Semigroups in which all strongly summable ultrafilters are sparse (Q713072)

The authors consider questions from [\textit{N. Hindman, I. Protasov} and \textit{D. Strauss}, Mat. Stud. 10, No. 2, 121--132 (1998; Zbl 0934.22005)] as a reaction to the recent paper [\textit{P. Krautzberger}, New York J. Math. 16, 629--649 (2010; Zbl 1234.03034)]. Enigmatic sparse strongly summable ultrafilters were introduced in [Hindman, Protasov, Strauss, op. cit.] to get ultrafilters on an Abelian group \(G\) with the following extremal algebraic property: the maximal group in \(\beta G\) with the identity \(p\) is \(G+p\). In the present paper the authors prove that each strongly summable ultrafilter on a countable Abelian group with a finite number of elements of order 2 is sparse. The question whether each strongly summable ultrafilter on a countable Abelian group is sparse remains open.