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 EShomotopy 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