On strongly freely decomposable and induced maps

Freely decomposable and strongly freely decomposable maps were introduced by G. R. Gordh and C. B. Hughes as a generalization of monotone maps with the property that these maps preserve local connectedness in inverse limits. We prove some relationships between f, Cn(f) and 2f, when f, Cn(f) or 2f belong to the following classes of maps: Almost monotone, quasi-monotone, weakly monotone, freely decomposable or strongly freely decomposable. We correct two corollaries formulated by Jaunusz J. Charatonik in ``On feebly monotone and related classes of maps\u27\u27. We also present an alternative reformulation of these results