Atom-generated planar lattices

Abstract

In this note, we discuss planar lattices generated by their atoms. We prove that if $L$ is a planar lattice generated by $n$ atoms, then both the left and the right boundaries of $L$ have at most $n+1$ elements. On the other hand, $L$ can be arbitrarily large. For every $k>1$, we construct a planar lattice $L$ generated by $4$ atoms such that $L$ has more than $k$ elements.