On the computation of order types of hammocks for domestic string algebras

This paper is the sequel of [9] where the authors embarked on the journey towards the computation of the order types of certain linear orders known as hammocks for domestic string algebras–that paper introduces a new finite combinatorial tool called the arch bridge quiver and investigates its properties. Recall that a string algebra $\Lambda$ over an algebraically closed field $\mathcal{K}$ is presented as a certain quotient $\mathcal{K}Q/\langle\rho\rangle$ of the path algebra $\mathcal{K}Q$ of a quiver $Q$, where $\rho$ is a set of monomial relations. The vertices of the Auslander-Reiten (A-R) quiver were classified essentially in [3] in terms of certain walks on the quiver known as strings and bands, whereas its arrows were classified by Ringel and Butler [2].

A string algebra is domestic if it has only finitely many bands. For two bands $\mathfrak{b}_{1}$ and $\mathfrak{b}_{2}$, say $\mathfrak{b}_{1}$ and $\mathfrak{b}_{2}$ commute if there are cyclic permutations $\mathfrak{b}^{\prime}_{1},\mathfrak{b}^{\prime\prime}_{1}$ of $\mathfrak{b}_{1}$ and $\mathfrak{b}^{\prime}_{2},\mathfrak{b}^{\prime\prime}_{2}$ of $\mathfrak{b}_{2}$ such that $\mathfrak{b}^{\prime}_{1}\mathfrak{b}^{\prime}_{2}$ and $\mathfrak{b}^{\prime\prime}_{2}\mathfrak{b}^{\prime\prime}_{1}$ are strings. If two bands commute in a string algebra then it is necessarily non-domestic by [4, Corollary 3.4.1].

For a vertex $v$ of the quiver $Q$ the simplest version of hammock, denoted $H_{l}(v)$, is the collection of all strings starting at the vertex. It can be equipped with an order $<_{l}$ such that $\mathfrak{x}<_{l}\mathfrak{y}$ in $H_{l}(v)$ guarantees the existence of a canonical graph map $M(\mathfrak{x})\to M(\mathfrak{y})$ between the corresponding string modules. Schröer [7, § 2.5] showed that $(H_{l}(v),<_{l})$ is a bounded discrete linear order. Prest and Schröer showed that it has finite m-dimension (essentially [5, Theorem 1.3]). For more flexibility we consider the hammock $H_{l}(\mathfrak{x}_{0})$ for an arbitrary string $\mathfrak{x}_{0}$, which is the collection of all strings which admit $\mathfrak{x}_{0}$ as a left substring–this too is a bounded discrete linear order with respect to the induced order $<_{l}$, say with $\mathfrak{m}_{1}(\mathfrak{x}_{0})$ and $\mathfrak{m}_{-1}(\mathfrak{x}_{0})$ as the maximal and minimal elements respectively. For technical reasons it is necessary to write the hammock as a union $H_{l}(\mathfrak{x}_{0})=H_{l}^{1}(\mathfrak{x}_{0})\cup H_{l}^{-1}(\mathfrak{x}_{0})$, where $H_{l}^{1}(\mathfrak{x}_{0})$ is the interval $[\mathfrak{x}_{0},\mathfrak{m}_{1}(\mathfrak{x}_{0})]$ and $H_{l}^{-1}(\mathfrak{x}_{0})=[\mathfrak{m}_{-1}(\mathfrak{x}_{0}),\mathfrak{x}_{0}]$. Given $\mathfrak{x}<_{l}\mathfrak{y}$ in $H_{l}(\mathfrak{x}_{0})$, the goal of this paper is to compute the order type of the interval $[\mathfrak{x},\mathfrak{y}]$ via the assignment of a label, which we call a ‘term’, to the path from $\mathfrak{x}$ to $\mathfrak{y}$. What we call a term in this paper is referred to as a ‘real term’ in [4, § 2.4].

