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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Isomorphism of lattices of recursively enumerable sets
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by Todd Hammond PDF
Trans. Amer. Math. Soc. 349 (1997), 2699-2719 Request permission

Abstract:

Let $\omega = \{ 0,1,2,\ldots \}$, and for $A\subseteq \omega$, let $\mathcal E^A$ be the lattice of subsets of $\omega$ which are recursively enumerable relative to the “oracle” $A$. Let $(\mathcal E^A)^*$ be $\mathcal E^A/\mathcal I$, where $\mathcal I$ is the ideal of finite subsets of $\omega$. It is established that for any $A,B\subseteq \omega$, $(\mathcal E^A)^*$ is effectively isomorphic to $(\mathcal E^B)^*$ if and only if $A’\equiv _T B’$, where $A’$ is the Turing jump of $A$. A consequence is that if $A’\equiv _T B’$, then $\mathcal E^A\cong \mathcal E^B$. A second consequence is that $(\mathcal E^A)^*$ can be effectively embedded into $(\mathcal E^B)^*$ preserving least and greatest elements if and only if $A’\leq _T B’$.
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Additional Information
  • Todd Hammond
  • Affiliation: Division of Mathematics and Computer Science, Truman State University, Kirksville, Missouri 63501
  • Email: thammond@math.nemostate.edu
  • Received by editor(s): December 17, 1991
  • Received by editor(s) in revised form: August 3, 1995
  • Additional Notes: This paper is based primarily on part of the author’s Ph.D. thesis, written under the supervision of Professor Robert Vaught.
  • © Copyright 1997 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 349 (1997), 2699-2719
  • MSC (1991): Primary 03D25
  • DOI: https://doi.org/10.1090/S0002-9947-97-01604-8
  • MathSciNet review: 1348861