than one label. It is clear that a downset $D$ of $P$ is uniquely defined by the tuple $e_\cC(D) = (x_1, \dots, x_d)$ where $x_i = |D \cap C_i|$. (Note that $D$ is thus the downset generated by the set $\{{{x_i}^{(i)}} \mid i \in [d] \}$.)

As Dilworth [@Di50] observed in the case that $\cC$ is a decomposition, we observed in [@Si15] that $e_\cC$ is in fact a lattice embedding of $\cD(P)$ into the product of chains $\cP_\cC = \prod_{i = 1}^d P_i$ where $P_i$ is the chain $0 \prec 1 \prec \dots \prec {n_i}$ with one more element than $C_i$. Thus by Birkoff’s result from [@Bi], every chain cover of $J_L$ gives an embedding $e_\cC$ of $L$ as a sublattice of a product $\cP_\cC$ of chains. In (Corollary 6.6 of) [@Si15] we showed that every embedding of $L$ as a sublattice of a product of chains such that $L$ contains the zero and unit of the product, is $e_\cC$ for some chain cover $\cC$ of $J_L$.

The following notation will also be useful, and explains the notation ${{[{{\alpha}_{[i]}}, {{\beta}^{[j]}}]}}$. Given a product of chains $\cP_\cC = \prod_{i = 1}^d P_i$, let $${{{\alpha}_{[i]}}}= (\underbrace{0,\dots,0,\alpha}_{i},0,\dots,0) \qquad \mbox{ and } \qquad {{{\beta}^{[j]}}}= 