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ComputabilityTheoretic and ProofTheoretic Aspects of Partial and Linear Orderings
 Israel Journal of mathematics
"... Szpilrajn's Theorem states that any partial order P = hS;
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Szpilrajn's Theorem states that any partial order P = hS; <P i has a linear extension L = hS; <L i. This is a central result in the theory of partial orderings, allowing one to de ne, for instance, the dimension of a partial ordering. It is now natural to ask questions like \Does a wellpartial ordering always have a wellordered linear extension?" Variations of Szpilrajn's Theorem state, for various (but not for all) linear order types , that if P does not contain a subchain of order type , then we can choose L so that L also does not contain a subchain of order type . In particular, a wellpartial ordering always has a wellordered extension.
The isomorphism problem for torsionfree abelian groups is analytic complete
 JOURNAL OF ALGEBRA
, 2008
"... We prove that the isomorphism problem for torsionfree Abelian groups is as complicated as any isomorphism problem could be in terms of the analytical hierarchy, namely Σ 1 1 complete. ..."
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We prove that the isomorphism problem for torsionfree Abelian groups is as complicated as any isomorphism problem could be in terms of the analytical hierarchy, namely Σ 1 1 complete.
Questions in Computable Algebra and Combinatorics
, 1999
"... this article, we will focus on two areas of computable mathematics, namely computable algebra and combinatorics. The goal of this article is to present a number of open questions in both computable algebra and computable combinatorics and to give the reader a sense of the research activity in these ..."
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this article, we will focus on two areas of computable mathematics, namely computable algebra and combinatorics. The goal of this article is to present a number of open questions in both computable algebra and computable combinatorics and to give the reader a sense of the research activity in these elds. Our philosophy is to try to highlight questions, whose solutions we feel will either give insight into algebra or combinatorics, or will require new technology in the computabilitytheoretical techniques needed. A good historical example of the rst phenomenom is the word problem for nitely presented groups which needed the development of a great deal of group theoretical machinery for its solution by Novikov [110] and Boone [10]. A good example of the latter phenomenon is the recent solution by Coles, Downey and Slaman [17] of the question of whether all rank one torsion free 1991 Mathematics Subject Classi cation. Primary 03D45; Secondary 03D25
Computability, Definability and Algebraic Structures
, 1999
"... In a later section, we will look at a result of Coles, Downey and Slaman [16] of pure computability theory. The result is that, for any set X, the set ..."
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In a later section, we will look at a result of Coles, Downey and Slaman [16] of pure computability theory. The result is that, for any set X, the set