ is isomorphic to the graph of orbits $AUT( \Gamma ' ) \backslash T( \Gamma ' )$ , where $\Gamma '$ denotes the t-subopuntoid subgraph of $\Gamma = S \Gamma ( S^*, e )$ that consists of all $\langle S \rangle$ -lobes of $\Gamma$ that are hosts and $T( \Gamma ' )$ is the $\langle S \rangle$ -lobe tree of $\Gamma '$ . Thus, $AUT( \Gamma ' ) \backslash T( \Gamma ' )$ , and hence $T( \Gamma ' )$ , has infinitely many vertices. This implies that $\Gamma$ has infinitely many $\langle S \rangle$ -lobes and we reach a contradiction, since $\Gamma$ has as many vertices as the $\mathcal{R}$ -class of $S^*$ containing $e$ . Hence, $Y_e$ is a finite tree. Any proper amalgamated free product of groups is necessarily infinite. Since $\Pi ( H_e(\!-\!), Y_e )$ is finite, it cannot be a proper tree product.

In contrast with the situation for an HNN extension of a finite group, which is always infinite, an HNN extension of a finite inverse semigroup can have finite $\mathcal{R}$ -classes.

Theorem 4.16. Let $S^* = [ S;\; U_1, U_2;\; \phi ]$ be a lower bounded HNN extension. Suppose $S$ is completely semisimple, with finite $\mathcal{R}$ -classes, $\prec \cap \succ _S \subseteq \prec _S$ holds and $E( U_1 ) \cup E( U_2 )$ is finite $\prec \cap \succ$ -above.

1. (i) If every component of $Y$ is a finite tree and has a tree product that is not proper, then $S^*$ has finite $\mathcal{R}$ -classes.

2. (ii) If $e_1 = ( e_1 ) \phi$ belongs to a trivial $\mathcal{D}$ -class of $S$ and every component of $Y$ , except for $Y_{ e_1 }$ , is a finite tree and has a tree product that is not proper, then each Schützenberger graph of $\langle S^* \rangle$ has finitely many $\langle S \rangle$ -lobes that have more than one vertex.

Proof. Suppose every component of $Y$ is a finite tree and has a tree product that is not proper. Every such tree product is isomorphic to a maximal subgroup of $S$ and thus is finite, since $S$ has finite $\mathcal{H}$ -classes. To prove that $S^*$ has finite $\mathcal{R}$ -classes, we show that every Schützenberger graph of $\langle S^* \rangle$ has finitely many $\langle S \rangle$ -lobes. We first show that every such graph has finitely many hosts.

Let $\Gamma$ be a Schützenberger graph of $\langle S^* \rangle$ . From Theorem 4.1, the graph $\Gamma$ is a complete $t$ -opuntoid graph that has a host, where the $\langle S \rangle$ -lobes are isomorphic to Schützenberger graphs of $\langle S \rangle$ . Let $\Gamma '$ denote the $t$ -subopuntoid subgraph of $\Gamma$ that consists of the $\langle S \rangle$ -lobes of every host of $\Gamma$ and all $t$ -edges connecting these hosts. If $n( \Gamma ) >1$ then, by Lemma 3.7, the graph $\Gamma$ has precisely one host and so, since a host has finitely many $\langle S \rangle$ -lobes, the subgraph $\Gamma '$ has finitely many $\langle S \rangle$ -lobes.

If $n( \Gamma ) = 1$ , then the graph of orbits $AUT( \Gamma ) \backslash T( \Gamma ' )$ is isomorphic to some connected component of $Y$ , from the proof of Theorem 4.4, and is thus finite, by assumption. The tree product of this connected component is not proper, by assumption, and so $AUT( \Gamma )$ is isomorphic to a maximal subgroup of $S$ and is finite. Thus, the set of orbits of any vertex or edge of $T( \Gamma ' )$ is also finite. It now follows that $T( \Gamma ' )$ has finitely many vertices and so $\Gamma '$ has finitely many $\langle S \rangle$ -lobes. Hence $\Gamma$ has finitely many hosts.

We now choose an $\langle S \rangle$ -lobe $\Delta$ of $\Gamma '$ and show that there is a bound on the length of any reduced $\langle S \rangle$ -lobe path in $\Gamma$ which starts in $\Delta$ . Let $\Delta = \Delta ^1, \Delta ^2, \ldots$ be a reduced $\langle S \rangle$ -lobe path in $\Gamma$ . Since $\Gamma '$ has finitely many $\langle S \rangle$ -lobes, there is a least positive integer $m > 1$ such that $\Delta ^m$ is external to $\Gamma '$ . Since either $\Gamma '$ or $\Delta ^{m-1}$ is a host of $\Gamma$ , we have $\Delta ^k \rightarrow \Delta ^{k+1}$ , for $k \geq m-1$ . The situation is illustrated in Fig. 7.

