- Δ1-completions of a poset
- Volume | Issue number
- 30 | 1
- Pages (from-to)
- Document type
- Interfacultary Research Institutes
- Institute for Logic, Language and Computation (ILLC)
A join-completion of a poset is a completion for which each element is obtainable as a supremum, or join, of elements from the original poset. It is well known that the join-completions of a poset are in one-to-one correspondence with the closure systems on the lattice of up-sets of the poset. A Δ1-completion of a poset is a completion for which, simultaneously, each element is obtainable as a join of meets of elements of the original poset and as a meet of joins of elements from the original poset. We show that Δ1-completions are in one-to-one correspondence with certain triples consisting of a closure system of down-sets of the poset, a closure system of up-sets of the poset, and a binary relation between these two systems. Certain Δ1-completions, which we call compact, may be described just by a collection of filters and a collection of ideals, taken as parameters. The compact Δ1-completions of a poset include its MacNeille completion and all its join- and all its meet-completions. These completions also include the canonical extension of the given poset, a completion that encodes the topological dual of the poset when it has one. Finally, we use our parametric description of Δ1-completions to compare the canonical extension to other compact Δ1-completions identifying its relative merits.
- go to publisher's site
If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library, or send a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible.