Are there $N_n$ lattices generalizing the $N_5$ lattice? $M_3$ and $N_5$ lattice, respectively, is a widely used notation for these two lattices.
I'm wondering what the indices mean.
For the M case, I found in the book of B. A. Davey, H. A. Priestley, Introduction to Lattices and Order, the definition of $M_n$ lattice (= lattice of height two with n atoms, $n \in \mathbb N$), which completely answers my question for $M_3$.
So my question is: Is there an analogous (standard) definition of $N_n$ lattice (or partially ordered set), such that $N_5$ is just the special case for $n = 5$?
My question is a little bit related to this one.
But in contrast to this question, I do not care where the "M" or "N" comes from.
 A: There is a definition for $N_n$ in the literature which reduces to the usual $N_5$ when $n=5$. See Diagram 11 on page 42 of
Equational bases and nonmodular lattice varieties by Ralph McKenzie, Transactions of the AMS,
Volume 174, December 1972.
I have a better drawing of a typical member of this sequence of lattices in one of my papers (see the front page of the article). I renumbered the sequence for my paper so that $N_0$ is the first lattice in the sequence, and it is the lattice that one usually calls $N_5$.  Therefore, if McKenzie's sequence of lattices is $N_n^{\textrm{McK}}$, $n\geq 5$, then my renumbered sequence is $N_{n-5}^{\textrm{KK}}$.
These lattices form a natural sequence of projective, subdirectly irreducible lattices. This combination of properties is interesting because if $L$ is a projective, subdirectly irreducible lattice, then the class of lattices that do not contain $L$ as a sublattice is closed under $H, S$, and $P$, hence is a variety. It is well-known that the class of lattices that do not contain $N_5$ as a sublattice is a variety of lattices, but less well-known that the same statement is true for $N_6^{\textrm{McK}}$, $N_7^{\textrm{McK}}$, $N_8^{\textrm{McK}}$, etc.
