Date of Degree
Algebra | Logic and Foundations
Diophantine Problem, bi-interpretability, first-order logic, metabelian groups, Baumslag-Solitar group, wreath products, free monoid
We first show that arithmetic is bi-interpretable (with parameters) with the free monoid and with partially commutative monoids with trivial center. This bi-interpretability implies that these monoids have the QFA property and that finitely generated submonoids of these monoids are definable. Moreover, we show that any recursively enumerable language in a finite alphabet X with two or more generators is definable in the free monoid. We also show that for metabelian Baumslag-Solitar groups and for a family of metabelian restricted wreath products, the Diophantine Problem is decidable. That is, we provide an algorithm that decides whether or not a given system of equations in these groups has a solution.
Lopez Cruz, Laura M., "Model Theory of Groups and Monoids" (2020). CUNY Academic Works.