  In this paper, first we introduce a new approach to the notion of
$F$-algebroids, which is a generalization of $F$-manifold algebras and
$F$-manifolds, and show that $F$-algebroids are the corresponding
semi-classical limits of pre-Lie formal deformations of commutative associative
algebroids. Then we use the deformation cohomology of pre-Lie algebroids to
study pre-Lie infinitesimal deformations and extension of pre-Lie
$n$-deformations to pre-Lie $(n+1)$-deformations of a commutative associative
algebroid. Next we develop the theory of Dubrovin's dualities of $F$-algebroids
with eventual identities and use Nijenhuis operators on $F$-algebroids to
construct new $F$-algebroids. Finally we introduce the notion of
pre-$F$-algebroids, which is a generalization of $F$-manifolds with compatible
flat connections. Dubrovin's dualities of pre-$F$-algebroids with eventual
identities, Nijenhuis operators on pre-$F$-algebroids and their applications to
integral systems are discussed.
