Weak post-Hopf algebra structures on Sweedler’s four-dimensional Hopf algebra

Inspired from the work of Li et al. [Post-Hopf algebras, relative Rota–Baxter operators and solutions to the Yang–Baxter equation, J. Noncommut. Geom. 18 (2024) 605–630], we shall introduce the notion of weak post-Hopf algebra, and describe all the weak post-Hopf algebra on Sweedler’s four-dimensional Hopf algebra. It is pointed that weak post-Hopf algebra is not necessarily post-Hopf algebra.