# Lengths of words in transformation semigroups generated by digraphs

## Abstract

Given a simple digraph D on n vertices (with $$n\ge 2$$), there is a natural construction of a semigroup of transformations $$\langle D\rangle$$. For any edge (ab) of D, let $$a\rightarrow b$$ be the idempotent of rank $$n-1$$ mapping a to b and fixing all vertices other than a; then, define $$\langle D\rangle$$ to be the semigroup generated by $$a \rightarrow b$$ for all $$(a,b) \in E(D)$$. For $$\alpha \in \langle D\rangle$$, let $$\ell (D,\alpha )$$ be the minimal length of a word in E(D) expressing $$\alpha$$. It is well known that the semigroup $$\mathrm {Sing}_n$$ of all transformations of rank at most $$n-1$$ is generated by its idempotents of rank $$n-1$$. When $$D=K_n$$ is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate $$\ell (K_n,\alpha )$$, for any $$\alpha \in \langle K_n\rangle = \mathrm {Sing}_n$$; however, no analogous non-trivial results are known when $$D \ne K_n$$. In this paper, we characterise all simple digraphs D such that either $$\ell (D,\alpha )$$ is equal to Howie–Iwahori’s formula for all $$\alpha \in \langle D\rangle$$, or $$\ell (D,\alpha ) = n - \mathrm {fix}(\alpha )$$ for all $$\alpha \in \langle D\rangle$$, or $$\ell (D,\alpha ) = n - \mathrm {rk}(\alpha )$$ for all $$\alpha \in \langle D\rangle$$. We also obtain bounds for $$\ell (D,\alpha )$$ when D is an acyclic digraph or a strong tournament (the latter case corresponds to a smallest generating set of idempotents of rank $$n-1$$ of $$\mathrm {Sing}_n$$). We finish the paper with a list of conjectures and open problems.

## Introduction

For any $$n \in \mathbb {N}$$, $$n \ge 2$$, let $$\mathrm {Sing}_n$$ be the semigroup of all singular (i.e. non-invertible) transformations on $$[n]:=\left\{ 1,\ldots ,n\right\}$$. It is well known (see ) that $$\mathrm {Sing}_n$$ is generated by its idempotents of defect 1 (i.e. the transformations $$\alpha \in \mathrm {Sing}_n$$ such that $$\alpha ^2 = \alpha$$ and $$\mathrm {rk}(\alpha ) := \vert \mathrm {Im}(\alpha ) \vert = n-1$$). There are exactly $$n(n-1)$$ such idempotents, and each one of them may be written as $$(a \rightarrow b)$$, for $$a,b \in [n]$$, $$a \ne b$$, where, for any $$v \in [n]$$,

\begin{aligned} (v) (a \rightarrow b) := {\left\{ \begin{array}{ll} b &{}\quad \text {if } v = a,\\ v &{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}

Motivated by this notation, we refer to these idempotents as arcs.

In this paper, we explore the natural connections between simple digraphs on [n] and subsemigroups of $$\mathrm {Sing}_n$$. For any subset $$U \subseteq \mathrm {Sing}_n$$, denote by $$\langle U \rangle$$ the semigroup generated by U. For any simple digraph D with vertex set $$V(D)=[n]$$ and edge set E(D), we associate the semigroup

\begin{aligned} \langle D \rangle := \left\langle (a \rightarrow b) \in \mathrm {Sing}_n : (a,b) \in E(D) \right\rangle . \end{aligned}

We say that a subsemigroup S of $$\mathrm {Sing}_n$$ is arc-generated by a simple digraph D if $$S=\langle D \rangle$$.

For the rest of the paper, we use the term ‘digraph’ to mean ‘simple digraph’ (i.e. a digraph with no loops or multiple edges). A digraph D is undirected if its edge set is a symmetric relation on V(D), and it is transitive if its edge set is a transitive relation on V(D). We shall always assume that D is connected (i.e. for every pair $$u, v \in V(D)$$ there is either a path from u to v, or a path from v to u) because otherwise $$\langle D \rangle \cong \langle D_1\rangle \times \dots \times \langle D_k\rangle$$, where $$D_1, \dots , D_k$$ are the connected components of D. We say that D is strong (or strongly connected) if for every pair $$u,v \in V(D)$$, there is a directed path from u to v. We say that D is a tournament if for every pair $$u,v \in V(D)$$ we have $$(u,v) \in E(D)$$ or $$(v,u) \in E(D)$$, but not both.

Many famous examples of semigroups are arc-generated. Clearly, by the discussion of the first paragraph, $$\mathrm {Sing}_n$$ is arc-generated by the complete undirected graph $$K_n$$. In fact, for $$n \ge 3$$, $$\mathrm {Sing}_n$$ is arc-generated by D if and only if D contains a strong tournament (see ). The semigroup of order-preserving transformations $$\text {O}_n := \{ \alpha \in \mathrm {Sing}_n : u \le v \Rightarrow u \alpha \le v \alpha \}$$ is arc-generated by an undirected path $$P_n$$ on [n], while the Catalan semigroup $$\text {C}_n := \{ \alpha \in \mathrm {Sing}_n : v \le v \alpha , u \le v \Rightarrow u \alpha \le v \alpha \}$$ is arc-generated by a directed path $$\vec {P}_n$$ on [n] (see [9, Corollary 4.11]). The semigroup of non-decreasing transformations $$\text {OI}_n := \{ \alpha \in \mathrm {Sing}_n : v \le v \alpha \}$$ is arc-generated by the transitive tournament $$\vec {T}_n$$ on [n] (Fig. 1 illustrates $$\vec {T}_5$$).

Connections between subsemigroups of $$\mathrm {Sing}_n$$ and digraphs have been studied before (see ). The following definition, which we shall adopt in the following sections, appeared in :

### Definition 1

For a digraph D, the closure $${\bar{D}}$$ of D is the digraph with vertex set $$V({\bar{D}}) := V\left( D\right)$$ and edge set $$E({\bar{D}}):=E\left( D\right) \cup \left\{ \left( a,b \right) :\left( b ,a \right) \in E\left( D\right) \text { is in a} \text {directed cycle of} D\right\}$$.

Say that D is closed if $$D = {\bar{D}}$$. Observe that $$\langle D \rangle = \langle {\bar{D}} \rangle$$ for any digraph D.

Recall that the orbits of $$\alpha \in \mathrm {Sing}_n$$ are the connected components of the digraph on [n] with edges $$\{ (x, x\alpha ) : x \in [n] \}$$. In particular, an orbit $$\Omega$$ of $$\alpha$$ is called cyclic if it is a cycle with at least two vertices. An element $$x \in [n]$$ is a fixed point of $$\alpha$$ if $$x\alpha =x$$. Denote by $$\mathrm {cycl}(\alpha )$$ and $$\mathrm {fix}(\alpha )$$ the number of cyclic orbits and fixed points of $$\alpha$$, respectively. Denote by $$\ker (\alpha )$$ the partition of [n] induced by the kernel of $$\alpha$$ (i.e. the equivalence relation $$\{ (x,y) \in [n]^2 : x\alpha = y \alpha \}$$).

We introduce some further notation. For any digraph D and $$v \in V(D)$$, define the in-neighbourhood and the out-neighbourhood of v by

\begin{aligned} N^-(v) := \{ u \in V(D) : (u, v) \in E(D)\} \text { and } N^+(v) := \{u \in V(D) : (v,u) \in E(D) \}, \end{aligned}

respectively. We extend these definitions to any subset $$C \subseteq V(D)$$ by letting $$N^\epsilon (C) := \bigcup _{c \in C} N^\epsilon (c)$$, where $$\epsilon \in \{+,- \}$$. The in-degree and out-degree of v are $$\deg ^-(v):=\vert N^-(v) \vert$$ and $$\deg ^+(v):=\vert N^+(v) \vert$$, respectively, while the degree of v is $$\deg (v) := \vert N^-(v) \cup N^+(v) \vert$$. For any two vertices $$u,v \in V(D)$$, the D-distance from u to v, denoted by $$d_D(u,v)$$, is the length of a shortest path from u to v in D, provided that such a path exists. The diameter of D is $$\mathrm {diam}(D) := \max \{ d_D(u,v) : u,v \in V(D), \ d_D(u,v) \text { is defined} \}$$.

Let D be any digraph on [n]. We are interested in the lengths of transformations of $$\langle D\rangle$$ viewed as words in the free monoid $$D^* := \{ (a \rightarrow b ) : (a,b)\in E(D)\}^*$$. Say that a word $$\omega \in D^*$$ expresses (or evaluates to) $$\alpha \in \langle D\rangle$$ if $$\alpha = \omega \phi$$, where $$\phi : D^* \rightarrow \langle D\rangle$$ is the evaluation semigroup morphism. For any $$\alpha \in \langle D \rangle$$, let $$\ell (D,\alpha )$$ be the minimum length of a word in $$D^*$$ expressing $$\alpha$$. For $$r \in [n-1]$$, denote

\begin{aligned} \ell (D,r)&:= \max \left\{ \ell (D,\alpha ) : \alpha \in \langle D \rangle , \mathrm {rk}(\alpha ) = r \right\} ,\\ \ell (D)&:= \max \left\{ \ell (D,\alpha ) : \alpha \in \langle D \rangle \right\} . \end{aligned}

The main result in the literature in the study of $$\ell (D,\alpha )$$ was obtained by Howie and Iwahori, independently, when $$D = K_n$$.

