The existential transversal property: a generalization of homogeneity and its impact on semigroups

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MetadadosDescriçãoIdioma
Autor(es): dc.creatorAraújo, João-
Autor(es): dc.creatorBentz, Wolfram-
Autor(es): dc.creatorCameron, Peter-
Data de aceite: dc.date.accessioned2025-08-21T15:13:33Z-
Data de disponibilização: dc.date.available2025-08-21T15:13:33Z-
Data de envio: dc.date.issued2023-01-29-
Data de envio: dc.date.issued2023-01-29-
Data de envio: dc.date.issued2020-
Fonte completa do material: dc.identifierhttp://hdl.handle.net/10400.2/13254-
Fonte: dc.identifier.urihttp://educapes.capes.gov.br/handle/10400.2/13254-
Descrição: dc.descriptionPreprint de J. Araújo, W. Bentz, and P.J. Cameron, “The Existential Transversal Property: A Generalization of Homogeneity and its Impact on Semigroups”, Transactions of the American Mathematical Society 374 (2021), 1155–1195.-
Descrição: dc.descriptionLet G be a permutation group of degree n, and k a positive integer with k ≤ n. We say that G has the k-existential transversal property, or k-et, if there exists a k-subset A (of the domain Ω) whose orbit un- der 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 and 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.-
Descrição: dc.descriptionThe first author was partially supported by the Funda ̧c ̃ao para a Ciˆencia e a Tecnologia (Portuguese Foundation for Science and Technology) through the projects UIDB/00297/2020 (Centro de Matemtica e Aplicaes), PTDC/MAT-PUR/31174/2017, UIDB/04621/2020 and UIDP/04621/2020. 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.-
Descrição: dc.descriptioninfo:eu-repo/semantics/publishedVersion-
Formato: dc.formatapplication/pdf-
Idioma: dc.languageen-
Publicador: dc.publisherAmerican Mathematical Society-
Relação: dc.relationCenter for Mathematics and Applications-
Relação: dc.relationCenter for Computational and Stochastic Mathematics-
Relação: dc.relationCenter for Computational and Stochastic Mathematics-
Palavras-chave: dc.subjectTransformation semigroups-
Palavras-chave: dc.subjectRegular semigroups-
Palavras-chave: dc.subjectPermutation groups-
Palavras-chave: dc.subjectPrimitive groups-
Palavras-chave: dc.subjectHomogeneous groups-
Título: dc.titleThe existential transversal property: a generalization of homogeneity and its impact on semigroups-
Aparece nas coleções:Repositório Aberto - Universidade Aberta (Portugal)

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