## Notre Dame Journal of Formal Logic

- Notre Dame J. Formal Logic
- Volume 47, Number 4 (2006), 443-599

### Filters on Computable Posets

Steffen Lempp and Carl Mummert

#### 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 \Delta^0_2 maximal filter, and there is a computable poset with no \Pi^0_1 or \Sigma^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 \Sigma^0_1 unbounded filter, and there is a computable poset with no \Pi^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 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.

#### Article information

**Source**

Notre Dame J. Formal Logic Volume 47, Number 4 (2006), 479-485.

**Dates**

First available: 9 January 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.ndjfl/1168352662

**Digital Object Identifier**

doi:10.1305/ndjfl/1168352662

**Mathematical Reviews number (MathSciNet)**

MR2272083

**Zentralblatt MATH identifier**

1128.03037

**Subjects**

Primary: 03D 03B30: Foundations of classical theories (including reverse mathematics) [See also 03F35]

Secondary: 06

**Keywords**

computable poset filter reverse mathematics

#### Citation

Lempp, Steffen; Mummert, Carl. Filters on Computable Posets. Notre Dame Journal of Formal Logic 47 (2006), no. 4, 479--485. doi:10.1305/ndjfl/1168352662. http://projecteuclid.org/euclid.ndjfl/1168352662.