Pleasant ideals (Q1182702)

Let \(\kappa\) be a regular, uncountable cardinal and let \(I\) be an ideal on \(\kappa\) (all ideals considered contain all singletons and are \(<\kappa\)-complete). If \(A\subseteq \kappa\) and \(f:A\to \kappa\) then \(f\) is called \(I\)-small if \(f^{-1}(\{\alpha\})\in I\) for every \(\alpha < \kappa\). In this paper the concept of a pleasant ideal is introduced. An ideal \(I\) on \(\kappa\) is pleasant if for every \(A\in I^ +\) and every regressive \(I\)-small \(f:A\to \kappa\) we have \(f(A)\in I^ +\), where \(I^ +=\{Y\subseteq \kappa: Y\notin \kappa\}\). It is shown that \(I\) is pleasant if and only if \(I\) is closed under \(I\)-diagonal unions. Several examples are provided, and ideal operators related to pleasantness are considered. Moreover, it is shown that an ideal is normal if and only if it is pleasant and extends the nonstationary ideal if and only if it is pleasant and selective.