  Hom-algebras are generalizations of algebras obtained using a twisting by a
linear map. But there is a priori a freedom on where to twist. We enumerate
here all the possible choices in the Lie and associative categories and study
the relations between the obtained algebras. The associative case is richer
since it admits the notion of unit element. We use this fact to find sufficient
conditions for hom-associative algebras to be associative and classify the
implications between the hom-associative types of unital algebras.
