The notion of Turing computable embedding is a computable analog of Borel embedding. It provides a way to compare classes of countable structures, effectively reducing the classification problem for one class to that for the other. Most of the known results on nonexistence of Turing computable embeddings reflect differences in the complexity of the sentences needed to distinguish among nonisomorphic members of the two classes. Here we consider structures obtained as sums. It is shown that the n-fold sums of members of certain classes lie strictly below the (n+1)-fold sums. The differences reflect model-theoretic considerations related to Morley degree, not differences in the complexity of the sentences that describe the structures. We consider three different kinds of sum structures: cardinal sums, in which the components are named by predicates; equivalence sums, in which the components are equivalence classes under an equivalence relation; and direct sums of certain groups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Friedman and L. Stanley, “On Borel reducibility theory for classes of computable structures,” J. Symb. Log., 54, No. 3, 894–914 (1989).
W. Calvert, D. Cummins, J. F. Knight, and S. Miller, “Comparing classes of finite structures,” Algebra and Logic, 43, No. 6, 374–392 (2004).
E. B. Fokina, J. F. Knight, A. Melnikov, S. M. Quinn, and C. M. Safranski, “Classes of Ulm type and coding rank-homogeneous trees in other structures,” J. Symb. Log., 76, No. 3, 846–869 (2011).
J. F. Knight, S. Miller, and M. Vanden Boom, “Turing computable embeddings,” J. Symb. Log., 72, No. 3, 901–918 (2007).
C. Maher, “On embeddings of computable structures, classes of structures, and computable isomorphism,” Ph. D. Thesis, Univ. Notre Dame (2009).
E. B. Fokina and S.-D. Friedman, “On Σ1 1 equivalence relations over the natural numbers,” Math. Log. Q., 58, Nos. 1/2, 113–124 (2012).
E. B. Fokina, S.-D. Friedman, V. Harizanov, J. F. Knight, C. McCoy, and A. Montalbán, “Isomorphism relations on computable structures,” J. Symb. Log., 77, No. 1, 122–132 (2012).
W. Calvert, “The isomorphism problem for computable Abelian p-groups of bounded length,” J. Symb. Log., 70, No. 1, 331–345 (2005).
W. Calvert, V. S. Harizanov, J. F. Knight, and S. Miller, “Index sets of computable structures,” Algebra and Logic, 45, No. 5, 306–325 (2006).
H. Becker, “Isomorphism of computable structures and Vaught’s Conjecture,” J. Symb. Log., 78, No. 4, 1328–1344 (2013).
A. Montalb’an, “A computability theoretic equivalent to Vaught’s Conjecture,” Preprint.
I. Kaplansky, Infinite Abelian Groups, Univ. Michigan Publ. Math., 2, Univ. Michigan Press, Ann Arbor (1954).
Author information
Authors and Affiliations
Corresponding author
Additional information
(U. Andrews, D. I. Dushenin, C. Hill, J. F. Knight and A. G. Melnikov) Supported by NSF, grant DMS-1101123.
(U. Andrews) Supported by NSF, grant DMS-1201338.
Translated from Algebra i Logika, Vol. 54, No. 6, pp. 748–768, November-December, 2015.
Rights and permissions
About this article
Cite this article
Andrews, U., Dushenin, D.I., Hill, C. et al. Comparing Classes of Finite Sums. Algebra Logic 54, 489–501 (2016). https://doi.org/10.1007/s10469-016-9368-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10469-016-9368-7