Structural aspects of semigroups based on digraphs
Algebraic Combinatorics, Volume 2 (2019) no. 5, p. 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 : 2018-03-02
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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