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dc.contributor.authorAraújo, João
dc.contributor.authorBentz, Wolfram
dc.contributor.authorCameron, Peter J.
dc.identifier.citationAraújo , J , Bentz , W & Cameron , P J 2021 , ' The existential transversal property : a generalization of homogeneity and its impact on semigroups ' , Transactions of the American Mathematical Society , vol. 374 , no. 2 , pp. 1155–1195 .
dc.identifier.otherPURE: 269765678
dc.identifier.otherPURE UUID: 2cf341e2-5351-44ef-8e5a-041f97b01315
dc.identifier.otherORCID: /0000-0003-3130-9505/work/80257837
dc.identifier.otherScopus: 85099364874
dc.identifier.otherWOS: 000607645000015
dc.descriptionFunding: The first author was partially supported by the Fundação para a Ciênciae a Tecnologia (Portuguese Foundation for Science and Technology) through the projects UIDB/00297/2020 (Centro de Matemtica e Aplicaes), PTDC/MAT-PUR/31174/2017, CEMAT-CIÊNCIAS UID/Multi/04621/2013, and through the project “Hilbert’s 24th problem” (PTDC/MHC-FIL/2583/2014). The second author was supported by travel grants from the University of Hull’s Faculty of Science and Engineering and the Center for Computational and Stochastic Mathematics.en
dc.description.abstractLet G be a permutation group of degree n, and k a positive integer with k ≤ n. We say that G has the k-existential property, or k-et, if there exists a k-subset A (of the domain Ω) whose orbit under G contains transversals for all k-partitions P of Ω. This property is a substantial weakening of the k-universal transversal property, or k-ut, investigated by the first and third author, which required this condition to hold for all k-subsets A of the domain Ω. Our first task in this paper is to investigate the k-et property a≤nd to decide which groups satisfy it. For example, it is known that for k< 6 there are several families of k-transitive groups, but for k ≥ 6 the only ones are alternating or symmetric groups; here we show that in the k-et context the threshold is 8, that is, for 8 ≤ k ≤ n/2, the only transitive groups with k-et are the symmetric and alternating groups; this is best possible since the Mathieu group M24 (degree 24) has 7-et. We determine all groups with k-et for 4 ≤ k ≤ n/2, up to some unresolved cases for k=4,5, and describe the property for k=2,3 in permutation group language. These considerations essentially answer Problem 5 proposed in the paper on k-ut referred to above; we also slightly improve the classification of groups possessing the k-ut property. In that earlier paper, the results were applied to semigroups, in particular, to the question of when the semigroup <G,t> is regular, where t is a map of rank k (with k < n/2); this turned out to be equivalent to the k-ut property. The question investigated here is when there is a k-subset A of the domain such that <G,t> is regular for all maps t with image A. This turns out to be much more delicate; the k-et property (with A as witnessing set) is a necessary condition, and the combination of k-et and (k-1)-ut is sufficient, but the truth lies somewhere between. Given the knowledge that a group under consideration has the necessary condition of k-et, the regularity question for k ≤ n/2 is solved except for one sporadic group. The paper ends with a number of problems on combinatorics, permutation groups and transformation semigroups, and their linear analogues.
dc.relation.ispartofTransactions of the American Mathematical Societyen
dc.rights© Copyright 2020, American Mathematical Society. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at
dc.subjectTransformation semigroupsen
dc.subjectRegular semigroupsen
dc.subjectPermutation groupsen
dc.subjectPrimitive groupsen
dc.subjectHomeogenous groupsen
dc.subjectQA Mathematicsen
dc.titleThe existential transversal property : a generalization of homogeneity and its impact on semigroupsen
dc.typeJournal articleen
dc.contributor.institutionUniversity of St Andrews.Pure Mathematicsen
dc.contributor.institutionUniversity of St Andrews.Centre for Interdisciplinary Research in Computational Algebraen
dc.description.statusPeer revieweden

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