New York Journal of Mathematics
Volume 24 (2018), 689-701


Neil Hindman and Dona Strauss

Sets and mappings in βS which are not Borel

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Published: September 13, 2018
Keywords: Borel sets, idempotents, Stone-Cech compactifications
Subject: 54D35, 54D80, 22A15

We extend theorems proved in [4] by showing that, if S is a countably infinite left cancellative semigroup and there is a finite bound on the size of sets of the form {x in S:xa=b} for a,b in S, then the following subsets of βS are not Borel: the set of idempotents, the smallest ideal, any semiprincipal right ideal defined by an element of S*, and S*S*. This has the imediate corollary that, if S is any infinite semigroup which either has the cancellation properties just described or has infinitely many cancellable elements, then the set of idempotents in βS is not Borel. We extend a theorem proved in [1], which states that for any infinite discrete group G and any p in G*, λp: βG → βG is not Borel, by showing that this theorem holds for all infinite semigroups which are right cancellative and very weakly left cancellative. We show that continuous maps between compact spaces map Baire sets to universally measurable sets, although this is far from being the case for Borel sets.


Author information

Neil Hindman:
Department of Mathematics, Howard University, Washington, DC 20059, USA.

Dona Strauss:
Department of Pure Mathematics, University of Leeds, Leeds LS2 9J2, UK.