text:
Supporting grassroots innovation: Facts and Figures
Local innovators have turned a second-hard scooter into a flour mill SRISTI, Ahmedabad
Adrian Smith and colleagues explore grassroots innovations, their potential for development and challenges facing practitioners.
Turning ideas into new products, services or practices often requires systematic steps in a process of innovation. Conventional policies for promoting innovation link entrepreneurial firms, science institutes, investors and policymakers, and aim to keep up with advancements in promising technologies. But innovation also takes place below the radar of national and regional innovation systems.
Grassroots-level, alternative innovation activities recognize the ingenious solutions continually developed locally to improve livelihoods and promote sustainability. Rooting innovation in local problems, resources, capabilities and socioeconomic conditions makes it meaningful to communities, which maintain control over the processes and outcomes. Grassroots innovation requires adaptable, locally inclusive policies — quite different to the mainstream.
Development organisations, funders, governments and nongovernmental organisations (NGOs) recently put grassroots innovation back on the agenda. But for practitioners, it never really went away. Many networks have supported it since the 1970s or earlier, when organisations like the Intermediate Technology Development Group (founded in 1966, now called Practical Action) and the Dag Hammarskjöld Foundation drew initial attention to grassroots innovation as part of an alternative to the high-technology and large-scale industrialisation dominating development theory and practice.
The challenge now is to bring the potential of grassroots innovation into debates about research, indicators, and pathways to sustainability at the UN Conference on Sustainable Development (Rio+20) and beyond.
Diversity and perspectives
The sheer diversity of grassroots innovation — in terms of technologies, people involved, organisations, purposes and contexts — makes it difficult to define. Overlapping terms such as user-led innovations, inclusive innovation, social innovation and hacking cultures can add to confusion. But there is one defining characteristic: grassroots innovation must emerge from, or be directed towards, local development.
Grassroots innovations develop through networks of activists, practitioners and organisations generating novel, bottom-up solutions for sustainable development — solutions that respond to the local situation and communities' knowledge, interests and values. Grassroots innovation can be viewed from three distinct perspectives, each focusing on a different part of the process.
The local ingenuity perspective focuses on innovations from local groups or individual inventors. They might be farmers developing irrigation systems [1], or drivers developing gear trains for their cycle rickshaws [2]. The emphasis is on people innovating for themselves and their community, perhaps drawing upon traditional and indigenous knowledge, and occasionally turning it into social enterprise, sometimes with outside help.
Organisations such as the Honey Bee Network and complementary institutions (see Table 1) seek out and help innovators documenting and developing knowledge, ideas and products, such as a pedal-powered washing machine [3] and a compost aerator device. After devastating earthquakes in Alto Mayo in Perú in 1990, Practical Action worked with local communities to develop 'improved quincha' — timber and lattice frame homes with an earth infill, based on traditional local housing technologies. Thousands of households adopted these after they withstood subsequent earthquakes. Practical Action has also helped develop the zeer pot [5], a homemade refrigerator that extends produce storage from four to twenty days, reducing waste and improving nutrition.
A second perspective focuses on local empowerment resulting when communities and technology developers interact. Local groups may not be the innovators, but developers make sure they are fully included in adopting and benefiting from technology, which may even be mass-produced devices such as solar photovoltaic systems. In other cases, the technology is deliberately developed with community engagement in the design, manufacture, maintenance and operation.
The emphasis here is on the innovation process as a tool or catalyst for developing capabilities, relationships and organisational partnerships useful to wider development processes — such as generating revenues, improving security and livelihoods, and building organisational capacity. The Social Technologies Network in Brazil (See Table 1), for instance, claims that a more socially just relationship can be built between technologists and local communities if the community controls both the process of innovation and the distribution of outcomes.
The third perspective is more critical sometimes, dismissing support for small-scale initiatives and simple technology as romanticising 'second-class' development. Grassroots innovation programmes are seen as overly optimistic and failing to address the root political, economic and social causes of poverty, social exclusion and unsustainability — causes that cannot be changed by working at the local level. Grassroots innovation does little to address the distribution of power in trade and investment, or the knowledge economies that channel scientific activities towards the interests of the wealthy and those controlling capital. But supporters of grassroots innovation counter that bottom-up activity makes these sometimes unjust economic and social structures all the more visible, and itself serves as criticism of the exclusions associated with conventional innovation systems.
Regardless of one's perspective, grassroots innovators are generating valuable knowledge and experience. The challenge for scientific research institutions or conventional innovators is to learn how to engage with that diversity.
Innovation networks
Given the frequent informality of grassroots innovation, data are patchy. Table 1, by no means exhaustive, lists networks and programmes that provide funds, visibility, guidance, training, services, dissemination, workshops, and advocacy for grassroots innovation.
Table 1: Grassroots Innovation Networks
Network/Institution Description
Activities and geographical focus Example Innovations
Prolinnova(Promoting Local Innovation in ecologically oriented agriculture and Natural Resource Management) Promotes local innovation in ecologically oriented agriculture and natural-resource management. It recognises indigenous knowledge and informal experimentation among farmers, forest dwellers, pastoralists, and fisherfolk. The intention is to develop methods, build capacity and scale up experience. International: 16 country platforms in Africa and Asia, and a regional Andes platform Farmer-led documentationusing participatory video in Ghana;participatory innovation development for climate change [913kB]adaptation in Nepal; linking innovation in agriculture and management of HIV/AIDS in Malawi;innovation in livestock-keeping by women [1.65MB] in South Africa
International Network on Appropriate Technology; andAnnual International Conference on Appropriate Technology Developed to continue the work of annual conferences on appropriate technology. Africa, Global South and USA
Five annual conferences since 2004 Earth construction to meet urban housing needs in Africa; ICTs for crop improvement and access to markets
Asia-Pacific National Innovation Systems Online Resource Centre
Provides access to resources and information amassed through projects that promote national innovation policy and practice in Asia-Pacific countries. Includes the Directory on Green Grassroots Innovation and Traditional Knowledge, which encourages policymakers in academia and research and development (R&D) institutions to focus on grassroots innovation.
Field visits and six workshops held 2007–2008 in China, Philippines, Malaysia, Sri Lanka and India Linked with Honey Bee Network (see below)
Honey Bee Network
Brings togetherindividuals and institutionscollecting, documenting, and disseminating innovations and practices at the grassroots level.
The network receives institutional support from the Society for Research and Initiatives for Sustainable Technologies and Institutions (SRISTI) and the National Innovation Foundation (NIF)
It has documented over 100,000 ideas, local innovations and traditional knowledge practices. Members can join the twice-annual Shodh Yatra journey, visiting rural communities to identify and document unrecognised ingenuity. Techniques for cultivating locally adapted traditional rice and fruit trees; labour/cost-saving machinese.g. for weaving sari cloth into low-cost sanitary napkins and processing bamboo;irrigation systems suited to local crops; gear trains forcycle rickshaws
Grassroots Innovation Augmentation Network (GIAN) A technology andbusiness incubator of grassroots innovations and traditional knowledge, linked to the Honey Bee Network and the NIF. Asia – India; six regions and many state-level incubators
GIAN has set up incubation centres across India to bring innovations to market. Camel bus; film projector; groundnut digger; trench digger
Digital Library (TKDL)
Bridges the gap between traditional knowledge information in local languages and international patent examiners. Asia – India
Over 150 books on traditional medicine have been transcribed so far Traditional Knowledge in Indian Systems of Medicineincluding Ayurveda, Unani, Siddha and Yoga
Centre of Science for Villages (CSV) Links research scientists and rural communities through training and other initiatives. Asia – India
Over 100 staff and volunteers at three demonstration campuses Rainwater harvesting; plant-based pesticides; honey bee apiary
China Innovation Network (CHIN) Tianjin University A twin centre to SRISTI. Plans to establish an innovation scholarship and an international grassroots innovation and traditional knowledge registry. Asia – China; involves 54 universities from 30 provinces of China
Scouted about 6000 innovations Cycle-based hoe; simple lift to bring agricultural produce to a rooftop for drying
The National Grassroots Innovation Databank – Malaysia Provides institutional support in identifying, sustaining and scaling up Malaysia's grassroots innovations and traditional knowledge
Asia-Pacific – Malaysia
228 innovations listed Preventing mosquito breeding in roof gutters;bioethanol produced from starch extracted from cassava
Uses technology to challenge poverty, working with communities on energy, agriculture, urban infrastructure, new technologies, and waste management
