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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?

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    $\begingroup$ Perhaps $M_3$ for three atoms? But I've seen call it $M_5$ (perhaps for five elements). The $M$ could be for modular. $N_5$ for five elements? The $N$ for not modular? Some of this may be older, from results about rings, modules... $\endgroup$
    – amrsa
    Jan 22, 2023 at 11:58
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    $\begingroup$ I tried in History of Science and Mathematics and couldn't find it. But it might be a good place to ask. $\endgroup$
    – amrsa
    Jan 22, 2023 at 12:13

1 Answer 1

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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 𝑀3 and 𝑁5 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''''.

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    $\begingroup$ That is indeed, very artificial intelligence! I heard of some guy who developed a program to write papers, and they were convincing enough to elude some referees, but essentially they said nothing at all :) $\endgroup$
    – amrsa
    Jan 22, 2023 at 17:23

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