Some classes of finite Hausdorff rank linear orders were studied in [1]. The class $\mathrm{LO}_{\mathrm{fp}}$ [1, § 5] is the smallest class of linear orders containing $1$ that is closed under $+$, $\omega\times\mbox{-}$ and $\omega^{*}\times\mbox{-}$ operations. Its subclass consisting of bounded discrete linear orders is denoted by $\mathrm{dLO}_{\mathrm{fp}}^{\mathrm{b}}$. The main results of this paper (Corollary 10.14 and Proposition 10.18) show that $\mathrm{dLO}_{\mathrm{fp}}^{\mathrm{b}}$ is precisely the class of order types of hammocks for domestic string algebras.

Fix a domestic string algebra $\Lambda$, a string $\mathfrak{x}_{0}$ and $i\in\{1,-1\}$. Motivated by the notion of the bridge quiver introduced by Schröer for the study of hammocks for domestic string algebras, the authors introduced the notion of the extended arch bridge quiver in [9], denoted $\mathcal{HQ}^{\mathrm{Ba}}_{i}(\mathfrak{x}_{0})$, that lies between the extended bridge quiver and the extended weak bridge quiver. For $i\in\{1,-1\}$, the paths in $\mathcal{HQ}^{\mathrm{Ba}}_{i}(\mathfrak{x}_{0})$ generate all strings in $H_{l}^{i}(\mathfrak{x}_{0})$ in the precise sense described in [4, Lemma 3.3.4].

In this paper, we introduce the notion of a ‘decorated tree’, denoted $\mathcal{T}^{\mathfrak{f}}_{i}(\mathfrak{x}_{0})$, that linearizes the extended arch bridge quiver $\mathcal{HQ}^{\mathrm{Ba}}_{i}(\mathfrak{x}_{0})$ from the viewpoint of the string $\mathfrak{x}_{0}$ placed at the root; here the adjective decorated refers to the assignment of either a $+$ or $-$ sign to each non-root vertex and, for each vertex, a linear ordering of the children with the same sign. The data present as part of the decoration of the tree allows us to inductively associate terms to individual vertices of $\mathcal{T}^{\mathfrak{f}}_{i}(\mathfrak{x}_{0})$ so that the term associated with the root, denoted $\mu_{i}(\mathfrak{x}_{0};\mathfrak{m}_{i}(\mathfrak{x}_{0}))$, labels $H_{l}^{i}(\mathfrak{x}_{0})$. The following flowchart concisely depicts the steps of the algorithm.

To further explain a rough idea behind the proof we need some terminology and notation. Recall from [9] that for a string $\mathfrak{x}$ of positive length, we say $\theta(\mathfrak{x})=1$ if its first syllable is an inverse syllable, otherwise we say $\theta(\mathfrak{x})=-1$.

For $i\in\{1,-1\},\mathfrak{x}\in H_{l}(\mathfrak{x}_{0}),\mathfrak{x}\neq\mathfrak{x}_{0}$, say that $\Theta(\mathfrak{x}_{0};\mathfrak{x})=C(i)$ if it is of the form $(i,\ldots,i)$. Say that $\Theta(\mathfrak{x}_{0};\mathfrak{x})=F(i)$ if it is of the form $(i,-i,\ldots,-i)$, where $-i$ may not appear; here $C$ and $F$ stand for the words constant and flipped respectively.

The simplest possible terms are associated with the flipped signature types, and hence lot of the effort of the main proof goes into constructing strings whose signature types are so. We use these as starting points for building strings with more complex signature types. The interaction between signature types, left $\mathbb{N}$-strings and terms is an essential feature of this proof.

Continuing from [9] we provide a set of (counter)examples of algebras with peculiar properties which could be useful elsewhere. Even though there is lot of literature and explicit proofs for gentle algebras, we believe that these two papers are the first systematic attempt to study the combinatorics of strings for all domestic string algebras.

