Given any digraph without loops or multiple arcs, there is a natural construction of a semigroup of transformations. To every arc of is associated the idempotent transformation mapping to and fixing all vertices other than . The semigroup is generated by the idempotent transformations for all arcs of .
In this paper, we consider the question of when there is a transformation in containing a large cycle, and, for fixed , we give a linear time algorithm to verify if contains a transformation with a cycle of length . We also classify those digraphs such that has one of the following properties: inverse, completely regular, commutative, simple, 0-simple, a semilattice, a rectangular band, congruence-free, is -trivial or -universal where is any of Green’s -, -, -, or -relation, and when 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}, language = {en}, url = {https://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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