International; UK head office; offices in Bangladesh, East Africa, Latin America, Nepal, South Asia, Southern Africa and Sudan Nanotechnology for water filtration; gravity ropeways for transporting produce to market in mountainous areas; cost-effective housing reconstructionpost-tsunami
Social Technologies Network – Brazil (Red de Tecnologias Sociales)
Supports products and techniques developed cooperatively with communities. It has inspired other networks, such asRed TISA (see below). 900 member organisations from Latin America
Annual Social Technology Prize builds adatabase of entrants and projects. Potable water storage (cisternas); bio-digesters using cattle dung for home energy;seed fair for exchange of traditional varieties in rural Argentina and Paraguay
Network on Technologies for Social Inclusion – Argentina (Red de Tecnologías para la Inclusion Social de Argentina – Red TISA) Helps create and exchange community and techno-scientific knowledge, and shares learning for inclusive and sustainable development. Latin America – Argentina
90 institutions and projects Cooperative recycling ventures;sugarcane harvesting machine for small-scale producers
Social Technologies Bank – Brazil (Fundacao Banco Tecnologias Sociais)
This database includes social technologies certified by the Social Technology Prize of the Bank of Brazil Foundation. Latin America –Brazil
Over 600 certified entries Dryland horticulture and processing of cashew nutsand fruits into pulp; urban agroecology projects; water conservation and recycling
Uruguayan Center for Appropriate Technology A non-profit organisation working closely with the Latin American Social Ecology Centre on energy, agroecology and medicinal plants.
Latin America – Uruguay
Low-cost sustainable energy production; knowledge maps of local and traditional medicinal plant uses
Grassroots Innovations
Provides research-based insights into grassroots innovation processes. UK; expanding to other countries Has documented grassroots innovations in energy, food, housing and complementary currencies.
Grassroots Invention Group (GIG) Develops low-cost personal computation and production technologies
20 active projects Prometheus, a Learning Independence Network being developed in Costa Rica; new approaches to teaching computer programming
D-Lab, MIT
Promotes appropriate, low-cost technologies for international development. USA and International
Hundreds of projects
Portable solar cooker; ceramic water filter; low cost, pedal-powered rickshaw lighting
Pioneered the term 'social entrepreneurs' for people solving pressing social needs, and changing society. The Ashoka Fellowship for social entrepreneurship, with over 2000 fellows, supports networking and learning to achieve social goals. International; programmes in over 60 countries; 25 regional offices in Africa, the Americas, Asia, Europe the Middle East and North Africa. Home-based nurse training in South Africa;youth involvement in community forest management in Peru; digital inclusion in Brazil
Contributing to sustainability
The grassroots approach is only one of many ways to promote more inclusive innovation, and just one of many approaches to sustainable development. That said, new initiatives looking to the grassroots should learn from established appropriate technology centres and science shops (organisations that respond directly to the needs of citizens and local groups for scientific research) [6]. Research has identified three enduring dilemmas characteristic of grassroots innovation. [7]
One is the dilemma of how to focus on local problems whilst simultaneously seeking wide‐scale diffusion and influence. Funders and support programmes often ask for ways to scale-up or replicate innovations across regions and markets, and this can undermine the goal of tailoring for local settings. For example, it takes more effort to expand the practice of building housing from local materials and through local capabilities than to expand the sale of standard housing.
The second dilemma is that locally appropriate innovations often conform to the social and economic imbalances they ultimately seek to transform. If the goal is local empowerment, grassroots innovations need to promote sustainable and socially just development, not perpetuate prevailing conditions. For example, biogas generators developed to provide energy access for India's poor produced technology products that only wealthier villagers can afford. [8] In effect, grassroots innovations need to be unsuitable in the short-term, and accompanied by processes that can induce changes that make them work in a more just future.
The biggest dilemma is how to work at finding project‐based solutions to problems that require changing the balance of social and economic power. Grassroots projects seek to incorporate sustainability and social justice without really addressing the wider social structures that cause unsustainability and injustice. Grassroots innovators tend not to have access to R&D institutions, struggle with insufficient resources and infrastructures, and lack the full range of capabilities that would help their activity flourish. And an earlier generation of science shops shows how activities can be misrepresented and marginalised by inappropriately conventional measures of scientific knowledge, such as peer-reviewed journal publications.
There are signs that grassroots innovators are learning to overcome these dilemmas (See Box 1). But without strategies for addressing broader structural problems, grassroots innovations will struggle.
Box 1. Learning to overcome enduring dilemmas in grassroots innovation
In Kenya, experience with developing decentralised solar photovoltaic electricity [9] suggests it is possible to both transform and conform to the local environment. But it also indicates the long-term challenges involved: the process of developing technological systems and business models, training installers, and countering indifference from policymakers took years, but helped change the socioeconomic environment in ways favourable to the solar home system design that was developed.
In India, the People's Science Movement [10] developed an approach that addresses the social relationships required to make technologies work. Innovators found that implementing a cleaner and safer process for tanning animal hides required parallel innovations to organisational processes. This included re-designing relationships between collectors of animal hides and tannery processing centres, and developing a cooperative business structure for the leather goods industry in the region.
And in Brazil, 'social technology incubators' are being developed through partnerships between universities, cooperatives, and fair trade schemes as a strategy for making the grassroots more central to research and development.
Connecting with the grassroots
From community food and energy initiatives to local (re-)manufacturing, community sanitation and water projects, there is a ferment of innovative grassroots activity for sustainable development. Local innovative capabilities generate 'user-led' ideas for sustainable development that stop communities becoming locked into development unsuitable to changing circumstances.
But taking grassroots innovation seriously will entail addressing relations with science and technology institutions, and this includes rethinking conventional innovation systems.[11]
Policies can play a part, by rewarding researchers who work with local communities, and recognising how this work complements more standard scientific outputs such as publications and patents. But discussions for the Rio+20 process and beyond, about research networks and indicators for sustainability, have yet to address how grassroots innovation can be factored into knowledge production processes.
Intellectual property is another thorny issue. Whilst some believe that grassroots innovation should be open and freely disseminated, others believe that innovators should benefit from their invention. Even open source enthusiasts recognise a need to 'protect' innovations from the patenting activities of more predatory firms. The Honey Bee Network is trying to do just that, through a partnership with the National Innovation Foundation, though there are many challenges.
Re-balanced innovation policies that take the grassroots seriously would provide genuine spaces and resources for ongoing engagement between local communities and research. Engagement needs to be a two-way process that ensures the grassroots informs future research agendas and funding priorities. A participatory planning process in Ecuador, consisting of roundtable meetings and discussions, is unusual in the way it is trying to incorporate grassroots and indigenous knowledge into a National Plan for Science, Technology, Innovation and Ancestral Knowledge. [12]
Yet distinctions between conventional and grassroots innovation systems are not clear-cut. Many people network through newsletters, face-to-face visits or events. But mobile phones and radios are also used to transfer cash, share market information, and disseminate novel practices in sectors such as agriculture and construction. So successful grassroots innovations partly depend on technologies produced in conventional innovation systems, just as domestication of high technologies for local use requires grassroots innovation to adapt to local needs.
Wealthier societies also contain many grassroots innovators and networks [13], representing a movement for sustainable innovation that differs from the mainstream of clean technology and green economy debates that have featured in the run up to the Rio+20 summit.
High-level events like Rio+20 open the way for the international community to connect much more effectively and equitably with the wealth of bottom-up activities. The 1992 UN Conference on Environment and Development in Rio tried to do this with Local Agenda 21. Renewed interest in grassroots innovation provides an opportunity to address enduring challenges and create the conditions needed for all countries to benefit from the resourcefulness of their communities.
Adrian Smith is a senior research fellow, and Elisa Arond is a research officer, both at the Science and Technology Policy Research Unit (SPRU)and the STEPS Centre (Social, Technological and Environmental Pathways to Sustainability) of the University of Sussex, UK.
Mariano Fressoli is a researcher, and Hernán Thomas is professor of science, technology and society, both working at the Instituto de Estudios sobre la Ciencia y la Tecnología at the Universidad Nacional de Quilmes, Argentina.
Dinesh Abrol is a chief scientist at the National Institute of Science, Technology and Development Studies, India.
This article is part of a Spotlight on Supporting grassroots innovation.
Source: SciDev.Net – 3 May 2012
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text:
\section{Introduction}

