The lattices $M_3$ and $N_5$ Why are $M_3$ and $N_5$ lattices in the theory of lattices called so?
They are well known for that their existence as a sublattice signifies lack of modularity/distributivity, yet their names are a mystery.
What's the origin of their name and why are they called so?
 A: I agree with amrsa's comment: the $M$ probably stands for "modular" and the $N$ probably stands for "nonmodular". In Birkhoff's 1940 Lattice Theory text he denotes them $M_5$ and $N_5$, presumably because they have five elements each. Later, as $M_n$ gained acceptance as the name for the lattice of height two with $n$ atoms, $M_5$ was replaced  by $M_3$.
It may be impossible to verify this, because authors do not always disclose their reasons for choosing notation. But, for fun, I decided to ask the AI "ChatGPT" to answer this question, and I got the following (not very useful!) answer:
[KK submits the original question to ChatGPT:]
Why are 3
and 5 lattices in the theory of lattices called so?
[ChatGPT answers:]

M3 and N5 lattices are two of the five two-dimensional lattices that are important in the theory of lattices. They are named for their two distinct lattice points, M and N. The M3 lattice has three lattice points, M, M', and M'', and the N5 lattice has five lattice points, N, N', N'', N''', and N''''.
