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Computably categorical structures and expansions by constants. (English) Zbl 0928.03040
It is an easy consequence of the Ryll-Nardzewski theorem that if the theory of an arbitrary structure $${\mathcal{A}}$$ is countably categorical then so is the theory of any expansion of $${\mathcal{A}}$$ by finitely many constants. The problem is to consider the situation for effective model theory. {T. Millar} [J. Symb. Log. 51, No. 2, 430-434 (1986; Zbl 0631.03018)] proved the analogous result (with an additional assumption on $${\mathcal{A}}$$) for the notion of computable (=recursive) categoricity, i.e. computable categoricity is preserved under the above mentioned expansions in case the existential theory of a computable (=recursive) structure $${\mathcal{A}}$$ is decidable. Here, the authors solve the general problem negatively. More precisely, it is proved that for each $$k\in\omega$$ there is a computably categorical structure $${\mathcal{A}}$$ whose expansion obtained by adding a constant naming any element of $${\mathcal{A}}$$ has dimension $$k$$, where the dimension of a structure $${\mathcal{A}}$$ is the number of computable isomorphism types of computable structures (classically) isomorphic to $${\mathcal{A}}$$.

##### MSC:
 03C57 Computable structure theory, computable model theory 03C35 Categoricity and completeness of theories 03D45 Theory of numerations, effectively presented structures
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##### References:
 [1] Philosophical Transactions of the Royal Society of London 248 pp 407–432– (1956) [2] Algebra i Logika 19 pp 621–639– (1980) [3] Aspects of effective algebra (1979) [4] Logical methods pp 1–91– (1993) [5] Transactions of the American Mathematical Society 95 pp 341–360– (1960) [6] Recursive categoricity and persistence 51 pp 430–434– (1986) [7] Handbook of recursion theory [8] Uspekhi Matemematischeskoe Nauk 16 pp 3–60– (1961) [9] Comptes Rendus de l’Académic des Sciences, Paris 240 pp 151–153– (1955) [10] Izvestiya kademiya Nauk Kazakhstan SSR 1 pp 83–94– (1974) [11] Annals of Pure and Applied Logic [12] Recursively presentable prime models 39 pp 305–309– (1974) [13] Annals of Pure and Applies Logic 60 pp 1–30– (1993) [14] Handbook of recursive mathematics · Zbl 0930.03037 [15] Fundamenta Mathematicae 44 pp 61–71– (1957) [16] Logic notebook: problems in mathematical logic (1986)
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