For any $n \in \mathbb{N}$, $n \geq 2$, let $\Sing_n$  be the semigroup of all singular (i.e. non-invertible) transformations on $[n]:=\left\{1,...,n\right\} $. It is well-known (see \cite{H66}) that $\Sing_n$ is generated by its idempotents of defect $1$ (i.e. the transformations $\alpha \in \Sing_n$ such that $\alpha^2 = \alpha$ and $\rk(\alpha) := \vert \Ima(\alpha) \vert = n-1$). There are exactly $n(n-1)$ such idempotents, and each one of them may be written as $(a \to b)$, for $a,b \in [n]$, $a \ne b$, where, for any $v \in [n]$,
\[ (v) (a \to b) := \begin{cases}
		b &\text{if } v = a,\\
		v &\text{otherwise}.
	\end{cases} \]
Motivated by this notation, we  refer to these idempotents as \emph{arcs}.

In this paper, we explore the natural connections between simple digraphs on $[n]$ and subsemigroups of $\Sing_n$; explicitly, for any simple digraph $D$ with vertex set $V(D)=[n]$ and edge set $E(D)$, we associate the semigroup
\[ \langle D \rangle := \left\langle (a \to b) \in \Sing_n : (a,b) \in E(D) \right\rangle. \]
We say that a subsemigroup $S$ of $\Sing_n$ is \emph{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 \emph{undirected} if its edge set is a symmetric relation on $V(D)$, and it is \emph{transitive} if its edge set is a transitive relation on $V(D)$. We shall always assume that $D$ is \emph{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 $\langle D \rangle \cong \genset{D_1} \times \dots \times \genset{D_k}$, where $D_1, \dots, D_k$ are the connected components of $D$. We say that $D$ is \emph{strong} (or \emph{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 \emph{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, $\Sing_n$ is arc-generated by the complete undirected graph $K_n$. In fact, for $n \geq 3$, $\Sing_n$ is arc-generated by $D$ if and only if $D$ contains a strong tournament (see \cite{H78}). The semigroup of order-preserving transformations $\text{O}_n := \{ \alpha \in \Sing_n : u \le v \Rightarrow u \alpha \le v \alpha \}$ is arc-generated by an undirected path $P_n$ on $[n]$, while the Catalan semigroup $\text{C}_n := \{ \alpha \in \Sing_n : v \le v \alpha, u \le v \Rightarrow u \alpha \le v \alpha \}$ is arc-generated by a directed path $\vec{P}_n$ on $[n]$ (see \cite[Corollary 4.11]{S96}). The semigroup of non-decreasing transformations $\text{OI}_n := \{ \alpha \in \Sing_n : v \le v \alpha \}$ is arc-generated by the transitive tournament $\vec{T}_n$ on $[n]$ (Figure \ref{fig:vecT5} illustrates $\vec{T}_5$).

\begin{figure}[!h]
\begin{center}
\begin{tikzpicture}
	\node (1) at (0,0) {1};
	\node (2) at (2,0) {2};
	\node (3) at (4,0) {3};
	\node (4) at (6,0) {4};
	\node (5) at (8,0) {5};

	\draw[-latex] (1) -- (2);
	\draw[-latex] (2) -- (3);
	\draw[-latex] (3) -- (4);
	\draw[-latex] (4) -- (5);

	\draw[-latex] (1) .. controls(2,1) .. (3);
	\draw[-latex] (1) .. controls(3,2) .. (4);
	\draw[-latex] (1) .. controls(4,3) .. (5);

	\draw[-latex] (2) .. controls(4,1) .. (4);
	\draw[-latex] (2) .. controls(5,2) .. (5);

	\draw[-latex] (3) .. controls(6,1) .. (5);
\end{tikzpicture}
\end{center}
\caption{$\vec{T}_5$} \label{fig:vecT5}
\end{figure}

Connections between subsemigroups of $\Sing_n$ and digraphs have been studied before (see \cite{S96,YY02,YY06,YY09}). The following definition, which we shall adopt in the following sections, appeared in \cite{YY09}:

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

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

Recall that the \emph{orbits} of $\alpha \in \Sing_n$ 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 \emph{cyclic} if it is a cycle with at least two vertices. An element $x \in [n]$ is a \emph{fixed point} of $\alpha$ if $x\alpha=x$. Denote by $\cycl(\alpha)$ and $\fix(\alpha)$ the number of cyclic orbits and fixed points of $\alpha$, respectively. Denote by $\ker(\alpha)$ the partition of $[n]$ induced by the \emph{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 \emph{in-neighbourhood} and the \emph{out-neighbourhood} of $v$ by
\[ N^-(v) := \{ u \in V(D) : (u, v) \in  E(D)\} \text{ and } N^+(v) := \{u \in V(D) : (v,u) \in E(D) \}, \]
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 \emph{in-degree} and \emph{out-degree} of $v$ are $\deg^-(v):=\vert N^-(v) \vert$ and $\deg^+(v):=\vert N^+(v) \vert$, respectively, while the \emph{degree} of $v$ is $\deg(v) := \vert N^-(v) \cup N^+(v) \vert$. For any two vertices $u,v \in V(D)$, the \emph{$D$-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 \emph{diameter} of $D$ is $\diam(D) := \max \{ d_D(u,v) : u,v \in V(D), \ d_D(u,v) \text{ is defined} \}$.

Let $D$ be any digraph on $[n]$. We are interested in the lengths of transformations of $\genset{D}$ viewed as words in the free monoid $D^* := \{ (a \to b ) : (a,b)\in E(D)\}^*$. Say that a word $\omega \in D^*$ \emph{expresses} (or \emph{evaluates to}) $\alpha \in \genset{D}$ if $\alpha = \omega \phi$, where $\phi : D^* \to \genset{D}$ 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{align*}
	\ell(D,r) &:= \max \left\{  \ell(D,\alpha) : \alpha \in \langle D \rangle, \rk(\alpha) = r \right\},\\
	\ell(D) &:= \max \left\{  \ell(D,\alpha) : \alpha \in \langle D \rangle \right\}.
\end{align*}

The main result in the literature in the study of $\ell(D,\alpha)$ was obtained by Howie and Iwahori, independently, when $D = K_n$.
\begin{theorem}[\cite{H80,I77}]  \label{Howie-Iwahori}
For any $\alpha \in \Sing_n$,
\[	\ell(K_n, \alpha) = n+ \cycl(\alpha) - \fix(\alpha). \]
Therefore, $\ell(K_n, r) = n +  \left\lfloor \frac{1}{2} (r-2) \right\rfloor$, for any $r \in [n-1]$, and $\ell(K_n)  = \ell(K_n, n-1) = \left\lfloor \frac{3}{2} (n-1) \right\rfloor$.
\end{theorem}

In the following sections, we study $\ell(D, \alpha)$, $\ell(D,r)$, and $\ell(D)$, for various classes of digraphs. In Section \ref{sec:short}, we characterise all digraphs $D$ on $[n]$ such that either $\ell(D, \alpha) = n + \cycl(\alpha) - \fix(\alpha)$ for all $\alpha \in \genset{D}$, or $\ell(D, \alpha) =n - \fix(\alpha)$ for all $\alpha \in \genset{D}$, or $\ell(D, \alpha) =n - \rk(\alpha)$ for all $\alpha \in \genset{D}$. In Section \ref{sec:long}, we are interested in the maximal possible length of a transformation in $\genset{D}$ 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 $\ell_{\min}^{\mathcal{C}}(n,r)$), and we provide some conjectures.

\section{Arc-generated semigroups with short words} \label{sec:short}

Let $D$ be a digraph on $[n]$, $n \geq 3$, and $\alpha \in \langle D \rangle$. Theorem \ref{Howie-Iwahori} implies the following three bounds:
\begin{equation}\label{eq:n-rk}
	\ell(D, \alpha) \ge n + \cycl(\alpha) - \fix(\alpha) \ge n - \fix(\alpha) \ge n- \rk(\alpha).
\end{equation}

The lowest bound is always achieved for constant transformations.

\begin{lemma}\label{le:constant}
For any digraph $D$ on $[n]$, if $\alpha \in \langle D \rangle$ has rank $1$, then $\ell(D, \alpha) = n - 1$.
\end{lemma}

\begin{proof}
It is clear that $\ell(D, \alpha) \geq n-1$ because $\alpha$ has $n-1$ non-fixed points. Let $\Ima(\alpha) = \{v_0\} \subseteq [n]$. 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
\[ 	C_d := \{ v \in [n] : d_D(v,v_0) = d\}. \]
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 $(v \to v') \in D$. For any distinct $v, u \in C_d$ and any choice of $v', u' \in C_{d-1}$, the arcs $(v \to v')$ and $(u \to u')$ commute; hence, we can decompose $\alpha$ as
\[ 	\alpha = \bigcirc_{d=m}^1 \bigcirc_{v \in C_d} (v \to v'), \]
where the composition of arcs is done from $m$ down to $1$.
\end{proof}

Inspired by the previous lower bounds, in this section we characterise the all connected digraphs $D$ on $[n]$ satisfying the following conditions:
\begin{align}
\forall \alpha \in \langle D \rangle, \ \ell(D, \alpha) & = n + \cycl(\alpha) - \fix(\alpha); \label{bound1} \tag{\textbf{C1}} \\
\forall \alpha \in  \langle D \rangle, \ \ell(D, \alpha) & =  n - \fix(\alpha); \label{bound2} \tag{\textbf{C2}} \\
\forall \alpha \in  \langle D \rangle, \ \ell(D, \alpha) & = n- \rk(\alpha). \label{bound3} \tag{\textbf{C3}}
\end{align}

\subsection{Digraphs satisfying condition (\ref{bound1})}

Theorem \ref{Howie-Iwahori} says that $K_n$ satisfies (\ref{bound1}). In order to characterise all digraphs satisfying (\ref{bound1}), we introduce the following property on a digraph $D$:
\begin{description}
\item[($\star$)] If $d_D(v_0, v_2) = 2$ and $v_0,v_1,v_2$ is a directed path in $D$, then $N^+\left(\{v_1,v_2\}\right) \subseteq \{v_0,v_1,v_2\}$.
\end{description}

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

\begin{remark} \label{re:star}
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 $\deg^+(v_2) >2$.
\end{remark}

\begin{remark} \label{re:star2}
Suppose that $D$ satisfies property ($\star$). If $v_0, v_1, v_2$ is a directed path in $D$ and either $v_1$ or $v_2$ have an out-neighbour not in $\{ v_0, v_1, v_2 \}$, then $(v_0,v_1) \in E(D)$.
\end{remark}

\begin{remark} \label{re:star3}
If $D$ satisfies property ($\star$), then $\diam(D) \leq 2$. Indeed, if $v_0, v_1, \dots, v_k$ is a directed path in $D$ with $d_D (v_0, v_k) = k \geq 3$, 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 \ref{re:star2}, $(v_0,v_2)\in E(D)$, which contradicts that $d_D (v_0, v_k) = k$.
\end{remark}

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

Let $D$ be a digraph and let $C_1$ and $C_2$ of be two strong components of $D$. We say that $C_1$ \emph{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 $C_1$ \emph{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 \emph{terminal} if there is no strong component $C \neq C_1$ of $D$ such that $C_1$ connects to $C$.

\begin{lemma}\label{le1:bound1}
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 $D$.
\end{lemma}
\begin{proof}
Let $C$ be a strong component of $D$. Since $D$ is closed, $C$ must be undirected. The lemma is clear if $\vert C \vert \leq 3$, so assume that $\vert C \vert \geq 4$. We have two cases:
\begin{description}
\item[Case 1:] Every vertex in $C$ has degree at most $2$. Then $C$ is a path or a cycle. Since $\vert C \vert \geq 4$ and $\diam(D) \leq 2$, then $C$ is a cycle of length $4$ or $5$; however, these cycles do no satisfy property ($\star$).

