Lengths of words in transformation semigroups generated by digraphs
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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 \(n1\) 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 \(n1\) is generated by its idempotents of rank \(n1\). 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 nontrivial 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 \(n1\) of \(\mathrm {Sing}_n\)). We finish the paper with a list of conjectures and open problems.
Keywords
Transformation semigroup Simple digraph Word length1 Introduction
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.
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 \(\{ (x,y) \in [n]^2 : x\alpha = y \alpha \}\)).
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 Arcgenerated semigroups with short words
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
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.
2.1 Digraphs satisfying condition (C1)
 (\(\star \))

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 outneighbour 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 outneighbour \(v_3 \not \in \{ v_0, v_1, v_2 \}\); 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
 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
 (i)
If \(C_2\) is nonterminal, 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 outdegree 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
 (i)
As \(C_2\) is nonterminal, 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
 (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
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 nontrivial 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
 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 outdegree 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. \(\square \)
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 \(\alpha \in \langle D\rangle \).
Lemma 2.8
 (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 \(u,u_1, \dots , u_{d1}, v\) is the unique path from u to v on the cycle \(\Theta _k\). Take$$\begin{aligned} (u \Rightarrow v) := (u \rightarrow u_1) \dots (u_{d1} \rightarrow v), \end{aligned}$$Then, \(\alpha = (k1)(k1) \dots (k1) \ k \ 1 \ (k2) \), where \((k1)\) appears \(k3\) times, has the cyclic component \((k2, k)\).$$\begin{aligned} \alpha:= & {} (1 \Rightarrow k3) (k \Rightarrow k4) (k1 \Rightarrow 1) (k2 \Rightarrow k)\\&(k3 \Rightarrow k1) (k4 \Rightarrow k2). \end{aligned}$$
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. \(\square \)
 (\(\star \star \))

For every strong component C of D, \(C \le 2\) if C is nonterminal, and \(C \le 3\) if C is terminal.
Lemma 2.10
 (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 nonterminal, 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
 (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
 (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 indegree zero or outdegree zero: this corresponds to the bipartition of [n] into \(V_1\) and \(V_2\).
3 Arcgenerated semigroups with long words
First values of \(\ell _{\max }(n,r)\)
r  

n  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, \(\ell _{\max }(n,1) = n1\) for all \(n \ge 2\); henceforth, we shall always assume that \(n \ge 3\) and \(r \in [n1] {\setminus } \{ 1\}\).
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
In this section, we establish the following theorem.
Theorem 3.1
First of all, we settle the case \(r = n1\), for which we have a finer result.
Lemma 3.2
Proof
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
Claim 3.5
If \(L \ge 2\), then \(\sum _{v \in [n]} \psi _A(v, v \alpha ) \le \frac{(nr)(n+r3)}{2}\).
Proof
Claim 3.6
If \(\vert L \vert = 1\), then \(\ell (A, \alpha ) \le \frac{(nr)(n+r3)}{2} + 1\).
Proof
 Case 1

\((n1)\) is fixed by \(\alpha \). Then, \(l(n1) = 0\) and \(l(v) \le \min \{n1, v \alpha \}  v\) for all \(v \in [n2]\). By the same reasoning as in Claim 3.5, we obtain \(\ell (A, \alpha ) \le \frac{(nr)(n+r3)}{2}\).
 Case 2

\((n1) \alpha = n\) and \(v \alpha \le n1\) for every \(v \in [n2]\). Then again \(l(v)\le \min \{n1, v\alpha \} v\), for all \(v \in [n2]\), and \(\ell (A, \alpha ) \le \frac{(nr)(n+r3)}{2}\).
 Case 3
 n has at least two preimages 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 v to \(n1\) is \(\epsilon _j\), then \((n1)\alpha _{j1}^{1} \subseteq S\) (v can only collapse with other preimages of \(\alpha \)). Then the arc \((n1 \rightarrow n)\) 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 \(l'(v) \le n1v\) arcs carrying v if \(v \in S\), \(l(v) \le n1v\) arcs carrying v if \(v \in T\), and \(l(v) \le v\alpha  v\) if \(v\alpha \ne n\). Again, we obtain \(\ell (A, \alpha ) \le \frac{(nr)(n+r3)}{2} + 1\). \(\square \)$$\begin{aligned} S = \{v \in n \alpha ^{1} : v_{l(v)  1} = n1\}, \quad T = n \alpha ^{1} {\setminus } S. \end{aligned}$$
Lemma 3.3 follows by the previous claims.\(\square \)
The following lemma completes the proof of Theorem 3.1.
Lemma 3.7
Proof
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 \(a_1 = n1\) 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 \(v \le n2\) and \(u \alpha _{j1} = v \alpha _{j1}\) 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 \(w \alpha _{j1} \le n2\), then \(u \alpha _{j1}< w \alpha _{j1} < v \alpha _{j1}\) since \(u< w < v\) and the graph induced by \([n2]\) in \(Q_n\) is the directed path \(\vec {P}_{n2}\); this contradicts that \(u \alpha _{j1} = v \alpha _{j1}\). Hence \(w \alpha _{j1} \ge n1\) and \(v \alpha _{j1} \ge n1\). If \(v \alpha _{j1} = n\) or \(v \beta _r = n1\), then \(\epsilon _j\) does not carry v. Thus, \(v\alpha _{j1} = n1\) and \(v \beta _r = n\). Then, in order to carry v to \(n1\), we have \(\epsilon _s = (n2 \rightarrow n1)\) for at least one \(s \in [l]\), and \(\epsilon _j = (n1 \rightarrow n)\). For \(s \in [j1]\), replace all occurrences \(\epsilon _s = (n2 \rightarrow n1)\) with \(\epsilon _s^\prime := (n2 \rightarrow n)\) 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
First values of \(\left( \ell _{\min }^{\mathrm {Tour}}(n,r) , \ell _{\max }^{\mathrm {Tour}}(n,r) \right) \)
r  

n  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 rsubset 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 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$$\begin{aligned} s(v) := \min \{ s \in S: d_T(s', v) \ge d_T(s,v), \forall s' \in S\}. \end{aligned}$$Then, D is acyclic and the set of vertices with outdegree 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$$\begin{aligned} E(D) := \{ (a,b) : (a,b) \in E(P_v) \text { for some } v \in [n]\}. \end{aligned}$$\(\square \)$$\begin{aligned} \alpha = \bigcirc _{v=n}^{r+1} (v \rightarrow v_1). \end{aligned}$$
Lemma 3.10
Proof
Theorem 3.11
Proof
Claim 3.12
\(r'(\mathrm {diam}(T)  r' + 1) + rr' \le \Delta (T,r) \le r \mathrm {diam}(T)\), 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, \(d_T(v_i,v_j) = ji\), for all \(0 \le i \le j \le D\). If \(1 \le r \le \lfloor (d+1)/2 \rfloor \), consider \(\mathbf{u}' = (v_0, \dots , v_{r1})\) and \(\mathbf{v}' = (v_{dr+1}, \dots , v_d)\), so we obtain \(\Delta (T, r) \ge r(dr+1)\). 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 \(\mathbf{v} = (u_n, u_1, \dots , u_{n1})\). 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
\(nr + \Delta (T, r1) \le \ell (T,r) \le n + 6 r \mathrm {diam}(T)  4r\).
Proof
\(\square \)
4 Conjectures and open problems
We finish the paper by proposing few conjectures and open problems.
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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