ORCID Profile
0000-0002-5489-1617
Current Organisation
University of St Andrews
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Publisher: Springer Science and Business Media LLC
Date: 03-04-2017
Publisher: Elsevier BV
Date: 03-2019
Publisher: American Mathematical Society (AMS)
Date: 18-11-2014
DOI: 10.1090/S0002-9947-2014-06110-2
Abstract: In this paper we classify the maximal subsemigroups of the full transformation semigroup Ω Ω \\Omega ^\\Omega , which consists of all mappings on the infinite set Ω \\Omega , containing certain subgroups of the symmetric group Sym ( Ω ) \\operatorname {Sym}(\\Omega ) on Ω \\Omega . In 1965 Gavrilov showed that there are five maximal subsemigroups of Ω Ω \\Omega ^\\Omega containing Sym ( Ω ) \\operatorname {Sym}(\\Omega ) when Ω \\Omega is countable, and in 2005 Pinsker extended Gavrilov’s result to sets of arbitrary cardinality. We classify the maximal subsemigroups of Ω Ω \\Omega ^\\Omega on a set Ω \\Omega of arbitrary infinite cardinality containing one of the following subgroups of Sym ( Ω ) \\operatorname {Sym}(\\Omega ) : the pointwise stabiliser of a non-empty finite subset of Ω \\Omega , the stabiliser of an ultrafilter on Ω \\Omega , or the stabiliser of a partition of Ω \\Omega into finitely many subsets of equal cardinality. If G G is any of these subgroups, then we deduce a characterisation of the mappings f , g ∈ Ω Ω f,g\\in \\Omega ^\\Omega such that the semigroup generated by G ∪ { f , g } G\\cup \\{f,g\\} equals Ω Ω \\Omega ^\\Omega .
Publisher: Springer Science and Business Media LLC
Date: 08-08-2016
Publisher: Cambridge University Press (CUP)
Date: 10-08-2015
DOI: 10.1017/S0004972715000751
Abstract: We calculate the rank and idempotent rank of the semigroup ${\\mathcal{E}}(X,{\\mathcal{P}})$ generated by the idempotents of the semigroup ${\\mathcal{T}}(X,{\\mathcal{P}})$ which consists of all transformations of the finite set $X$ preserving a nonuniform partition ${\\mathcal{P}}$ . We also classify and enumerate the idempotent generating sets of minimal possible size. This extends results of the first two authors in the uniform case.
Publisher: Elsevier BV
Date: 2013
Publisher: Springer Science and Business Media LLC
Date: 08-05-2017
Location: United Kingdom of Great Britain and Northern Ireland
No related grants have been discovered for James David Mitchell.