# 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.

### Keywords

Transformation semigroup Simple digraph Word length## 1 Introduction

*arcs*.

*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

*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.

*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 \(\vec {T}_5\)).

Connections between subsemigroups of \(\mathrm {Sing}_n\) and digraphs have been studied before (see [9, 10, 11, 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 \(\{ (x,y) \in [n]^2 : x\alpha = y \alpha \}\)).

*D*and \(v \in V(D)\), define the

*in-neighbourhood*and the

*out-neighbourhood*of

*v*by

*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 \(\mathrm {diam}(D) := \max \{ d_D(u,v) : u,v \in V(D), \ d_D(u,v) \text { is defined} \}\).

*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

### Theorem 1.1

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

*D*be a digraph on [

*n*], \(n \ge 3\), and \(\alpha \in \langle D \rangle \). Theorem 1.1 implies the following three bounds:

### 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

*D*from

*v*to \(v_0\) (as otherwise, \(\alpha \not \in \langle D \rangle \)). For any \(d \ge 1\), let

*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.

*D*on [

*n*] satisfying the following conditions:

### 2.1 Digraphs satisfying condition (C1)

**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

*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

*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

*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

*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

*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

- 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

*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*. \(\square \)

*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

*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

### 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 \(\alpha \in \langle D\rangle \).

*k*. Consider the digraphs \(\varGamma _1, \ \varGamma _2, \ \varGamma _3\) and \(\varGamma _4\) as illustrated below:

### Lemma 2.8

*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

- 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*]:where \(u,u_1, \dots , u_{d-1}, v\) is the unique path from$$\begin{aligned} (u \Rightarrow v) := (u \rightarrow u_1) \dots (u_{d-1} \rightarrow v), \end{aligned}$$*u*to*v*on the cycle \(\Theta _k\). TakeThen, \(\alpha = (k-1)(k-1) \dots (k-1) \ k \ 1 \ (k-2) \), where \((k-1)\) appears \(k-3\) times, has the cyclic component \((k-2, k)\).$$\begin{aligned} \alpha:= & {} (1 \Rightarrow k-3) (k \Rightarrow k-4) (k-1 \Rightarrow 1) (k-2 \Rightarrow k)\\&(k-3 \Rightarrow k-1) (k-4 \Rightarrow k-2). \end{aligned}$$

*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. \(\square \)

*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

*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

*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 \((\star \star )\).

### 2.3 Digraphs satisfying condition (C3)

The following result characterises digraphs satisfying condition (**C3**).

### Theorem 2.12

*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 \(V_2\).

*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

## 3 Arc-generated semigroups with long words

*D*that maximise \(\ell (D,r)\) and \(\ell (D)\). For \(r \in [n-1]\), define

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

| |||||
---|---|---|---|---|---|

| 1 | 2 | 3 | 4 | 5 |

2 | 1 | ||||

3 | 2 | 6 | |||

4 | 3 | 11 | 13 | ||

5 | 4 | 18 | 24 | 33 | |

6 | 5 | 26 | 42 | 51 | 66 |

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

*n*], and, for any \(r \in [n-1]\), define

*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

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

### Lemma 3.2

*A*. Therefore,

### Proof

*A*. Then \(\alpha \in \langle A \rangle \) defined by

*l*arcs, since it moves

*l*vertices.

*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

### 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

*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

### 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

*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

### Claim 3.6

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

### Proof

*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*:For all \(v \in S\), if the arc carrying$$\begin{aligned} S = \{v \in n \alpha ^{-1} : v_{l(v) - 1} = n-1\}, \quad T = n \alpha ^{-1} {\setminus } S. \end{aligned}$$*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

*n*] and a transformation \(\beta _r \in \langle Q_n\rangle \) of rank

*r*such that

### Proof

*n*] with edge set

### 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. \(\square \)

### 3.2 Strong tournaments

*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

*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.

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

| |||||
---|---|---|---|---|---|

| 2 | 3 | 4 | 5 | 6 |

3 | (6, 6) | ||||

4 | (8, 8) | (11, 11) | |||

5 | (6, 11) | (8, 14) | (10, 17) | ||

6 | (8, 13) | (10, 18) | (11, 21) | (13, 24) | |

7 | (8, 16) | (10, 22) | (11, 26) | (13, 29) | (15, 32) |

### Lemma 3.9

- 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 \(v \in [n]\), defineIn 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$$\begin{aligned} s(v) := \min \{ s \in S: d_T(s', v) \ge d_T(s,v), \forall s' \in S\}. \end{aligned}$$
*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 edgesThen,$$\begin{aligned} E(D) := \{ (a,b) : (a,b) \in E(P_v) \text { for some } v \in [n]\}. \end{aligned}$$*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\(\square \)$$\begin{aligned} \alpha = \bigcirc _{v=n}^{r+1} (v \rightarrow v_1). \end{aligned}$$

### Lemma 3.10

### Proof

*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 \(T^*\):

*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

*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

*n*odd, we have

### Proof

### 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

*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,

*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

*T*; therefore,

\(\square \)

## 4 Conjectures and open problems

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

*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

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

### 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 \(\langle D \rangle \).

## Notes

### 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.

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