The rest of the paper is organized as follows. The concept of terms together with their associated order types is introduced in §2, the example of a particular domestic string algebra, namely $\Lambda_{2}$, is discussed in detail. Even though the bridge quiver of a domestic string algebra does not contain any directed cycle, its underlying undirected graph could still contain cycles. After fixing a string $\mathfrak{x}_{0}$ and parity $i$, we linearize/separate all paths in the extended arch bridge quiver $\mathcal{HQ}^{\mathrm{Ba}}_{i}(\mathfrak{x}_{0})$ with the help of the concept of decorated trees introduced in §4. The results in §3, §5 and §6 help to classify strings in $H^{l}_{i}(\mathfrak{x}_{0})$ into various classes. Recall that there are only finitely many H-reduced strings in a domestic string algebra [9, Corollary 5.8]. In §7, we characterize all H-reduced forking strings using the vertices of the decorated tree and then use them to build all H-reduced strings in §8. The signature types of these H-reduced strings are computed in §9. Finally the algorithm (Algorithm 10.9) for computation of the term associated with a closed interval in the hammock, and hence that of the order type, is completed in §10.

The reader is advised to refer to [9] for any terminology/notation not explained in this paper.

The class $\mathrm{dLO}_{\mathrm{fp}}^{\mathrm{b}}$ consisting of bounded discrete linear orders is of particular interest because of the next result.

Consider any $\mathfrak{v}\in H_{l}(\mathfrak{x}_{0})$. Recall from [4, §2.4] that $l(\mathfrak{v})$ is the immediate successor of $\mathfrak{v}$ with respect to $<_{l}$, if exists, that satisfies $|\mathfrak{v}|<|l(\mathfrak{v})|$. We can inductively define $l^{n+1}(\mathfrak{v}):=l(l^{n}(\mathfrak{v}))$, if exists. If $l^{n}(\mathfrak{v})$ exists for all $n\geq 0$ then the limit of the increasing sequence $\langle 1,l\rangle(\mathfrak{v}):=\lim_{n\to\infty}l^{n}(\mathfrak{v})$ is a left $\mathbb{N}$-string. On the other hand, if there is a maximum $N\geq 1$ such that $l^{N}(\mathfrak{v})$ exists but $l^{N+1}(\mathfrak{v})$ does not exist then we define $[1,l](\mathfrak{v}):=l^{N}(\mathfrak{v})$. Note that $\langle 1,l\rangle(\mathfrak{v})\notin H_{l}(\mathfrak{x}_{0})$ but $[1,l](\mathfrak{v})\in H_{l}(\mathfrak{x}_{0})$.

Recall that any left $\mathbb{N}$-string in a domestic string algebra is an almost periodic string, i.e., is of the form ${}^{\infty}\mathfrak{b}\mathfrak{u}$ for some primitive cyclic word $\mathfrak{b}$ and some finite string $\mathfrak{u}$ such that the composition $\mathfrak{b}\mathfrak{u}$ is defined. Let $\hat{H}_{l}(\mathfrak{x}_{0})$ denote the completion of $H_{l}(\mathfrak{x}_{0})$ by all left $\mathbb{N}$-strings ${}^{\infty}\mathfrak{b}\mathfrak{u}$ such that $\mathfrak{b}\mathfrak{u}\in H_{l}(\mathfrak{x}_{0})$. The definition of the ordering $<_{l}$ can be easily extended to include infinite left $\mathbb{N}$-strings so that $\hat{H}_{l}(\mathfrak{x}_{0})$ is also a bounded total order with same extremal elements as that of $H_{l}(\mathfrak{x}_{0})$.