### Theorem 1.1

[4, 5] For any $$\alpha \in \mathrm {Sing}_n$$,

\begin{aligned} \ell (K_n, \alpha ) = n+ \mathrm {cycl}(\alpha ) - \mathrm {fix}(\alpha ). \end{aligned}

Therefore, $$\ell (K_n, r) = n + \left\lfloor \frac{1}{2} (r-2) \right\rfloor$$, for any $$r \in [n-1]$$, and $$\ell (K_n) = \ell (K_n, n-1) = \left\lfloor \frac{3}{2} (n-1) \right\rfloor$$.

In the following sections, we study $$\ell (D, \alpha )$$, $$\ell (D,r)$$, and $$\ell (D)$$, for various classes of digraphs. In Sect. 2, we characterise all digraphs D on [n] such that either $$\ell (D, \alpha ) = n + \mathrm {cycl}(\alpha ) - \mathrm {fix}(\alpha )$$ for all $$\alpha \in \langle D\rangle$$, or $$\ell (D, \alpha ) =n - \mathrm {fix}(\alpha )$$ for all $$\alpha \in \langle D\rangle$$, or $$\ell (D, \alpha ) =n - \mathrm {rk}(\alpha )$$ for all $$\alpha \in \langle D\rangle$$. In Sect. 3, we are interested in the maximal possible length of a transformation in $$\langle D\rangle$$ of rank r among all digraphs D on [n] of certain class $$\mathcal {C}$$; we denote this number by $$\ell _{\max }^{\mathcal {C}}(n,r)$$. In particular, when $$\mathcal {C}$$ is the class of acyclic digraphs, we find an explicit formula for $$\ell _{\max }^{\mathcal {C}}(n,r)$$. When $$\mathcal {C}$$ is the class of strong tournaments, we find upper and lower bounds for $$\ell _{\max }^{\mathcal {C}}(n,r)$$ (and for the analogously defined $$\ell _{\min }^{\mathcal {C}}(n,r)$$). Finally, in Sect. 4 we provide a list of conjectures and open problems.

## Arc-generated semigroups with short words

Let D be a digraph on [n], $$n \ge 3$$, and $$\alpha \in \langle D \rangle$$. Theorem 1.1 implies the following three bounds:

\begin{aligned} \ell (D, \alpha ) \ge n + \mathrm {cycl}(\alpha ) - \mathrm {fix}(\alpha ) \ge n - \mathrm {fix}(\alpha ) \ge n- \mathrm {rk}(\alpha ). \end{aligned}
(1)

The lowest bound is always achieved for constant transformations (i.e. transformations of rank 1).

### Lemma 2.1

For any digraph D on [n], if $$\alpha \in \langle D \rangle$$ has rank 1, then $$\ell (D, \alpha ) = n - 1$$.

### Proof

It is clear that $$\ell (D, \alpha ) \ge n-1$$ because $$\alpha$$ has $$n-1$$ non-fixed points. Let $$\mathrm {Im}(\alpha ) = \{v_0\} \subseteq [n]$$. Note that, for any $$v \in [n]$$, there is a directed path in D from v to $$v_0$$ (as otherwise, $$\alpha \not \in \langle D \rangle$$). For any $$d \ge 1$$, let

\begin{aligned} C_d := \{ v \in [n] : d_D(v,v_0) = d\}. \end{aligned}

Clearly, $$[n] {\setminus } \{ v_0 \}= \bigcup _{d=1}^{m} C_d$$, where $$m := \max _{v\in [n]}\{ d_{D}(v,v_0) \}$$ and the union is disjoint. For any $$v \in C_d$$, let $$v'$$ be a vertex in $$C_{d-1}$$ such that $$(v \rightarrow v') \in D$$. For any distinct $$v, u \in C_d$$ and any choice of $$v', u' \in C_{d-1}$$, the arcs $$(v \rightarrow v')$$ and $$(u \rightarrow u')$$ commute; hence, we can decompose $$\alpha$$ as

\begin{aligned} \alpha = \bigcirc _{d=m}^1 \bigcirc _{v \in C_d} (v \rightarrow v'), \end{aligned}

where the composition of arcs is done from m down to 1. $$\square$$

### Remark 1

Using a similar argument as in the previous proof, we may show that $$\langle D \rangle$$ contains all constant transformations if and only if D is strongly connected.

Inspired by the bounds given in (1), we characterise all the connected digraphs D on [n] satisfying the following conditions:

### Digraphs satisfying condition (C1)

Theorem 1.1 says that $$K_n$$ satisfies (C1). In order to characterise all digraphs satisfying (C1), we introduce the following property on a digraph D:

($$\star$$):

If $$d_D(v_0, v_2) = 2$$ and $$v_0,v_1,v_2$$ is a directed path in D, then $$N^+\left( \{v_1,v_2\}\right) \subseteq \{v_0,v_1,v_2\}$$.

We shall study the strong components of digraphs satisfying property $$(\star )$$. We state few observations that we use repeatedly in this section.

### Remark 2

Suppose that D satisfies property $$(\star )$$. If $$v_0, v_1, v_2$$ is a directed path in D and $$\deg ^+(v_1) >2$$, or $$\deg ^+(v_2) >2$$, then $$(v_0, v_2) \in E(D)$$. Indeed, if $$(v_0, v_2) \not \in E(D)$$, then $$d_D(v_0, v_2) = 2$$, so, by property $$(\star )$$, $$N^+\left( \{v_1,v_2\}\right) \subseteq \{v_0,v_1,v_2\}$$; this contradicts that $$\deg ^+(v_1) >2$$, or $$\deg ^+(v_2) >2$$.

### Remark 3

Suppose that D satisfies property $$(\star )$$. If $$v_0, v_1, v_2$$ is a directed path in D and either $$v_1$$ or $$v_2$$ has an out-neighbour not in $$\{ v_0, v_1, v_2 \}$$, then $$(v_0,v_1) \in E(D)$$.

### Remark 4

If D satisfies property $$(\star )$$, then $$\mathrm {diam}(D) \le 2$$. Indeed, if $$v_0, v_1, \dots , v_k$$ is a directed path in D with $$d_D (v_0, v_k) = k \ge 3$$, then $$v_0, v_1, v_2$$ is a directed path in D and $$v_2$$ has an out-neighbour $$v_3 \not \in \{ v_0, v_1, v_2 \}$$; by Remark 3, $$(v_0,v_2)\in E(D)$$, which contradicts that $$d_D (v_0, v_k) = k$$.

Note that digraphs satisfying property $$(\star )$$ are a slight generalisation of transitive digraphs.

Let D be a digraph and let $$C_1$$ and $$C_2$$ of be two strong components of D. We say that $$C_1$$ connects to $$C_2$$ if $$(v_1, v_2) \in E(D)$$ for some $$v_1 \in C_1$$, $$v_2 \in C_2$$; similarly, we say that $$C_1$$ fully connects to $$C_2$$ if $$(v_1, v_2) \in E(D)$$ for all $$v_1 \in C_1$$, $$v_2 \in C_2$$. The strong component $$C_1$$ is called terminal if there is no strong component $$C \ne C_1$$ of D such that $$C_1$$ connects to C.

### Lemma 2.2

Let D be a closed digraph satisfying property $$(\star )$$. Then, any strong component of D is either an undirected path $$P_3$$ or complete. Furthermore, $$P_3$$ may only appear as a terminal strong component of D.

### Proof

Let C be a strong component of D. Since D is closed, C must be undirected. The lemma is clear if $$\vert C \vert \le 3$$, so assume that $$\vert C \vert \ge 4$$. We have two cases:

Case 1 :

Every vertex in C has degree at most 2. Then C is a path or a cycle. Since $$\vert C \vert \ge 4$$ and $$\mathrm {diam}(D) \le 2$$, then C is a cycle of length 4 or 5; however, these cycles do not satisfy property $$(\star )$$.

Case 2 :

There exists a vertex $$a \in C$$ of degree 3 or more. Any two neighbours of a are adjacent: indeed, for any $$u,v \in N(a)$$, uav is a path and $$\deg ^+(a) > 2$$, so $$(u,v) \in E(D)$$ by Remark 2. Hence, the neighbourhood of a is complete and every neighbour of a has degree 3 or more. Applying this rule recursively, we obtain that every vertex in C has degree 3 or more, and the neighbourhood of every vertex is complete. Therefore, C is complete because $$\mathrm {diam}(D) \le 2$$.

Finally, if $$P_3$$ is a strong component of D, there cannot be any edge coming out of it because of property $$(\star )$$, so it must be a terminal component. $$\square$$

### Lemma 2.3

Let D be a closed digraph satisfying property $$(\star )$$. Let $$C_1$$ and $$C_2$$ be strong components of D, and suppose that $$C_1$$ connects to $$C_2$$.

1. (i)

If $$C_2$$ is non-terminal, then $$C_1$$ fully connects to $$C_2$$.

2. (ii)

Let $$|C_2| = 1$$. If either $$|C_1| \ne 2$$, or the vertex in $$C_1$$ that connects to $$C_2$$ has out-degree at least 3, then $$C_1$$ fully connects to $$C_2$$.

3. (iii)

Let $$|C_2| = 2$$. If not all vertices in $$C_1$$ connect to the same vertex in $$C_2$$, then $$C_1$$ fully connects to $$C_2$$.

4. (iv)

If $$|C_2| \ge 3$$, then $$C_1$$ fully connects to $$C_2$$.

### Proof

Recall that $$C_1$$ and $$C_2$$ are undirected because D is closed. If $$|C_1| = 1$$ and $$|C_2| = 1$$, clearly $$C_1$$ fully connects to $$C_2$$. Henceforth, we assume $$|C_1| \ge 2$$ or $$|C_2| \ge 2$$. Let $$c_1 \in C_1$$ and $$c_2 \in C_2$$ be such that $$(c_1,c_2) \in E(D)$$. As $$C_1$$ is a non-terminal, Lemma 2.2 implies that $$C_1$$ is complete.

