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dc.contributor.authorPeresse, Yann
dc.contributor.authorHyde, James
dc.contributor.authorMitchell, James D.
dc.contributor.authorJonusas, Julius
dc.date.accessioned2020-09-24T00:08:05Z
dc.date.available2020-09-24T00:08:05Z
dc.date.issued2020-01-28
dc.identifier.citationPeresse , Y , Hyde , J , Mitchell , J D & Jonusas , J 2020 , ' Sets of universal sequences for the symmetric group and analogous semigroups ' , Proceedings of the American Mathematical Society . https://doi.org/10.1090/proc/14881
dc.identifier.issn0002-9939
dc.identifier.otherPURE: 17414873
dc.identifier.otherPURE UUID: bf49fd02-7276-479b-b61e-83cd9690a6fb
dc.identifier.otherScopus: 85083099731
dc.identifier.urihttp://hdl.handle.net/2299/23166
dc.description.abstractA universal sequence for a group or semigroup S is a sequence of words w1,w2,… such that for any sequence s1,s2,…∈S, the equations wi=si, i∈N, can be solved simultaneously in S. For example, Galvin showed that the sequence {a−1(aiba−i)b−1(aib−1a−i)ba:i∈N} is universal for the symmetric group Sym(X) when X is infinite, and Sierpiński showed that (a2b3(abab3)n+1ab2ab3)n∈N is universal for the monoid XX of functions from the infinite set X to itself. In this paper, we show that under some conditions, the set of universal sequences for the symmetric group on an infinite set X is independent of the cardinality of X. More precisely, we show that if Y is any set such that |Y|≥|X|, then every universal sequence for Sym(X) is also universal for Sym(Y). If |X|>2ℵ0, then the converse also holds. It is shown that an analogue of this theorem holds in the context of inverse semigroups, where the role of the symmetric group is played by the symmetric inverse monoid. In the general context of semigroups, the full transformation monoid XX is the natural analogue of the symmetric group and the symmetric inverse monoid. If X and Y are arbitrary infinite sets, then it is an open question as to whether or not every sequence that is universal for XX is also universal for YY. However, we obtain a sufficient condition for a sequence to be universal for XX which does not depend on the cardinality of X. A large class of sequences satisfy this condition, and hence are universal for XX for every infinite set Xen
dc.format.extent17
dc.language.isoeng
dc.relation.ispartofProceedings of the American Mathematical Society
dc.subjectSEMIGROUPS
dc.subjectuniversal sequences
dc.subjectGroup Theory
dc.subjectsymmetric group
dc.titleSets of universal sequences for the symmetric group and analogous semigroupsen
dc.contributor.institutionMathematics Research Group
dc.contributor.institutionSchool of Physics, Astronomy and Mathematics
dc.description.statusPeer reviewed
rioxxterms.versionAM
rioxxterms.versionofrecordhttps://doi.org/10.1090/proc/14881
rioxxterms.typeJournal Article/Review
herts.preservation.rarelyaccessedtrue


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