### Spectra of structures and relations

Valentina S. Harizanov and Russel G. Miller
Source: J. Symbolic Logic Volume 72, Issue 1 (2007), 324-348.

#### Abstract

We consider embeddings of structures which preserve spectra: if g:ℳ →𝒮 with 𝒮 computable, then ℳ should have the same Turing degree spectrum (as a structure) that g(ℳ) has (as a relation on 𝒮). We show that the computable dense linear order ℒ is universal for all countable linear orders under this notion of embedding, and we establish a similar result for the computable random graph 𝔖. Such structures are said to be spectrally universal. We use our results to answer a question of Goncharov, and also to characterize the possible spectra of structures as precisely the spectra of unary relations on 𝔖. Finally, we consider the extent to which all spectra of unary relations on the structure ℒ may be realized by such embeddings, offering partial results and building the first known example of a structure whose spectrum contains precisely those degrees c with c'T 0''.

First Page:
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

Permanent link to this document: http://projecteuclid.org/euclid.jsl/1174668398
Digital Object Identifier: doi:10.2178/jsl/1174668398
Mathematical Reviews number (MathSciNet): MR2298485
Zentralblatt MATH identifier: 1116.03029

### References

C. J. Ash and J. Knight, Computable structures and the hyperarithmetical hierarchy, Elsevier, Amsterdam, 2000.
Mathematical Reviews (MathSciNet): MR1767842
Zentralblatt MATH: 0960.03001
J. Chisholm, The complexity of intrinsically r.e. subsets of existentially decidable models, Journal of Symbolic Logic, vol. 55 (1990), pp. 1213--1232.
Mathematical Reviews (MathSciNet): MR1071324
Digital Object Identifier: doi:10.2307/2274483
Project Euclid: euclid.jsl/1183743415
Zentralblatt MATH: 0722.03031
B. F. Csima, V. S. Harizanov, R. G. Miller, and A. Montalbán, Computability of Fraïssé limits, to appear.
R. G. Downey, S. S. Goncharov, and D. R. Hirschfeldt, Degree spectra for relations on Boolean algebras, Algebra and Logic, vol. 42 (2003), pp. 105--111.
Mathematical Reviews (MathSciNet): MR2003628
R. G. Downey and C. G. Jockusch, Every low Boolean algebra is isomorphic to a recursive one, Proceedings of the American Mathematical Society, vol. 122 (1994), pp. 871--880.
Mathematical Reviews (MathSciNet): MR1203984
Digital Object Identifier: doi:10.2307/2160766
Zentralblatt MATH: 0820.03019
R. G. Downey and J. F. Knight, Orderings with $\alpha$-th jump degree $\bf 0\sp (\alpha)$, Proceedings of the American Mathematical Society, vol. 114 (1992), pp. 545--552.
Mathematical Reviews (MathSciNet): MR1065942
Digital Object Identifier: doi:10.2307/2159679
Zentralblatt MATH: 0748.03027
S. S. Goncharov, V. S. Harizanov, J. F. Knight, C. McCoy, R. G. Miller, and R. Solomon, Enumerations in computable structure theory, Annals of Pure and Applied Logic, vol. 136 (2005), pp. 219--246.
Mathematical Reviews (MathSciNet): MR2169684
Digital Object Identifier: doi:10.1016/j.apal.2005.02.001
Zentralblatt MATH: 1081.03033
V. S. Harizanov, Uncountable degree spectra, Annals of Pure Applied Logic, vol. 54 (1991), pp. 255--263.
Mathematical Reviews (MathSciNet): MR1133007
Digital Object Identifier: doi:10.1016/0168-0072(91)90049-R
Zentralblatt MATH: 0744.03035
--------, Pure computable model theory, Handbook of recursive mathematics, vol. 1, Studies in Logic and the Foundations of Mathematics, vol. 138, Elsevier, Amsterdam, 1998, pp. 3--114.
Mathematical Reviews (MathSciNet): MR1673621
Zentralblatt MATH: 0952.03037
Digital Object Identifier: doi:10.1016/S0049-237X(98)80002-5
--------, Relations on computable structures, Contemporary mathematics (N. Bokan, editor), University of Belgrade, 2000, pp. 65--81.
D. R. Hirschfeldt, B. Khoussainov, R. A. Shore, and A. M. Slinko, Degree spectra and computable dimensions in algebraic structures, Annals of Pure and Applied Logic, vol. 115 (2002), pp. 71--113.
Mathematical Reviews (MathSciNet): MR1897023
Digital Object Identifier: doi:10.1016/S0168-0072(01)00087-2
Zentralblatt MATH: 1016.03034
W. Hodges, A shorter model theory, Cambridge University Press, Cambridge, 1997.
Mathematical Reviews (MathSciNet): MR1462612
Zentralblatt MATH: 0873.03036
C. G. Jockusch, Jr. and R. I. Soare, Degrees of orderings not isomorphic to recursive linear orderings, Annals of Pure and Applied Logic, vol. 52 (1991), pp. 39--64.
Mathematical Reviews (MathSciNet): MR1104053
Digital Object Identifier: doi:10.1016/0168-0072(91)90038-N
Zentralblatt MATH: 0734.03026
B. Khoussainov and R. A. Shore, Computable isomorphisms, degree spectra of relations, and Scott families, Annals of Pure and Applied Logic, vol. 93 (1998), pp. 153--193.
Mathematical Reviews (MathSciNet): MR1635605
Digital Object Identifier: doi:10.1016/S0168-0072(97)00059-6
Zentralblatt MATH: 0927.03072
--------, Effective model theory: the number of models and their complexity, Models and computability: Invited papers from logic colloquium $'$97 (S.B. Cooper and J.K. Truss, editors), London Mathematical Society Lecture Note Series, vol. 259, Cambridge University Press, Cambridge, 1999, pp. 193--240.
Mathematical Reviews (MathSciNet): MR1721168
Zentralblatt MATH: 0940.03045
J. F. Knight, Degrees coded in jumps of orderings, Journal of Symbolic Logic, vol. 51 (1986), pp. 1034--1042.
Mathematical Reviews (MathSciNet): MR865929
Digital Object Identifier: doi:10.2307/2273915
Project Euclid: euclid.jsl/1183742241
Zentralblatt MATH: 0633.03038
R. Miller, The $\Delta\sp 0\sb 2$-spectrum of a linear order, Journal of Symbolic Logic, vol. 66 (2001), pp. 470--486.
Mathematical Reviews (MathSciNet): MR1833459
Digital Object Identifier: doi:10.2307/2695025
Project Euclid: euclid.jsl/1183746454
Zentralblatt MATH: 0992.03050
M. Moses, Relations intrinsically recursive in linear orders, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 32 (1986), pp. 467--472.
Mathematical Reviews (MathSciNet): MR860036
L. J. Richter, Degrees of structures, Journal of Symbolic Logic, vol. 46 (1981), pp. 723--731.
Mathematical Reviews (MathSciNet): MR641486
Digital Object Identifier: doi:10.2307/2273222
Project Euclid: euclid.jsl/1183740883
Zentralblatt MATH: 0512.03024
R. I. Soare, Recursively enumerable sets and degrees, Springer-Verlag, New York, 1987.
Mathematical Reviews (MathSciNet): MR882921