1. (i)

As $$C_2$$ is non-terminal, there exists $$d \in D {\setminus } (C_1 \cup C_2)$$ such that $$(c_2,d) \in E(D)$$. Suppose that $$|C_1| \ge 2$$. Then, for any $$c'_1 \in C_1 {\setminus } \{ c_1 \}$$, $$c'_1, c_1, c_2$$ is a directed path in D with $$d \in N^{+}( c_2 )$$, so Remark 3 implies $$(c'_1, c_2) \in E(D)$$. Suppose now that $$|C_2| \ge 2$$. Then, for any $$c'_2 \in C_2 {\setminus } \{ c_2\}$$, $$c_1, c_2, c'_2$$ is a directed path in D with $$d \in N^{+}(c_2)$$, so again $$(c_1, c'_2) \in E(D)$$. Therefore, $$C_1$$ fully connects to $$C_2$$.

2. (ii)

Suppose that $$|C_1| \ge 2$$. If $$\vert C_1 \vert >2$$, then $$\deg ^+(c_1) > 2$$, because $$C_1$$ is complete. Thus, for each $$c'_1 \in C_1 {\setminus } \{ c_1 \}$$, $$c'_1, c_1, c_2$$ is a directed path in D with $$\deg ^+(c_1) > 2$$, so $$(c'_1, c_2) \in E(D)$$ by Remark 2. As $$|C_2| = 1$$, this shows that $$C_1$$ fully connects to $$C_2$$.

3. (iii)

Let $$C_2 = \{ c_2, c'_2 \}$$ and let $$c'_1 \in C_1 {\setminus } \{ c_1 \}$$ be such that $$(c'_1, c'_2) \in E(D)$$. For any $$b ,d\in C_1$$, $$b \ne c_1$$, $$d \ne c'_1$$, both $$b, c_1, c_2$$ and $$d, c'_1, c'_2$$ are directed paths in D with $$c'_2 \in N^+(c_2)$$ and $$c_2 \in N^+(c'_2)$$; hence, $$(b,c_2) , (d, c'_2) \in E(D)$$ by Remark 3.

4. (iv)

Suppose that $$C_2 = P_3$$. Say $$C_2 = \{ c_2, c'_2, c''_2\}$$ with either $$d_{D}(c_2, c''_2)=2$$ or $$d_{D}(c'_2, c''_2)=2$$. In any case, $$c_1, c_2, c'_2$$ is a directed path in D with $$c''_2 \in N^+ (\{c_2, c'_2 \})$$, so $$(c_1, c'_2) \in E(D)$$ by Remark 3; now, $$c_1, c'_2, c''_2$$ is a directed path in D with $$c_2 \in N^+ (\{c'_2, c''_2 \})$$, so $$(c_1, c''_2) \in E(D)$$. Hence, $$c_1$$ is connected to all vertices of $$C_2$$. As $$C_1$$ is complete, a similar argument shows that every $$c'_1 \in C_1 {\setminus } \{ c_1\}$$ connects to every vertex in $$C_2$$.

Suppose now that $$C_2 = K_m$$ for $$m \ge 3$$. By a similar reasoning as the previous paragraph, we show that $$(c_1, v) \in E(D)$$ for all $$v \in C_2$$. Now, for any $$c'_1 \in C_1 {\setminus } \{ c_1 \}$$, $$v \in C_2$$, $$c'_1, c_1, v$$ is a directed path in D so $$(c'_1, v) \in E(D)$$ by Remark 3. $$\square$$

### Lemma 2.4

Let D be a closed digraph satisfying property $$(\star )$$. Let $$C_i$$, $$i=1,2,3$$, be strong components of D, and suppose that $$C_1$$ connects to $$C_2$$ and $$C_2$$ connects to $$C_3$$. If $$C_1$$ does not connect to $$C_3$$, then $$|C_2| = |C_3| = 1$$, $$C_3$$ is terminal in D, and $$C_2$$ is terminal in $$D {\setminus } C_3$$.

### Proof

By Lemma 2.3 (i), $$C_1$$ fully connects to $$C_2$$. Assume that $$C_1$$ does not connect to $$C_3$$. Let $$c_i \in C_i$$, $$i=1,2,3$$, be such that $$(c_1, c_2), (c_2, c_3) \in E(D)$$. If $$C_2$$ has a vertex different from $$c_2$$, Remark 3 ensures that $$(c_1, c_3) \in E(D)$$, which contradicts our hypothesis. Then $$\vert C_2 \vert =1$$. The same argument applies if $$C_3$$ has a vertex different from $$c_3$$, so $$\vert C_3 \vert =1$$. Finally, Remark 3 applied to the path $$c_1, c_2, c_3$$ also implies that $$C_3$$ is terminal in D and $$C_2$$ is terminal in $$D {\setminus } C_3$$. $$\square$$

The following result characterises all digraphs satisfying condition (C1).

### Theorem 2.5

Let D be a connected digraph on [n]. The following are equivalent:

1. (i)

For all $$\alpha \in \langle D \rangle$$, $$\ell (D, \alpha ) = n + \mathrm {cycl}(\alpha ) - \mathrm {fix}(\alpha )$$.

2. (ii)

D is closed satisfying property $$(\star )$$.

### Proof

In order to simplify notation, denote

\begin{aligned} g(\alpha ) := n + \mathrm {cycl}(\alpha ) - \mathrm {fix}(\alpha ). \end{aligned}

First, we show that (i) implies (ii). Suppose $$\ell (D,\alpha ) = g(\alpha )$$ for all $$\alpha \in \langle D\rangle$$. We use the one-line notation for transformations: $$\alpha = (1)\alpha \ (2)\alpha \ \dots \ (k)\alpha$$, where $$x=(x)\alpha$$ for all $$x >k$$, $$x \in [n]$$. Clearly, if D is not closed, there exists an arc $$\alpha \in \langle D\rangle \backslash D$$, so $$1 < \ell (D, \alpha ) \ne g(\alpha ) = 1$$. In order to prove that property $$(\star )$$ holds, let 1, 2, 3 be a shortest path in D. If $$(2 \rightarrow v) \in \langle D\rangle$$, for some $$v \in [n]{\setminus }\{1,2,3\}$$, then $$\alpha = 3v3v \in \langle D\rangle$$, but $$g(\alpha ) = 2 \ne \ell (D,\alpha ) = 3$$. If $$(3 \rightarrow v) \in \langle D\rangle$$, then $$\alpha = 3vvv \in \langle D\rangle$$, but $$g(\alpha ) = 3 \ne \ell (D,\alpha ) = 4$$. Therefore, $$N^+(\{2,3 \}) \subseteq \{ 1,2,3 \}$$, and $$(\star )$$ holds.

Conversely, we show that (ii) implies (i). Let $$\alpha \in \langle D\rangle$$. We remark that any cycle of $$\alpha$$ belongs to a strong component of D.

### Claim 2.6

Let C be a strong component of D. Then either $$\alpha$$ fixes all vertices of C or $$|(C \alpha ) \cap C| < |C|$$.

### Proof

Suppose that $$\alpha \vert _C$$, the restriction of $$\alpha$$ to C, is non-trivial and $$|(C \alpha ) \cap C| = |C|$$. Then $$\alpha \vert _C$$ is a permutation of C. Let $$u \in C$$ and suppose that $$(u \rightarrow v)$$ is the first arc moving u in a word expressing $$\alpha$$ in $$D^*$$. If $$v \in C$$, we have $$u \alpha = v \alpha$$, which contradicts that $$\alpha \vert _C$$ is a permutation. If $$v \in C'$$ for some other strong component $$C'$$ of D, then $$u \alpha \notin C$$ which again contradicts our assumption. $$\square$$

### Claim 2.7

Let $$u,v \in [n]$$ be such that $$u \alpha = v$$. If $$d_D(u, v) = 2$$, then:

1. 1.

v is in a terminal component of D.

2. 2.

There is a path uwv of length 2 in D such that $$w \alpha = v \alpha = v$$; for any other path uxv of length 2 in D, we have $$x \alpha \in \{x, v\}$$.

### Proof

Let $$C_1$$ and $$C_2$$ be strong components of D such that $$u \in C_1$$ and $$v \in C_2$$. We analyse the four possible cases in which $$d_D(u,v) = 2$$. In the first three cases, we use the fact that $$\langle P_3\rangle \cong {\mathrm {O}}_3$$, hence we can order $$u< w < v$$ and $$\alpha$$ is an increasing transformation of the ordered set $$\{u,w,v\}$$; thus $$u \alpha = w \alpha = v \alpha = v$$.

Case 1 :

$$C_1 = C_2$$. By Lemma 2.2, $$C_1 \cong P_3$$ and it is a terminal component. Therefore, 2. holds as there is a unique path from u to v.

Case 2 :

$$C_1$$ connects to $$C_2$$ and $$|C_2| \ne 2$$. As $$d_D(u, v) = 2$$, $$C_1$$ does not fully connect $$C_2$$, so, by Lemma 2.3, $$|C_2| = 1$$, $$C_2$$ is terminal, $$|C_1| = 2$$, and the vertex $$w \in C_1$$ connecting to $$C_2=\{ v\}$$ has out-degree 2. Then, by property $$(\star )$$, uwv is the unique path from u to v.

