Condensational equivalence, equimorphism, elementary equivalence and similar similarities

We investigate the interplay between several similarities of relational structures: the condensational equivalence (defined by X∼cY iff there are bijective homomorphisms f:X→Y and g:Y→X), the isomorphism, the equimorphism (bi-embedability), the elementary equivalence and the similarities of structures determined by some similarities of their self-embedding monoids. It turns out that the Hasse diagram describing the hierarchy of these equivalence relations restricted to the set ModL(κ) of all L-structures of size κ collapses significantly for a finite cardinal κ or for a unary language L, while for infinite structures of non-unary languages we have a large diversity.