Abstract
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.
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References
Amjid, V., Yousafzai, F., Iqbal, Q.: Generalized ideals in ordered left almost semigroups. Afrika Math. 28, 605–614 (2017)
Čerin, Z.: Homotopy theory of topological semigroups. Topol. Appl. 123, 57–68 (2002)
Cho, J.R., Ježek, J., Kepka, T.: Paramedial groupoids. Czechoslovak Math. J. 49, 277–290 (1999)
Engelking, R.: General Topology. 2nd edn., Sigma Series in Pure Mathematics, vol. 6, Heldermann, Berlin (1989)
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)
Hurewicz, W.: On the concept of fiber space. Proc. Nat. Acad. Sci. USA 41, 956–961 (1955)
Kazim, M.A., Naseeruddin, M.: On almost semigroups. The Aligarh Bull. Math. 2, 1–7 (1972)
Kočinac, L.D.R., Othman, H.A.: On homotopical properties of topological left almost semigroups. TWMS J. Pure Appl. Math., accepted
Mushtaq, Q., Inam, M.: Left almost semigroups defined by a free algebra. Quasigroups Relat. Syst. 16, 69–76 (2008)
Mushtaq, Q., Khan, M., Shum, K.P.: Topological structures on LA-semigroups. Bull. Malays. Math. Sci. Soc. 36, 901–906 (2013)
Mushtaq, Q., Yusuf, L.M.: On LA-semigroups. Aligarh Bull. Math. 8, 65–70 (1978)
Othman, H.A.: On loop functions in the theory of topological LA-semigroups. Filomat 36, 6957–6967 (2022)
Saif, A., Othman, H.A.: Retractions in homotopy theory for finite topological semigroups. Tbilisi Math. J. 14, 219–232 (2021)
Shah, T., Yousaf, K.: Topological LA-groups and LA-rings. Quasigroups Relat. Syst. 18, 95–104 (2010)
Spanier, E.H.: Algebraic Topology. Springer, New York (1966)
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The first author would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant Code: 22UQU4330052DSR15.
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Communicated by Jimmie D. Lawson.
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Hakeem A. Othman contributed equally to this work.
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Kočinac, L.D.R., Othman, H.A. Fibration mappings of topological left almost semigroups. Semigroup Forum 107, 188–199 (2023). https://doi.org/10.1007/s00233-023-10370-1
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DOI: https://doi.org/10.1007/s00233-023-10370-1