Case 3 :

$$C_1$$ connects to $$C_2$$ and $$|C_2| = 2$$. As $$d_D(u, v) = 2$$, $$C_1$$ does not fully connect $$C_2$$, so, by Lemma 2.3, $$C_2$$ is terminal and uwv is the unique path of length two from u to v, where w is the other vertex of $$C_2$$.

Case 4 :

$$C_1$$ does not connect to $$C_2$$. Since $$d_D(u, v) = 2$$, there exist strong components $$C^{(1)}, \dots , C^{(k)}$$ such that $$C_1$$ connects to $$C^{(i)}$$ and $$C^{(i)}$$ connects to $$C_2$$, for all $$1 \le i \le k$$. By Lemma 2.4, $$C^{(i)} = \{ x_i \}$$, $$C_2 = \{v\}$$ is terminal and $$N^+(x_i) = \{v\}$$ for all i. Thus $$u, x_i, v$$ are the only paths of length two from u to v; in particular, $$x_i \alpha \in \{x_i, v\}$$ for all $$x_i$$. As $$u \alpha = v$$, there must exist $$1 \le j \le k$$ such that $$w := x_j$$ is mapped to v. $$\square$$

Now we produce a word $$\omega \in D^*$$ expressing $$\alpha$$ of length $$g(\alpha )$$. Define

\begin{aligned} U := \{ u \in D : d_D(u, u\alpha ) = 2 \}. \end{aligned}

For every $$u \in U$$, let $$u^\prime$$ be a vertex in D such that $$u, u^\prime , u \alpha$$ is a path and $$u^\prime \alpha = u \alpha$$. The existence of $$u'$$ is guaranteed by Claim 2.7. Define a word $$\omega _0 \in D^*$$ by

\begin{aligned} \omega _0 := \bigcirc _{u \in U} (u \rightarrow u^\prime ) (u^\prime \rightarrow u \alpha ). \end{aligned}

Sort the strong components of D in topological order: $$C_1, \dots , C_k$$, i.e. for $$i \ne j$$, $$C_i$$ connects to $$C_j$$ only if $$j > i$$. For each $$1 \le i \le k$$, define

\begin{aligned} S_i := \{v \in C_i {\setminus } (U \cup U^\prime ) : v \alpha \in C_i\}, \end{aligned}

where $$U^\prime := \{ u^\prime : u \in U \}$$, and consider the transformation $$\beta _i : C_i \rightarrow C_i$$ defined by

\begin{aligned} x \beta _i = {\left\{ \begin{array}{ll} x \alpha &{}\quad \text {if } x \in S_i\\ x &{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}

If $$|C_i| \le 2$$ or $$C_i \cong P_3$$, then $$\mathrm {cycl}(\beta _i) = 0$$ and $$\beta _i$$ can be computed with $$\vert C_i \vert - \mathrm {fix}(\beta _i)$$ arcs. Otherwise, $$C_i$$ is a complete undirected graph. If $$\beta _i \in \mathrm {Sing}(C_i)$$, then by Theorem 1.1, there is a word $$\omega _i \in C_i^* \subseteq D^*$$ of length $$\vert C_i \vert + \mathrm {cycl}(\beta _i) - \mathrm {fix}(\beta _i)$$ expressing $$\beta _i$$. Suppose now that $$\beta _i$$ is a non-identity permutation of $$C_i$$. By Claim 2.6, $$\alpha$$ does not permute $$C_i$$ and there exists $$h_i \in C_i {\setminus } (C_i \alpha )$$. Note that $$h_i \in C_i {\setminus } S_i$$. Define $${\hat{\beta _i}} \in \mathrm {Sing}(C_i)$$ by

\begin{aligned} x {\hat{\beta _i}} = {\left\{ \begin{array}{ll} x \alpha &{}\quad \text {if }x\in S_i\\ a_i &{} \quad \text {if } x = h_i \\ x &{}\quad \text {otherwise}, \end{array}\right. } \end{aligned}

where $$a_i$$ is any vertex in $$S_i$$. Then $$\alpha \vert _{S_i} = {\hat{\beta }}\vert _{S_i}$$. Again by Theorem 1.1, there is a word $$\omega _i \in C_i^* \subseteq D^*$$ of length $$\vert C_i \vert + \mathrm {cycl}({\hat{\beta }}_i) - \mathrm {fix}({\hat{\beta }}) =\vert C_i \vert + \mathrm {cycl}(\beta _i) - \mathrm {fix}(\beta _i)$$ expressing $${\hat{\beta }}_i$$.

The following word maps all the vertices in $$[n] {\setminus } (U \cup U^\prime \cup C_i)$$ that have image in $$C_i$$:

\begin{aligned} \omega _i' = \bigcirc \left\{ (a \rightarrow a\alpha ) : a \in [n] {\setminus } (U \cup U^\prime \cup C_i), a\alpha \in C_i \right\} . \end{aligned}

Finally, let

\begin{aligned} \omega := \omega _0 \omega _k \omega _k' \dots \omega _1 \omega _1' \in D^*. \end{aligned}

It is easy to check that $$\omega$$ indeed expresses $$\alpha$$. Since $$\sum _{i=1}^k \mathrm {fix}(\beta _i) = \mathrm {fix}(\alpha ) + \sum _{i=1}^k \vert C_i {\setminus } S_i \vert$$ and $$\sum _{i=1}^k \ell (\omega ^\prime _i) = \sum _{i=1}^k \vert C_i {\setminus } ( U \cup U^\prime \cup S_i) \vert$$, we have

\begin{aligned} \ell (\omega ) = 2 \vert U \vert + \sum _{i=1}^k (\ell (\omega _i) + \ell (\omega _i^\prime )) = n + \sum _{i=1}^k \mathrm {cycl}(\beta _i) - \mathrm {fix}(\alpha ) = g(\alpha ). \end{aligned}

$$\square$$

### Digraphs satisfying condition (C2)

The characterisation of connected digraphs satisfying condition (C2) is based on the classification of connected digraphs D such that $$\mathrm {cycl}(\alpha ) = 0$$, for all $$\alpha \in \langle D\rangle$$.

For $$k \ge 3$$, let $$\Theta _k$$ be the directed cycle of length k. Consider the digraphs $$\varGamma _1, \ \varGamma _2, \ \varGamma _3$$ and $$\varGamma _4$$ as illustrated below:

### Lemma 2.8

Let D be a connected digraph on [n]. The following are equivalent:

1. (i)

For all $$\alpha \in \langle D \rangle$$, $$\mathrm {cycl}(\alpha ) = 0$$.

2. (ii)

D has no subdigraph isomorphic to $$\varGamma _1$$, $$\varGamma _2$$, $$\varGamma _3$$, $$\varGamma _4$$, or $$\Theta _k$$, for all $$k \ge 5$$.

### Proof

In order to prove that (i) implies (ii), we show that if $$\varGamma$$ is equal to $$\varGamma _i$$ or $$\Theta _k$$, for $$i\in $$, $$k \ge 5$$, then there exists $$\alpha \in \langle \varGamma \rangle$$ such that $$\mathrm {cycl}(\alpha ) \ne 0$$.

• If $$\varGamma = \varGamma _1$$, take

\begin{aligned} \alpha:= & {} (3 \rightarrow 4) (4 \rightarrow 5) (1 \rightarrow 4) (4 \rightarrow 3) (2 \rightarrow 4) (4 \rightarrow 1) (3 \rightarrow 4) (4 \rightarrow 2) \\= & {} 21555. \end{aligned}
• If $$\varGamma = \varGamma _2$$, take

\begin{aligned} \alpha:= & {} (3 \rightarrow 4) (4 \rightarrow 5) (1 \rightarrow 3) (3 \rightarrow 4) (2 \rightarrow 3) (3 \rightarrow 1) (4 \rightarrow 3) (3 \rightarrow 2) \\= & {} 21555. \end{aligned}
• If $$\varGamma = \varGamma _3$$, take

\begin{aligned} \alpha := (3 \rightarrow 4) (2 \rightarrow 3) (1 \rightarrow 2) (3 \rightarrow 1) = 2144. \end{aligned}
• If $$\varGamma = \varGamma _4$$, take

\begin{aligned} \alpha = (3 \rightarrow 4) (4 \rightarrow 5) (2 \rightarrow 3) (3 \rightarrow 4) (1 \rightarrow 2) (4 \rightarrow 1) = 21555. \end{aligned}
• Assume $$\varGamma = \Theta _k$$ for $$k \ge 5$$. Consider the following transformation of [k]:

\begin{aligned} (u \Rightarrow v) := (u \rightarrow u_1) \dots (u_{d-1} \rightarrow v), \end{aligned}

where $$u,u_1, \dots , u_{d-1}, v$$ is the unique path from u to v on the cycle $$\Theta _k$$. Take

\begin{aligned} \alpha:= & {} (1 \Rightarrow k-3) (k \Rightarrow k-4) (k-1 \Rightarrow 1) (k-2 \Rightarrow k)\\&(k-3 \Rightarrow k-1) (k-4 \Rightarrow k-2). \end{aligned}

Then, $$\alpha = (k-1)(k-1) \dots (k-1) \ k \ 1 \ (k-2)$$, where $$(k-1)$$ appears $$k-3$$ times, has the cyclic component $$(k-2, k)$$.

Conversely, assume that D satisfies (ii). If $$n \le 3$$, it is clear that $$\mathrm {cycl}(\alpha ) = 0$$, for all $$\alpha \in \langle D\rangle$$, so suppose $$n \ge 4$$. We first obtain some key properties about the strong components of $${\bar{D}}$$.

### Claim 2.9

Any strong component of $${\bar{D}}$$ is an undirected path, an undirected cycle of length 3 or 4, or a claw $$K_{3,1}$$ (i.e. a bipartite undirected graph on $$ =  \cup \{4\}$$). Moreover, if a strong component of D is not an undirected path, then it is terminal.

