Let κ be an infinite cardinal. For every ordinal α<κ, let G_α be a nontrivial group written additively, let G=\bigoplus_α<κG_α, and let H_α={x∈ G:x(γ)=0 for all γ<α}. Let βG be the Stone-Čech compactification of G as a discrete semigroup and define a closed subsemigroup T⊆βG by T=\bigcap_α<κcl_βG(H_α\setminus 0). We show that, for every p,q∈ T, if (β G+p)∩(β G+q)≠∅, then either p∈βG+q or q∈βG+p.

Supported by NRF grant IFR1202220164, the John Knopfmacher Centre for Applicable Analysis and Number Theory, and the Friedel Sellschop Award.