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A computably categorical structure whose expansion by a constant has infinite computable dimension. (English) Zbl 1055.03026
Summary: P. Cholak, S. Goncharov, B. Khoussainov, and R. A. Shore [ibid. 64, 13–37 (1999; Zbl 0928.03040)] showed that for each $$k>0$$ there is a computably categorical structure whose expansion by a constant has computable dimension $$k$$. We show that the same is true with $$k$$ replaced by $$\omega$$. Our proof uses a version of Goncharov’s method of left and right operations.

##### MSC:
 03C57 Computable structure theory, computable model theory
##### Keywords:
computably categorical structure; computable dimension
Zbl 0928.03040
Full Text:
##### References:
 [1] P. Cholak, S. S. Goncharov, B. Khoussainov, and R. A. Shore Computably categorical structures and expansions by constants , Journal of Symbolic Logic, vol. 64 (1999), pp. 13–37. JSTOR: · Zbl 0928.03040 [2] Y. L. Ershov, S. S. Goncharov, A. Nerode, and J. B. Remmel (editors) Handbook of recursive mathematics , Studies in Logic and the Foundations of Mathematics, vol. 138–139, Elsevier Science, Amsterdam,1998. · Zbl 0930.03037 [3] S. S. Goncharov Computable single-valued numerations , Algebra and Logic , vol. 19 (1980), pp. 325–356. · Zbl 0514.03029 [4] V. S. Harizanov Pure computable model theory , in Ershov et al. [?], pp. 3–114. · Zbl 0952.03037 [5] D. R. Hirschfeldt Degree spectra of intrinsically c. e. relations , Journal of Symbolic Logic, vol. 66 (2001), pp. 441–469. JSTOR: · Zbl 0988.03065 [6] D. R. Hirschfeldt, B. Khoussainov, R. A. Shore, and A. M. Slinko Degree spectra and computable dimension in algebraic structures , Annals of Pure and Applied Logic , vol. 115 (2002), pp. 71–113. · Zbl 1016.03034 [7] 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. · Zbl 0927.03072 [8] T. Millar Recursive categoricity and persistence , Journal of Symbolic Logic, vol. 51 (1986), pp. 430–434. JSTOR: · Zbl 0631.03018 [9] R. I. Soare Recursively enumerable sets and degrees , Perspectives in Mathematical Logic, Springer-Verlag, Heidelberg,1987.
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