According to the nLab article on locales, a frame has all meets by the adjoint functor theorem:
This seems a bit strange to me, since it's well-known that an infinite intersection of open subsets is not necessarily open.
Is the assertion on nLab true? If so, what open set do we get from $\wedge_{n=1}^\infty (-1/n,1/n)$ in the frame of opens on $\mathbb{R}$? Is it just the initial object (empty set)?
Let's forget about frames and look at spaces. If $\{U_i:i\in I\}$ is a collection of open sets in a space $\mathcal{X}$, then there is a largest open set contained in each $U_i$, namely $int(\bigcap_{i\in I}U_i)$. So viewed as a poset, the set of open subsets of $\mathcal{X}$ is a complete lattice with joins being given by unions and meets being given by interiors of intersections. The first half of this is trivial, and I call the second half of this "Arby's theorem."
But in some sense the presence of infinite meets is "coincidental" - we only really care about finite meets. This gets to the issue of how we define "homomorphism" - arbitrary meets happen to exist, but we don't care about preserving them.
