Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  investigate the interplay beten several similarities of relational structures: the condensational equivalence (defined by   X∼ c Y     X    ∼    c    Y        iff there are bijective homomorphisms   f:X→Y     f  :  X  →  Y        and   g:Y→X     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   Mod L (κ)       Mod    L    (  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  have a large diversity.
['Miloš S. Kurilić', 'Nenad Morača']
I am writing a research paper on a idea Condensational equivalence, equimorphism, elementary equivalence and similar similarities
[{"from": "human", "value": "I am writing a research paper on a idea Condensational equivalence, equimorphism, elementary equivalence and similar similarities"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  investigate the interplay beten several similarities of relational structures: the condensational equivalence (defined by   X\u223c c Y     X    \u223c    c    Y        iff there are bijective homomorphisms   f:X\u2192Y     f  :  X  \u2192  Y        and   g:Y\u2192X     g  :  Y  \u2192  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   Mod L (\u03ba)       Mod    L    (  \u03ba  )        of all  L -structures of size  \u03ba  collapses significantly for a finite cardinal  \u03ba  or for a unary language  L , while for infinite structures of non-unary languages  have a large diversity."}]