Longer chains of idempotents in βG

Neil Hindman, Dona Strauss, and Yevhen Zelenyuk. "Longer chains of idempotents in βG." Fundamenta Mathematicae 220.3 (2013): 243-261. <http://eudml.org/doc/283119>.

@article{NeilHindman2013,
abstract = {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.},
author = {Neil Hindman, Dona Strauss, Yevhen Zelenyuk},
journal = {Fundamenta Mathematicae},
keywords = {idempotents; chains; Stone-Čech compactification},
language = {eng},
number = {3},
pages = {243-261},
title = {Longer chains of idempotents in βG},
url = {http://eudml.org/doc/283119},
volume = {220},
year = {2013},
}

TY - JOUR
AU - Neil Hindman
AU - Dona Strauss
AU - Yevhen Zelenyuk
TI - Longer chains of idempotents in βG
JO - Fundamenta Mathematicae
PY - 2013
VL - 220
IS - 3
SP - 243
EP - 261
AB - 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.
LA - eng
KW - idempotents; chains; Stone-Čech compactification
UR - http://eudml.org/doc/283119
ER -