Infinite meet semilattice with maximal element $\hat{1}$ that is not a lattice

Question

I can find examples of this where there is no maximal element, see here for example. However, does there exist an infinite meet semilattice with a maximal element that is not a lattice? I cannot seem to find an example of this anywhere. Clarification: infinite here just means the cardinality of the lattice must be infinite.

Answer 1

Let

$$L=\{\langle -1,0\rangle,\langle 0,1\rangle,\langle 0,-1\rangle\}\cup\left\{\left\langle\frac1n,0\right\rangle:n\in\Bbb Z^+\right\}\;,$$

and define a strict partial order $\prec$ on $L$ by $\langle x_0,y_0\rangle\prec\langle x_1,y_1\rangle$ iff $x_0<x_1$. Then $\langle 1,0\rangle$ is the maximum element of $L$, $L$ is a meet semilattice, and $\langle 0,1\rangle$ and $\langle 0,-1\rangle$ have no join.