Figure 7. The $\langle S \rangle$ -lobes $\Delta ^{ m-1 }, \Delta ^m, \ldots, \Delta ^n$ of $\Gamma$ .

Let $x_m \rightarrow ^t y_m$ be a $t$ -edge from a vertex $x_m$ of $\Delta ^{m-1}$ to a vertex $y_m$ of $\Delta ^m$ . The case when we have a $t$ -edge $y_m \rightarrow ^t x_m$ from a vertex $y_m$ of $\Delta ^m$ to a vertex $x_m$ of $\Delta ^{m-1}$ is similar. Since the $\langle S \rangle$ -lobes are isomorphic to Schützenberger graphs of $\langle S \rangle$ , by Theorem 4.1, and $\Delta ^{ m-1} \rightarrow \Delta ^m$ , we have $( y_m, \Delta ^m, y_m ) \cong \mathcal{A} ( S, f_m )$ , for some $f_m \in E( U_2 )$ .

Next, suppose we have $\Delta ^k \leftrightarrow \Delta ^{k+1}$ , for all $k \geq m$ . For each $k \geq m$ , the reduced $\langle S \rangle$ -lobe path $\Delta ^m, \Delta ^{m+1}, \ldots, \Delta ^k$ , including the $t$ -edges connecting the $\langle S \rangle$ -lobes, forms a $t$ -opuntoid graph $\Sigma _k$ , where each $\langle S \rangle$ -lobe is a host of $\Sigma _k$ . Since $\Delta ^k \leftrightarrow \Delta ^{k+1}$ , for all $k \geq m$ , the graph $\Sigma _k$ can be obtained from $\Delta ^m$ by repeated applications of Construction 3.3.

Since $( y_m, \Delta ^m, y_m ) \cong \mathcal{A} ( S, f_m )$ , we have $( y_m, \Sigma _k, y_m ) \leadsto \mathcal{A} ( S^*, f_m )$ , by Lemma 3.4. Using Lemmas 3.4 and 3.8, it follows that $( y_m, \Sigma _k, y_m )$ is embedded onto a $t$ -subopuntoid subautomaton of $\mathcal{A} ( S^*, f_m )$ , where the image of each $\langle S \rangle$ -lobe of $\Sigma _k$ is also a host of $\mathcal{A} ( S^*, f_m )$ . Since $S\Gamma ( S^*, f_m )$ has finitely many hosts, as proved above, the sequence of graphs $\Sigma _k$ is bounded. Thus there exists a least positive integer $n > m$ such that $\Delta ^{ n-1 } \nleftarrow \Delta ^n$ .

Let $x_n \rightarrow ^t y_n$ be a $t$ -edge from a vertex $x_n$ of $\Delta ^{n-1}$ to a vertex $y_n$ of $\Delta ^n$ . The case when we have a $t$ -edge $y_n \rightarrow ^t x_n$ from a vertex $y_n$ of $\Delta ^n$ to a vertex $y_n$ of $\Delta ^{n-1}$ is similar. Since the $\langle S \rangle$ -lobes are isomorphic to Schützenberger graphs of $\langle S \rangle$ , by Theorem 4.1, we have $( y_n, \Delta ^n, y_n ) \cong \mathcal{A} ( S, f_n )$ , for some $f_n \in E( U_2 )$ . We show that $[ f_m ] < [ f_n ]$ . where $[ f_k ]$ denotes the $\prec \cap \succ$ -class of $f_k$ , for $k = m, n$ .

Without loss of generality, we assume that we also have a $t$ -edge $x_k \rightarrow ^t y_k$ from a vertex $x_k$ of $\Delta ^{k-1}$ to a vertex $y_k$ of $\Delta ^k$ and let $f_k \in E( U_2 )$ such that $( y_k, \Delta ^k, y_k ) \cong \mathcal{A} ( S, f_k )$ , for $m+1 \leq k \leq n$ . Put $g_k = ( f_k ) \phi ^{-1}$ , for $m \leq k \leq n$ . Since $\Delta ^k \leftrightarrow \Delta ^{ k+1 }$ , for $m \leq k \leq n-2$ , we have $( x_{ k+1 }, \Delta ^k, x_{ k+1 } ) \cong \mathcal{A} ( S, g_{ k+1 } )$ , for $m \leq k \leq n-2$ . Since $\Delta ^{ n-1 } \nleftarrow \Delta ^n$ , we have $( x_n, \Delta ^{ n-1 }, x_n ) \cong \mathcal{A} ( S, g'_n )$ such that $g'_n < g_n$ in $S$ , for some $g'_n \in E( S )$ . Now $f_k \mathcal{D} g_{ k+1 }$ in $S$ , for $m \leq k \leq n-2$ , and $f_{ n-1 } \mathcal{D} g'_n < g_n$ in $S$ .