\item[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 \ref{re:star}. Hence, the neighbourhood of $a$ is complete and 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 $\diam(D) \leq 2$.
\end{description}

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.
\end{proof}

\begin{lemma}\label{le2:bound1}
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 $C_2$.
\begin{description}
\item[(i)] If $C_2$ is nonterminal, then $C_1$ fully connects to $C_2$.
\item[(ii)] Let $|C_2|  = 1$. If either $|C_1| \neq 2$, or the vertex in $C_1$ that connects to $C_2$ has out-degree at least $3$, then $C_1$ fully connects to $C_2$.
\item[(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 $C_2$.
\item[(iv)] If $|C_2| \ge 3$, then $C_1$ fully connects to $C_2$.
\end{description}
\end{lemma}
\begin{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 $|C_1| \geq 2$ or $|C_2| \geq 2$. 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 nonterminal, Lemma \ref{le1:bound1} implies that $C_1$ is complete.
\begin{description}
\item[(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 $|C_1| \geq 2$. 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 \ref{re:star2} implies $(c'_1, c_2) \in E(D)$. Suppose now that $|C_2| \geq 2$. 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 $C_2$.

\item[(ii)] Suppose that $|C_1| \geq 2$. 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 \ref{re:star}. As $|C_2|  = 1$, this shows that $C_1$ fully connects to $C_2$.

\item[(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 $, $b \neq c_1$, $d \neq c'_1$, 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 \ref{re:star2}.

\item[(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 \ref{re:star2}; 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 \ref{re:star2}.
\end{description}
\end{proof}

\begin{lemma}\label{le3:bound1}
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 $D \setminus C_3$.
\end{lemma}
\begin{proof}
By Lemma \ref{le2:bound1} \textbf{(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 \ref{re:star2} 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 \ref{re:star2} 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 $D \setminus C_3$.
\end{proof}

The following result characterises all digraphs satisfying condition (\ref{bound1}).

\begin{theorem} \label{th:l=g}
Let $D$ be a connected digraph on $[n]$. The following are equivalent:
\begin{description}
	\item[(i)] \label{it:l=g1} For all $\alpha \in \langle D \rangle$, $\ell(D, \alpha) = n + \cycl(\alpha) - \fix(\alpha)$.

	\item[(ii)] \label{it:l=g2} $D$ is closed satisfying property ($\star$).
\end{description}
\end{theorem}

\begin{proof}
In order to simplify notation, denote
\[ g(\alpha) := n + \cycl(\alpha) - \fix(\alpha). \]
First, we show that \textbf{(i)} implies \textbf{(ii)}. Suppose $\ell(D,\alpha) = g(\alpha)$ for all $\alpha \in \genset{D}$. 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 $\alpha \in \genset{D} \backslash D$, so $1 < \ell(D, \alpha) \neq g(\alpha) = 1$. In order to prove that property ($\star$) holds, let $1,2,3$ be a shortest path in $D$. If $(2 \to v) \in \genset{D}$, for some $v \in [n]\setminus\{1,2,3\}$, then $\alpha = 3v3v \in \genset{D}$, but $g(\alpha) = 2 \neq \ell(D,\alpha) = 3$. If $(3 \to v) \in \genset{D}$, then $\alpha = 3vvv \in \genset{D}$, but $g(\alpha) = 3 \neq \ell(D,\alpha) = 4$. Therefore, $N^+(\{2,3 \}) \subseteq \{ 1,2,3 \}$, and ($\star$) holds. \\

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

\begin{claim} \label{claim:alpha_SC}
Let $C$ be a strong component of $D$. Then either $\alpha$ fixes all vertices of $C$ or $|(C \alpha) \cap C| < |C|$.
\end{claim}
\begin{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 $(u \to v)$  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.
\end{proof}

\begin{claim} \label{claim:d=2}
Let $u,v \in [n]$ be such that $u \alpha = v$. If $d_D(u, v) = 2$, then:
\begin{enumerate}
	\item $v$ is in a terminal component of $D$.

	\item 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 $x \alpha \in \{x, v\}$.
\end{enumerate}
\end{claim}
\begin{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 $\genset{P_3} \cong \mathrm{O}_3$, hence we can order $u < w < v$ and $\alpha$ is an increasing transformation of the ordered set $\{u,w,v\}$; thus $u \alpha = w \alpha = v \alpha = v$.
\begin{description}
\item[Case 1:] $C_1 = C_2 $. By Lemma \ref{le1:bound1}, $C_1 \cong P_3$ and it is a terminal component. Therefore, \emph{2.} holds as there is a unique path from $u$ to $v$.

\item[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 \ref{le2:bound1}, $|C_2| = 1$, $C_2$ 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 $v$.

\item[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 \ref{le2:bound1}, $C_2$ is terminal and $u,w,v$ is the unique path of length two from $u$ to $v$, where $w$ is the other vertex of $C_2$.

\item[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 \ref{le3:bound1}, $C^{(i)} = \{ x_i \}$, $C_2 = \{v\}$ 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 $v$.
\end{description}
\end{proof}

Now we produce a word $\omega \in D^*$ expressing $\alpha$ of length $g(\alpha)$. Define
\[ U := \{ u \in D : d_D(u, u\alpha) = 2 \}. \]
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 \ref{claim:d=2}. Define a word $\omega_0 \in D^*$ by
\[ 	\omega_0 := \bigcirc_{u \in U} (u \to u^\prime) (u^\prime \to u \alpha ). \]

Sort the strong components of $D$ in topological order: $C_1, \dots, C_k$, i.e. for $i \neq j$, $C_i$ connects to $C_j$ only if $j > i$. For each $1 \leq i \leq k$, define
\[ 	S_i := \{v \in C_i \setminus (U \cup U^\prime) : v \alpha \in C_i\}, \]
where $U^\prime := \{ u^\prime : u \in U \}$, and consider the transformation $\beta_i : C_i \to C_i$ defined by
\[ 	x \beta_i = \begin{cases}
		x \alpha &\text{if } x \in S_i\\
		x &\text{otherwise}.
	\end{cases} \]
If $|C_i| \le 2$ or $C_i \cong P_3$, then $\cycl(\beta_i) = 0$ and $\beta_i$ can be computed with $\vert C_i \vert - \fix(\beta_i)$ arcs. Otherwise, $C_i$ is a complete undirected graph. If $\beta_i \in \Sing(C_i)$, then by Theorem \ref{Howie-Iwahori}, there is a word $\omega_i \in C_i^* \subseteq D^*$ of length $\vert C_i \vert + \cycl(\beta_i) - \fix(\beta_i)$ expressing $\beta_i$. Suppose now that $\beta_i$ is a non-identity permutation of $C_i$. By Claim \ref{claim:alpha_SC}, $\alpha$ 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 $\hat{\beta_i} \in \Sing(C_i)$ by
\[ 	x \hat{\beta_i} = \begin{cases}
		x \alpha &\text{if $x\in S_i$}\\
		a_i & \text{if } x = h_i \\
		x &\text{otherwise},
	\end{cases}\]
where $a_i$ is any vertex in $S_i$. Then $\alpha \vert_{S_i} = \hat{\beta}\vert_{S_i}$. Again by Theorem \ref{Howie-Iwahori}, there is a word $\omega_i \in C_i^* \subseteq D^*$ of length $\vert C_i \vert + \cycl(\hat{\beta}_i) - \fix(\hat{\beta}) =\vert C_i \vert + \cycl(\beta_i) - \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 $C_i$:
\[ 	\omega_i' = \bigcirc \left\{ (a \to a\alpha) : a \in [n] \setminus (U \cup U^\prime \cup C_i), a\alpha \in C_i \right\}. \]

Finally, let
\[ 	\omega := \omega_0 \omega_k \omega_k' \dots \omega_1 \omega_1' \in D^*. \]
It is easy to check that $\omega$ indeed expresses $\alpha$. Since $\sum_{i=1}^k \fix(\beta_i) = \fix(\alpha) + \sum_{i=1}^k \vert C_i \setminus  S_i \vert$ 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
\[ 	\ell(\omega) = 2 \vert U \vert + \sum_{i=1}^k (\ell(\omega_i) + \ell(\omega_i^\prime)) =  n + \sum_{i=1}^k \cycl(\beta_i) -  \fix(\alpha) =  g(\alpha). \]
\end{proof}

\subsection{Digraphs satisfying condition (\ref{bound2})}

The characterisation of connected digraphs satisfying condition (\ref{bound2}) is based on the classification of connected digraphs $D$ such that $\cycl(\alpha) = 0$, for all $\alpha \in \genset{D}$.

