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dc.contributor.authorCameron, P. J.
dc.contributor.authorCastillo-Ramirez, A.
dc.contributor.authorGadouleau, M.
dc.contributor.authorMitchell, J. D.
dc.identifier.citationCameron , 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 .
dc.identifier.otherPURE: 241708296
dc.identifier.otherPURE UUID: 36be5d79-324f-42ff-a8c9-6a8dee8cc415
dc.identifier.otherScopus: 84981239684
dc.identifier.otherORCID: /0000-0003-3130-9505/work/58055581
dc.identifier.otherWOS: 000392426800005
dc.descriptionThe second and third authors were supported by the EPSRC grant EP/K033956/1.en
dc.description.abstractGiven 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.relation.ispartofJournal of Algebraic Combinatoricsen
dc.rights© The Author(s) 2016. This article is published with open access at This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, 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.subjectTransformation semigroupen
dc.subjectSimple digraphen
dc.subjectWord lengthen
dc.subjectQA Mathematicsen
dc.titleLengths of words in transformation semigroups generated by digraphsen
dc.typeJournal articleen
dc.description.versionPublisher PDFen
dc.contributor.institutionUniversity of St Andrews.Pure Mathematicsen
dc.contributor.institutionUniversity of St Andrews.Statisticsen
dc.contributor.institutionUniversity of St Andrews.Centre for Interdisciplinary Research in Computational Algebraen
dc.description.statusPeer revieweden

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