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Computable shuffle sums of ordinals. (English) Zbl 1149.03034
The shuffle sum $$\sigma (S)$$ of a countable set of linear orders $$S = \{ L_i \} _{i \in \omega}$$ is the unique linear order obtained by partitioning the rationals into dense sets $$\{Q_i \}_{i \in \omega}$$ and replacing each rational in $$Q_i$$ by $$L_i$$.
The main result of this paper characterizes those $$S \subset \omega+1$$ for which $$\sigma (S)$$ is computable in terms of limit infimum sets (LimInf sets) and limitwise monotonic sets relative to $$0^\prime$$ (LimMon($$0^\prime$$) sets). In particular, the author proves that for $$S \subset \omega +1$$, “$$\sigma (S)$$ is computable” is equivalent to “$$S$$ is a LimInf set” and also equivalent to “$$S$$ is a LimMon($$0^\prime$$) set.”
This can be applied to give a succinct proof that if $$S \subset \omega$$ is $$\Sigma^0_3$$, then $$\sigma (S \cup \{ \omega \} )$$ is computable, a result originally proved by C. J. Ash, C. G. Jockusch, and J. F. Knight [“Jumps of orderings”, Trans. Am. Math. Soc. 319, No. 2, 573–599 (1990; Zbl 0705.03022)]. More about LimMon($$0^\prime$$) sets can be found (for example) in [D. R. Hirschfeldt, “Prime models of theories of computable linear orderings”, Proc. Am. Math. Soc. 129, No. 10, 3079–3083 (2001; Zbl 0974.03040)].

##### MSC:
 03D45 Theory of numerations, effectively presented structures
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##### References:
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