Thus, the idempotents $f_m, g_{ m+1 }, \ldots, f_{ n-2 }, g_{ n-1 }, f_{ n-1 }$ are all $\prec \cap \succ$ -related and we also have $f_{ n-1 } \prec _S g_n$ , and so $f_{ n-1 } \prec g_n$ , by the definitions of $\prec$ and $\prec _S$ . Suppose we have $f_{ n-1 } \succ g_n$ . Then $f_{ n-1 } \succ _S g_n$ , since it is assumed that $\prec \cap \succ _S \subseteq \prec _S$ . As $S$ is assumed completely semisimple, we then have $f_{ n-1 } \mathcal{D} g'_n = g_n$ , a contradiction. Thus, we do not have $f_{ n-1 } \succ g_n$ and so $[ f_m ] = [ f_{ n-1 } ] < [ g_n ] = [ f_n ]$ .

Similarly, there is a least positive integer $q > n$ with $\Delta ^{q-1} \nleftarrow \Delta ^q$ . Continuing in this manner, we obtain a strictly ascending sequence $[ f_m ] < [ f_n ] < [ f_q ] \cdots$ . Since $S$ is finite $\prec \cap \succ$ -above, the above sequence must be finite and terminates in $[ e_1 ]$ . Thus, there is a bound on the length of any reduced $\langle S \rangle$ -lobe path in $\Gamma$ starting in $\Delta$ . Since $S$ has finite $\mathcal{R}$ -classes, the number of $\langle S \rangle$ -lobes in $\Gamma$ that are adjacent to any given $\langle S \rangle$ -lobe is also finite. It now follows that $\Gamma$ has finitely many $\langle S \rangle$ -lobes and part (i) is proved.

Assuming $e_1 = ( e_1 ) \phi$ belongs to a trivial $\mathcal{D}$ -class, the connected component $Y_{ e_1 }$ of $Y$ consists of one vertex and one loop, with all vertex and edge groups trivial. Every $\langle S \rangle$ -lobe of the Schützenberger graph $S \Gamma ( S^*, e_1 )$ has precisely one vertex. Using a proof similar to that in part (i), any Schützenberger graph $\Gamma$ of $\langle S^* \rangle$ , other than $S \Gamma ( S^*, e_1 )$ , has finitely many hosts. Then, since any $\langle S \rangle$ -lobe that feeds off a trivial $\langle S \rangle$ -lobe must also be trivial, the proof that $\Gamma$ has finitely many non-trivial $\langle S \rangle$ -lobes is also similar to that in (i).

In Jajcayova [13], it was shown that an HNN extension of a free inverse semigroup $S$ is lower bounded, and if $U_1$ and $U_2$ are finitely generated, then the HNN extension has decidable word problem.

Corollary 4.17. Let $S^* = [ S;\; U_1, U_2;\; \phi ]$ be an HNN extension of a free inverse monoid and suppose the following hold, for all $f, g \in E( U_1 )$ :

1. (i) We do not have $f \prec _S ( g ) \phi$ in $S$ and so $E( U_1 ) \cap E( U_2 ) = \emptyset$ .

2. (ii) $f \prec _S g$ implies $f \mathcal{R} u \mathcal{L} u_1^{-1} u_1 \leq g$ , for some $u \in U_1$ .

3. (iii) $( f ) \phi \prec _S ( g ) \phi$ implies $( f ) \phi \mathcal{R} ( u ) \phi \mathcal{L} ( u^{-1} u ) \phi \leq ( g ) \phi$ , for some $u \in U_1$ .

Then, $S^*$ is completely semisimple, combinatorial, and with finite $\mathcal{R}$ -classes. If $E( U_1 ) \cap E( U_2 ) = \{ 1 \}$ , the identity of $S$ , and (i), (ii), and (iii) hold for $f, g \in E( U_1 ) \backslash \{ 1 \}$ , then $S^*$ is completely semisimple, combinatorial, and there is a bound on the number of elements of $S$ needed to express the elements as a product, within each $\mathcal{R}$ -class of $S^*$ .

