On Tabor's problem concerning a certain quasi-ordering of iterative roots of functions (Q914868)

Given a selfmapping f of a set E denote by F(f) the set of all iterative roots of f and endow it with the following quasi-ordering: \(\phi\leq \psi\) iff \(\phi\) is an iterative root of \(\psi\). J. Tabor asked when this relation is antisymmetric, i.e. when it is an ordering. Answering this question the authors proved what follows. If the set E is infinite then the set of all selfmappings of E can be decomposed into three parts \(\Phi_ 1\), \(\Phi_ 2\), \(\Phi_ 3\) which have the same cardinality and satisfy the following conditions: 1) Any \(f\in \Phi_ 1\) has no proper iterative root. 2) Every \(f\in \Phi_ 2\cup \Phi_ 3\) has a proper iterative root and \(f\in \Phi_ 2\) implies that \(\leq\) is not antisymmetric on F(f), while \(\leq\) is antisymmetric on F(f) for every \(f\in \Phi_ 3\).