### Proof

Let C be a strong component of $${\bar{D}}$$. Clearly, C is undirected and, by (ii), it cannot contain a cycle of length at least 5. If C has a cycle of length 3 or 4, then the whole of C must be that cycle and C is terminal (otherwise, it would contain $$\varGamma _3$$ or $$\varGamma _4$$, respectively). If C has no cycle of length 3 and 4, then C is a tree. It can only be a path or $$K_{3,1}$$, for otherwise it would contain $$\varGamma _1$$ or $$\varGamma _2$$; clearly, $$K_{3,1}$$ may only appear as a terminal component. $$\square$$

Suppose there is $$\alpha \in \langle D\rangle$$ that has a cyclic orbit (so $$\mathrm {cycl}(\alpha ) \ne 0$$). This cyclic orbit must be contained in a strong component C of $${\bar{D}}$$, and Claim 2.9 implies that $$C \cong \varGamma$$, where $$\varGamma \in \{ K_{3,1}, {\bar{\Theta }}_s, P_r : s \in \{3,4\}, r \in \mathbb {N} \}$$. If $$\varGamma = K_{3,1}$$ or $$\varGamma = {\bar{\Theta }}_s$$, then C is a terminal component, so $$\alpha$$ acts on C as some transformation $$\beta \in \langle \varGamma \rangle$$; however, it is easy to check that no transformation in $$\langle \varGamma \rangle$$ has a cyclic orbit. If $$\varGamma = P_r$$, for some r, then $$\alpha$$ acts on C as a partial transformation $$\beta$$ of $$P_r$$. Since $$\langle P_r\rangle = {\mathrm {O}}_r$$, $$\beta$$ has no cyclic orbit. $$\square$$

We introduce a new property of a connected digraph D:

($$\star \star$$):

For every strong component C of D, $$|C| \le 2$$ if C is non-terminal, and $$|C| \le 3$$ if C is terminal.

### Lemma 2.10

Let D be a closed connected digraph on [n] satisfying property $$(\star )$$. The following are equivalent:

1. (i)

D satisfies property $$(\star \star )$$.

2. (ii)

D has no subdigraph isomorphic to $$\varGamma _1$$, $$\varGamma _2$$, $$\varGamma _3$$, $$\varGamma _4$$, or $$\Theta _k$$, for some $$k \ge 5$$.

### Proof

If (i) holds, it is easy to check that D does not contain any subdigraphs isomorphic to $$\varGamma _1$$, $$\varGamma _2$$, $$\varGamma _3$$, $$\varGamma _4$$, or $$\Theta _k$$ for some $$k \ge 5$$.

Conversely, suppose that (ii) holds. Let C be a strong component of D. If C is non-terminal, Lemma 2.2 implies that C is complete; hence, $$|C| \le 2$$ as otherwise D would contain $$\varGamma _4$$ as a subdigraph. If C is terminal, Lemma 2.2 implies that C is complete or $$P_3$$; hence, $$|C| \le 3$$ as otherwise D would contain $$\varGamma _3$$ as a subdigraph. $$\square$$

### Theorem 2.11

Let D be a connected digraph on [n]. The following are equivalent:

1. (i)

For all $$\alpha \in \langle D \rangle$$, $$\ell (D, \alpha ) = n - \mathrm {fix}(\alpha )$$.

2. (ii)

D is closed satisfying properties $$(\star )$$ and $$(\star \star )$$.

### Proof

Clearly, D satisfies (i) if and only if it satisfies condition (C1) and $$\mathrm {cycl}(\alpha ) = 0$$, for all $$\alpha \in \langle D \rangle$$. By Theorem 2.5 and Lemmas 2.8 and 2.10, D satisfies (i) if and only if D satisfies (ii). $$\square$$

### Digraphs satisfying condition (C3)

The following result characterises digraphs satisfying condition (C3).

### Theorem 2.12

Let D be a connected digraph on [n]. The following are equivalent:

1. (i)

For every $$\alpha \in \langle D \rangle$$, $$\ell (D, \alpha ) = n - \mathrm {rk}(\alpha )$$.

2. (ii)

$$\langle D \rangle$$ is a band, i.e. every $$\alpha \in \langle D \rangle$$ is idempotent.

3. (iii)

Either $$n=2$$ and $$D \cong K_2$$, or there exists a bipartition $$V_1 \cup V_2$$ of [n] such that $$(i_1,i_2) \in E(D)$$ only if $$i_1 \in V_1$$, $$i_2 \in V_2$$.

### Proof

Clearly (i) implies (ii): if $$\ell (D,\alpha ) = n - \mathrm {rk}(\alpha )$$, then $$\mathrm {rk}(\alpha ) = \mathrm {fix}(\alpha )$$ by inequality (1), so $$\alpha$$ is idempotent.

Now we prove that (ii) implies (iii). If there exist $$u, v, w \in [n]$$ pairwise distinct such that $$(u,v), (v,w) \in E(D)$$, then $$\alpha = (v \rightarrow w) (u \rightarrow v)$$ is not an idempotent. Therefore, for $$n \ge 3$$, if every $$\alpha \in \langle D \rangle$$ is idempotent, then a vertex in D either has in-degree zero or out-degree zero: this corresponds to the bipartition of [n] into $$V_1$$ and $$V_2$$.

We finally prove that (iii) implies (i). Let $$n \ge 3$$ and suppose that there exists a bipartition $$V_1 \cup V_2$$ of [n] such that $$(i_1,i_2) \in E(D)$$ only if $$i_1 \in V_1$$, $$i_2 \in V_2$$. Then for any $$\alpha \in \langle D \rangle$$, all elements of $$V_2$$ are fixed by $$\alpha$$ and $$i_1 \alpha \in \{i_1\} \cup N^+(i_1)$$ for any $$i_1 \in V_1$$. In particular, any non-fixed point of $$\alpha$$ is mapped to a fixed point, so $$r:=\mathrm {rk}(\alpha ) = \mathrm {fix}(\alpha )$$. Let $$J := \{v_1, \dots , v_{n-r}\} \subseteq V_1$$ be the set of non-fixed points of $$\alpha$$; therefore

\begin{aligned} \alpha = (v_1 \rightarrow v_1 \alpha ) \dots (v_{n-r} \rightarrow v_{n-r} \alpha ), \end{aligned}

where each one of the $$n-r$$ arcs above belongs to $$\langle D\rangle$$. The result follows by inequality (1). $$\square$$

## Arc-generated semigroups with long words

Fix $$n \ge 2$$. In this section, we consider digraphs D that maximise $$\ell (D,r)$$ and $$\ell (D)$$. For $$r \in [n-1]$$, define

\begin{aligned} \ell _{\max }(n,r)&:= \max \left\{ \ell (D, r) : V(D) = [n] \right\} , \\ \ell _{\max }(n)&:= \max \left\{ \ell (D) : V(D) = [n] \right\} . \end{aligned}

The first few values of $$\ell _{\max }(n,r)$$, calculated with the GAP package Semigroups , are given in Table 1. By Lemma 2.1, $$\ell _{\max }(n,1) = n-1$$ for all $$n \ge 2$$; henceforth, we shall always assume that $$n \ge 3$$ and $$r \in [n-1] {\setminus } \{ 1\}$$.

In the following sections, we restrict the class of digraphs that we consider in the definition of $$\ell _{\max }(n,r)$$ and $$\ell _{\max }(n)$$ to two important cases: acyclic digraphs and strong tournaments.

### Acyclic digraphs

For any $$n \ge 3$$, let $$\mathrm {Acyclic}_n$$ be the set of all acyclic digraphs on [n], and, for any $$r \in [n-1]$$, define

\begin{aligned} \ell _{\max }^{\mathrm {Acyclic}}(n,r)&:= \max \left\{ \ell (A, r) : A \in \mathrm {Acyclic}_n \right\} ,\\ \ell _{\max }^{\mathrm {Acyclic}}(n)&:= \max \left\{ \ell (A) : A \in \mathrm {Acyclic}_n \right\} . \end{aligned}

Without loss of generality, we assume that any acyclic digraph A on [n] is topologically sorted, i.e. $$(u, v) \in E(A)$$ only if $$v > u$$.

In this section, we establish the following theorem.

### Theorem 3.1

For any $$n\ge 3$$ and $$r \in [n-1] {\setminus } \{ 1\}$$,

\begin{aligned} \ell _{\max }^{\mathrm {Acyclic}}(n,r)&= \frac{(n-r)(n+r-3)}{2} + 1, \\ \ell _{\max }^{\mathrm {Acyclic}}(n)&= \ell _{\max }^{\mathrm {Acyclic}}(n,2) = \frac{1}{2} (n^2 -3n + 4). \end{aligned}

First of all, we settle the case $$r = n-1$$, for which we have a finer result.

### Lemma 3.2

Let $$n \ge 3$$ and $$A \in \mathrm {Acyclic}_n$$. Then, $$\ell (A,n-1)$$ is equal to the length of a longest path in A. Therefore,

\begin{aligned} \ell _{\max }^{\mathrm {Acyclic}}(n,n-1) = n-1. \end{aligned}

### Proof

Let $$v_1, \dots , v_{l+1}$$ be a longest path in A. Then $$\alpha \in \langle A \rangle$$ defined by

\begin{aligned} v \alpha := {\left\{ \begin{array}{ll} v_{i+1} &{}\quad \text {if } v = v_i, \ i \in [l], \\ v &{}\quad \text {otherwise}, \end{array}\right. } \end{aligned}

has rank $$n-1$$ and requires at least l arcs, since it moves l vertices.