Proof. We recall why the lower bounded properties hold. If $u \geq e$ , where $u \in U_i$ and $e \in E( S )$ , then we have $u \in E( U_i )$ , for $i = 1, 2$ , since $S$ is free. For $e \in E( S )$ , there are finitely many idempotents $f \in E( U_i )$ with $f \geq e$ , for $i = 1, 2$ . Then, for $e \in E( S )$ , the set $\{ u \in U_i \;:\; u \geq e \}$ is either empty or has a least element $f_i ( e )$ , for $i = 1, 2$ . Thus, the first condition of a lower bounded HNN extension is satisfied.

Let $e \in E( S )$ , $i \in \{ 1, 2 \}$ and $\{ u_k \}$ be a sequence of idempotents in $E( U_i )$ such that $u_k \geq f_i ( e u_k ) \geq u_{ k+1 }$ , for all $k$ . We have monomorphisms from $\mathcal{A}( S, e )$ , $\mathcal{A}( S, u_k )$ and $\mathcal{A}( S, f( e u_k ) )$ into $\mathcal{A}( S, e u_k )$ , for each $k$ , which we regard as inclusions. Let $\Sigma _k = S \Gamma ( S, e ) \cap S \Gamma ( S, f( e u_k ) )$ . Suppose $\Sigma _k = \Sigma _{ k+1 }$ , for some $k$ . Now $u_k \geq f( u e_k ) \geq u_k \geq f( u e_{ k+1 } )$ . Then if $w \in E( U_I )$ and $w \geq e u_{ k+1 }$ in $S$ , then we must have $w \geq u_{ k+1 }$ . Thus, we have $f( e u_{ k+1 } ) = u_{ k+1 }$ . Conversely, since $S \Gamma ( S, e )$ is finite, we can have $\Sigma _k \subsetneq \Sigma _{ k+1 }$ at most a finite number of times. Thus, the second condition of a lower bounded HNN extension is satisfied. Hence, the HNN extension $S^* = [ S;\; U_1, U_2;\; \phi ]$ is lower bounded.

Since $S$ is a free inverse semigroup, it is completely semisimple and has finite $\mathcal{R}$ -classes. From Corollary 4.12, we have $\prec \cap \succ _S \subseteq \prec _S$ and $S^*$ is completely semisimple. Further, the relation $\prec \cap \succ$ is the $\mathcal{D}$ -relation in $U_1$ on $E( U_1 )$ and the $\mathcal{D}$ -relation in $U_2$ on $E( U_2 )$ . If $f, g \in E( U_1 )$ and $f \mathcal{R} u \mathcal{L} u^{-1}u < g$ , for some $u \in U_1$ , then $[ f ] < [ g ]$ , since $S$ is completely semisimple. Since a free inverse monoid is finite $\mathcal{J}$ -above, we then have that $E( U_1 ) \cup E( U_2 )$ is finite $\prec \cap \succ$ -above.

Conditions (i), (ii), and (iii) imply that every component $Y_f$ of $Y$ consists of two vertices, the $\mathcal{D}$ -class of $S$ containing $f$ and the $\mathcal{D}$ -class of $S$ containing $( f ) \phi$ , and one edge, the $\mathcal{D}$ -class of $U_1$ containing $f$ , for $f \in E( U_1 )$ . Since a free inverse monoid is combinatorial, we now have that $S^*$ has finite $\mathcal{R}$ -classes, by Theorem 4.16 (i).

If $E( U_1 ) \cap E( U_2 )$ consists of the identity of $S$ , and (i), (ii), and (iii) hold for $f, g \in E( U_1 ) \backslash \{ 1 \}$ , then $S^*$ is completely semisimple, combinatorial and has finite $\mathcal{R}$ -classes, by the above. Since $e_1 = 1 = e_2$ , each Schützenberger graph of $\langle S^* \rangle$ has finitely many $\langle S \rangle$ -lobes that have more than one vertex, from Theorem 4.16 (ii). Thus, if $r \in S^*$ then all the elements of the $\mathcal{R}$ -class of $S^*$ containing $r$ can be expressed as a product involving fewer than $N$ elements of $S$ , for some $N \geq 1$ .

An inverse semigroup $S$ is residually finite if for every finite non-empty subset $F \subseteq S$ there exists a homomorphism from $S$ into some finite inverse semigroup $T$ which separates the elements of $F$ . Any inverse semigroup with finite $\mathcal{R}$ -classes is residually finite, from [15, Lemma 5.3].