Note that any left $\mathbb{N}$-string in $\hat{H}_{l}(\mathfrak{x}_{0})$ is “far away” from its endpoints. To be more precise if $\langle 1,l\rangle(\mathfrak{v})$ exists then the interval $[\langle 1,l\rangle(\mathfrak{v}),\mathfrak{m}_{1}(\mathfrak{x}_{0})]$ in $\hat{H}_{l}(\mathfrak{x}_{0})$ contains infinitely many elements of $H_{l}(\mathfrak{x}_{0})$, where $\langle 1,l\rangle(\mathfrak{v})<_{l}\mathfrak{m}_{1}(\mathfrak{x}_{0})$, where $\mathfrak{m}_{1}(\mathfrak{x}_{0})$ is a finite string. Thus the left $\mathbb{N}$-string $\langle 1,l\rangle(\mathfrak{v})$ can also be approached from the right, with respect to $<_{l}$, as a limit of some decreasing sequence in $H_{l}(\mathfrak{x}_{0})$.

Now we recall the concept of fundamental solution of the following equation from [4, §2.4]:

If one solution to the above equation exists then infinitely many solutions exist for if $\mathfrak{y}^{\prime}$ is a solution then so is $\bar{l}(\mathfrak{y}^{\prime})$ and the fundamental solution is a solution of minimal length. If $\mathfrak{v}$ is a left substring of $\mathfrak{y}$, then we rewrite Equation (1) as $\mathfrak{y}=\langle\bar{l},l\rangle(\mathfrak{v})$, otherwise we write $\mathfrak{v}=\langle l,\bar{l}\rangle(\mathfrak{y})$. If $\mathfrak{y}=\langle\bar{l},l\rangle(\mathfrak{v})$, then we also say that $\mathfrak{y}=\langle\bar{l},l\rangle(l^{n}(\mathfrak{v}))$ for any $n\in\mathbb{N}$.

Similarly if $\langle 1,l\rangle(\mathfrak{v})$ does not exist but $l(\mathfrak{v})$ exists then we may ask if the equation

has a solution. If one solution exists then finitely many such solutions exist for if $\mathfrak{y}^{\prime}$ is a solution and $[1,\bar{l}](\mathfrak{v})\neq\bar{l}(\mathfrak{y}^{\prime})$ then $\bar{l}(\mathfrak{y}^{\prime})$ is also a solution. As explained above we can define a fundamental solution of Equation (2) and introduce the notations $[l,\bar{l}]$ or $[\bar{l},l]$ depending on the comparison of lengths.

In the above two paragraphs we introduced new expressions using two kinds of bracket operations, namely $\langle\bar{l},l\rangle$ and $[\bar{l},l]$. We say that such expressions are ‘terms’–the precise definition is given below.

If $\mathfrak{y}=\langle\bar{l},l\rangle(\mathfrak{v})$ then the term $\langle\bar{l},l\rangle$ labels the path from $\mathfrak{v}$ to $\mathfrak{y}$ in $\hat{H}_{l}(\mathfrak{x}_{0})$. Let us describe the path from $\mathfrak{v}$ to any element $\mathfrak{u}$ in the interval $[\mathfrak{v},\mathfrak{y}]$. For $\mathfrak{u}$ there exists some $n\geq 1$ such that either $\mathfrak{u}=l^{n}(\mathfrak{v})$ or $\mathfrak{u}=\bar{l}^{n}(\mathfrak{y})$. In the former case, we know the label of the required path, namely $l^{n}$, whereas in the latter case, we use concatenation operation on terms, written juxtaposition, to label the path as $\bar{l}^{n}\langle\bar{l},l\rangle$. Here we acknowledge that the only path to reach such $\mathfrak{u}$ from $\mathfrak{v}$ has to pass through $\mathfrak{y}$ which is captured using the bracket term. In other words, the bracket term $\langle\bar{l},l\rangle$ describes a ‘ghost path’ which passes through each $l^{n}(\mathfrak{v})$ to reach the limit $\langle 1,l\rangle(\mathfrak{v})$, and then directly jumps to $\mathfrak{y}$ avoiding all elements of the interval $[\langle 1,l\rangle(\mathfrak{v}),\mathfrak{y}]$–such elements can only be accessed through reverse paths from $\mathfrak{y}$.

