  Various embedding problems of lattices into complete lattices are solved. We
prove that for any join-semilattice S with the minimal join-cover refinement
property, the ideal lattice IdS of S is both algebraic and dually algebraic.
Furthermore, if there are no infinite D-sequences in J(S), then IdS can be
embedded into a direct product of finite lower bounded lattices. We also find a
system of infinitary identities that characterize sublattices of complete,
lower continuous, and join-semidistributive lattices. These conditions are
satisfied by any (not necessarily finitely generated) lower bounded lattice and
by any locally finite, join-semidistributive lattice. Furthermore, they imply
M. Ern\'e's dual staircase distributivity. On the other hand, we prove that the
subspace lattice of any infinite-dimensional vector space cannot be embedded
into any countably complete, countably upper continuous, and countably lower
continuous lattice. A similar result holds for the lattice of all order-convex
subsets of any infinite chain.
