Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Transactions of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $A'\leq _T B'$.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 03D25

Retrieve articles in all journals with MSC (1991): 03D25

Additional Information

Todd Hammond
Affiliation: Division of Mathematics and Computer Science, Truman State University, Kirksville, Missouri 63501

PII: S 0002-9947(97)01604-8
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

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia