Set Theory and Topology seminar (BGU)
On Wednesday, November 12, 16:45 – 18:30, Room -101 of the Mathematics Department
Speaker: Nadav Meir (BGU)
Title: On products of elementarily indivisible structures
Abstract: A structure M in a first order language L is indivisible if for every coloring of its universe in two colors, there is a monochromatic substructure M’ of M such that M’ is isomorphic to M. Additionally, we say that M is symmetrically indivisible if M’ can be chosen to be symmetrically embedded in M (That is, every automorphism of M’ can be extended to an automorphism of M), and that M is elementarily indivisible if M’ can be chosen to be an elementary substructure.
The notion of indivisibility is a long-studied subject. We will present these strengthenings of the notion, examples and some basic properties. in [HKO] several questions regarding these new notions arose: If M is symmetrically indivisible are all of its reducts to a sublanguage symmetrically indivisible? Is an elementarily indivisible structure necessarily homogeneous? Does elementary indivisibility imply symmetric indivisibility?
We will define a new “product” of structures, generalizing the notions of lexicographic order and lexicographic product of graphs, which preserves indivisibility properties and use it to answer the questions above.
[HKO] Assaf Hasson, Menachem Kojman and Alf Onshuus, On symmetric indivisibility of countable structures, Model Theoretic Methods in Finite Combinatorics, AMS, 2011, pp.417–452.