Mathematics – General Mathematics
Scientific paper
2005-01-20
Algebra Universalis 53, no. 2-3 (2005) 149--173
Mathematics
General Mathematics
To appear in Algebra Universalis
Scientific paper
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.
No associations
LandOfFree
Sublattices of complete lattices with continuity conditions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Sublattices of complete lattices with continuity conditions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Sublattices of complete lattices with continuity conditions will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-641625