Ultraproducts, \(p\)-limits and antichains on the Comfort group order
The authors of the paper under review consider the following question: Does \({\mathbf M}{\mathbf A}\) imply that the Comfort group order is not downward directed? The answer ``yes'' is equivalent to the existence of a \(p\)-compact group \(G\) and a \(q\)-compact group \(H\) for which \(G\times H\) is not \(r\)-compact for any free ultrafilter \(r\) on \(\omega\). The authors construct \(p\)-compact groups as follows: Let \(p\), \(q\) be incomparable selective ultrafilters. Then there exists a \(p\)-compact group \(G\) and a \(q\)-compact group \(H\) for which \(G\times H\) is not countably compact. This implies the following important results: if there are \(2^c\) incomparable selective ultrafilters, then the Comfort group order has an antichain of size \(2^c\). \({\mathbf M}{\mathbf A}\) implies the Comfort group order has an antichain of size \(2^c\).
scientific article; zbMATH DE number 2104477
scientific article