Filters on Computable Posets

Abstract

We explore the problem of constructing maximal and unbounded filters on computable posets. We obtain both computability results and reverse mathematics results. A maximal filter is one that does not extend to a larger filter. We show that every computable poset has a ∆ 0 2 maximal filter, and there is a computable poset with no Π 0 1 or Σ 0 1 maximal filter. There is a computable poset on which every maximal filter is Turing complete. We obtain the reverse mathematics result that the principle "every countable poset has a maximal filter" is equivalent to ACA 0 over RCA 0. An unbounded filter is a filter which achieves each of its lower bounds in the poset. We show that every computable poset has a Σ 0 1 unbounded filter, and there is a computable poset with no Π 0 1 unbounded filter. We show that there is a computable poset on which every unbounded filter is Turing complete, and the principle "every countable poset has an unbounded filter" is equivalent to ACA 0 over RCA 0. We obtain additional reverse mathematics results related to extending arbitrary filters to unbounded filters and forming the upward closures of subsets of computable posets.

Reverse Mathematics of Mf Spaces

“…The second, more serious, difficulty is that RCA 0 does not prove that every countable poset has a maximal filter, or even that for every countable poset there is an enumeration of a maximal filter on the poset. These results follow from joint work with Steffen Lempp that will appear in [8]. The two difficulties just discussed indicate that RCA 0 is not strong enough to give an honest formalization of countably based MF spaces.…”

Reverse Mathematics and Uniformity in Proofs without Excluded Middle

“…These lemmas show that (4.1.3) is provable in RCA 0 but not E-HA ω + AC + IP ω ef . [10,Theorem 3.6]). The following statement is equivalent to arithmetical comprehension over RCA 0 (and hence over RCA).…”

“…These lemmas show that (4.1.3) is provable in RCA 0 but not E-HA ω + AC + IP ω ef . Lemma 4.7 (Lempp and Mummert [10,Theorem 3.5]). RCA 0 proves that any enumerated filter on a countable poset can be extended to an unbounded enumerated filter.…”

Reverse mathematics in lattice theory

“…The reverse mathematics in partially ordered sets has been studied in the last few years, in [14], [22], etc. Frittaion and Marcone studied scattered partial orders and partial orders satisfying finite antichain condition from a reverse mathematics point of view, and Lempp and Mummert explored the problem of constructing maximal filters on countable partially ordered sets.…”

“…In this chapter, we will consider properties of lattices provable within RCA 0 . For posets, Lempp and Mummert proved in [22] that the "existence of maximal filters" is equivalent to ACA 0 over RCA 0 , and Mummert proved in his PhD thesis [25] that "every filter can be extended to a maximal filter" is equivalent to Π 1 1 -CA 0 over RCA 0 . For lattices, we have two additional operations, meet and join, which help a lot in the constructions of objects wanted.…”