Sublattices of complete lattices with continuity conditions

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. Erné’s dual staircase distributivity. On the other hand, we prove that the subspace lattice of any infinitedimensional vector space cannot be embedded into any א0-complete, א0-upper continuous, and א0-lower continuous lattice. A similar result holds for the lattice of all order-convex subsets of any infinite chain.