# ALGEBRAIC COMBINATORICS

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 $\left(a,b\right)$ of $D$ is associated the idempotent transformation $\left(a\to b\right)$ mapping $a$ to $b$ and fixing all vertices other than $a$. The semigroup $〈D〉$ is generated by the idempotent transformations $\left(a\to b\right)$ for all arcs $\left(a,b\right)$ 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.

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.
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/

[1] Aĭzenshtat, A. Ya. The defining relations of the endomorphism semigroup of a finite linearly ordered set, Sib. Mat. Zh., Volume 3 (1962), pp. 161-169 | MR | Zbl

[2] Bang-Jensen, Jørgen; Gutin, Gregory Z. Digraphs, Springer Monographs in Mathematics, Springer London, 2009 | DOI | Zbl

[3] Bankevich, A. V.; Karpov, Dmitriy V. Bounds of the number of leaves of spanning trees, Journal of Mathematical Sciences, Volume 184 (2012) no. 5, pp. 564-572 | DOI | MR | Zbl

[4] Bondy, Adrian Graph Theory, Graduate Texts in Mathematics, Springer, 2008

[5] Cameron, P. J.; Castillo-Ramirez, A.; Gadouleau, M.; Mitchell, J. D. Lengths of words in transformation semigroups generated by digraphs, Journal of Algebraic Combinatorics, Volume 45 (2017) no. 1, pp. 149-170 | DOI | MR | Zbl

[6] Dolinka, Igor; East, James Idempotent Generation in the Endomorphism Monoid of a Uniform Partition, Communications in Algebra, Volume 44 (2016) no. 12, pp. 5179-5198 | DOI | MR | Zbl

[7] Ganyushkin, Olexandr; Mazorchuk, Volodymyr Classical Finite Transformation Semigroups, Algebra and Applications, 9, Springer London, 2009 | DOI | MR | Zbl

[8] Gomes, Gracinda M. S.; Howie, John M. On the ranks of certain finite semigroups of transformations, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 101 (1987) no. 3, pp. 395-403 | DOI | MR | Zbl

[9] Higgins, Peter M. Combinatorial results for semigroups of order-preserving mappings, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 113 (1993) no. 2, pp. 281-296 | DOI | MR | Zbl

[10] Hopcroft, John; Tarjan, Robert Algorithm 447: efficient algorithms for graph manipulation, Communications of the ACM, Volume 16 (1973) no. 6, pp. 372-378 | DOI

[11] Horváth, Gábor; Nehaniv, Chrystopher L.; Podoski, Károly The maximal subgroups and the complexity of the flow semigroup of finite (di)graphs, International Journal of Algebra and Computation, Volume 27 (2017) no. 7, pp. 863-886 | DOI | MR | Zbl

[12] Howie, John M. The subsemigroup generated by the idempotents of a full transformation semigroup, J. London Math. Soc., Volume 41 (1966) no. 1, pp. 707-716 | DOI | MR | Zbl

[13] Howie, John M. Idempotent generators in finite full transformation semigroups, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 81 (1978) no. 3-4, pp. 317-323 | DOI | MR | Zbl

[14] Howie, John M. Fundamentals of semigroup theory, London Mathematical Society Monographs. New Series, 12, The Clarendon Press Oxford University Press, New York, 1995, x+351 pages (Oxford Science Publications) | MR | Zbl

[15] Kawarabayashi, Ken-ichi; Kobayashi, Yusuke; Reed, Bruce The disjoint paths problem in quadratic time, Journal of Combinatorial Theory, Series B, Volume 102 (2012) no. 2, pp. 424-435 | DOI | MR | Zbl

[16] Kornhauser, Daniel; Miller, Gary; Spirakis, Paul Coordinating Pebble Motion On Graphs, The Diameter Of Permutation Groups, And Applications, 25th Annual Symposium on Foundations of Computer Science, 1984. (1984), pp. 241-250 | DOI

[17] Mitchell, James D. et al. Semigroups - GAP package, Version 3.1.1 (2019) | DOI

[18] Pin, Jean Eric Varieties of Formal Languages, Foundations of Computer Science, Springer, 1986

[19] Ratner, Daniel; Warmuth, Manfred The (${n}^{2}-1$)-puzzle and related relocation problems, Journal of Symbolic Computation, Volume 10 (1990) no. 2, pp. 111-137 | DOI | MR | Zbl

[20] Rhodes, John Applications of Automata Theory and Algebra: Via the Mathematical Theory of Complexity to Biology, Physics, Psychology, Philosophy, and Games, World Scientific Pub Co Inc, 2009

[21] Solomon, Andrew Catalan monoids, monoids of local endomorphisms, and their presentations, Semigroup Forum, Volume 53 (1996) no. 1, pp. 351-368 | DOI | MR | Zbl

[22] GAP – Groups, Algorithms, and Programming, Version 4.10.1 (2019) (https://www.gap-system.org)

[23] Wilson, Richard M. Graph puzzles, homotopy, and the alternating group, Journal of Combinatorial Theory, Series B, Volume 16 (1974) no. 1, pp. 86-96 | DOI | MR | Zbl

[24] Wright, E. M. The number of irreducible tournaments, Glasgow Mathematical Journal, Volume 11 (1970) no. 2, pp. 97-101 | DOI | MR | Zbl

[25] Yang, Xiuliang; Yang, Haobo Maximal Regular Subsemibands of ${\mathrm{Sing}}_{n}$, Semigroup Forum, Volume 72 (2005) no. 1, pp. 75-93 | DOI | MR | Zbl

[26] Yang, Xiuliang; Yang, Haobo Isomorphisms of transformation semigroups associated with simple digraphs, Asian-European Journal of Mathematics, Volume 2 (2009) no. 4, pp. 727-737 | DOI | MR | Zbl

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