For $k \ge 3$, let $\Theta_k$ be the directed cycle of length $k$. Consider the digraphs $\Gamma_1, \ \Gamma_2, \ \Gamma_3$ and $\Gamma_4$ as illustrated below:

 \[ \begin{tikzpicture}[vertex/.style={circle, draw, fill=none, inner sep=0.2cm}]
    \vertex{1}{0}{2}
    \node at (0,3.4) {1};
    \vertex{2}{1}{2}
    \node at (1,3.4) {2};
    \vertex{3}{2}{2}
    \node at (2,3.4) {3};
   \vertex{4}{1}{1}
   \node at (1,1.7) {4};
   \vertex{5}{1}{0}
   \node at (1,0) {5};

    \edge{1}{4}
    \edge{2}{4}
    \edge{3}{4}
    \arc{4}{5}

\node at (1,-1) {$\Gamma_1$};
  \end{tikzpicture}

  \qquad\qquad

  \begin{tikzpicture}[vertex/.style={circle, draw, fill=none, inner sep=0.2cm}]
\node at (1,-1) {$\Gamma_2$};

    \vertex{1}{1}{1}
    \node at (1,1.7) {2};
    \vertex{2}{2}{1}
    \node at (2,1.7) {1};
   \vertex{3}{1}{0}
    \node at (1,0) {3};
    \vertex{4}{0}{1}
     \node at (0,1.7) {4};
   \vertex{5}{0}{2}
       \node at (0,3.4) {5};

    \edge{1}{3}
    \edge{2}{3}
    \edge{3}{4}
    \arc{4}{5}
  \end{tikzpicture}

    \qquad\qquad

  \begin{tikzpicture}[vertex/.style={circle, draw, fill=none, inner sep=0.2cm}]
\node at (1,-1) {$\Gamma_3$};

    \vertex{1}{2}{0}
    \node at (2,0) {1};
    \vertex{2}{0}{0}
        \node at (0,0) {2};
    \vertex{3}{1}{1}
        \node at (1,1.7) {3};
    \vertex{4}{1}{2}
        \node at (1,3.4) {4};

    \arc{1}{2}
    \arc{2}{3}
    \arc{3}{1}
    \arc{3}{4}
  \end{tikzpicture}

  \qquad\qquad

  \begin{tikzpicture}[vertex/.style={circle, draw, fill=none, inner sep=0.2cm}]
\node at (1,-1) {$\Gamma_4$};

    \vertex{1}{0}{1}
    \node at (0,1.7) {1};
    \vertex{2}{0}{0}
     \node at (0,0) {2};
    \vertex{3}{2}{0}
    \node at (2,0) {3};
    \vertex{4}{2}{1}
    \node at (2,1.7) {4};
   \vertex{5}{2}{2}
   \node at (2,3.4) {5};

    \arc{1}{2}
    \arc{2}{3}
    \arc{3}{4}
    \arc{4}{1}
	\arc{4}{5}
  \end{tikzpicture} \]

\begin{lemma} \label{prop:acyclic_alpha}
Let $D$ be a connected digraph on $[n]$. The following are equivalent:
\begin{description}
	\item[(i)] For all $\alpha \in \langle D \rangle$, $\cycl(\alpha) = 0$.

	\item[(ii)] $D$ has no subdigraph isomorphic to $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $\Gamma_4$, or $\Theta_k$, for all $k \ge 5$.
\end{description}
\end{lemma}
\begin{proof}
In order to prove that \textbf{(i)} implies \textbf{(ii)}, we show that if $\Gamma$ is equal to $\Gamma_i$ or $\Theta_k$, for $i\in [4]$, $k \ge 5$, then there exists $\alpha \in \genset{\Gamma}$ such that $\cycl(\alpha) \neq 0$.
\begin{itemize}
\item If $\Gamma = \Gamma_1$, take
\[ 	\alpha := (3 \to 4) (4 \to 5) (1 \to 4) (4 \to 3) (2 \to 4) (4 \to 1) (3 \to 4) (4 \to 2) = 21555. \]

\item If $\Gamma = \Gamma_2$, take
\[ 	\alpha := (3 \to 4) (4 \to 5) (1 \to 3) (3 \to 4) (2 \to 3) (3 \to 1) (4 \to 3) (3 \to 2) = 21555. \]

\item If $\Gamma = \Gamma_3$, take
\[ 	\alpha := (3 \to 4) (2 \to 3) (1 \to 2) (3 \to 1) = 2144. \]

\item If $\Gamma = \Gamma_4$, take
\[	\alpha = (3 \to 4) (4 \to 5) (2 \to 3) (3 \to 4) (1 \to 2) (4 \to 1) = 21555. \]

\item Assume $\Gamma = \Theta_k$ for $k \ge 5$. Consider the following transformation of $[k]$:
	\[ (u \Rightarrow v) := (u \to u_1) \dots (u_{d-1} \to v), \]
where $u,u_1, \dots, u_{d-1}, v$ is the unique path from $u$ to $v$ on the cycle $\Theta_k$. Take
\[ 	\alpha := (1 \Rightarrow k-3) (k \Rightarrow k-4) (k-1 \Rightarrow 1) (k-2 \Rightarrow k) (k-3 \Rightarrow k-1) (k-4 \Rightarrow k-2). \]
Then, $\alpha = (k-1)(k-1) \dots (k-1) \ k \ 1 \ (k-2) $, where $(k-1)$ appears $k-3$ times, has the cyclic component $(k-2, k)$.
\end{itemize}

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

\begin{claim} \label{claim:SC_path}
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.
\end{claim}
\begin{proof}
Let $C$ be a strong component of $\bar{D}$. Clearly, $C$ is undirected and, by \textbf{(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 $\Gamma_3$ or $\Gamma_4$, 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 $\Gamma_1$ or $\Gamma_2$; clearly, $K_{3,1}$ may only appear as a terminal component.
\end{proof}

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

We introduce a new property of a connected digraph $D$:
\begin{description}
	   \item[($\star\star$)] \label{it:SC_2} For every strong component $C$ of $D$, $|C| \le 2$ if $C$ is nonterminal, and $|C| \le 3$ if $C$ is terminal.
\end{description}

\begin{lemma}\label{le:bound2}
Let $D$ be a closed connected digraph on $[n]$ satisfying property ($\star$). The following are equivalent:
\begin{description}
\item[(i)] $D$ satisfies property ($\star\star$).
\item[(ii)] $D$ has no subdigraph isomorphic to $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $\Gamma_4$, or $\Theta_k$, for some $k \ge 5$.
\end{description}
\end{lemma}
\begin{proof}
If \textbf{(i)} holds, it is easy to check that $D$ does not contain any subdigraphs isomorphic to $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $\Gamma_4$, or $\Theta_k$ for some $k \ge 5$.

Conversely, suppose that \textbf{(ii)} holds. Let $C$ be a strong component of $D$. If $C$ is non-terminal, Lemma \ref{le1:bound1} implies that $C$ is complete; hence, $|C| \le 2$ as otherwise $D$ would contain $\Gamma_4$ as a subdigraph. If $C$ is terminal, Lemma \ref{le1:bound1} implies that $C$ is complete or $P_3$; hence, $|C| \le 3$ as otherwise $D$ would contain $\Gamma_3$ as a subdigraph.
\end{proof}

\begin{theorem} \label{th:l=n-fix}
Let $D$ be a connected digraph on $[n]$. The following are equivalent:
\begin{description}
	\item[(i)] For all $\alpha \in \langle D \rangle$, $\ell(D, \alpha) = n - \fix(\alpha)$.

	\item[(ii)] $D$ is closed satisfying properties ($\star$) and ($\star\star$).
\end{description}
\end{theorem}
\begin{proof}
Clearly, $D$ satisfies \textbf{(i)} if and only if it satisfies condition (\ref{bound1}) and $\cycl(\alpha) = 0$, for all $\alpha \in \langle D \rangle$. By Theorem \ref{th:l=g}, Lemma \ref{prop:acyclic_alpha} and Lemma \ref{le:bound2}, $D$ satisfies \textbf{(i)} if and only if  $D$ satisfies \textbf{(ii)}.
\end{proof}

\subsection{Digraphs satisfying condition (\ref{bound3})}

The following result characterises digraphs satisfying condition (\ref{bound3}).

\begin{theorem}\label{thm-bands}
Let $D$ be a connected digraph on $[n]$. The following are equivalent:
\begin{description}
	\item[(i)] For every $\alpha \in \langle D \rangle$, $\ell(D, \alpha) = n - \rk(\alpha)$.

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

	\item[(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 $i_1 \in V_1$, $i_2 \in V_2$.
\end{description}
\end{theorem}

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

Now we prove that \textbf{(ii)} implies \textbf{(iii)}. If there exist $u, v, w \in [n]$ pairwise distinct such that $(u,v), (v,w) \in E(D)$ , then $\alpha = (v \to w) (u \to v)$ 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 \textbf{(iii)} implies \textbf{(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 $r:=\rk(\alpha) = \fix(\alpha)$. Let $J := \{v_1, \dots, v_{n-r}\} \subseteq V_1$ be the set of non-fixed points of $\alpha$; therefore
\[ 	\alpha = (v_1 \to v_1 \alpha) \dots (v_{n-r} \to v_{n-r} \alpha), \]
where each one of the $n-r$ arcs above belongs to $\genset{D}$. The result follows by inequality (\ref{eq:n-rk}).
\end{proof}

\section{Arc-generated semigroups with long words} \label{sec:long}

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

\begin{problem}
Is $\ell_{\max}(n)$ upper bounded by a polynomial in $n$?
\end{problem}

\begin{table}[!h]
\begin{center}
\begin{tabular}{c|c| c|c|c|c}
  $n \backslash r$ & $1$ & $2$ & $3$ & $4$ & $5$ \\
  \hline
  $2$  		& $1$ & & & &  \\
  $3$               & $2$ & $6$ &   &   &    \\
  $4$                & $3$ & $11$ & $13$ &   &    \\
  $5$                & $4$ & $18$ & $24$ & $33$ &    \\
  $6$ 		& $5$ & $26$ & $42$ & $51$ & $66$
\end{tabular}
\caption{First values of $\ell_{\max}(n,r)$} \label{table:lmax}
\end{center}
\end{table}

The first few values of $\ell_{\max}(n,r)$, calculated with the GAP package \emph{Semigroups} \cite{M15}, are given in Table \ref{table:lmax}. By Lemma \ref{le:constant}, $\ell_{\max}(n,1) =  n-1$ for all $n \geq 2$; henceforth, we shall always assume that $n \geq 3$ 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.

