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Transactions of the American Mathematical Society

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



Isomorphism of lattices of recursively enumerable sets

Author: Todd Hammond
Journal: Trans. Amer. Math. Soc. 349 (1997), 2699-2719
MSC (1991): Primary 03D25
MathSciNet review: 1348861
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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

Keywords: Effective isomorphism, effectively isomorphic, recursively enumerable, oracle, Turing jump, effective embedding, effectively embeddable.
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.
Article copyright: © Copyright 1997 American Mathematical Society