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Lengths of words in transformation semigroups generated by digraphs.

Cameron, Peter J. and Castillo-Ramirez, Alonso and Gadouleau, Maximilien and Mitchell, James D. (2017) 'Lengths of words in transformation semigroups generated by digraphs.', Journal of algebraic combinatorics., 45 (1). pp. 149-170.

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.

Item Type:Article
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1007/s10801-016-0703-9
Publisher 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.
Date accepted:25 July 2016
Date deposited:31 August 2016
Date of first online publication:08 August 2016
Date first made open access:No date available

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