Corollary 4.18. Let $S^* = [ S;\; U_1, U_2;\; \phi ]$ be an HNN extension, where $S$ is finite, combinatorial, and conditions (i), (ii), (iii) of Corollary 4.17 hold. Then, $S^*$ has finite $\mathcal{R}$ -classes and so is residually finite.

Proof. Let $i \in \{ 1, 2 \}$ . Since $U_i$ is finite, if $e \in E( S )$ then there exists a least idempotent $f \in E( U_i )$ with $e \leq f$ . If $u \in U_i$ with $u \geq e$ . then $f \mathcal{R} fu \mathcal{L} u^{-1} f u$ in $U_i$ and $u^{-1} f u \geq f$ , since $u^{-1} f u \in E( U_i )$ and $u^{-1} f u \geq e$ . As $S$ is finite, and so completely semisimple, we must have $u^{-1} f u = f$ . Then, $f u$ belongs to the maximal subgroup of $U_1$ containing $f$ , which is trivial. Hence $f u = f$ and it follows that the HNN extension $S^* = [ S;\; U_1, U_2;\; \phi ]$ is lower bounded.

From Corollary 4.12, we have $\prec \cap \succ _S \subseteq \prec _S$ and $S^*$ is completely semisimple. As in the proof of Corollary 4.17, the relation $\prec \cap \succ$ is the $\mathcal{D}$ -relation in $U_1$ on $E( U_1 )$ and the $\mathcal{D}$ -relation in $U_2$ on $E( U_2 )$ . If $f, g \in E( U_1 )$ and $f \mathcal{R} u \mathcal{L} u^{-1}u < g$ , for some $u \in U_1$ , then $[ f ] < [ g ]$ , since $S$ is completely semisimple. Since $S$ finite, we then have that $E( U_1 ) \cup E( U_2 )$ is finite $\prec \cap \succ$ -above.

Conditions (i), (ii), and (iii) imply that every component $Y_f$ of $Y$ consists of two vertices, the $\mathcal{D}$ -class of $S$ containing $f$ and the $\mathcal{D}$ -class of $S$ containing $( f ) \phi$ , and one edge, the $\mathcal{D}$ -class of $U_1$ containing $f$ , for $f \in E( U_1 )$ . Since $S$ is combinatorial, we now have that $S^*$ has finite $\mathcal{R}$ -classes, by Theorem 4.16 (i).

An inverse semigroup $S$ is $E$ -unitary if $s \geq e$ implies $s \in E( S )$ , for all $s \in S$ and $e \in E( S )$ . From [19, Theorem 3.8], we have that $S$ is $E$ -unitary if and only if there exists a monomorphism from $\mathcal{A}( S, s_1 )$ into $\mathcal{A}( S, s_2 )$ , whenever $s_1 \geq s_2$ in $S$ . Equivalently, the inverse semigroup $S$ is $E$ -unitary if and only if homomorphisms between Schützenberger graphs are monomorphic. For $S^*$ to be $E$ -unitary, the homomorphisms between Schützenberger graphs of $S^*$ must induce embeddings of the respective lobe trees.

Theorem 4.19. Let $S^* = [ S;\; U_1, U_2;\; \phi ]$ be a lower bounded HNN extension where $S$ is $E$ -unitary, $s u \in E( S )$ implies $s s^{-1} \sim _1 s$ , and $s \cdot ( u ) \phi \in E( S )$ implies $s s^{-1} \sim _2 s$ , for all $s \in S$ and $u \in U_1$ . Then $S^*$ is $E$ -unitary.

Proof. Let $\Gamma$ and $\Gamma '$ be Schützenberger graphs of $S^*$ and let $\alpha \;:\; \Gamma \rightarrow \Gamma '$ be a homomorphism. Let $\beta$ denote the homomorphism $T( \Gamma ) \rightarrow T( \Gamma ' )$ between the lobe trees induced by $\alpha$ . We first show that $\beta$ is an embedding.