Once we have concatenation operation on terms as well as the bracket operation, it is natural to ask if more complex terms could be constructed by their combination. For example, we may ask if the expression $\langle\bar{l},l\rangle^{2}$ labels a path between two points in some domestic string algebra. In case the answer is affirmative, we can also ask the same question for the expression $\langle 1,\langle\bar{l},l\rangle\rangle:=\lim_{n\to\infty}\langle\bar{l},l\rangle^{n}$. Further we can also ask if there are solutions to equations of the form

In summary, we need to close the collection of terms under concatenation and both bracket operations, whenever such a term labels a path between two elements in some hammock in some domestic string algebra.

Now we are ready to formally define terms. Starting with symbols $l,\bar{l}$ we simultaneously inductively define the class of simple $l$-terms, denoted $\mathbf{T}_{l}$ and the class of simple $\bar{l}$-terms, denoted $\mathbf{T}_{\bar{l}}$ as follows:

• •

  $[\bar{l},l]\in\mathbf{T}_{l}$ and $[l,\bar{l}]\in\mathbf{T}_{\bar{l}}$;

• •

  if $\tau\in\mathbf{T}_{l}$ and $\mu\in\mathbf{T}_{\bar{l}}$ then $\langle\mu,\tau\rangle\in\mathbf{T}_{l}$ and $\langle\tau,\mu\rangle\in\mathbf{T}_{\bar{l}}$;

• •

  if $\tau,\tau^{\prime}\in\mathbf{T}_{l}$ and $\mu,\mu^{\prime}\in\mathbf{T}_{\bar{l}}$ then $\tau^{\prime}\tau\in\mathbf{T}_{l}$ and $\mu^{\prime}\mu\in\mathbf{T}_{\bar{l}}$.

Set $\mathbf{T}:=\mathbf{T}_{l}\cup\mathbf{T}_{\bar{l}}$ and call an element of $\mathbf{T}$ a simple term.

Define a map $\mathcal{I}:\mathbf{T}\to\mathbf{T}$ as follows:

• •

  $\mathcal{I}(l):=\bar{l}$ and $\mathcal{I}(\bar{l}):=l$;

• •

  $\mathcal{I}(\langle\mu,\tau\rangle):=\langle\mathcal{I}(\tau),\mathcal{I}(\mu)\rangle$;

• •

  $\mathcal{I}([\bar{l},l]):=[l,\bar{l}]$ and $\mathcal{I}([l,\bar{l}]):=[\bar{l},l]$;

• •

  $\mathcal{I}(\nu^{\prime}\nu):=\mathcal{I}(\nu^{\prime})\mathcal{I}(\nu)$.

Suppose $\tau,\tau_{1},\tau_{2}\in\mathbf{T}_{l}$ and $\mu,\mu_{1},\mu_{2}\in\mathbf{T}_{\bar{l}}$. Let $\approx_{l}$ be the equivalence relation on $\mathbf{T}_{l}$ generated by

and $\approx_{\bar{l}}$ be the equivalence relation on $\mathbf{T}_{\bar{l}}$ generated by

for each $m\geq 2$.

To make these into congruence relations, whenever $\tau_{1}\approx_{l}\tau_{2}$ and $\mu_{1}\approx_{\bar{l}}\mu_{2}$ we further need the following identifications:

Say that $\nu\in\mathbf{T}$ is irreducible if either $\nu\in\{l,\bar{l}\}$ or $\nu$ is of the form $\langle\mu,\tau\rangle$.

Given $\tau\in\mathbf{T}_{l}$ and $\mu\in\mathbf{T}_{\bar{l}}$, say that

• •

  $\mu\tau$ is a valid concatenation if $\mathcal{I}(\tau)\mu\approx_{\bar{l}}\mathcal{I}(\tau)$;

• •

  $\tau\mu$ is a valid concatenation if $\mathcal{I}(\mu)\tau\approx_{l}\mathcal{I}(\mu)$.

