F-algebra–Rinehart pairs and super F-algebroids

We define F-algebra–Rinehart pairs and super F-algebroids and study the connection between them. In particular, we prove that every super F-algebroid can be seen as an F-algebra–Rinehart pair. As by product, we provide an algebraic formalization of the notion of F-manifold following Dubrovin’s formalization in the case of Frobenius manifolds.