May 10
Roman Kossak,
CUNY
The lattice problem for models of arithmetic
The lattice problem for models of PA is to determine which lattices can be represented either as lattices of elementary substructures of a model of PA or, more generally, which can be represented as lattices of elementary substructures of a model N that contain a given elementary substructure M of N.
Since the 1970's, the problem generated much research with highly nontrivial results with proofs combining specific methods in the model theory of arithmetic with lattice theory and various combinatorial theorems. The problem has a definite answer in the case of distributive lattices, and, despite much effort, there are still many open questions in the nondistributive case. I will briefly survey some early results and present a few proofs that illustrate the difference between the distributive and nondistributive cases.