The question is as follows:
Let f be a monomorphism from a lattice $L$ to a lattice $M$.Show that $L$ is isomorphic to a sublattice of $M$.
My attempt:
Since $f$ is a monomorphism from a lattice $L$ to a lattice $M$ therefore it will be a lattice homomorphism which is injective. Since, $f$ is injective therefore every element of $L$ will be mapped into a unique element in $f(L)$ and hence $M$,and also every element of $f(L)$ will have a unique preimage.
Thus $f:L\to f(L)$ is bijective and hence a lattice homomorphism. Further $f(L)$ is a sublattice of $M$ and hence $L$ is isomorphic to a sublattice of $M$.
Am I correct ?
One thing that can be done is First show that f(L) is sublattice of M After that Show that L is isomorphic to f(L) which is nothing but the sublattice of M