  Smooth actions of the multiplicative monoid $(\mathbb{R},\cdot)$ of real
numbers on manifolds lead to an alternative, and for some reasons simpler,
definition of a vector bundle, a double vector bundle and related structures
like a graded bundle [Grabowski and Rotkiewicz, J. Geom. Phys. 2011]. For these
reasons it is natural to study smooth actions of certain monoids closely
related with the monoid $(\mathbb{R},\cdot)$ . Namely, we discuss geometric
structures naturally related with: smooth and holomorphic actions of the monoid
of multiplicative complex numbers, smooth actions of the monoid of second jets
of punctured maps $(\mathbb{R},0)\rightarrow (\mathbb{R},0)$, smooth action of
the monoid of real 2 by 2 matrices and smooth actions of multiplicative reals
on a supermanifold. In particular cases we recover the notions of a holomorphic
vector bundle, a complex vector bundle and a non-negatively graded manifold.
