Let $S$ be a non-empty set of filters on a meet-semilattice.
If our semilattice is a distributive lattice, then the supremum (on the poset of filters ordered by set-theoretic inclusion) of $S$ is the filter corresponding to the filter base generated by $\bigcup S$. You can finds a proof of such a theorem for example in my article: http://ijpam.eu/contents/2012-74-1/6/index.html
Can this statement be strengthened for a more general case than distributive lattices?
I would like to see a counter-example for the case is our semilattice is not a lattice, or better for the more specific case when it is a lattice but not a distributive lattice.