On retracting properties and covering homotopy theorem for S-maps into Sχ-cofibrations and Sχ-fibrations

In this paper we generalize the retracting property in homotopy theory for topological semigroups by introducing the notions of deformation S-retraction with its weaker forms and ES-homotopy extension property. Furthermore, the covering homotopy theorems for S-maps into Sχ-fibrations and Sχ-cofibrations are introduced and pullbacks for Sχ-fibrations behave properly.