Conversely, let $$\alpha \in A$$ be a transformation of rank $$n-1$$, and consider a word expressing $$\alpha$$ in $$A^*$$:

\begin{aligned} \alpha = (u_1 \rightarrow v_1) (u_2 \rightarrow v_2) \dots (u_s \rightarrow v_s). \end{aligned}

Since $$\alpha$$ has rank $$n-1$$, we must have $$v_2 = u_1$$ and by induction $$v_i = u_{i-1}$$ for $$2 \le i \le s$$. As A is acyclic, $$u_s, u_{s-1}, \dots , u_1, v_1$$ forms a path in A, so $$s \le l$$. $$\square$$

The following lemma shows that the formula of Theorem 3.1 is an upper bound for $$\ell _{\max }^{\mathrm {Acyclic}}(n,r)$$.

### Lemma 3.3

For any $$n\ge 3$$ and $$r \in [n-1] {\setminus } \{ 1\}$$,

\begin{aligned} \ell _{\max }^{\mathrm {Acyclic}}(n,r) \le \frac{(n-r)(n+r-3)}{2} + 1. \end{aligned}

### Proof

Let A be an acyclic digraph on [n], let $$\alpha \in \langle A\rangle$$ be a transformation of rank $$r \ge 2$$, and let $$L \subset V(A)$$ be the set of terminal vertices of A. For any $$u,v \in [n]$$, denote the length of a longest path from u to v in A as $$\psi _A(u,v)$$.

### Claim 3.4

$$\ell (A, \alpha ) \le \sum _{v \in [n]} \psi _A(v, v \alpha )$$.

### Proof

Let $$\omega = (a_1 \rightarrow b_1) \dots (a_l \rightarrow b_l)$$ be a shortest word expressing $$\alpha$$ in $$A^*$$, with $$l = \ell (A, \alpha )$$. Say that the arc $$(a_i \rightarrow b_i)$$, $$i \ge 2$$, carries $$v \in [n]$$ if $$v (a_1 \rightarrow b_1) \dots (a_{i-1} \rightarrow b_{i-1}) = a_i$$ (assume that $$a_1 \rightarrow b_1$$ only carries $$a_1$$). Every arc $$(a_i \rightarrow b_i)$$ carries at least one vertex, for otherwise we could remove that arc form the word $$\omega$$ and obtain a shorter word still expressing $$\alpha$$. Let $$v \in [n]$$, and denote $$v_0 = v$$ and $$v_i = v (a_1 \rightarrow b_1) \dots (a_i \rightarrow b_i)$$ (and hence $$v_l = v \alpha$$). Let us remove the repetitions in this sequence: let $$j_0 = 0$$ and for $$i \ge 1$$, $$j_i = \min \{j : v_j \ne v_{j_{i-1}}\}$$. Then the sequence $$v = v_{j_0}, v_{j_1}, \dots , v_{j_{l(v)}} = v \alpha$$ forms a path in A of length l(v), and hence $$l(v) \le \psi (v, v\alpha )$$. For each $$v \in [n]$$, there are l(v) arcs in $$\omega$$ carrying v, so the length of $$\omega$$ satisfies

\begin{aligned} l \le \sum _{v =1}^n l(v) \le \sum _{v \in [n]} \psi _A(v, v \alpha ). \end{aligned}

$$\square$$

### Claim 3.5

If $$|L| \ge 2$$, then $$\sum _{v \in [n]} \psi _A(v, v \alpha ) \le \frac{(n-r)(n+r-3)}{2}$$.

### Proof

As $$|L| \ge 2$$, and A is topologically sorted, we have $$\{ n, n-1 \} \subseteq L$$, and any $$\alpha \in \langle A\rangle$$ fixes both $$n-1$$ and n, i.e. $$\psi _A(v, v \alpha ) = 0$$ for $$v \in \{n-1, n\}$$. For any $$v \in [n-2]$$, we have

\begin{aligned} \psi _A(v, v\alpha ) \le \min \{ n-1, v\alpha \} - v. \end{aligned}

Hence

\begin{aligned} \sum _{v \in [n]} \psi _A(v, v \alpha )&= \sum _{v \in [n-2]} \psi _A(v, v \alpha )\\&\le \sum _{v \in [n-2]} \left( \min \{ n-1, v\alpha \} - v \right) \\&= \sum _{w \in [n-2]\alpha } \left( \min \{ n-1, w\} |w \alpha ^{-1}| \right) - T_{n-2}, \end{aligned}

where $$T_k = \frac{k(k+1)}{2}$$. The summation is maximised when $$|n \alpha ^{-1}| = n-r$$ and $$|w \alpha ^{-1}| = 1$$ for $$n-r+1 \le w \le n-2$$, thus yielding

\begin{aligned} \sum _{v \in [n]} \psi _A(v, v \alpha )&\le (n-1)(n-r) + (T_{n-2} - T_{n-r}) - T_{n-2}\\&= \frac{(n-r)(n+r-3)}{2}. \end{aligned}

$$\square$$

### Claim 3.6

If $$\vert L \vert = 1$$, then $$\ell (A, \alpha ) \le \frac{(n-r)(n+r-3)}{2} + 1$$.

### Proof

As A is topologically sorted, $$L = \{ n\}$$. We use the notation from the proof of Claim 3.4. We then have $$l(n) = 0$$. We have three cases:

Case 1 :

$$(n-1)$$ is fixed by $$\alpha$$. Then, $$l(n-1) = 0$$ and $$l(v) \le \min \{n-1, v \alpha \} - v$$ for all $$v \in [n-2]$$. By the same reasoning as in Claim 3.5, we obtain $$\ell (A, \alpha ) \le \frac{(n-r)(n+r-3)}{2}$$.

Case 2 :

$$(n-1) \alpha = n$$ and $$v \alpha \le n-1$$ for every $$v \in [n-2]$$. Then again $$l(v)\le \min \{n-1, v\alpha \} -v$$, for all $$v \in [n-2]$$, and $$\ell (A, \alpha ) \le \frac{(n-r)(n+r-3)}{2}$$.

Case 3 :

n has at least two pre-images under $$\alpha$$. Let $$\omega = (a_1 \rightarrow b_1) \dots (a_l \rightarrow b_l)$$ be a shortest word expressing $$\alpha$$ in $$A^*$$, and denote $$\alpha _0 = \mathrm {id}$$ and $$\epsilon _i = (a_i \rightarrow b_i)$$, $$\alpha _i = \epsilon _1 \dots \epsilon _i$$ for $$i \in [l]$$. We partition $$n \alpha ^{-1}$$ into two parts S and T:

\begin{aligned} S = \{v \in n \alpha ^{-1} : v_{l(v) - 1} = n-1\}, \quad T = n \alpha ^{-1} {\setminus } S. \end{aligned}

For all $$v \in S$$, if the arc carrying v to $$n-1$$ is $$\epsilon _j$$, then $$(n-1)\alpha _{j-1}^{-1} \subseteq S$$ (v can only collapse with other pre-images of $$\alpha$$). Then the arc $$(n-1 \rightarrow n)$$ occurs only once in the word $$\omega$$ (if it occurs multiple times, then remove all but the last occurrence of that arc to obtain a shorter word expressing $$\alpha$$). If we do not count that arc, we have $$l'(v) \le n-1-v$$ arcs carrying v if $$v \in S$$, $$l(v) \le n-1-v$$ arcs carrying v if $$v \in T$$, and $$l(v) \le v\alpha - v$$ if $$v\alpha \ne n$$. Again, we obtain $$\ell (A, \alpha ) \le \frac{(n-r)(n+r-3)}{2} + 1$$. $$\square$$

Lemma 3.3 follows by the previous claims.$$\square$$

The following lemma completes the proof of Theorem 3.1.

### Lemma 3.7

For any $$n \ge 3$$ and $$r \in [n-1] {\setminus } \{ 1\}$$, there exists an acyclic digraph $$Q_n$$ on [n] and a transformation $$\beta _r \in \langle Q_n\rangle$$ of rank r such that

\begin{aligned} \ell (Q_n, \beta _r) \ge \frac{(n-r)(n+r-3)}{2} + 1. \end{aligned}

### Proof

Let $$Q_n$$ be the acyclic digraph on [n] with edge set

\begin{aligned} E(Q_n) := \left\{ (u, u+1) : u \in [n-1] \right\} \cup \left\{ (n-2, n) \right\} . \end{aligned}

For any $$r \in [n-1] {\setminus } \{ 1\}$$, define $$\beta _r \in \langle Q_n\rangle$$ by

\begin{aligned} v\beta _r := {\left\{ \begin{array}{ll} n-r+v &{}\text {if } v \in [r-2], \\ n - 1 &{}\text {if } v \in [n-1] {\setminus } [r-2], \ n-v \equiv 0 \mod 2,\\ n &{}\text {if }v \in [n-1] {\setminus } [r-2], \ n-v \equiv 1 \mod 2,\\ n &{}\text {if } v = n. \end{array}\right. } \end{aligned}

Let $$\beta _r$$ be expressed as a word in $$Q_n^*$$ of minimum length as

\begin{aligned} \beta _r = (a_1 \rightarrow b_1) \dots (a_l \rightarrow b_l), \end{aligned}

where $$l= \ell (Q_n, \beta _r)$$. Denote $$\alpha _0 := \mathrm {id}$$, $$\epsilon _i := (a_i \rightarrow b_i)$$, and $$\alpha _i := \epsilon _1 \dots \epsilon _i$$, for $$i \in [l]$$. Say that $$\epsilon _i$$ carries $$u \in [n]$$ if $$u \alpha _{i-1} = a_i$$ and hence $$u \alpha _i \ne u \alpha _{i-1}$$.

