# ALGEBRAIC COMBINATORICS

Structural aspects of semigroups based on digraphs
Algebraic Combinatorics, Volume 2 (2019) no. 5, pp. 711-733.

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

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

Revised: 2018-11-26
Accepted: 2018-12-21
Published online: 2019-10-08
DOI: https://doi.org/10.5802/alco.56
Classification: 20M20,  05C20,  05C25
Keywords: digraphs, flow semigroup of digraph, semigroups, monoids
@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},
publisher = {MathOA foundation},
volume = {2},
number = {5},
year = {2019},
pages = {711-733},
doi = {10.5802/alco.56},
mrnumber = {4023563},
zbl = {07115038},
language = {en},
url = {alco.centre-mersenne.org/item/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. https://alco.centre-mersenne.org/item/ALCO_2019__2_5_711_0/

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