\(\alpha\)-inflatable semigroups (Q1184177)

Let \(S\) be a discrete semigroup. Then the multiplication of \(S\) extends uniquely to the Stone-Čech compactification \(\beta S\), such that \(\beta S\) becomes a right topological semigroup with dense topological center. In an earlier paper the author has shown that if \(S\) is `inflatable', (i.e. \(S\) is countable, right cancellative, and satisfies a weak (but complicated) left cancellation law) then the product \((\beta S\setminus S)^ 2\) of the remainder \(\beta S\setminus S\) with itself does not contain the closure of the minimal ideal \(M(\beta S)\) of \(\beta S\). In the present paper the author extends this result to the newly defined class of `\(\alpha\)-inflatable semigroups' where \(\alpha\) is an integer \(\geq 2\) (the old `inflatable' is equivalent to 2-inflatable). For every \(\alpha\geq 2\) he constructs a semigroup which is \(\alpha+1\)- inflatable, but not \(\alpha\)-inflatable.