Sublattices of complete lattices with continuity conditions

Abstract

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 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.