Structural aspects of semigroups based on digraphs

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

doi:10.5802/alco.56

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

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

Abstract

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 $(a \to b)$ mapping $a$ to $b$ and fixing all vertices other than $a$ . The semigroup $〈 D 〉$ 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 $〈 D 〉$ 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.

Received: 2018-03-02
Revised: 2018-11-26
Accepted: 2018-12-21
Published online: 2019-10-08

MR 4023563 | Zbl 07115038

DOI: https://doi.org/10.5802/alco.56
Classification: 20M20, 05C20, 05C25
Keywords: digraphs, flow semigroup of digraph, semigroups, monoids

BibTeX
RIS

@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},
     pages = {711--733},
     publisher = {MathOA foundation},
     volume = {2},
     number = {5},
     year = {2019},
     doi = {10.5802/alco.56},
     mrnumber = {4023563},
     zbl = {07115038},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.56/}
}

TY  - JOUR
AU  - East, James
AU  - Gadouleau, Maximilien
AU  - Mitchell, James D.
TI  - Structural aspects of semigroups based on digraphs
JO  - Algebraic Combinatorics
PY  - 2019
DA  - 2019///
SP  - 711
EP  - 733
VL  - 2
IS  - 5
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.56/
UR  - https://www.ams.org/mathscinet-getitem?mr=4023563
UR  - https://zbmath.org/?q=an%3A07115038
UR  - https://doi.org/10.5802/alco.56
DO  - 10.5802/alco.56
LA  - en
ID  - ALCO_2019__2_5_711_0
ER  -

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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