\subsection{Acyclic digraphs}

For any $n \geq 3$, let $\Acy_n$ be the set of all acyclic digraphs on $[n]$, and, for any $r \in [n-1]$, define
\begin{align*}
	\ell_{\max}^{\Acy}(n,r) &:= \max \left\{ \ell(A, r) :  A \in \Acy_n \right\},\\
	\ell_{\max}^{\Acy}(n) &:= \max \left\{ \ell(A) : A \in \Acy_n  \right\}.
\end{align*}
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.

\begin{theorem} \label{th:lAcy}
For any $n\geq 3$ and $r \in [n-1] \setminus \{ 1\}$,
\begin{align*}
\ell_{\max}^{\Acy}(n,r) &= \frac{(n-r)(n+r-3)}{2} + 1, \\
\ell_{\max}^{\Acy}(n) & = \ell_{\max}^{\Acy}(n,2) = \frac{1}{2} (n^2 -3n + 4).
\end{align*}
\end{theorem}

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

\begin{lemma}
Let $n \geq 3$ and $A \in \Acy_n$. Then, $\ell(A,n-1)$ is equal to the length of a longest path in $A$. Therefore,
\[ 	\ell_{\max}^{\Acy}(n,n-1) = n-1. \]
\end{lemma}
\begin{proof}
Let $v_1, \dots, v_{l+1}$ be a longest path in $A$. Then $\alpha \in \langle A \rangle$ defined by
\[ 	v \alpha := \begin{cases}
	v_{i+1} &\text{if } v = v_i, \ i \in [l], \\
	v &\text{otherwise},
	\end{cases} \]
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 $A^*$:
\[ 	\alpha = (u_1 \to v_1) (u_2 \to v_2) \dots (u_s \to v_s). \]
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 $s \le l$.
\end{proof}

The following lemma shows that the formula of Theorem \ref{th:lAcy} is an upper bound for $\ell_{\max}^{\Acy}(n,r)$.

\begin{lemma}
For any $n\geq 3$ and $r \in [n-1] \setminus \{ 1\}$,
\[ 	\ell_{\max}^{\Acy}(n,r) \leq \frac{(n-r)(n+r-3)}{2} + 1. \]
\end{lemma}
\begin{proof}
Let $A$ be an acyclic digraph on $[n]$, let $\alpha \in \genset{A}$ be a transformation of rank $r \geq 2$, 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 $\psi_A(u,v)$.

\begin{claim}\label{lem:l_A}
$\ell(A, \alpha) \le \sum_{v \in [n]} \psi_A(v,  v \alpha)$.
\end{claim}
\begin{proof}
Let $\omega = (a_1 \to b_1) \dots (a_l \to b_l)$ be a shortest word expressing $\alpha$ in $A^*$, with $l = \ell(A, \alpha)$. Say that the arc $(a_i \to b_i)$, $i \geq 2$, \emph{carries} $v \in [n]$ if $v (a_1 \to b_1) \dots (a_{i-1} \to b_{i-1}) = a_i$ (assume that $a_1 \to b_1$ only carries $a_1$). Every arc $(a_i \to b_i)$ 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 $v_i = v (a_1 \to b_1) \dots (a_i \to b_i)$ (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
\[ 	 l \le \sum_{v =1}^n l(v) \le \sum_{v \in [n]} \psi_A(v,  v \alpha). \]
\end{proof}

\begin{claim}\label{lem:psi(A)1}
If $|L| \ge 2$, then $\sum_{v \in [n]} \psi_A(v,  v \alpha) \le \frac{(n-r)(n+r-3)}{2}$.
\end{claim}
\begin{proof}
As $|L| \ge 2$, and $A$ is topologically sorted, we have $\{ n, n-1 \} \subseteq  L$, and any $\alpha \in \genset{A}$ 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
\[ 	\psi_A(v, v\alpha) \le \min\{ n-1, v\alpha\} - v. \]
Hence
\begin{align*}
	\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{align*}
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{align*}
	\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{align*}
\end{proof}

\begin{claim}\label{lem:psi(A)2}
If $\vert L \vert = 1$, then $\ell(A, \alpha) \le \frac{(n-r)(n+r-3)}{2} + 1$.
\end{claim}
\begin{proof}
As $A$ is topologically sorted, $L = \{ n\}$. We use the notation from the proof of Claim \ref{lem:l_A}. We then have $l(n) = 0$. We have three cases:
\begin{description}
\item[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 \ref{lem:psi(A)1}, we obtain $\ell(A, \alpha) \le \frac{(n-r)(n+r-3)}{2}$.

\item[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 $\ell(A, \alpha) \le \frac{(n-r)(n+r-3)}{2}$.

\item[Case 3:] $n$ has at least two pre-images under $\alpha$. Let $\omega = (a_1 \to b_1) \dots (a_l \to b_l)$ be a shortest word expressing $\alpha$ in $A^*$, and denote $\alpha_0 = \id$ and $\epsilon_i = (a_i \to b_i)$, $\alpha_i = \epsilon_1 \dots \epsilon_i$ for $i \in [l]$. We partition $n \alpha^{-1}$ into two parts $S$ and $T$:
\[ 	S = \{v \in n \alpha^{-1} : v_{l(v) - 1} = n-1\}, \quad T = n \alpha^{-1} \setminus S. \]
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 $(n-1 \to n)$ 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 $\ell(A, \alpha) \le \frac{(n-r)(n+r-3)}{2} + 1$.
\end{description}
\end{proof}
\end{proof}

The following lemma completes the proof of Theorem \ref{th:lAcy}.

\begin{lemma}
For any $n \geq 3$ and $r \in [n-1] \setminus \{ 1\}$, there exists an acyclic digraph $Q_n$ on $[n]$ and a transformation $\beta_r \in \genset{Q_n}$ of rank $r$ such that
\[ \ell(Q_n, \beta_r) \geq  \frac{(n-r)(n+r-3)}{2} + 1. \]
\end{lemma}
\begin{proof}
Let $Q_n$ be the acyclic digraph on $[n]$ with edge set
\[ E(Q_n) := \left\{ (u, u+1) : u \in [n-1] \right\} \cup \left\{ (n-2, n) \right\}. \]
For any $r \in [n-1] \setminus \{ 1\}$, define $\beta_r \in \genset{Q_n}$ by
\[	v\beta_r := \begin{cases}
		n-r+v &\text{if } v \in [r-2], \\
		n - 1 &\text{if } v \in [n-1] \setminus [r-2], \  n-v \equiv 0 \mod 2,\\
		n &\text{if }v \in [n-1] \setminus [r-2], \  n-v \equiv 1 \mod 2,\\
		n &\text{if } v = n.
	\end{cases} \]

Let $\beta_r$ be expressed as a word in $Q_n^*$ of minimum length as
\[ 	\beta_r = (a_1 \to b_1) \dots (a_l \to b_l), \]
where $l= \ell(Q_n, \beta_r)$. Denote $\alpha_0 := \id$, $\epsilon_i := (a_i \to b_i)$, 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 $u \alpha_i \ne u \alpha_{i-1}$.

\begin{claim}
For each $i \in [l]$, the arc $\epsilon_i$ carries exactly one vertex.
\end{claim}
\begin{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 $\epsilon_s = (n-2 \to n-1)$ for at least one $s \in [l]$, and $\epsilon_j = (n-1 \to n)$. For $s \in [j-1]$, replace all occurrences $\epsilon_s = (n-2 \to n-1)$ with $\epsilon_s^\prime := (n-2 \to n)$ and delete $\epsilon_j$: this yields a word in $Q_n^*$ of length $l^\prime<l$ expressing $\beta_r$, which is a contradiction.
\end{proof}

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{align*}
	\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{align*}
\end{proof}

\subsection{Strong tournaments}

Let $n \geq 3$. Recall that if $T$ is a strong tournament on $[n]$, then $\{ a \to b : (a,b) \in E(T) \}$ is a minimal generating set of $\Sing_n$. Let $\Tour_n$ denote the set of all strong tournaments on $[n]$. For $r \in [n-1]$, define
\begin{align*}
	\ell_{\max}^{\Tour}(n, r) &:= \max \{ \ell(T, r) : T \in \Tour_n  \},\\
	\ell_{\max}^{\Tour}(n) &:= \max \{ \ell(T) : T \in \Tour_n  \}.
\end{align*}

Define analogously $\ell_{\min}^{\Tour}(n,r)$ and $\ell_{\min}^{\Tour}(n)$. The first few values of $\ell_{\min}^{\Tour}(n,r)$ and $\ell_{\max}^{\Tour}(n,r)$, calculated with the GAP package \emph{Semigroups} \cite{M15} using data from \cite{MK15}, are given by Table \ref{table:lmax_tour}. The calculation of these values has been the inspiration for the results and conjectures of this section.

\begin{table}[!hb]
\begin{center}
\begin{tabular}{c|c|c|c|c|c}
	$n \backslash r$ & $2$ & $3$ & $4$ & $5$ & $6$ \\
	\hline
	$3$ & $(6, 6)$ & & & & \\
	$4$ & $(8,8)$ & $(11,11)$ & & & \\
	$5$ & $(6, 11)$ & $(8, 14)$ & $(10, 17)$ & & \\
	$6$ & $(8, 13)$ & $(10, 18)$ & $(11, 21)$ & $(13, 24)$ & \\
	$7$ & $(8, 16)$ & $(10, 22)$ & $(11, 26)$ & $(13, 29)$ & $(15, 32)$
\end{tabular}
\caption{First values of $\left( \ell_{\min}^{\Tour}(n,r) , \ell_{\max}^{\Tour}(n,r) \right)$.} \label{table:lmax_tour}
\end{center}
\end{table}

\begin{lemma}\label{lem:same_kernel}
Let $n \geq 3$ and $T \in \Tour_n$.
\begin{enumerate}
	\item For any partition $P$ of $[n]$ into $r$ parts, there exists an idempotent $\alpha \in \Sing_n$ with $\ker(\alpha) = P$ such that $\ell(T, \alpha)= n - r$.

