  It is well-known that the theories of semi-vector spaces and semi-algebras --
which were not much studied over time -- are utilized/applied in Fuzzy Set
Theory in order to obtain extensions of the concept of fuzzy numbers as well as
to provide new mathematical tools to investigate properties and new results on
fuzzy systems. In this paper we investigate the theory of semi-vector spaces
over the semi-field of nonnegative real numbers ${\mathbb R}_{0}^{+}$. We prove
several results concerning semi-vector spaces and semi-linear transformations.
Moreover, we introduce in the literature the concept of eigenvalues and
eigenvectors of a semi-linear operator, describing in some cases how to compute
them. Topological properties of semi-vector spaces such as completeness and
separability are also investigated. New families of semi-vector spaces derived
from semi-metric, semi-norm, semi-inner product, among others are exhibited.
Additionally, some results on semi-algebras are presented.
