single element.

\(1) at (1,0) [$w_1$]{}; (2) at (0,1) [$w_2$]{}; (3) at (2,1) [$w_3$]{}; (4) at (1,2) [$w_4$]{}; (5) at (3,2) [$w_5$]{}; (6) at (2,3) [$w_6$]{}; (7) at (4,3) [$w_7$]{}; (8) at (3,4) [$w_8$]{};

\(1) edge\[-, dashed, shorten &lt;= 2pt, shorten &gt;= 2pt\] (2) (1) edge\[-,shorten &lt;= 2pt, shorten &gt;= 2pt\] (3) (2) edge\[-,shorten &lt;= 2pt, shorten &gt;= 2pt\] (4) (3) edge\[-,dashed,shorten &lt;= 2pt, shorten &gt;= 2pt\] (4) (3) edge\[-,shorten &lt;= 2pt, shorten &gt;= 2pt\] (5) (4) edge\[-,shorten &lt;= 2pt, shorten &gt;= 2pt\] (6) (5) edge\[-,dashed,shorten &lt;= 2pt, shorten &gt;= 2pt\] (6) (5) edge\[-,shorten &lt;= 2pt, shorten &gt;= 2pt\] (7) (6) edge\[-,shorten &lt;= 2pt, shorten &gt;= 2pt\] (8) (7) edge\[-,dashed,shorten &lt;= 2pt, shorten &gt;= 2pt\] (8) ;

\[ex:bbL34\] The simplest nontrivial example of a set of multi-indices is $$\begin{aligned}
\mathbb{L}_{4}^3 & = {\left\{ {(w_1,w_2,w_3,w_4) : w_1\leq w_2, w_3\leq w_4, w_1 < w_3, w_2 < w_4}\right\}} \\
& = {\left\{ {(1122),(1123),(1133),(1223),(1233),(2233)}\right\}}\end{aligned}$$

Define $\wedge \colon \mathbb{L}_{2n}^N\times \mathbb{L}_{2n}^N \to \mathbb{L}_{2n}^N$ by $w\wedge w' = ((w\wedge w')_1, \ldots ,(w\wedge w')_{2n})$ where $$(w\wedge w') _ k = \begin{cases}
\min\{ w_k,w'_k\} & \text{if $k$ odd,} \\
\max\{ w_k,w'_k\} & \text{if $k$ even.} 
\end{cases}$$

\[thm:Lsemilattice\] $(\mathbb{L}_{2n}^N ,\wedge)$ is a meet-semilattice.

We begin by showing that if $w,w'\in\mathbb{L}_{2n}^N$, 