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Predictions of the top-pair production cross section in electron-proton (ep)collisions at the future circular collider hadron-electron (FCC-he) and thelarge hadron electron collider (LHeC) in the leading order (LO) and thenext-to-leading order (NLO) approximations in perturbative quantumchromodynamics (QCD) are presented. Numerical results for the top structurefunctions and the ratio of structure functions are computed at therenormalization scale $\mu^{2}=Q^{2}+4m^{2}_{t}$ in the collinear generalizeddouble asymptotic scaling (DAS) approach. These results are presented in termsof the effective parameters of the parameterization of $F_{2}(x,Q^{2})$, whererelying on a Froissart-bounded. These quantitative results can be important forfuture collider experiments at the center-of-mass energy frontier and theimprovement of the phenomenological models for the development of the cosmicray cascades in ultra-high-energy domain. We obtained the uncertainties becauseof the power-law behavior $x^{-\Delta}$ of the collinear parton distributionfunctions (PDFs), the factorization, and renormalization scales, on the topproperties at high inelasticity for the FCC-he center-of-mass (CM) energy atthe NLO approximation.

Top cross section in the LHeC and FCC-he energy range