Extensions of invariant ideals (Q1105599)

An ideal I on the set X is said to have the property S(\(\kappa)\) if for every partition of X into sets of cardinality at least \(\kappa\) (an uncountable regular cardinal) there is a selector in I. The function \(f: \kappa\to \kappa\) is said to be essential if the range of f has cardinality \(\kappa\) ; it is said to be almost one to one if \(| f^{-1}[\{\alpha \}]| <\kappa\) for all \(\alpha <\kappa\). The ideal J on \(\kappa\) is invariant under f if f[A]\(\in J\) for all \(A\in J\). The main theorem of this paper answers a question of \textit{B. Weglorz} [Algebra Univers. 13, 41-55 (1981; Zbl 0481.04004)] by proving that if \({\mathcal F}\) is a family of \(\kappa\) functions from \(\kappa\) to \(\kappa\) with each essential function in \({\mathcal F}\) almost one to one, then every \({\mathcal F}\)-invariant ideal on \(\kappa\) can be extended to an \({\mathcal F}\)- invariant ideal J on \(\kappa\) such that J has the property S(\(\kappa)\).