A Useful Undecidable Theory

We show that many so called discrete weak semilatticesconsidered earlier in a series of author's publications havehereditary undecidable first-order theories. Since such structuresappear naturally in some parts of computability theory, we obtainseveral new undecidability results. This applies e.g. to thestructures of complete numberings, of m-degrees of indexsets and of the Wadge degrees of partitions in the Baire space andω-algebraic domains.