Publication
The existential transversal property: a generalization of homogeneity and its impact on semigroups
| dc.contributor.author | Araújo, João | |
| dc.contributor.author | Bentz, Wolfram | |
| dc.contributor.author | Cameron, Peter | |
| dc.date.accessioned | 2023-01-30T16:07:53Z | |
| dc.date.available | 2023-01-30T16:07:53Z | |
| dc.date.issued | 2021 | |
| dc.description | Preprint 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. | pt_PT |
| dc.description.abstract | Let 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. | pt_PT |
| dc.description.sponsorship | The 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. | pt_PT |
| dc.description.version | info:eu-repo/semantics/publishedVersion | pt_PT |
| dc.identifier.citation | J. Araújo, W. Bentz, and P.J. Cameron, “The Existential Transversal Property: A Generalization of Homogeneity and its Impact on Semigroups”, Trans. Amer. Math. Soc. 374 (2021), 1155-1195. | pt_PT |
| dc.identifier.doi | 10.1090/tran/8285 | pt_PT |
| dc.identifier.eissn | 1088-6850 | |
| dc.identifier.issn | 0002-9947 | |
| dc.identifier.uri | http://hdl.handle.net/10400.2/13254 | |
| dc.language.iso | eng | pt_PT |
| dc.peerreviewed | yes | pt_PT |
| dc.publisher | American Mathematical Society | pt_PT |
| dc.relation | Center for Mathematics and Applications | |
| dc.relation | Center for Computational and Stochastic Mathematics | |
| dc.relation | Center for Computational and Stochastic Mathematics | |
| dc.subject | Transformation semigroups | pt_PT |
| dc.subject | Regular semigroups | pt_PT |
| dc.subject | Permutation groups | pt_PT |
| dc.subject | Primitive groups | pt_PT |
| dc.subject | Homogeneous groups | pt_PT |
| dc.title | The existential transversal property: a generalization of homogeneity and its impact on semigroups | pt_PT |
| dc.type | journal article | |
| dspace.entity.type | Publication | |
| oaire.awardTitle | Center for Mathematics and Applications | |
| oaire.awardTitle | Center for Computational and Stochastic Mathematics | |
| oaire.awardTitle | Center for Computational and Stochastic Mathematics | |
| oaire.awardURI | info:eu-repo/grantAgreement/FCT/6817 - DCRRNI ID/UIDB%2F00297%2F2020/PT | |
| oaire.awardURI | info:eu-repo/grantAgreement/FCT/3599-PPCDT/PTDC%2FMAT-PUR%2F31174%2F2017/PT | |
| oaire.awardURI | info:eu-repo/grantAgreement/FCT/6817 - DCRRNI ID/UIDB%2F04621%2F2020/PT | |
| oaire.awardURI | info:eu-repo/grantAgreement/FCT/6817 - DCRRNI ID/UIDP%2F04621%2F2020/PT | |
| oaire.citation.endPage | 1195 | pt_PT |
| oaire.citation.startPage | 1155 | pt_PT |
| oaire.citation.title | Transactions of the American Mathematical Society | pt_PT |
| oaire.fundingStream | 6817 - DCRRNI ID | |
| oaire.fundingStream | 3599-PPCDT | |
| oaire.fundingStream | 6817 - DCRRNI ID | |
| oaire.fundingStream | 6817 - DCRRNI ID | |
| person.familyName | Ribeiro Soares Gonçalves de Araújo | |
| person.familyName | Bentz | |
| person.familyName | Cameron | |
| person.givenName | João Jorge | |
| person.givenName | Wolfram | |
| person.givenName | Peter | |
| person.identifier.ciencia-id | EC1F-273A-9F24 | |
| person.identifier.orcid | 0000-0001-6655-2172 | |
| person.identifier.orcid | 0000-0003-0002-1277 | |
| person.identifier.orcid | 0000-0003-3130-9505 | |
| person.identifier.scopus-author-id | 7202869893 | |
| project.funder.identifier | http://doi.org/10.13039/501100001871 | |
| project.funder.identifier | http://doi.org/10.13039/501100001871 | |
| project.funder.identifier | http://doi.org/10.13039/501100001871 | |
| project.funder.identifier | http://doi.org/10.13039/501100001871 | |
| project.funder.name | Fundação para a Ciência e a Tecnologia | |
| project.funder.name | Fundação para a Ciência e a Tecnologia | |
| project.funder.name | Fundação para a Ciência e a Tecnologia | |
| project.funder.name | Fundação para a Ciência e a Tecnologia | |
| rcaap.rights | restrictedAccess | pt_PT |
| rcaap.type | article | pt_PT |
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