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Lengths of words in transformation semigroups generated by digraphs

Cameron, Peter J.; Castillo-Ramirez, Alonso; Gadouleau, Maximilien; Mitchell, James D.

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Authors

Peter J. Cameron

Alonso Castillo-Ramirez

James D. Mitchell



Abstract

Given a simple digraph D on n vertices (with n≥2n≥2 ), there is a natural construction of a semigroup of transformations ⟨D⟩⟨D⟩ . For any edge (a, b) of D, let a→ba→b be the idempotent of rank n−1n−1 mapping a to b and fixing all vertices other than a; then, define ⟨D⟩⟨D⟩ to be the semigroup generated by a→ba→b for all (a,b)∈E(D)(a,b)∈E(D) . For α∈⟨D⟩α∈⟨D⟩ , let ℓ(D,α)ℓ(D,α) be the minimal length of a word in E(D) expressing αα . It is well known that the semigroup SingnSingn of all transformations of rank at most n−1n−1 is generated by its idempotents of rank n−1n−1 . When D=KnD=Kn is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate ℓ(Kn,α)ℓ(Kn,α) , for any α∈⟨Kn⟩=Singnα∈⟨Kn⟩=Singn ; however, no analogous non-trivial results are known when D≠KnD≠Kn . In this paper, we characterise all simple digraphs D such that either ℓ(D,α)ℓ(D,α) is equal to Howie–Iwahori’s formula for all α∈⟨D⟩α∈⟨D⟩ , or ℓ(D,α)=n−fix(α)ℓ(D,α)=n−fix(α) for all α∈⟨D⟩α∈⟨D⟩ , or ℓ(D,α)=n−rk(α)ℓ(D,α)=n−rk(α) for all α∈⟨D⟩α∈⟨D⟩ . We also obtain bounds for ℓ(D,α)ℓ(D,α) when D is an acyclic digraph or a strong tournament (the latter case corresponds to a smallest generating set of idempotents of rank n−1n−1 of SingnSingn ). We finish the paper with a list of conjectures and open problems.

Citation

Cameron, P. J., Castillo-Ramirez, A., Gadouleau, M., & Mitchell, J. D. (2017). Lengths of words in transformation semigroups generated by digraphs. Journal of Algebraic Combinatorics, 45(1), 149-170. https://doi.org/10.1007/s10801-016-0703-9

Journal Article Type Article
Acceptance Date Jul 25, 2016
Online Publication Date Aug 8, 2016
Publication Date Feb 1, 2017
Deposit Date Aug 26, 2016
Publicly Available Date Aug 31, 2016
Journal Journal of Algebraic Combinatorics
Print ISSN 0925-9899
Electronic ISSN 1572-9192
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 45
Issue 1
Pages 149-170
DOI https://doi.org/10.1007/s10801-016-0703-9

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Copyright Statement
© The Author(s) 2016. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.





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