The real number line $[-\infty,\infty]=\mathbb{R}\union\{-\infty,\infty\}$ is complete in its usual ordering of numbers. Furthermore, the meet of a subset $S$ of $\mathbb{R}$ is the infimum of the set $S$ .

Now define the map $f:[-\infty,\infty]\to [-\infty,\infty]$ as

In the second place, if $S$ is a finite subset of $\mathbb{R}$ then $S$ contains a minimum element $s\in S$ . So $f(s)\in f(S)$ and $f(s)\leq f(t)$ for all $t\in S$ , so $f(\min S)=f(s)=\min f(S)$ . Hence $f$ is a lattice homomorphism.

However, $f$ is not a complete lattice homomorphism. To see this let $S=\{x\in \mathbb{R}: 0< x\}$ . Then $\inf S=0$ . However, $f(\inf S)=f(0)=0$ while $\inf f(S)=\inf \{1\}=1$ .