Longer chains of idempotents in βG

Abstract

Given idempotents e and f in a semigroup, e ≤ f if and only if e = f e = 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 qσ σ<λ of idempotents in Cp, 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 qσ 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.

Ideals and idempotents in the uniform ultrafilters

“…It is clear that if conditions (5) and (7) are satisfied for every µ, then F will be an ultrafilter on G. It is also clear that if condition (6) is satisfied at every stage thenF will be a set. To finish the proof of the theorem, we just need to check that it is actually possible to carry out such a construction.…”

“…However, the existence of left-maximal idempotents has been a difficult problem. Even for S = (Z, +), the existence of left-maximal idempotents in βS was open for years, despite the fact that much work was done on this problem (see Questions 9.25 and 9.26 in [6]; see also Questions 5.5( 2),( 3) in [7], Problems 4.6 and 4.7 in [4], and [10]). The following Proposition (a known folklore result) tells us one way to find left-maximal idempotents in βS.…”

“…The following list of conditions needs to be satisfied at every stage of our recursion. Note that a similar list is given in [1], but we have had to modify their condition (5) and add a new condition (7).…”

“…This is the same as in [1], and it is verified there that conditions (1) − (4) and ( 6) are satisfied at F µ+1 . Condition ( 5) is vacuously satisfied at the odd step, so we just need to check that condition (7) is satisfied. For this, note that every element of F µ+1 contains an element of the form D p ∩ C for some p ∈ [κ + ] <ω and C ∈ F µ .…”

Increasing sequences of principal left ideals of $\beta \mathbb{Z}$ are finite

“…There are infinite decreasing chains of idempotents in βZ [3]. In [5] an infinite increasing right chain of idempotents in βZ was constructed. However, the question whether there is any infinite increasing chain (left chain) of idempotents in βZ remained open [4,Question 9.27].…”