Any binary relation $R\subseteq X\times Y$ gives rise to a Galois connection between the powersets of $X$ and $Y$ in a well known way (on MO you can see it e. g. in this answer; in fact, such Galois connections are in one-to-one correspondence with binary relations).

It follows that any $R$ defines closure operators on subsets of both $X$ and $Y$, with anti-isomorphic complete lattices of closed sets.

Does any complete lattice occur in this way? If not, which ones do occur? Which lattices can occur for given fixed cardinalities of $X$ and $Y$? Can one characterize those relations which give rise to "nice" (modular, distributive, Heyting, Boolean, ...) lattices?

I am aware that this is most probably very well studied, so this is a reference request more than anything else; still I would be glad to have an explained answer without any references too :D

Any binary relation $R\subseteq X\times Y$ gives rise to a Galois connection between the powersets of $X$ and $Y$ in a well known way (on MO you can see it e. g. in this answer; in fact, such Galois connections are in one-to-one correspondence with binary relations).

It follows that any $R$ defines closure operators on subsets of both $X$ and $Y$, with anti-isomorphic complete lattices of closed sets.

Does any complete lattice occur in this way? If not, which ones do occur? Which lattices can occur for given fixed cardinalities of $X$ and $Y$? Can one characterize those relations which give rise to "nice" (modular, distributive, Heyting, Boolean, ...) lattices?

I am aware that this is most probably very well studied, so this is a reference request more than anything else; still I would be glad to have an explained answer without any references too :D

Any binary relation $R\subseteq X\times Y$ gives rise to a Galois connection between the powersets of $X$ and $Y$ in a well known way (on MO you can see it e. g. in this answer; in fact, such Galois connections are in one-to-one correspondence with binary relations).

It follows that any $R$ defines closure operators on subsets of both $X$ and $Y$, with anti-isomorphic complete lattices of closed sets.

Does any complete lattice occur in this way? If not, which ones do occur? Which lattices can occur for given fixed cardinalities of $X$ and $Y$?

I am aware that this is most probably very well studied, so this is a reference request more than anything else; still I would be glad to have an explained answer without any references too :D

Any binary relation $R\subseteq X\times Y$ gives rise to a Galois connection between the powersets of $X$ and $Y$ in a well known way (on MO you can see it e. g. in this answer; in fact, such Galois connections are in one-to-one correspondence with binary relations).

It follows that any $R$ defines closure operators on subsets of both $X$ and $Y$, with anti-isomorphic complete lattices of closed sets.

Does any complete lattice occur in this way? If not, which ones do occur? Which lattices can occur for given fixed cardinalities of $X$ and $Y$? Can one characterize those relations which give rise to "nice" (modular, distributive, Heyting, Boolean, ...) lattices?

I am aware that this is most probably very well studied, so this is a reference request more than anything else; still I would be glad to have an explained answer without any references too :D