Let $\Delta ^1, \Delta ^2, \ldots, \Delta ^n$ be a reduced $\langle S \rangle$ -lobe path in $\Gamma$ where $( \Delta ^1 ) \beta = ( \Delta ^n ) \beta$ . Then, $( \Delta ^1 ) \beta, ( \Delta ^2 ) \beta, \ldots, ( \Delta ^n ) \beta$ denote the vertices of a loop in $T( \Gamma ' )$ . Since $T( \Gamma ' )$ is a tree, there exist some $k \leq n-1$ such that $( \Delta ^{k-1} ) \beta = ( \Delta ^{k+1} ) \beta$ . The $t$ -edges between $( \Delta ^{ k-1 } ) \beta$ and $( \Delta ^k ) \beta$ all start in one $\langle S \rangle$ -lobe and end in the other. Without loss of generality, assume all these $t$ -edges start in $( \Delta ^{ k-1 } ) \beta$ and end in $( \Delta ^k ) \beta$ . Then, there is a $t$ -edge $x_1 \rightarrow ^t y_1$ from a vertex $x_1$ of $\Delta ^{k-1}$ to a vertex $y_1$ of $\Delta ^k$ and a $t$ -edge $x_2 \rightarrow ^t y_2$ from a vertex $x_2$ of $\Delta ^{k+1}$ to a vertex $y_2$ of $\Delta ^k$ . The situation is illustrated in Fig. 8.

Figure 8. The Schützenberger graphs $\Gamma$ and $\Gamma '$ .

Let $s \in S$ such that $( y_1, \Delta ^k, y_2 ) \cong \mathcal{A}( S, s )$ . Then, we have $t$ -edges $( x_1 ) \alpha \rightarrow ^t ( y_1 ) \alpha$ and $( x_2 ) \alpha \rightarrow ^t ( y_2 ) \alpha$ from $( \Delta ^{k-1} ) \beta$ to $( \Delta ^k ) \beta$ in $\Gamma '$ . By the $t$ -saturation property of $t$ -opuntoid graphs, there exists a path $( y_2 ) \alpha \rightarrow ^{( u )\phi } ( y_1 ) \alpha$ in $( \Delta ^k ) \beta$ , for some $u \in U_1$ . Let $r \in S$ such that $( ( y_1 ) \alpha, ( \Delta ^k ) \beta, ( y_2 ) \alpha ) \cong \mathcal{A}( S, r )$ . Then, we have $s \geq r$ and $( u^{-1} ) \phi \geq r$ in $S$ . Thus $r r^{-1} \leq s \cdot ( u )\phi$ .

Since $S$ is $E$ -unitary we that $s \cdot ( u )\phi$ is idempotent. By the conditions of the statement of the theorem, we then have $ss^{-1} \sim _2 s$ . This implies that either $ss^{-1} = s$ or $ss^{-1} = s v$ , for some $v \in U_2$ . The first case implies $y_1 = y_2$ and so $\Delta ^{k-1} = \Delta ^{ k+1 }$ , a contradiction since the original $\langle S \rangle$ -lobe path was reduced. The second case implies that there is a path $y_1 \rightarrow ^v y_2$ in $\Delta ^k$ and so, by the $t$ -saturation property, we must have $\Delta ^{k-1} = \Delta ^{ k+1 }$ , again a contradiction.

Hence, $\beta$ must be one-one on the vertices of $T( \Gamma )$ . Since $T( \Gamma )$ is a tree, this implies that $\beta$ is an embedding. Since $S$ is $E$ -unitary, the homomorphisms between Schützenberger graphs of $\langle S \rangle$ are monomorphisms. Thus, each $\langle S \rangle$ -lobe of $\Gamma$ is embedded, under $\alpha$ , into some $\langle S \rangle$ -lobe of $\Gamma '$ . It now follows that $\alpha$ must be monomorphic. Hence, the HNN extension $S^*$ is $E$ -unitary.

A subsemigroup $U$ of an inverse semigroup $S$ is a unitary subsemigroup if we have $u s \in U$ implies $s \in U$ , and $s u \in U$ implies $s \in U$ , for all $s \in S$ and $u \in U$ . We note a few observations in the following corollary.

Corollary 4.20. Let $S^* = [ S;\; U_1, U_2;\; \phi ]$ be an HNN extension where $S$ is an $E$ -unitary inverse semigroup. If $S$ is a monoid and $U_1$ , $U_2$ are subgroups of the groups of units of $S$ then $S^*$ is $E$ -unitary. If $U_1$ and $U_2$ are semilattices satisfying the descending chain condition or are full unitary inverse subsemigroups of $S$ then $S^*$ is $E$ -unitary.

