Simple constructions of $\mathrm{F}\mathrm{B}\mathrm{L}(A)$ and $\mathrm{F}\mathrm{B}\mathrm{L}[E]$

We show that the free Banach lattice $\mathrm{F}\mathrm{B}\mathrm{L}(A)$ may be constructed as the completion of $\mathrm{F}\mathrm{V}\mathrm{L}(A)$ with respect to the maximal lattice seminorm $\nu$ on $\mathrm{F}\mathrm{V}\mathrm{L}(A)$ with $\nu (a)\le 1$ for all $a\in A$. We present a similar construction for the free Banach lattice $\mathrm{F}\mathrm{B}\mathrm{L}[E]$ generated by a Banach space $E$.