	\item For any $r$-subset $S$ of $[n]$, there exists an idempotent $\alpha \in \Sing_n$ with $\Ima(\alpha) = S$ such that $\ell(T, \alpha)= n - r$.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
\item 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 $\ell(T, \alpha) = n - r$.

\item Without loss of generality, let $S = [r] \subseteq [n]$. For every $v \in [n]$, define
\[ 	s(v) := \min \{ s \in S: d_T(s', v) \geq d_T(s,v), \forall s' \in S\}. \]
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
\[ E(D) := \{ (a,b) : (a,b) \in E(P_v) \text{ for some } v \in [n]\}. \]
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
\[  \alpha = \bigcirc_{v=n}^{r+1} (v \to v_1). \]
\end{enumerate}
\end{proof}

\begin{lemma} \label{le:idempotents}
Let $n \geq 3$, $T \in \Tour_n$, and $\alpha:=(u \to v) \in \Sing_n$, for $(u,v) \not \in E(T)$. Then
\[ \ell(T, \alpha) = 4 d_T(u,v) - 2. \]
\end{lemma}
\begin{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{align*}
	(v_0 \to v_d)  = & (v_d \to v_0) (v_{d-1}  \to v_d) (v_{d-2} \to v_{d-1}) \cdots (v_1 \to  v_2) (v_0 \to v_1 ) \\
	& \left( (v_2 \to v_0) (v_1  \to v_2) \right) \left( (v_3 \to v_1) (v_2 \to v_3) \right) \cdots \left( (v_d \to v_{d-2}) (v_{d-1} \to v_d) \right) \\
	& (v_{d-2} \to v_{d-1}) \cdots (v_0 \to v_1).
\end{align*}

So $\ell(T, \alpha) \leq 4d-2$. For the lower bound, we note that any word in $T^*$ expressing $(u \to v)$ must begin with $(v \to u)$. 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
\[ 	1 + l + 3(l-1) = 4l-2 \ge 4 d - 2. \]
\end{proof}

Let $n = 2m+1 \geq 3$ be odd, and let $\kappa_n$ be the \emph{circulant tournament} on $[n]$ with edges $E(\kappa_n):=\{ (i, (i+j) \mod n): i \in [n], j \in [m] \}$. Figure \ref{fig:kappa5} illustrates $\kappa_5$. In the following theorem, we use $\kappa_n$ to provide upper and lower bounds for $\ell_{\min}^{\Tour}(n,r)$ and $\ell_{\max}^{\Tour}(n,r)$ when $n$ is odd.

\begin{figure}[!h]
\begin{center}
\begin{tikzpicture}

	\node (1) at (90-72*1:2) {1};
	\node (2) at (90-72*2:2) {2};
	\node (3) at (90-72*3:2) {3};
	\node (4) at (90-72*4:2) {4};
	\node (5) at (90-72*5:2) {5};

    \draw[-latex] (1) -- (2);
    \draw[-latex] (2) -- (3);
    \draw[-latex] (3) -- (4);
    \draw[-latex] (4) -- (5);
    \draw[-latex] (5) -- (1);

	\draw[-latex] (1) -- (3);
    \draw[-latex] (2) -- (4);
    \draw[-latex] (3) -- (5);
    \draw[-latex] (4) -- (1);
    \draw[-latex] (5) -- (2);

\end{tikzpicture}
\end{center}
\caption{The circulant tournament $\kappa_5$.} \label{fig:kappa5}
\end{figure}

\begin{theorem}
For any $n$ odd, we have
\begin{align*}
	n + r - 2 &\le \ell_{\min}^{\Tour}(n,r) \le n + 8r,\\
	(\hat{r} + 1)(n - \hat{r}) - 1 &\le \ell_{\max}^{\Tour}(n,r) \le 6rn + n -10r.
\end{align*}
where $\hat{r} = \min\{r-1, \lfloor n/2 \rfloor\}$.
\end{theorem}
\begin{proof}
Let  $T \in \Tour_n$ and $2 \le r \le n-1$. We introduce the following notation:
\begin{align*}
 	[n]_r & := \{ {\bf u} := (u_1, \dots, u_r) : u_i \neq u_j, \forall i, j \},  \\
	\Delta(T, r) &:= \max \left\{  \sum_{i=1}^r d_T(u_i, v_i) : {\bf u}, {\bf v} \in [n]_r \right\}.
\end{align*}
The result follows by the next claims.
\begin{claim}
$r'(\diam(T) - r' + 1) + r-r' \le \Delta(T,r) \le r \diam(T)$, where $r' = \min\{r, \lfloor ( \diam(T)+1)/2 \rfloor\}$.
\end{claim}
\begin{proof}
The upper bound is clear. For the lower bound, let $u, v \in [n]$ be such that $d_T(u,v) = \diam(T)$, and let $u = v_0, v_1, \dots, v_d = v$ be a shortest path from $u$ to $v$, where $d = \diam(T)$. 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 ${\bf u}' = (v_0, \dots, v_{r-1})$ and ${\bf v}' = (v_{d-r+1}, \dots, v_d)$, 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 $(u'_j, v'_j) \notin T$.
\end{proof}

\begin{claim}
 $\min \{ \Delta(T,r) : T \in \Tour(n) \} = \Delta(\kappa_n,r) = 2r$.
\end{claim}
\begin{proof}
Let ${\bf u} = (u_1, \dots, u_n)$ form a Hamiltonian cycle, and choose ${\bf v} = (u_n, u_1, \dots, u_{n-1})$. Then $d_T(u_i, v_i) \ge 2$ for all $i$. Conversely, since $\diam(\kappa_n) = 2$, we have $\Delta(\kappa_n,r) = 2r$.
\end{proof}

\begin{claim}
$n-r + \Delta(T, r-1) \le \ell(T,r) \le n + 6 r \diam(T) - 4r$.
\end{claim}
\begin{proof}
For the lower bound, consider $\alpha \in \Sing_n$ as follows. Let ${\bf u} = (u_1, \dots, u_{r-1})$ and ${\bf v} = (v_1, \dots, v_{r-1})$ achieve $\Delta(T,r-1)$, and let $v \notin \{v_1, \dots, v_{r-1}\}$; define
\[ 	x \alpha = \begin{cases}
		v_i &\text{if } x = u_i, \\
		v &\text{otherwise}.
	\end{cases} \]
Let $\omega = e_1 \dots e_l$ (where $e_i =(a_i \to b_i)$) 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 $c(e_i) \in \Ima(\alpha)$ if $e_i$ carries vertices eventually mapping to $c(e_i)$. Denote the number of arcs labelled $c$ as $l(c)$, we then have $l = \sum_{c \in \Ima(\alpha)} l(c)$. For any $u \in V$, there are at least $d_T(u, u \alpha)$ arcs carrying $u$. Therefore,
\[ 	l = \sum_{c \in \Ima(\alpha)} l(c)
	\ge \sum_{i=1}^{r-1} d_T(u_i, v_i) + \sum_{a \notin {\bf u}} d_T(a, v)
	\ge \Delta(T, r-1) + n-r.\]

For the upper bound, we can express any $\alpha \in \Sing_n$ of rank $r$ in the following fashion. By Lemma \ref{lem:same_kernel}, there exists $\beta \in \Sing_n$ with the same kernel as $\alpha$ such that $\ell(T, \beta) = n-r$. Suppose that $\Ima(\alpha) = \{ v_1, \dots, v_r \}$ and $\Ima(\beta) = \{u_1, \dots, u_r \}$, where $u_i \beta^{-1}  = v_i \alpha^{-1}$, for $i \in [r]$. Let $h \in [n] \setminus \Ima(\beta)$. Define a transformation $\gamma$ of $[n]$ by
\[ 	x \gamma = \begin{cases}
		v_i &\text{if } x = u_i,\\
		 v_1 & \text{if } x= h, \\
		x &\text{otherwise}.
	\end{cases}\]
Then $\alpha = \beta \gamma$, where $\gamma \in \Sing_n$, and by Theorem \ref{Howie-Iwahori}
\[ 	\ell(K_n, \gamma) = n - \fix(\gamma) + \cycl(\gamma) \le r + \frac{r}{2} = \frac{3r}{2}. \]
By Lemma \ref{le:idempotents}, each arc associated to $K_n$ may be expressed in at most $4 \diam(T) - 2$ arcs associated to $T$; therefore,
\[ 	\ell(T, \gamma) \leq \frac{3r}{2}( 4 \diam(T) - 2) = 6r \diam(T) - 3r. \]
Thus,
\[ \ell(T,\alpha) \leq \ell(T,\beta) + \ell(T,\gamma) \leq n + 6r\diam(T) -4r. \]
\end{proof}
\end{proof}

