Finite transferable lattices are sharply transferable

Abstract:

A lattice $\mathfrak {L}$ is called transferable if and only if, whenever $\mathfrak {L}$ can be embedded into the lattice $I(\mathcal {K})$ of all ideals of a lattice $\mathcal {K}$, $\mathfrak {L}$ can be embedded into $\mathcal {K}$ itself. If for every lattice embedding $f$ of $\mathfrak {L}$ into $I(\mathcal {K})$ there exists an embedding $g$ of $\mathfrak {L}$ into $\mathcal {K}$ satisfying the further condition that for $x$ and $y$ in $L$, $f(x) \in g(y)$ holds if and only if $x \leqslant y$, then $\mathfrak {L}$ is called sharply transferable. It is shown that every finite transferable lattice is sharply transferable.