Proof. Suppose $S$ is a monoid and $U_1$ , $U_2$ are subgroups of the groups of units. If $u \geq e$ , for some $u \in U_1$ and $e \in E( S )$ , then $u = 1$ , the identity of the monoid, since $S$ is $E$ -unitary. Similarly, if $( u ) \phi \geq e$ , for some $u \in U_1$ and $e \in E( S )$ , then $( u ) \phi = 1$ . It follows that $S^* = [ S;\; U_1, U_2;\; \phi ]$ is a lower bounded HNN extension. Suppose $s u \in E( S )$ , for some $s \in S$ and $u \in U_1$ . Then, $s u = s uu^{-1} s^{-1} = s s^{-1}$ , as $u u^{-1} = 1$ , and so $s s^{-1} \sim _1 s$ . Similarly, if $s \cdot ( u ) \phi \in E( S )$ , for some $s \in S$ and $u \in U_1$ , then $s s^{-1} \sim _2 s$ . Thus, $S^*$ is $E$ -unitary, from Theorem 4.19.

Suppose $U_1$ and $U_2$ are semilattices satisfying the descending chain condition. It is immediate that $S^* = [ S;\; U_1, U_2;\; \phi ]$ is a lower bounded HNN extension. If $s u \in E( S )$ , for some $s \in S$ and $u \in U_1$ , then $u \in U_1 = E( U_1 )$ implies $s \in E( S )$ , since $S$ is $E$ -unitary. Then $s s^{-1} = s$ implies $s s^{-1} \sim _1 s$ . Similarly, if $s \cdot ( u ) \phi \in E( S )$ , for some $s \in S$ and $u \in U_1$ , then $s s^{-1} \sim _2 s$ . Thus $S^*$ is $E$ -unitary, from Theorem 4.19.

Suppose $U_1$ and $U_2$ are full unitary inverse subsemigroups of $S$ . Since $U_1$ and $U_2$ are full in $S$ , we have $E( U_1 ) = E( U_2 ) = E( S )$ , and it is then immediate that $S^* = [ S;\; U_1, U_2;\; \phi ]$ is a lower bounded HNN extension. If $s u \in E( S ) = E( U_1 )$ , for some $s \in S$ and $u \in U_1$ , then $s \in U_1$ , since $U_1$ is a unitary subsemigroup. Then, $s s^{-1} = s \cdot ( s^{-1} )$ , where $s^{-1} \in U_1$ , and so $s s^{-1} \sim _1 s$ . Similarly, if we have $s \cdot ( u ) \phi \in E( S )$ , for some $s \in S$ and $u \in U_1$ , then $s s^{-1} \sim _2 s$ . Hence, $S^*$ is $E$ -unitary, from Theorem 4.19.

An inverse semigroup $S$ is $0$ - $E$ -unitary if $s \geq e$ implies $s \in E( S )$ , for all $s \in S \backslash \{ 0 \}$ and $e \in E( S ) \backslash \{ 0 \}$ . The inverse semigroup $S$ is strongly $0$ - $E$ -unitary if it admits an idempotent pure partial homomorphism to a group.

Corollary 4.21. Let $S^* = [ S;\; U_1, U_2;\; \phi ]$ be a lower bounded HNN extension where $S$ , $U_1$ , and $U_2$ are $0$ - $E$ -unitary, sharing a common $0$ . If $s u \in E( S )$ implies $s s^{-1} \sim _1 s$ , and $s \cdot ( u ) \phi \in E( S )$ implies $s s^{-1} \sim _2 s$ , for all $s \in S \backslash \{ 0 \}$ and $u \in U_1 \backslash \{ 0 \}$ , then $S^*$ is $0$ - $E$ -unitary.

Proof. The proof is similar to that of Theorem 4.19.

The polycyclic monoid $P_n$ is the inverse monoid with zero that has the following presentation $\langle a_1, a_2, \dots, a_n, 0, 1 \mid a_i^{-1} a_i = 1, a_i^{-1} a_j = 0, i \neq j \rangle$ , as an inverse monoid with zero. Non-zero elements can be written in the unique form $xy^{-1}$ , where $x, y$ are elements of $A^*_n$ , the free monoid on $A_n = \{ a_1, \ldots, a_n \}$ . Multiplication is then defined by:

\begin{align*} x y^{-1} \cdot u v^{-1} = \begin{cases} x z v^{-1} & \text{if } u = yz \text{, for some word } z\\[5pt] x ( vz )^{-1} & \text{if } y = uz \text{, for some word } z\\[5pt] 0 & \text{otherwise} \end{cases} \end{align*}

Idempotents are given by $xx^{ -1 }$ , where $x \in A^*_n$ , and $x y^{-1} \mathcal{R} x x^{-1}$ . The monoid $P_n$ is $0$ - $E$ -unitary, $0$ -bisimple, and combinatorial. For $m \leq n$ , we have a natural embedding of $P_m$ into $P_n$ , induced by the injection from $A_m$ into $A_n$ . Polycyclic inverse monoids are used to construct $C^*$ -algebras [8]. Nearly all the inverse semigroup studied in $C^*$ -algebra theory are strongly $0$ - $E$ -unitary [17, Section 5].

