Given any digraph $D$ without loops or multiple arcs, there is a natural construction of a semigroup $\langle D\rangle $ of transformations. To every arc $(a,b)$ of $D$ is associated the idempotent transformation $(a\to b)$ mapping $a$ to $b$ and fixing all vertices other than $a$. The semigroup $\langle D\rangle $ is generated by the idempotent transformations $(a\to b)$ for all arcs $(a,b)$ of $D$.

In this paper, we consider the question of when there is a transformation in $\langle D\rangle $ containing a large cycle, and, for fixed $k\in \mathbb{N}$, we give a linear time algorithm to verify if $\langle D\rangle $ contains a transformation with a cycle of length $k$. We also classify those digraphs $D$ such that $\langle D\rangle $ has one of the following properties: inverse, completely regular, commutative, simple, 0-simple, a semilattice, a rectangular band, congruence-free, is $\mathcal{K}$-trivial or $\mathcal{K}$-universal where $\mathcal{K}$ is any of Green’s $\mathscr{H}$-, $\mathcal{L}$-, $\mathcal{R}$-, or $\mathcal{J}$-relation, and when $\langle D\rangle $ has a left, right, or two-sided zero.

Revised:

Accepted:

Published online:

DOI: 10.5802/alco.56

Keywords: digraphs, flow semigroup of digraph, semigroups, monoids

^{1}; Gadouleau, Maximilien

^{2}; Mitchell, James D.

^{3}

@article{ALCO_2019__2_5_711_0, author = {East, James and Gadouleau, Maximilien and Mitchell, James D.}, title = {Structural aspects of semigroups based on digraphs}, journal = {Algebraic Combinatorics}, pages = {711--733}, publisher = {MathOA foundation}, volume = {2}, number = {5}, year = {2019}, doi = {10.5802/alco.56}, mrnumber = {4023563}, zbl = {07115038}, language = {en}, url = {http://www.numdam.org/articles/10.5802/alco.56/} }

TY - JOUR AU - East, James AU - Gadouleau, Maximilien AU - Mitchell, James D. TI - Structural aspects of semigroups based on digraphs JO - Algebraic Combinatorics PY - 2019 SP - 711 EP - 733 VL - 2 IS - 5 PB - MathOA foundation UR - http://www.numdam.org/articles/10.5802/alco.56/ DO - 10.5802/alco.56 LA - en ID - ALCO_2019__2_5_711_0 ER -

%0 Journal Article %A East, James %A Gadouleau, Maximilien %A Mitchell, James D. %T Structural aspects of semigroups based on digraphs %J Algebraic Combinatorics %D 2019 %P 711-733 %V 2 %N 5 %I MathOA foundation %U http://www.numdam.org/articles/10.5802/alco.56/ %R 10.5802/alco.56 %G en %F ALCO_2019__2_5_711_0

East, James; Gadouleau, Maximilien; Mitchell, James D. Structural aspects of semigroups based on digraphs. Algebraic Combinatorics, Volume 2 (2019) no. 5, pp. 711-733. doi : 10.5802/alco.56. http://www.numdam.org/articles/10.5802/alco.56/

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