Lengths of words in transformation semigroups generated by digraphs
Date
02/2017Metadata
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Given a simple digraph D on n vertices (with n≥2), there is a natural construction of a semigroup of transformations ⟨D⟩. For any edge (a, b) of D, let a→b be the idempotent of rank n−1 mapping a to b and fixing all vertices other than a; then, define ⟨D⟩ to be the semigroup generated by a→b for all (a,b)∈E(D). For α∈⟨D⟩, let ℓ(D,α) be the minimal length of a word in E(D) expressing α. It is well known that the semigroup Singn of all transformations of rank at most n−1 is generated by its idempotents of rank n−1. When D=Kn is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate ℓ(Kn,α), for any α∈⟨Kn⟩=Singn; however, no analogous non-trivial results are known when D≠Kn. In this paper, we characterise all simple digraphs D such that either ℓ(D,α) is equal to Howie–Iwahori’s formula for all α∈⟨D⟩, or ℓ(D,α)=n−fix(α) for all α∈⟨D⟩, or ℓ(D,α)=n−rk(α) for all α∈⟨D⟩. We also obtain bounds for 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−1 of Singn). 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 , vol. 45 , no. 1 , pp. 149-170 . https://doi.org/10.1007/s10801-016-0703-9
Publication
Journal of Algebraic Combinatorics
Status
Peer reviewed
ISSN
0925-9899Type
Journal article
Rights
© The Author(s) 2016. This article is published with open access at Springerlink.com. 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.
Description
The second and third authors were supported by the EPSRC grant EP/K033956/1.Collections
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