We finish this section by proposing two conjectures. 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 \ref{fig:pi5} illustrares $\pi_5$.

\begin{figure}[!ht]
\begin{center}
\begin{tikzpicture}
	\node (1) at (0,0) {1};
	\node (2) at (2,0) {2};
	\node (3) at (4,0) {3};
	\node (4) at (6,0) {4};
	\node (5) at (8,0) {5};

	\draw[-latex] (1) -- (2);
	\draw[-latex] (2) -- (3);
	\draw[-latex] (3) -- (4);
	\draw[-latex] (4) -- (5);

	\draw[-latex] (3) .. controls(2,1) .. (1);
	\draw[-latex] (4) .. controls(3,2) .. (1);
	\draw[-latex] (5) .. controls(4,3) .. (1);

	\draw[-latex] (4) .. controls(4,1) .. (2);
	\draw[-latex] (5) .. controls(5,2) .. (2);

	\draw[-latex] (5) .. controls(6,1) .. (3);
\end{tikzpicture}
\end{center}
\caption{$\pi_5$} \label{fig:pi5}
\end{figure}

\begin{conjecture}\label{conj}
For every $n \geq 3$, $r \in [n-1]$, and $T \in \Tour_n$, we have
\[  \ell(T, r) \leq \ell(\pi_n, r) = \ell_{\max}^{\Tour}(n, r) ,
\]
with equality if and only if $T \cong \pi_n$. Furthermore,
\[ \ell(\pi_n) = \ell_{\max}^{\Tour}(n) = \frac{n^2 + 3n - 6}{2},\]
which is achieved for $\alpha : = n \ (n-1) \ \dots \ 2 \ n$.
\end{conjecture}

Tournament $\pi_n$ has appeared in the literature before: it is shown in \cite{M66} that $\pi_n$ has the minimum number of strong subtournaments among all strong tournaments on $[n]$. On the other hand, it was shown in \cite{BH65} that, for $n$ odd, the circulant tournament $\kappa_n$ has the maximal number of strong subtournaments among all strong tournaments on $[n]$.

\begin{conjecture}
For every $n \geq 3$ odd, $r \in [n-1]$, and $T \in \Tour_n$, we have
\[  \ell_{\min}^{\Tour}(n, r) = \ell(\kappa_n, r).\]
Furthermore,
\[ 	\ell_{\min}^{\Tour}(n,2) = n+1 \ \text{ and } \ \ell_{\min}^{\Tour}(n,r) = n+r, \]
for all $3 \le r \le \frac{n+1}{2}$.
\end{conjecture}

redpajama_set_name: RedPajamaArXiv


First thing's first, and as with most events, planning is where we begin. Treat yourself to a personalised agenda or notebook for the most personal event of your year! Get yourself all organised for your big day with a personalised diary.
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WASHINGTON, DC — Between 2017 and 2018, the number of women working in construction trades increased by 17.6 percent, rising to well over a quarter of a million women (276,000), according to a new analysis from the Institute for Women's Policy Research (IWPR). This growth is nearly five times faster than job growth of 3.7 percent in construction occupations overall.
Read the fact sheet, "Women Gain Jobs in Construction Trades but Remain Underrepresented in the Field," at IWPR.org.


text:
Wullschlaegelia es un género de orquídeas perteneciente a la subfamilia Epidendroideae dentro de la familia Orchidaceae. Tiene cuatro especies.

Descripción

Las especies de este género se caracterizan por tener raíces  finas y largas, a menudo ligeramente tuberiformes y tallo de color amarillo pálido, delgadísimo y anémico con las hojas rectas o curvas, cónicas.

Sus pequeñas  flores aparecen en la primavera en un pálido racimo terminal más o menos largo, algo grande y espaciado. El labio es largo, articulado a los pies de la columna, pero la base se amplia en el desarrollo formado por el lateral crecimiento de los sépalos. La columna es corta y gruesa.

Distribución

El género está formado por solo dos especies de hábitos terrestres, por lo general saprofitas, que vegetan en la descomposición de la materia orgánica en los bosques húmedos. Una especie se originó en América Central y la Selva Amazónica, la otra se puede encontrar en casi toda América Latina, probablemente la única excepción de Chile.

Taxonomía
El género es muy similar al género Uleiorchis, que se distingue por ser algo pubescente, mientras que en Uleiorchis son glabras, por no presentar la raíz del tubérculo en las raíces de la primavera Uleiorchis, por su  menor tamaño y las brácteas florales uninervadas  que son plurinervadas en Uleiorchis.

El género fue presentado por Reichenbach y publicado en Botanische Zeitung.  Berlín 21: 131, en 1863. Publicado originalmente por Peter Olof Swartz como Cranichis aphylla, en 1788.
Etimología
El nombre del género  es un homenaje a Heinrich Rudolph Wullschlaegel, alemán coleccionista de plantas.

Especies
 Wullschlaegelia aphylla Rchb.f.
 Wullschlaegelia calcarata Benth.

Referencias

Enlaces externos

 Imágenes en Google

redpajama_set_name: RedPajamaWikipedia


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Alor is located at the eastern part of East Nusa Tenggara Province. If you wish to fly to Alor, Kupang is the main hub to Alor, where you can get a flight to Mali Airport in Kalabahi, the administrative capital of Alor Regency. In Kalabahi, accommodations and small eateries line up along the road nearby the airport and also along the road to the pier. Diving in Alor is remarkable and full of surprise. The water has become a regular cetacean migration route. Hence, you may encounter some whales' species passing along, these giant mammals mostly appear in Alor waters rather than in any other area in Indonesia. Divers can also spot the Ocean Sunfish during mid year when sea temperature turns low, and another big things to watch are dolphins that often spotted jumping out of the water. You will not only encounter the big stuffs but also the creatures that need macro lens to be captured, such as nudibranch, leaffish, frogfish, the famous 'nemo', and other exotic macro biotas. The regency consist of 20 big and small islands is Located around Pantar Strait, the meeting point of the Flores Sea on the north and the Ombay Strait on the south. The location has given the Alor waters unique current patterns. The strait between Pantar and Alor often becomes the convergence zone from Australia and the Flores Sea, where in the middle of the year cold current coming from Australia carries abundant nutrients which attracts big and small fish to Alor waters. Though in this period is the best time to dive in Alor, but it is necessary to remember that water temperature could drop to around 20°C.
The Banda Sea stretches from Alor to the coast of West Papua. A string of islands in the south Maluku archipelago forms the southern border of the sea. Like Nusa Tenggara, these islands are part of the seismically active 'Ring of Fire.' The Banda Islands, formerly known as the Spice Islands, are in the central Banda Sea. This little-visited area has some of the world's richest reefs bursting with unspoiled corals and pelagic fishes. Diving in the remote Banda Sea is possible only by liveaboard. The route to and from West Papua is convenient for sailing and diving along the way. Stopovers at Wetar Island, the Damar, Banda and Lucipara Islands, and Koon Island off the coast of Seram, guarantee a pristine diving experience. This little-visited area has some of the world's richest reefs bursting with unspoiled corals and pelagic fishes. The weather in this part of Indonesia is on a different cycle from the rest of the country. The best time to dive here is March-April and October-November when the water is calm and warm, 27º-30ºC (84-90ºF), and the visibility is 30-40m (100-130ft). The southeast monsoon (May-September) is wet, windy, and the water temperature drops to 25ºC (80ºF) due to strong upwellings in the area. Plankton blooms at this time reduce visibility to 15m (50ft). From December to February the wind is not as strong but it is strong enough to make boat travel uncomfortable. The Banda Islands have beautiful corals, big tuna and overhangs with big sponges and sea fans are the highlight. Walls, pinnacles and steep slopes, sometimes with strong currents, characterize the diving here. Rays and turtles are common visitors here. There is good macro life in the area. The Lucipara Islands area has good walls, schooling fishes, a few sharks and lots of soft corals. Mola-molas have been spotted here too. Koon Island, also called 'Too Many Fish,' this site is amazing because of all the fishes that hang out at the reef corner in the current.


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text:
Igor Sutyagin
Anonymous Claims Ex-Russian Spy Scholar Enlisted to UK Info Warfare Project
LONDON (Sputnik) – The documents on the UK Integrity Initiative covert operation, published by the Anonymous hacker group, showed that Russian researcher Igor Sutyagin, who was accused of espionage and subsequently pardoned, is a member of the so-called UK Cluster of London's hybrid warfare program.
Anonymous Exposes UK Hybrid Warfare Project
Two Accidents With Admiral Kuznetsov in a Month: How the Russian Carrier Compares to US Counterparts
On December 5, a Russian Su-33 fighter jet skidded off the Admiral Kuznetsov aircraft carrier in the Mediterranean while landing. On November 14, a MiG-29K fighter bomber crashed while landing on the carrier's flight deck. The two mishaps raised a barrage of often unjustified criticism both in Russia and abroad.

redpajama_set_name: RedPajamaCommonCrawl