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Lengths of words in transformation semigroups generated by digraphs
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dc.contributor.author | Cameron, P. J. | |
dc.contributor.author | Castillo-Ramirez, A. | |
dc.contributor.author | Gadouleau, M. | |
dc.contributor.author | Mitchell, J. D. | |
dc.date.accessioned | 2016-08-09T10:30:16Z | |
dc.date.available | 2016-08-09T10:30:16Z | |
dc.date.issued | 2017-02 | |
dc.identifier.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 | en |
dc.identifier.issn | 0925-9899 | |
dc.identifier.other | PURE: 241708296 | |
dc.identifier.other | PURE UUID: 36be5d79-324f-42ff-a8c9-6a8dee8cc415 | |
dc.identifier.other | ArXiv: http://arxiv.org/abs/1602.00935v1 | |
dc.identifier.other | Scopus: 84981239684 | |
dc.identifier.other | ORCID: /0000-0003-3130-9505/work/58055581 | |
dc.identifier.other | WOS: 000392426800005 | |
dc.identifier.other | ORCID: /0000-0002-5489-1617/work/73700796 | |
dc.identifier.uri | https://hdl.handle.net/10023/9277 | |
dc.description | The second and third authors were supported by the EPSRC grant EP/K033956/1. | en |
dc.description.abstract | 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 | |
dc.language.iso | eng | |
dc.relation.ispartof | Journal of Algebraic Combinatorics | en |
dc.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. | en |
dc.subject | Transformation semigroup | en |
dc.subject | Simple digraph | en |
dc.subject | Word length | en |
dc.subject | QA Mathematics | en |
dc.subject | Mathematics(all) | en |
dc.subject | T-NDAS | en |
dc.subject.lcc | QA | en |
dc.title | Lengths of words in transformation semigroups generated by digraphs | en |
dc.type | Journal article | en |
dc.description.version | Publisher PDF | en |
dc.contributor.institution | University of St Andrews. Pure Mathematics | en |
dc.contributor.institution | University of St Andrews. Statistics | en |
dc.contributor.institution | University of St Andrews. Centre for Interdisciplinary Research in Computational Algebra | en |
dc.identifier.doi | https://doi.org/10.1007/s10801-016-0703-9 | |
dc.description.status | Peer reviewed | en |
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