Subprincipal closed ideals in \(\beta N\)
Let \((N,+)\) be the semigroup of positive integers with usual addition and \((\beta N,+)\) be the Stone-Cech compactification of \((N,+)\), where the addition in \(\beta N\) is the usual left-continuous extension of the ordinary addition on N. The algebraic structure of \(\beta N\) was studied by \textit{N. Hindman} in 1980 [Semigroup Forum 21, 221-255 (1980; Zbl 0448.22004) and 25, 291-310 (1985; Zbl 0499.22002)] and has proved somewhat useful in solving problems in Ramsey Theory). In this paper, subprincipal closed ideals of \(\beta N\) generated by N are studied. The upper, lower and Banach density of a subset \(A\subset N\) are defined to be \(\overline{d(A)}=\limsup_{n\to \infty}\{(| A\cap \{1,2,...,k\}|)/k\); \(k\geq n\}\); \underbar{d(A)}\(=\lim_{n\to \infty}\inf \{(| A\cap \{1,2,...,k\}|)/k\); \(k\geq n\}\) and \(d^*(A)=\sup \{\alpha \in Real:\) \(\exists\) increasing sequences \(<t_ n>^{\infty}_{n=1}\) and \(<x_ n>^{\infty}_{n=1}\) in N such that, for each \(n\in N\), \(| A\cap \{t_ n+1,t_ n+2,...t_ n+x_ n\}| \geq \alpha x_ n\}\) respectively. Also, define \(\Delta =\{p\in \beta N:\) for all \(A\in p\), \(\bar d(A)>0\}\); \(\Delta^*=\{p\in \beta N:\) for all \(A\in p\), \(d^*(A)>0\}\) and \(\Delta_ 1=\{p\in \beta N:\) for all \(A\in p\) there exists \(k\in N\) with \(d^*(\cup^{k}_{t=1}A- t)=1\}\). The authors show that any closed ideal of \(N^*=\beta N\setminus N\) which is not equal to \(\Delta\), has non-empty intersection with \(\Delta^*\setminus \Delta\). Also, they point out that it is possible to find \(p\in \Delta \setminus \Delta_ 1\), such that \(\Delta_ 1\subsetneqq cl(N^*+p)\subsetneqq \Delta^*\) and \(\Delta\) is the intersection of the subprincipal closed ideals lying strictly between \(\Delta_ 1\) and \(\Delta^*\).
scientific article; zbMATH DE number 4077580
scientific article