  Recently, a geometrical characterization of vector spaces served to
generalize them into a new class of algebras. Instead of the algebraic
properties of the underlying fields, we generalized the recently discovered
property of such spaces that generates these fields. This property also
concerned the semi-linear transformations, which are necessary to define
geometrical invariance.Yet, the class of such geometrical generalizations was
practically unknown. We only knew that it differs from the class of modules,
because of a simple example (the sum monoid of natural numbers). Here, we
partly clarify the extent of this class: we prove that it contains at least the
one of all commutative based universal algebras and we provide it with four new
examples. Again, our further examples are not modules. They exhibit a wider
choice of both types and algebraic properties, though they keep the
representations by "coordinates".
