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Maximal subsemigroups of the semigroup of all mappings on an infinite set

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Maximal subsemigroups of the semigroup of all mappings on an infinite set. / East, James; Mitchell, James D.; Péresse, Y.

In: Transactions of the American Mathematical Society, Vol. 367, No. 3, 18.11.2014, p. 1911-1944.

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@article{1808b2d188784c3fb91f455e02e91b5a,
title = "Maximal subsemigroups of the semigroup of all mappings on an infinite set",
abstract = "In this paper we classify the maximal subsemigroups of the full transformation semigroup ΩΩ, which consists of all mappings on the infinite set Ω, containing certain subgroups of the symmetric group Sym(Ω) on Ω. In 1965 Gavrilov showed that there are five maximal subsemigroups of ΩΩ containing Sym(Ω) when Ω is countable and in 2005 Pinsker extended Gavrilov{\textquoteright}s result to sets of arbitrary cardinality. We classify the maximal subsemigroups of ΩΩ on a set Ω of arbitrary infinite cardinality containing one of the following subgroups of Sym(Ω): the pointwise stabiliser of a non-empty finite subset of Ω, the stabiliser of an ultrafilter on Ω, or the stabiliser of a partition of Ω into finitely many subsets of equal cardinality. If G is any of these subgroups, then we deduce acharacterisation of the mappings f, g ∈ ΩΩ such that the semigroup generated by G ∪ {f, g} equals ΩΩ.",
keywords = "SEMIGROUPS, Abstract Algebra, TRANSFORMATIONS, Infinite Combinatorics",
author = "James East and Mitchell, {James D.} and Y. P{\'e}resse",
note = "This is the accepted manuscript of the following article: J. East, J. D. Mitchell and Y. P{\'e}resse, “Maximal subsemigroupsof the semigroup of all mappings on an infinite set”, Transactions of the American Mathematical Society, Vol. 367(3), November 2014. The final published version is available online at: http://www.ams.org/journals/tran/2015-367-03/S0002-9947-2014-06110-2/ {\textcopyright} Copyright 2014 American Mathematical Society ",
year = "2014",
month = nov,
day = "18",
doi = "10.1090/S0002-9947-2014-06110-2",
language = "English",
volume = "367",
pages = "1911--1944",
journal = "Transactions of the American Mathematical Society",
issn = "0002-9947",
publisher = "American Mathematical Society",
number = "3",

}

RIS

TY - JOUR

T1 - Maximal subsemigroups of the semigroup of all mappings on an infinite set

AU - East, James

AU - Mitchell, James D.

AU - Péresse, Y.

N1 - This is the accepted manuscript of the following article: J. East, J. D. Mitchell and Y. Péresse, “Maximal subsemigroupsof the semigroup of all mappings on an infinite set”, Transactions of the American Mathematical Society, Vol. 367(3), November 2014. The final published version is available online at: http://www.ams.org/journals/tran/2015-367-03/S0002-9947-2014-06110-2/ © Copyright 2014 American Mathematical Society

PY - 2014/11/18

Y1 - 2014/11/18

N2 - In this paper we classify the maximal subsemigroups of the full transformation semigroup ΩΩ, which consists of all mappings on the infinite set Ω, containing certain subgroups of the symmetric group Sym(Ω) on Ω. In 1965 Gavrilov showed that there are five maximal subsemigroups of ΩΩ containing Sym(Ω) when Ω is countable and in 2005 Pinsker extended Gavrilov’s result to sets of arbitrary cardinality. We classify the maximal subsemigroups of ΩΩ on a set Ω of arbitrary infinite cardinality containing one of the following subgroups of Sym(Ω): the pointwise stabiliser of a non-empty finite subset of Ω, the stabiliser of an ultrafilter on Ω, or the stabiliser of a partition of Ω into finitely many subsets of equal cardinality. If G is any of these subgroups, then we deduce acharacterisation of the mappings f, g ∈ ΩΩ such that the semigroup generated by G ∪ {f, g} equals ΩΩ.

AB - In this paper we classify the maximal subsemigroups of the full transformation semigroup ΩΩ, which consists of all mappings on the infinite set Ω, containing certain subgroups of the symmetric group Sym(Ω) on Ω. In 1965 Gavrilov showed that there are five maximal subsemigroups of ΩΩ containing Sym(Ω) when Ω is countable and in 2005 Pinsker extended Gavrilov’s result to sets of arbitrary cardinality. We classify the maximal subsemigroups of ΩΩ on a set Ω of arbitrary infinite cardinality containing one of the following subgroups of Sym(Ω): the pointwise stabiliser of a non-empty finite subset of Ω, the stabiliser of an ultrafilter on Ω, or the stabiliser of a partition of Ω into finitely many subsets of equal cardinality. If G is any of these subgroups, then we deduce acharacterisation of the mappings f, g ∈ ΩΩ such that the semigroup generated by G ∪ {f, g} equals ΩΩ.

KW - SEMIGROUPS

KW - Abstract Algebra

KW - TRANSFORMATIONS

KW - Infinite Combinatorics

U2 - 10.1090/S0002-9947-2014-06110-2

DO - 10.1090/S0002-9947-2014-06110-2

M3 - Article

VL - 367

SP - 1911

EP - 1944

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 3

ER -