Corollary 4.22. Let $S^* = [ S;\; U_1, U_2;\; \phi ]$ be an HNN extension where $S = P_n$ , $U_1 = P_m$ , for $m \leq n$ , are the polycyclic inverse monoids and $\phi$ is induced by any injection from $A_m$ into $A_b$ . Then, $S^*$ is $0$ - $E$ -unitary with group of units isomorphic to a free group on a singleton and all other maximal subgroups are trivial.

Proof. Let $x x^{-1} \in E( S )$ . Since $S$ is $0$ - $E$ -unitary, if $x x^{ -1 } \leq z y^{-1}$ , where $z y^{-1} \in S$ , then $z y^{-1}$ is idempotent. If $x x^{ -1 } \leq y y^{ -1 }$ in $S$ then $x = y z$ , for some word $z$ . Thus, there are finitely many idempotents $y y^{ -1 }$ with $x x^{ -1 } \leq y y^{ -1 }$ . Hence, the set $\{ y y^{-1} \in U_i \;:\; y y^{-1} \geq x x^{-1} \}$ has a least element $f_1 ( x x^{-1} )$ , possibly $1$ , for $i = 1, 2$ . We have:

\begin{align*} x x^{-1} \cdot y y^{-1} = \begin{cases} x x^{-1} & \text{if } x = y z \text{, for some word } z\\[5pt] y y^{-1} & \text{if } y = x z \text{, for some word } z\\[5pt] 0 & \text{otherwise} \end{cases} \end{align*}

Let $y y^{-1} \in E( U_i )$ with $x x^{ -1 } \nleq y y^{ -1 }$ . Then either $x x^{-1} \cdot y y^{-1} = 0$ or we have $x x^{-1} \cdot y y^{ -1} = y y^{-1}$ . Assume $x x^{-1} \geq y y^{-1}$ and so $f_i ( x x^{-1} \cdot y y^{ -1} ) = y y^{-1}$ . If $y_1 y^{-1}_1 \in E( U_i )$ with $y y^{ -1} \geq y_1 y^{ -1 }_1$ then $f_i ( x x^{-1} \cdot y_1 y^{ -1}_1 )$ $= f_i ( x x^{-1} \cdot y y^{ -1 } \cdot y_1 y^{ -1}_1 )$ $= y y^{ -1 } \cdot y_1 y^{ -1}_1 = y_1 y^{-1}_1$ . It now follows that $S^* = [ S;\; U_1, U_2;\; \phi ]$ is a lower bounded HNN extension.

Let $x y^{ -1 } \in S$ and $u v^{ -1 } \in U_i$ such that $x y^{ -1 } \cdot u v^{ -1 }$ is idempotent, for $i \in \{ 1, 2 \}$ . Suppose $u = y z$ , for some word $z$ . Then, $y \in U_i$ . Since $x y^{ -1 } \cdot u v^{ -1 } = ( x z ) v^{ -1 }$ is idempotent, we have $x z = v$ and so $x \in U_i$ . Thus $x y^{-1} \in U_i$ , $x x^{ -1 } \cdot x y^{ -1} = x y^{ -1 }$ and so $x x^{ -1 } \sim _i x y^{-1}$ . Suppose we have $y = u z$ , for some word $z$ . Then, since $x y^{ -1 } \cdot u v^{ -1 }$ $= x ( v z )^{ -1 }$ is idempotent, we have $x = v z$ . Thus we have $x x^{ -1 } \cdot v u^{-1}$ $= vz ( v z )^{ -1 } \cdot v u^{ -1 }$ $=vz ( u z )^{ -1 } = x y^{ -1 }$ and so $x x^{ -1 } \sim _i x y^{-1}$ . It follows that $S^*$ is $0$ - $E$ -unitary, by Corollary 4.21.

Since $S$ is $0$ -bisimple, the component $Y_1$ of the graph of groups $Y$ , as defined in Notation 4.3, consists of one vertex and one edge, where $1$ is the identity of $S$ . Since $S$ is combinatorial, the fundamental group of the graph of groups $( H_1 (\!-\!), Y_1 )$ is isomorphic to the free group on a singleton. The maximal subgroup of $S^*$ containing $1$ is isomorphic to the fundamental group of the graph of groups $( H_1 (\!-\!), Y_1 )$ , by Theorem 4.4. All other maximal subgroups of $S^*$ are trivial, by Theorem 4.6.