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Extending partial orders to dense linear orders

✍ Scribed by Theodore A. Slaman; W.Hugh Woodin


Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
652 KB
Volume
94
Category
Article
ISSN
0168-0072

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## Abstract It is shown that if (__M__, <, ⃛) is an o‐minimal structure such that (__M__, <) is a dense total order and β‰Ύ is a parameter‐definable partial order on __M__, then β‰Ύ has an extension to a definable total order.

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We show that for a fuzzy partial order R on a ΓΏnite universe , there is a ΓΏnite family of fuzzy linear orders {Li: 16i6k} such that R(x; y) = min{ L i (x; y): 16i6k} for all x and y. This generalizes a well-known result on crisp partial orders, which states that each partial order on a ΓΏnite set is

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## Abstract Let βͺ―~R~ be the preorder of embeddability between countable linear orders colored with elements of Rado's partial order (a standard example of a wqo which is not a bqo). We show that βͺ―~R~ has fairly high complexity with respect to Borel reducibility (e.g. if __P__ is a Borel preorder, t