Filters on Computable Posets

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 ∆2 maximal filter, and there is a computable poset with no Π1 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 ACA0 over RCA0. An unbounded filter is a filter which achieves each of its lower bounds in the poset. We show that every computable poset has a Σ1 unbounded filter, and there is a computable poset with no Π1 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 ACA0 over RCA0. We obtain additional reverse mathematics results related to extending arbitrary filters to unbounded filters and forming the upward closures of subsets of computable posets.