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2006 Filters on Computable Posets
Steffen Lempp, Carl Mummert
Notre Dame J. Formal Logic 47(4): 479-485 (2006). DOI: 10.1305/ndjfl/1168352662

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.

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Steffen Lempp. Carl Mummert. "Filters on Computable Posets." Notre Dame J. Formal Logic 47 (4) 479 - 485, 2006. https://doi.org/10.1305/ndjfl/1168352662

Information

Published: 2006
First available in Project Euclid: 9 January 2007

zbMATH: 1128.03037
MathSciNet: MR2272083
Digital Object Identifier: 10.1305/ndjfl/1168352662

Subjects:
Primary: 03B30 , 03D
Secondary: 06

Keywords: computable poset , filter‎ , reverse mathematics

Rights: Copyright © 2006 University of Notre Dame

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