Say that a finite concatenation $\tau_{n}\ldots\tau_{2}\tau_{1}$ of simple terms is a mixed $l$-term (resp. mixed $\bar{l}$-term) if

• •

  $\tau_{k}\in\mathbf{T}_{l}$ if and only if $k$ is odd (resp. even);

• •

  $\tau_{k+1}\tau_{k}$ is a valid concatenation for each $1\leq k<n$.

Let $\mathbf{M}_{l}$ and $\mathbf{M}_{\bar{l}}$ denote the classes of all mixed $l$ and $\bar{l}$-terms respectively. Set $\mathbf{M}:=\mathbf{M}_{l}\cup\mathbf{M}_{\bar{l}}$ and say that a term is an element of $\mathbf{M}$.

The congruence relations $\approx_{l}$ and $\approx_{\bar{l}}$ are naturally extended to $\mathbf{M}_{l}$ and $\mathbf{M}_{\bar{l}}$ respectively.

Associate to each $\nu\in\mathbf{T}$ its order type, denoted $\mathcal{O}(\nu)$, as follows:

• •

  $\mathcal{O}(l)=\mathcal{O}(\bar{l})=\mathcal{O}([\bar{l},l])=\mathcal{O}([l,\bar{l}]):=\mathbf{1}$;

• •

  $\mathcal{O}(\langle\mu,\tau\rangle):=\omega\times\mathcal{O}(\tau)+\omega^{*}\times\mathcal{O}(\mu)$;

• •

  $\mathcal{O}(\tau^{\prime}\tau):=\mathcal{O}(\tau)+\mathcal{O}(\tau^{\prime})$.

Clearly $\mathcal{O}$ takes values in $\mathrm{dLO}_{\mathrm{fp}}^{\mathrm{b}}$.

In view of the above remark, the map $\mathcal{O}$ extends to $\mathbf{M}$ by setting

In view of Remarks 2.3 and 2.5 the next result follows from [1, Corollary 9.4].

Before we close the section, let us define a natural notation, $\ast$, on terms.

1. (1)

   $l\ast 1:=l$ and $l\ast(-1):=\bar{l}$.

2. (2)

   $\langle\tau,\eta\rangle\ast 1:=\langle\tau,\eta\rangle$ and $\langle\tau,\eta\rangle\ast(-1):=\langle\eta,\tau\rangle$.

3. (3)

   $[\tau,\eta]\ast 1:=[\tau,\eta]$ and $[\tau,\eta]\ast(-1):=[\eta,\tau]$.

Recall the notion of an H-pair from [9, § 5].

Suppose $\mathfrak{b}$ is a band, $\mathfrak{y}$ is a string and $i\in\{1,-1\}$. Say that the pair $(\mathfrak{b},\mathfrak{y})$ is an $i$-left valid pair ($i$-LVP for short) if $\mathfrak{b}_{\mathfrak{y}}\mathfrak{y}$ is a string for some cyclic permutation $\mathfrak{b}_{\mathfrak{y}}$ of $\mathfrak{b}$ such that $\theta(\mathfrak{b}_{\mathfrak{y}})=i$. Dually say that the pair $(\mathfrak{b},\mathfrak{y})$ is a $i$-right valid pair ($i$-RVP for short) if $(\mathfrak{b}^{-1},\mathfrak{y}^{-1})$ is a $(-i)$-LVP.

We drop the reference to $i$ in $i$-LVP if either it is clear from the context or if $i$ does not play any role in the statement.

For an LVP $(\mathfrak{b},\mathfrak{y})$, let $\bar{\mathfrak{w}}_{l}(\mathfrak{b},\mathfrak{y})$ be the longest common right substring (possibly of length $0$) of $\mathfrak{y}$ and $\mathfrak{b}_{\mathfrak{y}}\mathfrak{y}$, and $\mathfrak{w}_{l}(\mathfrak{b},\mathfrak{y}):=\rho_{r}(\bar{\mathfrak{w}}_{l}(\mathfrak{b},\mathfrak{y}))$. Dually for an RVP $(\mathfrak{b},\mathfrak{y})$, let $\bar{\mathfrak{w}}_{r}(\mathfrak{b},\mathfrak{y})$ be the longest common left substring (possibly of length $0$) of $\mathfrak{y}$ and $\mathfrak{y}\mathfrak{b}_{\mathfrak{y}}$, and $\mathfrak{w}_{r}(\mathfrak{b},\mathfrak{y}):=\rho_{l}(\bar{\mathfrak{w}}_{r}(\mathfrak{b},\mathfrak{y}))$.

