 

Neil Hindman and Lakeshia Legette Jones
Idempotents in βS that are only products trivially view print


Published: 
January 21, 2014 
Keywords: 
ultrafilters, strongly summable, strongly productive, StoneČech compactification, idempotents, Martin's Axiom, sparse 
Subject: 
03E50, 22A15, 54D35, 54D80 


Abstract
All results mentioned in this abstract assume Martin's Axiom.
(Some of them are known to not be derivable in ZFC.)
It is known that if S is the free semigroup on countably many
generators, then there exists an idempotent p∈βS such that
if q,r∈βS and qr=p, then q=r=p. We show that the
same conclusion holds for the semigroups (N,⋅) and
(F,∪) where F is the set of finite
nonempty subsets of N. Such a strong conclusion is not possible
if S is the free group on countably many generators or is the free
semigroup on finitely many (but more than one) generators, since then
any idempotent can be written as a product involving elements of S.
But we show that in these cases we can produce p such that if
q,r∈βS and qr=p, then either q=r=p or q and r satisfy
one of the trivial exceptions that must exist. Finally, we show
that for the free semigroup on countably many generators, the conclusion
can be derived from a set theoretical assumption that is at least
potentially weaker than what had previously been required.


Acknowledgements
The first author acknowledges support received from the National Science Foundation via Grant DMS1160566. The second author acknowledges support received from the Simons Foundation via Grant 210296


Author information
Neil Hindman:
Department of Mathematics, Howard University, Washington, DC 20059, USA
nhindman@aol.com
Lakeshia Legette Jones:
Department of Mathematics and Statistics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA
lljones3@ualr.edu

