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 [Reference Howie11], Petrich [Reference Petrich18], and Lawson [Reference Lawson16].

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 [Reference Stephen19].

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 [Reference Bennett and Jajcayová5]. 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 [Reference Bennett and Jajcayová5].

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 [Reference Yamamura20], 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 [Reference Bennett and Jajcayová5] 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. [Reference Bennett and Jajcayová5, 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. [Reference Bennett and Jajcayová5, 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. [Reference Bennett and Jajcayová5, 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.

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 $\Gamma '$ .

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 $n( \Gamma ) = 1$ .

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.

Proof. The result is a restatement of Jajcayová [Reference Jajcayová14, 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 [Reference Cohen7] and Dicks and Dunwoody [Reference Dicks and Dunwoody9, Chapters 1, 2, 3] for notation and definitions.

The following result generalizes Yamamura [Reference Yamamura21, Theorem 5.2], on locally full HNN extensions, and overlaps with Ayyash [Reference Ayyash1, Theorem $5.4.1$ ], on HNN extensions of finite inverse semigroups.

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.

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.