Say that an LVP $(\mathfrak{b},\mathfrak{y})$ is a left tight valid pair (LTVP, for short) if $|\bar{\mathfrak{w}}_{l}(\mathfrak{b},\mathfrak{y})|=0$; otherwise say it is a left loose valid pair (LLVP, for short). Dually we define RTVP and RLVP.

For a long LVP $(\mathfrak{b},\mathfrak{y})$ the above proposition gives that $N(\mathfrak{b},\mathfrak{y})=0$.

Suppose $\mathfrak{x}\mathfrak{y}$ is a string. Since $(\mathfrak{b},\mathfrak{y})$ is $s$-long, $\mathfrak{w}_{l}(\mathfrak{b},\mathfrak{y})$ is a proper right substring of $\rho_{r}(\mathfrak{y})$ which states that $\theta(\gamma^{\prime})=-i$ where $\gamma^{\prime}\in\mathfrak{b}_{\mathfrak{y}}$ is the syllable such that $\mathfrak{w}_{l}(\mathfrak{b},\mathfrak{y})\gamma^{\prime}$ is a string. Since $\mathfrak{x}\mathfrak{w}_{l}(\mathfrak{b},\mathfrak{y})$ is a string and $\delta(\mathfrak{w}_{l}(\mathfrak{b},\mathfrak{y})\gamma^{\prime})=0$, $\mathfrak{x}\mathfrak{b}_{\mathfrak{y}}\mathfrak{y}$ is also a string. This shows that $\mathfrak{x}\mathfrak{y}\mapsto\mathfrak{x}\mathfrak{b}_{\mathfrak{y}}\mathfrak{y}$ is indeed a well-defined map from $H^{i}_{l}(\mathfrak{y})\to H^{i}_{l}(\mathfrak{b}_{\mathfrak{y}}\mathfrak{y})$. This map is clearly injective.

Let $\mathfrak{x}$ be the minimal string such that $\mathfrak{x}\rho_{r}(\mathfrak{y})\in\rho\cup\rho^{-1}$. Then clearly the word $\mathfrak{x}\mathfrak{y}$ is not a string but the argument as in the above paragraph ensures that $\mathfrak{x}\mathfrak{b}_{\mathfrak{y}}\mathfrak{y}$ is a string. Therefore the map is not surjective.

For the converse, non-surjectivity of the map guarantees the existence of a string $\mathfrak{x}$ of minimal positive length with $\theta(\mathfrak{x})=i$ such that $\mathfrak{x}\mathfrak{b}_{\mathfrak{y}}\mathfrak{y}$ is a string whereas the word $\mathfrak{x}\mathfrak{y}$ is not. Then there is some right substring $\mathfrak{y}^{\prime}$ of $\mathfrak{y}$ such that $\mathfrak{x}\mathfrak{y}^{\prime}\in\rho\cup\rho^{-1}$. The definition of $\rho_{r}(\mathfrak{y})$ ensures that $|\mathfrak{y}^{\prime}|\leq|\rho_{r}(\mathfrak{y})|$. Since $\mathfrak{x}\mathfrak{b}_{\mathfrak{y}}\mathfrak{y}^{\prime}$ is a string we see that $|\mathfrak{w}_{l}(\mathfrak{b},\mathfrak{y})|<|\mathfrak{y}^{\prime}|$ thus completing the proof that the pair $(\mathfrak{b},\mathfrak{y})$ is $s$-long.    $\Box$

Say that a long LVP $(\mathfrak{b},\mathfrak{y})$ is double long if it is both $s$-long or $b$-long; otherwise say that it is single long.

