  We establish the hierarchy among twelve equivalence relations (similarities)
on the class of relational structures: the equality, the isomorphism, the
equimorphism, the full relation, four similarities of structures induced by
similarities of their self-embedding monoids and intersections of these
equivalence relations. In particular, fixing a language $L$ and a cardinal
$\kappa$, we consider the interplay between the restrictions of these
similarities to the class $Mod _L (\kappa )$ of all $L$-structures of size
$\kappa$. It turns out that, concerning the number of different similarities
and the shape of the corresponding Hasse diagram, the class of all structures
naturally splits into three parts: finite structures, infinite structures of
unary languages, and infinite structures of non-unary languages (where all
these similarities are different).
