Definability in the Infix Order on Words

We develop a theory of (first-order) definability in the infix partial order on words in parallel with a similar theory for the h-quasiorder of finite k-labeled forests. In particular, any element is definable (provided that words of length 1 or 2 are taken as parameters), the first-order theory of the structure is atomic and computably isomorphic to the first-order arithmetic. We also characterize the automorphism group of the structure and show that any arithmetical predicate invariant under the automorphisms of the structure is definable in the structure.

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