that $$L_{2n}^N\subset\mathcal{L}_{2n}^N\subset \mathbb{L}_{2n}^N.$$ Again, $L_{2n}^N$ is not a sub meet-semilattice of $\mathbb{L}_{2n}^N$.

We now define functions that takes arbitrary multi-indices and produces sets of pairs of strictly ordered multi-indices, i.e., maps $\varphi_i\colon \mathbb{L}^{N-1}_{2(n-1)}\to 2^{L^N_{n}\times L^N_{n}}$, $i=1,\ldots, n-2$. Set $$\label{varphik1}
\varphi_1(w) =
\begin{cases}
\left\{ (y,y)\in L^N_n\times L^N_n : y_1 \leq w_{1},\ w_{2} < y_{2} \leq w_{3}\right\} & \text{if $w_{2} < w_{3}$,} \\
\Big\{ (y,y')\in L^N_n\times L^N_n : y_1 \leq w_{1},\ y_{2} = w_{3},\ y'_{2} = w_{2}+1,\ \text{$y_j=y'_j$ for $j\neq 2$}\Big\} & \text{if $w_{3}\leq w_{2}$.}
\end{cases}$$ For $k=2,\dots,n-3$, define $\varphi_k= \varphi_{k,N,n}\colon \mathbb{L}_{2(n-1)}^{N-1} \to 2^{L_{n}^N}\times 2^{L_{n}^N}$ by $$\label{varphi}
\varphi_k(w) =
\begin{cases}
\left\{ (y,y)\in L^N_n\times L^N_n : w_{2k} < y_{k+1} \leq w_{2k+1}\right\} & \text{if $w_{2k}<w_{2k+1}$,} \\
\Big\{ (y,y')\in L^N_n\times L^N_n : y_{k+1} = w_{2k+1},\ y'_{k+1} = w_{2k}+1,\  \text{$y_j=y'_j$ for $j\neq k+1$}\  \Big\} & \text{if $w_{2k+1}\leq w_{2k}$.}
\end{cases}$$ And $$\label{varphiknm2}
\varphi_{n-2}(w) =
\begin{cases}
\left\{ (y,y)\in L^N_n\times L^N_n : w_{2(n-2)} < y_{n-1} \leq w_{2(n-2)+1},\ w_{2(n-1)} < y_{n} \right\} & \text{if $w_{2(n-2)}<w_{2(n-2)+1}$,} \\
\Big\{ (y,y')\in L^N_n\times L^N_n : y_{n-1} = w_{2(n-2)+1},\ y'_{n-1} = w_{2(n-2)}+1,\ w_{2(n-1)} < y_{n}\\
\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad\quad\quad\quad\quad\text{$y_j=y'_j$ for $j\neq n-1$}\  \Big\} & \text{if $w_{2(n-2)+1}\leq w_{2(n-2)}$.}
\end{cases}$$

[ *If $n=3$ and $N=7$, then $\varphi_1\colon \mathbb{L}_{4}^{6} \to 2^{L_{3}^7}\times 2^{L_{3}^7}$. Furthermore, if $w=(2,4,3,6)$, then $$\varphi_1(w) = \big\{\big((1,3,7),(1,5,7) \big),\big( (2,3,7),(2,5,7) 