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 (a,b) of D is associated the idempotent transformation (ab) mapping a to b and fixing all vertices other than a. The semigroup D is generated by the idempotent transformations (ab) for all arcs (a,b) 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, 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 -, -, -, or 𝒥-relation, and when D has a left, right, or two-sided zero.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.56
Classification: 20M20,  05C20,  05C25
Keywords: digraphs, flow semigroup of digraph, semigroups, monoids
East, James 1; Gadouleau, Maximilien 2; Mitchell, James D. 3

1 Centre for Research in Mathematics School of Computing Engineering and Mathematics Western Sydney University Locked Bag 1797, Penrith NSW 2751, Australia.
2 Department of Computer Science Durham University South Road, Durham DH1 3LE, UK.
3 School of Mathematics and Statistics University of St Andrews St Andrews, Fife KY16 9SS, UK.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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/articles/10.5802/alco.56/

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