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 (a, b) 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.

1 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 [2]) 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 [3]). 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 $\vec {T}_5$).

Fig. 1

$\vec {T}_5$

Connections between subsemigroups of $\mathrm {Sing}_n$ and digraphs have been studied before (see [9–12]). The following definition, which we shall adopt in the following sections, appeared in [12]:

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.

2 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:

2.1 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 $u,v \in N(a)$, u, a, v 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 $C_2$.

• (i) If $C_2$ is non-terminal, then $C_1$ fully connects to $C_2$.

• (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 $C_2$.

• (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 $C_2$.

• (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.

• (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 $C_2$.

• (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 $C_2$.

• (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.

• (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:

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

• (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. v is in a terminal component of D.

• 2. There is a path u, w, v of length 2 in D such that $w \alpha = v \alpha = v$; for any other path u, x, v 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 $(\star )$, u, w, v 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 u, w, v 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

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

$\square$

2.2 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:

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

• (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 $i\in [4]$, $k \ge 5$, 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 $[4] = [3] \cup \{4\}$). 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:

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

• (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:

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

• (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). $\square$

2.3 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:

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

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

• (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). $\square$

3 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

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

Table 1 First values of $\ell _{\max }(n,r)$

The first few values of $\ell _{\max }(n,r)$, calculated with the GAP package Semigroups [7], 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.

3.1 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

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

$\square$

3.2 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 [7] using data from [6], 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.

Table 2 First values of $\left( \ell _{\min }^{\mathrm {Tour}}(n,r) , \ell _{\max }^{\mathrm {Tour}}(n,r) \right)$

Lemma 3.9

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

• 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. 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. 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. 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.

Fig. 2

Circulant tournament $\kappa _5$

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,

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

$\square$

$\square$

4 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 $\pi _5$.

Fig. 3

$\pi _5$

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 [8] that $\pi _n$ has the minimum number of strong subtournaments among all strong tournaments on [n]. On the other hand, it was shown in [1] 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 $\langle D \rangle$.