Structural aspects of semigroups based on digraphs

East, James; Gadouleau, Maximilien; Mitchell, James D.

Algebraic Combinatorics, Tome 2 (2019) no. 5, pp. 711-733.

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 $𝒦$-trivial or $𝒦$-universal where $𝒦$ is any of Green’s $\mathscr{H}$-, $ℒ$-, $ℛ$-, or $𝒥$-relation, and when $\langle D\rangle$ has a left, right, or two-sided zero.

Classification : 20M20, 05C20, 05C25
Mots clés : digraphs, flow semigroup of digraph, semigroups, monoids

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