### Claim 3.8

For each $$i \in [l]$$, the arc $$\epsilon _i$$ carries exactly one vertex.

### Proof

First, $$(a_1, b_1) \in E(Q_n)$$ and $$a_1 \beta _r = b_1 \beta _r$$ imply that $$a_1 = n-1$$ and $$b_1 = n$$. Suppose that there is an arc $$\epsilon _j$$, $$j \in [l]$$, that carries two vertices $$u < v$$; take j to be minimal index with this property. We remark that $$v \le n-2$$ and $$u \alpha _{j-1} = v \alpha _{j-1}$$ imply $$u \beta _r = v \beta _r$$. Then $$w := u+1$$ satisfies $$w \beta _r \ne u \beta _r$$, so w is not carried by $$\epsilon _j$$. If $$w \alpha _{j-1} \le n-2$$, then $$u \alpha _{j-1}< w \alpha _{j-1} < v \alpha _{j-1}$$ since $$u< w < v$$ and the graph induced by $$[n-2]$$ in $$Q_n$$ is the directed path $$\vec {P}_{n-2}$$; this contradicts that $$u \alpha _{j-1} = v \alpha _{j-1}$$. Hence $$w \alpha _{j-1} \ge n-1$$ and $$v \alpha _{j-1} \ge n-1$$. If $$v \alpha _{j-1} = n$$ or $$v \beta _r = n-1$$, then $$\epsilon _j$$ does not carry v. Thus, $$v\alpha _{j-1} = n-1$$ and $$v \beta _r = n$$. Then, in order to carry v to $$n-1$$, we have $$\epsilon _s = (n-2 \rightarrow n-1)$$ for at least one $$s \in [l]$$, and $$\epsilon _j = (n-1 \rightarrow n)$$. For $$s \in [j-1]$$, replace all occurrences $$\epsilon _s = (n-2 \rightarrow n-1)$$ with $$\epsilon _s^\prime := (n-2 \rightarrow n)$$ and delete $$\epsilon _j$$: this yields a word in $$Q_n^*$$ of length $$l^\prime <l$$ expressing $$\beta _r$$, which is a contradiction. $$\square$$

For all $$i \in [l]$$, denote $$\delta (i) := \sum _{v \in [n]} d_{Q_n}(v \alpha _i, v \beta _r)$$. We then have $$\delta (l) = 0$$, and by the claim, $$\delta (i) \ge \delta (i-1) - 1$$ for all $$i \in [l]$$. Thus $$l \ge \delta (0)$$, where

\begin{aligned} \delta (0)&= \sum _{v \in [n]} d_{Q_n}(v, v \beta _r)\\&= \sum _{v=1}^{r-2} (n-r) + \sum _{v=r-1}^{n-2} (n-1 - v) + 1\\&= \frac{(n-r)(n+r-3)}{2} + 1. \end{aligned}

$$\square$$

### Strong tournaments

Let $$n \ge 3$$. Recall that if T is a strong tournament on [n], then $$\{ a \rightarrow b : (a,b) \in E(T) \}$$ is a minimal generating set of $$\mathrm {Sing}_n$$. Let $$\mathrm {Tour}_n$$ denote the set of all strong tournaments on [n]. For $$r \in [n-1]$$, define

\begin{aligned} \ell _{\max }^{\mathrm {Tour}}(n, r)&:= \max \{ \ell (T, r) : T \in \mathrm {Tour}_n \},\\ \ell _{\max }^{\mathrm {Tour}}(n)&:= \max \{ \ell (T) : T \in \mathrm {Tour}_n \}. \end{aligned}

Define analogously $$\ell _{\min }^{\mathrm {Tour}}(n,r)$$ and $$\ell _{\min }^{\mathrm {Tour}}(n)$$. The first few values of $$\ell _{\min }^{\mathrm {Tour}}(n,r)$$ and $$\ell _{\max }^{\mathrm {Tour}}(n,r)$$, calculated with the GAP package Semigroups  using data from , are given in Table 2. The calculation of these values has been the inspiration for the results of this section and the conjectures of the next one.

### Lemma 3.9

Let $$n \ge 3$$ and $$T \in \mathrm {Tour}_n$$.

1. 1.

For any partition P of [n] into r parts, there exists an idempotent $$\alpha \in \mathrm {Sing}_n$$ with $$\ker (\alpha ) = P$$ such that $$\ell (T, \alpha )= n - r$$.

2. 2.

For any r-subset S of [n], there exists an idempotent $$\alpha \in \mathrm {Sing}_n$$ with $$\mathrm {Im}(\alpha ) = S$$ such that $$\ell (T, \alpha )= n - r$$.

### Proof

1. 1.

Let $$P = \{ P_1, \dots , P_r \}$$. For all $$1 \le i \le r$$, the digraph $$T[P_i]$$ induced by $$P_i$$ is a tournament, so it is connected and there exists a vertex $$v_i$$ reachable by any other vertex in $$P_i$$: let $$\alpha$$ map the whole of $$P_i$$ to $$v_i$$. Then $$\alpha$$, when restricted to $$P_i$$, is a constant map, which can be computed using $$|P_i| - 1$$ arcs. Summing for i from 1 to r, we obtain that $$\ell (T, \alpha ) = n - r$$.

2. 2.

Without loss of generality, let $$S = [r] \subseteq [n]$$. For every $$v \in [n]$$, define

\begin{aligned} s(v) := \min \{ s \in S: d_T(s', v) \ge d_T(s,v), \forall s' \in S\}. \end{aligned}

In particular, if $$v \in S$$, then $$s(v) = v$$. Moreover, if $$v = v_0, v_1, \dots , v_d = s(v)$$ is a shortest path from v to s(v), with $$d = d_T(v,s(v))$$, then $$s(v_i) = s(v)$$ for all $$0 \le i \le d$$. For each $$v \in [n ]$$, fix a shortest path $$P_v$$ from v to s(v), and consider the digraph D on [n] with edges

\begin{aligned} E(D) := \{ (a,b) : (a,b) \in E(P_v) \text { for some } v \in [n]\}. \end{aligned}

Then, D is acyclic and the set of vertices with out-degree zero in D is exactly S. Let sort [n] so that D has reverse topological order: $$(a,b) \in E(D)$$ only if $$a >b$$. Note that S is fixed by this sorting. Let $$\alpha$$ be given by $$v \alpha := s(v)$$; hence, with the above sorting

\begin{aligned} \alpha = \bigcirc _{v=n}^{r+1} (v \rightarrow v_1). \end{aligned}

$$\square$$

### Lemma 3.10

Let $$n \ge 3$$, $$T \in \mathrm {Tour}_n$$, and $$\alpha :=(u \rightarrow v) \in \mathrm {Sing}_n$$, for $$(u,v) \not \in E(T)$$. Then

\begin{aligned} \ell (T, \alpha ) = 4 d_T(u,v) - 2. \end{aligned}

### Proof

Let $$u = v_0, v_1, \dots , v_d = v$$ be a shortest path from u to v in T, where $$d := d_T(u,v)$$. As $$(u,v) \not \in E(T)$$ and T is a tournament, we must have $$(v,u) \in E(T)$$. By the minimality of the path, for any $$j +1 < i$$, we have $$(v_j, v_i) \not \in E(T)$$, so $$(v_i, v_j) \in E(T)$$. Then, the following expresses $$\alpha$$ with arcs in $$T^*$$:

\begin{aligned} (v_0 \rightarrow v_d)&= (v_d \rightarrow v_0) (v_{d-1} \rightarrow v_d) (v_{d-2} \rightarrow v_{d-1}) \cdots (v_1 \rightarrow v_2) (v_0 \rightarrow v_1 ) \\&\qquad \left( (v_2 \rightarrow v_0) (v_1 \rightarrow v_2) \right) \left( (v_3 \rightarrow v_1) (v_2 \rightarrow v_3) \right) \\&\qquad \cdots \left( (v_d \rightarrow v_{d-2}) (v_{d-1} \rightarrow v_d) \right) \\&\qquad (v_{d-2} \rightarrow v_{d-1}) \cdots (v_0 \rightarrow v_1). \end{aligned}

So $$\ell (T, \alpha ) \le 4d-2$$. For the lower bound, we note that any word in $$T^*$$ expressing $$(u \rightarrow v)$$ must begin with $$(v \rightarrow u)$$. Then, u has to follow a walk in T towards v; say this walk has length $$l \ge d$$. All the vertices on the walk must be moved away (as otherwise they would collapse with u) and have to come back to their original position (since $$\alpha$$ fixes them all); as the shortest cycle in a tournament has length 3, this process adds at least $$3(l-1)$$ symbols to the word. Altogether, this yields a word of length at least

\begin{aligned} 1 + l + 3(l-1) = 4l-2 \ge 4 d - 2. \end{aligned}

$$\square$$

Let $$n = 2m+1 \ge 3$$ be odd, and let $$\kappa _n$$ be the circulant tournament on [n] with edges $$E(\kappa _n):=\{ (i, (i+j) \mod n): i \in [n], j \in [m] \}$$. Figure 2 illustrates $$\kappa _5$$. In the following theorem, we use $$\kappa _n$$ to provide upper and lower bounds for $$\ell _{\min }^{\mathrm {Tour}}(n,r)$$ and $$\ell _{\max }^{\mathrm {Tour}}(n,r)$$ when n is odd.

