Δ1-completions of a Poset

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