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2013 | 220 | 3 | 243-261
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Longer chains of idempotents in βG

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Given idempotents e and f in a semigroup, e ≤ f if and only if e = fe = ef. We show that if G is a countable discrete group, p is a right cancelable element of G* = βG∖G, and λ is a countable ordinal, then there is a strictly decreasing chain $⟨q_{σ}⟩_{σ<λ}$ of idempotents in $C_p$, the smallest compact subsemigroup of G* with p as a member. We also show that if S is any infinite subsemigroup of a countable group, then any nonminimal idempotent in S* is the largest element of such a strictly decreasing chain of idempotents. (It had been an open question whether there was a strictly decreasing chain $⟨q_{σ}⟩_{σ<ω+1}$ in ℕ*.) As other corollaries we show that if S is an infinite right cancellative and weakly left cancellative discrete semigroup, then βS contains a decreasing chain of idempotents of reverse order type λ for every countable ordinal λ and that if S is an infinite cancellative semigroup then the set U(S) of uniform ultrafilters contains such decreasing chains.
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  • Department of Mathematics, Howard University, Washington, DC 20059, U.S.A.
autor
  • Department of Pure Mathematics, University of Leeds, Leeds LS2 9J2, UK
  • School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm220-3-5
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