Longer chains of idempotents in βG

Neil Hindman ; Dona Strauss ; Yevhen Zelenyuk

Fundamenta Mathematicae, Tome 220 (2013), p. 243-261 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2013-01-01

Zbl 1269.54010

EUDML-ID : urn:eudml:doc:283119