Fibration mappings of topological left almost semigroups

We discuss the notion of fibration property of continuous functions in the theory of topological left almost semigroups. The concept of L-fibration is introduced and investigated. The restriction and composition properties of L-fibrations and the lifting property of L-fibrations are studied through the concepts of L-lifting function and L-regular lifting function. The study shows the uniqueness of L-regular lifting function of any L-fibration. Besides, the homotopically equivalent and preserving projection of L-regular lifting functions are also investigated. Finally, by using the idea of homotopy extension property in homotopy theory of topological spaces, the study provides the notion of L-homotopy extension property in the theory of topological left almost semigroups by giving L-homotopy extension theorem and the concepts of L-absolute retract and L-absolute neighborhood retract. Also, the role of L-absolute retract, L-absolute neighborhood retract and L-regular lifting functions in the theory of L-homotopy extension are revealed.