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Register a new userAutomatic Ordinals
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Olivier Finkel
Publication date
2012
fields
Informatics Engineering
and research's language is
English
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We prove that the injectively omega-tree-automatic ordinals are the ordinals smaller than $omega^{omega^omega}$. Then we show that the injectively $omega^n$-automatic ordinals, where $n>0$ is an integer, are the ordinals smaller than $omega^{omega^n}$. This strengthens a recent result of Schlicht and Stephan who considered in [Schlicht-Stephan11] the subclasses of finite word $omega^n$-automatic ordinals. As a by-product we obtain that the hierarchy of injectively $omega^n$-automatic structures, n>0, which was considered in [Finkel-Todorcevic12], is strict.
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We consider $omega^n$-automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of length $omega^n$ for some integer $ngeq 1$. We show that all these structures are $omega$-tree-automatic structures presentable by Muller or Rabin tree automata. We prove that the isomorphism relation for $omega^2$-automatic (resp. $omega^n$-automatic for $n>2$) boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups) is not determined by the axiomatic system ZFC. We infer from the proof of the above result that the isomorphism problem for $omega^n$-automatic boolean algebras, $n > 1$, (respectively, rings, commutative rings, non commutative rings, non commutative groups) is neither a $Sigma_2^1$-set nor a $Pi_2^1$-set. We obtain that there exist infinitely many $omega^n$-automatic, hence also $omega$-tree-automatic, atomless boolean algebras $B_n$, $ngeq 1$, which are pairwise isomorphic under the continuum hypothesis CH and pairwise non isomorphic under an alternate axiom AT, strengthening a result of [FT10].
An $omega$-tree-automatic structure is a relational structure whose domain and relations are accepted by Muller or Rabin tree automata. We investigate in this paper the isomorphism problem for $omega$-tree-automatic structures. We prove first that the isomorphism relation for $omega$-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n >1) is not determined by the axiomatic system ZFC. Then we prove that the isomorphism problem for $omega$-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n >1) is neither a $Sigma_2^1$-set nor a $Pi_2^1$-set.
We study the topological version of the partition calculus in the setting of countable ordinals. Let $alpha$ and $beta$ be ordinals and let $k$ be a positive integer. We write $betato_{top}(alpha,k)^2$ to mean that, for every red-blue coloring of the collection of 2-sized subsets of $beta$, there is either a red-homogeneous set homeomorphic to $alpha$ or a blue-homogeneous set of size $k$. The least such $beta$ is the topological Ramsey number $R^{top}(alpha,k)$. We prove a topological version of the ErdH{o}s-Milner theorem, namely that $R^{top}(alpha,k)$ is countable whenever $alpha$ is countable. More precisely, we prove that $R^{top}(omega^{omega^beta},k+1)leqomega^{omega^{betacdot k}}$ for all countable ordinals $beta$ and finite $k$. Our proof is modeled on a new easy proof of a weak version of the ErdH{o}s-Milner theorem that may be of independent interest. We also provide more careful upper bounds for certain small values of $alpha$, proving among other results that $R^{top}(omega+1,k+1)=omega^k+1$, $R^{top}(alpha,k)< omega^omega$ whenever $alpha<omega^2$, $R^{top}(omega^2,k)leqomega^omega$ and $R^{top}(omega^2+1,k+2)leqomega^{omegacdot k}+1$ for all finite $k$. Our computations use a variety of techniques, including a topological pigeonhole principle for ordinals, considerations of a tree ordering based on the Cantor normal form of ordinals, and some ultrafilter arguments.
Given a cardinal $kappa$ and a sequence $left(alpha_iright)_{iinkappa}$ of ordinals, we determine the least ordinal $beta$ (when one exists) such that the topological partition relation [betarightarrowleft(top,alpha_iright)^1_{iinkappa}] holds, including an independence result for one class of cases. Here the prefix $top$ means that the homogeneous set must have the correct topology rather than the correct order type. The answer is linked to the non-topological pigeonhole principle of Milner and Rado.
A computable structure $mathcal{A}$ has degree of categoricity $mathbf{d}$ if $mathbf{d}$ is exactly the degree of difficulty of computing isomorphisms between isomorphic computable copies of $mathcal{A}$. Fokina, Kalimullin, and Miller showed that every degree d.c.e. in and above $mathbf{0}^{(n)}$, for any $n < omega$, and also the degree $mathbf{0}^{(omega)}$, are degrees of categoricity. Later, Csima, Franklin, and Shore showed that every degree $mathbf{0}^{(alpha)}$ for any computable ordinal $alpha$, and every degree d.c.e. in and above $mathbf{0}^{(alpha)}$ for any successor ordinal $alpha$, is a degree of categoricity. We show that every degree c.e. in and above $mathbf{0}^{(alpha)}$, for $alpha$ a limit ordinal, is a degree of categoricity. We also show that every degree c.e. in and above $mathbf{0}^{(omega)}$ is the degree of categoricity of a prime model, making progress towards a question of Bazhenov and Marchuk.
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