Fix $\mathfrak{x}_{0}\in\mathrm{Str}(\Lambda)$ and $i\in\{1,-1\}$. We wish to associate a rooted tree, $\mathcal{T}^{\mathfrak{f}}_{i}(\mathfrak{x}_{0})$, with some additional structure described in the definition below to the triple $(\Lambda,i,\mathfrak{x}_{0})$.

Here $T$ and $NT$ stands for ‘torsion’ and ‘non-torsion’ respectively. Recall that a string $\mathfrak{y}$ is a (left) torsion string if $\alpha\mathfrak{y}$ is not a string for any syllable $\alpha$.

Consider the set $\mathcal{A}^{f}_{i}(\mathfrak{x}_{0})$ of all arrows $\mathfrak{u}$ appearing in a path in $\mathcal{HQ}^{\mathrm{Ba}}_{i}(\mathfrak{x}_{0})$ starting from $\mathfrak{x}_{0}$. Recall from [9, Remark 6.2] that $\mathcal{A}^{f}_{i}(\mathfrak{x}_{0})$ is a finite set.

Since $\Lambda$ is domestic $\mathcal{T}^{\mathfrak{f}}_{i}(\mathfrak{x}_{0})$ is finite. Let $\mathcal{V}^{f}_{i}(\mathfrak{x}_{0})$ denote the set of vertices of the tree $\mathcal{T}^{\mathfrak{f}}_{i}(\mathfrak{x}_{0})$. Let $\mathcal{U}^{f}_{i}(\mathfrak{x}_{0}),\mathcal{H}^{f}_{i}(\mathfrak{x}_{0}),\mathcal{Z}^{f}_{i}(\mathfrak{x}_{0}),\mathcal{R}^{f}_{i}(\mathfrak{x}_{0})$ denote those elements of $\mathcal{V}^{f}_{i}(\mathfrak{x}_{0})$ that are arch-bridges, half i-arch bridges, torsion zero i-arch bridges and torsion reverse arch bridges respectively. These sets together with the root give a partition of $\mathcal{V}^{f}_{i}(\mathfrak{x}_{0})$.

Let $\xi^{f}_{i}(\mathfrak{x}_{0})$ denote the set of all children of $\mathfrak{x}_{0}$. For each non-root $\mathfrak{u}\in\mathcal{V}^{f}_{i}(\mathfrak{x}_{0})$ let $\xi^{f}(\mathfrak{u})$ denote the set of children of $\mathfrak{u}$. Partition $\xi^{f}(\mathfrak{u})$ as $\xi^{f}_{1}(\mathfrak{u})\sqcup\xi^{f}_{-1}(\mathfrak{u})$ by defining $\xi^{f}_{j}(\mathfrak{u}):=\{\mathfrak{u}^{\prime}\in\xi^{f}(\mathfrak{u})\mid\theta(\beta(\mathfrak{u}^{\prime}))=j\}$. For a non-root vertex $\mathfrak{u}$ let $\pi(\mathfrak{u})$ denote the parent of $\mathfrak{u}$. The height of $\mathfrak{u}\in\mathcal{V}^{f}_{i}(\mathfrak{x}_{0})$ is the non-negative integer $h(\mathfrak{u})$ such that $\pi^{h(\mathfrak{u})}(\mathfrak{u})=\mathfrak{x}_{0}$.

Define a function $\phi:\mathcal{V}^{f}_{i}(\mathfrak{x}_{0})\to\{1,0,-1\}$ by $\phi(\mathfrak{u}):=\begin{cases}0&\mbox{if }h(\mathfrak{u})=0;\\ -i&\mbox{if }h(\mathfrak{u})=1;\\ \theta(\beta(\mathfrak{u}))&\mbox{if }h(\mathfrak{u})>1.\end{cases}$