We prove that the homomorphic quasiorder of finite k-labeled forests has undecidable elementary theory for k ≥3, in contrast to the known decidability result for k=2. We establish also undecidablity (again for every k ≥3) of elementary theories of two other relevant structures: the homomorphic quasiorder of finite k-labeled trees, and of finite k-labeled trees with a fixed label of the root element.
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