Computably extendible order types
In this thesis we consider, from a computability perspective, the question of what order-theoretic properties of a partial order can be preserved under linear extension. It is well-known that such properties as well-foundedness or scatteredness can be preserved, that is, given any well-founded parti...
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ndltd-bl.uk-oai-ethos.bl.uk-6941122018-04-04T03:32:46ZComputably extendible order typesGay, James Robert KishoreHalupczok, Immanuel ; Cooper, S.Barry ; Macpherson, H.Dugald2016In this thesis we consider, from a computability perspective, the question of what order-theoretic properties of a partial order can be preserved under linear extension. It is well-known that such properties as well-foundedness or scatteredness can be preserved, that is, given any well-founded partial order you can find a well-founded linear extension and mutatis mutandis for scattered partial orders. An order type σ is extendible if a partial order that does not embed σ can always be extended to a linear order that does not extend σ. So for example “given any well-founded partial order, you can find a well-founded linear extension” is equivalent to saying that ω^∗ is extendible. The extendible order types were classified by Bonnet [3] in 1969. We define notions of computable extendibility and then apply them to investigate the computable extendibility of three commonly used order types, ω^∗ , ω^∗ + ω and η. In Chapter 2 we prove that given a computably well-founded computable partial order, you can find a computably well-founded ω-c.e. linear extension, and further that this result doesn’t hold for n-c.e. for any finite n. In Chapter 3 we show how to extend these results for linearisations of computable partial orders which do not embed ζ = ω^∗ + ω. In Chapter 4 we prove the analogous results for scattered partial orders.519.7University of Leedshttp://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.694112http://etheses.whiterose.ac.uk/13976/Electronic Thesis or Dissertation |
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519.7 Gay, James Robert Kishore Computably extendible order types |
description |
In this thesis we consider, from a computability perspective, the question of what order-theoretic properties of a partial order can be preserved under linear extension. It is well-known that such properties as well-foundedness or scatteredness can be preserved, that is, given any well-founded partial order you can find a well-founded linear extension and mutatis mutandis for scattered partial orders. An order type σ is extendible if a partial order that does not embed σ can always be extended to a linear order that does not extend σ. So for example “given any well-founded partial order, you can find a well-founded linear extension” is equivalent to saying that ω^∗ is extendible. The extendible order types were classified by Bonnet [3] in 1969. We define notions of computable extendibility and then apply them to investigate the computable extendibility of three commonly used order types, ω^∗ , ω^∗ + ω and η. In Chapter 2 we prove that given a computably well-founded computable partial order, you can find a computably well-founded ω-c.e. linear extension, and further that this result doesn’t hold for n-c.e. for any finite n. In Chapter 3 we show how to extend these results for linearisations of computable partial orders which do not embed ζ = ω^∗ + ω. In Chapter 4 we prove the analogous results for scattered partial orders. |
author2 |
Halupczok, Immanuel ; Cooper, S.Barry ; Macpherson, H.Dugald |
author_facet |
Halupczok, Immanuel ; Cooper, S.Barry ; Macpherson, H.Dugald Gay, James Robert Kishore |
author |
Gay, James Robert Kishore |
author_sort |
Gay, James Robert Kishore |
title |
Computably extendible order types |
title_short |
Computably extendible order types |
title_full |
Computably extendible order types |
title_fullStr |
Computably extendible order types |
title_full_unstemmed |
Computably extendible order types |
title_sort |
computably extendible order types |
publisher |
University of Leeds |
publishDate |
2016 |
url |
http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.694112 |
work_keys_str_mv |
AT gayjamesrobertkishore computablyextendibleordertypes |
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1718619720173748224 |