### Theorem 3.11

For any n odd, we have

\begin{aligned} n + r - 2&\le \ell _{\min }^{\mathrm {Tour}}(n,r) \le n + 8r,\\ ({\hat{r}} + 1)(n - {\hat{r}}) - 1&\le \ell _{\max }^{\mathrm {Tour}}(n,r) \le 6rn + n -10r. \end{aligned}

where $${\hat{r}} = \min \{r-1, \lfloor n/2 \rfloor \}$$.

### Proof

Let $$T \in \mathrm {Tour}_n$$ and $$2 \le r \le n-1$$. We introduce the following notation:

\begin{aligned}{}[n]_r&:= \{ \mathbf{u} := (u_1, \dots , u_r) : u_i \ne u_j, \forall i, j \}, \\ \Delta (T, r)&:= \max \left\{ \sum _{i=1}^r d_T(u_i, v_i) : \mathbf{u}, \mathbf{v} \in [n]_r \right\} . \end{aligned}

The result follows by the next claims.

### Claim 3.12

$$r'(\mathrm {diam}(T) - r' + 1) + r-r' \le \Delta (T,r) \le r \mathrm {diam}(T)$$, where $$r' = \min \{r, \lfloor ( \mathrm {diam}(T)+1)/2 \rfloor \}$$.

### Proof

The upper bound is clear. For the lower bound, let $$u, v \in [n]$$ be such that $$d_T(u,v) = \mathrm {diam}(T)$$, and let $$u = v_0, v_1, \dots , v_d = v$$ be a shortest path from u to v, where $$d = \mathrm {diam}(T)$$. Then, $$d_T(v_i,v_j) = j-i$$, for all $$0 \le i \le j \le D$$. If $$1 \le r \le \lfloor (d+1)/2 \rfloor$$, consider $$\mathbf{u}' = (v_0, \dots , v_{r-1})$$ and $$\mathbf{v}' = (v_{d-r+1}, \dots , v_d)$$, so we obtain $$\Delta (T, r) \ge r(d-r+1)$$. If $$r \ge \lfloor (d+1)/2 \rfloor$$, simply add vertices $$u'_j$$ and $$v'_j$$ such that $$(u'_j, v'_j) \notin T$$. $$\square$$

### Claim 3.13

$$\min \{ \Delta (T,r) : T \in \mathrm {Tour}(n) \} = \Delta (\kappa _n,r) = 2r$$.

### Proof

Let $$\mathbf{u} = (u_1, \dots , u_n)$$ form a Hamiltonian cycle, and choose $$\mathbf{v} = (u_n, u_1, \dots , u_{n-1})$$. Then $$d_T(u_i, v_i) \ge 2$$ for all i. Conversely, since $$\mathrm {diam}(\kappa _n) = 2$$, we have $$\Delta (\kappa _n,r) = 2r$$. $$\square$$

### Claim 3.14

$$n-r + \Delta (T, r-1) \le \ell (T,r) \le n + 6 r \mathrm {diam}(T) - 4r$$.

### Proof

For the lower bound, consider $$\alpha \in \mathrm {Sing}_n$$ as follows. Let $$\mathbf{u} = (u_1, \dots , u_{r-1})$$ and $$\mathbf{v} = (v_1, \dots , v_{r-1})$$ achieve $$\Delta (T,r-1)$$, and let $$v \notin \{v_1, \dots , v_{r-1}\}$$; define

\begin{aligned} x \alpha = {\left\{ \begin{array}{ll} v_i &{}\quad \text {if } x = u_i, \\ v &{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}

Let $$\omega = e_1 \dots e_l$$ (where $$e_i =(a_i \rightarrow b_i)$$) be a shortest word expressing $$\alpha$$, where $$l := \ell (T,\alpha )$$. Recall that an arc $$e_i$$ carries a vertex c if $$c e_1 \dots e_{i-1} = a_i$$. By the minimality of $$\omega$$, every arc carries at least one vertex. Moreover, if c and d are carried by $$e_i$$, then $$c \alpha = d \alpha$$; therefore, we can label every arc $$e_i$$ of $$\omega$$ by an element $$c(e_i) \in \mathrm {Im}(\alpha )$$ if $$e_i$$ carries vertices eventually mapping to $$c(e_i)$$. Denote the number of arcs labelled c as l(c), we then have $$l = \sum _{c \in \mathrm {Im}(\alpha )} l(c)$$. For any $$u \in V$$, there are at least $$d_T(u, u \alpha )$$ arcs carrying u. Therefore,

\begin{aligned} l = \sum _{c \in \mathrm {Im}(\alpha )} l(c) \ge \sum _{i=1}^{r-1} d_T(u_i, v_i) + \sum _{a \notin \mathbf{u}} d_T(a, v) \ge \Delta (T, r-1) + n-r. \end{aligned}

For the upper bound, we can express any $$\alpha \in \mathrm {Sing}_n$$ of rank r in the following fashion. By Lemma 3.9, there exists $$\beta \in \mathrm {Sing}_n$$ with the same kernel as $$\alpha$$ such that $$\ell (T, \beta ) = n-r$$. Suppose that $$\mathrm {Im}(\alpha ) = \{ v_1, \dots , v_r \}$$ and $$\mathrm {Im}(\beta ) = \{u_1, \dots , u_r \}$$, where $$u_i \beta ^{-1} = v_i \alpha ^{-1}$$, for $$i \in [r]$$. Let $$h \in [n] {\setminus } \mathrm {Im}(\beta )$$. Define a transformation $$\gamma$$ of [n] by

\begin{aligned} x \gamma = {\left\{ \begin{array}{ll} v_i &{}\quad \text {if } x = u_i,\\ v_1 &{} \quad \text {if } x= h, \\ x &{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}

Then $$\alpha = \beta \gamma$$, where $$\gamma \in \mathrm {Sing}_n$$, and by Theorem 1.1

\begin{aligned} \ell (K_n, \gamma ) = n - \mathrm {fix}(\gamma ) + \mathrm {cycl}(\gamma ) \le r + \frac{r}{2} = \frac{3r}{2}. \end{aligned}

By Lemma 3.10, each arc associated with $$K_n$$ may be expressed in at most $$4 \mathrm {diam}(T) - 2$$ arcs associated with T; therefore,

\begin{aligned} \ell (T, \gamma ) \le \frac{3r}{2}( 4 \mathrm {diam}(T) - 2) = 6r \mathrm {diam}(T) - 3r. \end{aligned}

Thus,

\begin{aligned} \ell (T,\alpha ) \le \ell (T,\beta ) + \ell (T,\gamma ) \le n + 6r\mathrm {diam}(T) -4r. \end{aligned}

$$\square$$

$$\square$$

## Conjectures and open problems

We finish the paper by proposing few conjectures and open problems.

Let $$\pi _n$$ be the tournament on [n] with edges $$E(\pi _n):=\{ ( i, (i+1) \mod n ) : i \in [n]\} \cup \{ (i, j) : j +1 < i \}$$. Figure 3 illustrates $$\pi _5$$.

### Conjecture 4.1

For every $$n \ge 3$$, $$r \in [n-1]$$, and $$T \in \mathrm {Tour}_n$$, we have

\begin{aligned} \ell (T, r) \le \ell (\pi _n, r) = \ell _{\max }^{\mathrm {Tour}}(n, r) , \end{aligned}

with equality if and only if $$T \cong \pi _n$$. Furthermore,

\begin{aligned} \ell (\pi _n) = \ell _{\max }^{\mathrm {Tour}}(n) = \frac{n^2 + 3n - 6}{2}, \end{aligned}

which is achieved for $$\alpha : = n \ (n-1) \ \dots \ 2 \ n$$.

Tournament $$\pi _n$$ has appeared in the literature before: it is shown in  that $$\pi _n$$ has the minimum number of strong subtournaments among all strong tournaments on [n]. On the other hand, it was shown in  that, for n odd, the circulant tournament $$\kappa _n$$ has the maximal number of strong subtournaments among all strong tournaments on [n].

### Conjecture 4.2

For every $$n \ge 3$$ odd, $$r \in [n-1]$$, and $$T \in \mathrm {Tour}_n$$, we have

\begin{aligned} \ell _{\min }^{\mathrm {Tour}}(n, r) = \ell (\kappa _n, r). \end{aligned}

Furthermore,

\begin{aligned} \ell _{\min }^{\mathrm {Tour}}(n,2) = n+1 \ \text { and } \ \ell _{\min }^{\mathrm {Tour}}(n,r) = n+r, \end{aligned}

for all $$3 \le r \le \frac{n+1}{2}$$.

### Conjecture 4.3

There exists $$c > 0$$ such that for every simple digraph D on [n], $$\ell (D) = O(n^c)$$.

The referee of this paper noted that the automorphism groups of $$K_n$$ and $$\langle K_n \rangle = \mathrm {Sing}_n$$ are both isomorphic to $$\mathrm {Sym}_n$$ and proposed the following problems.

### Problem 1

Investigate connections between the automorphism groups of D and $$\langle D \rangle$$. Is it possible to classify all digraphs D such that the automorphism group of D and of $$\langle D \rangle$$ are isomorphic?

### Problem 2

Generalise the ideas of this paper to oriented matroids. Is there a natural way to associate (not necessarily idempotent) transformations to each signed circuit of an oriented matroid?

In a forthcoming paper, we investigate the relationship between the graph theoretic properties of D and the semigroup properties of $$\langle D \rangle$$.

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## Acknowledgments

The second and third authors were supported by the EPSRC grant EP/K033956/1. We kindly thank the insightful comments and suggestions for open problems of the anonymous referee of this paper.

## Author information

Authors

### Corresponding author

Correspondence to A. Castillo-Ramirez.