Nadav Meir: Almost eliminating quantifiers in compositions and applications to elementary indivisibility

BGU logic seminar.
Nadav Meir will continue his talk from last week.

When: Dec. 30, 10h30

Where: building 72 classroom 123

Speaker: Nadav Meir (BGU)

Title: Almost eliminating quantifiers in compositions and applications to elementary indivisibility.
Abstract:
For a n L-structure M , a substructure M is symmetrically embedded if every automorphism of N can be extended to an automorphism of M.
We say M is symmetrically indivisible if for every colouring of it’s universe in 2 colors, there is a symmetrically embedded monochromatic substructure M ‘⊂ M such that M ‘ ≅ M .
We say M is elementarily indivisible if for every colouring of it’s universe in 2 colors, there is a monochromatic elementary substructure M ‘ ⊂ M such that M ‘ ≅ M .
 
In [HKO11] the following questions were presented:
Is there a rigid elementarily indivisible structure? Is an elementarily indivisible structure homogeneous?
Does elementary indivisibility imply symmetric indivisibility?
In this talk we will present a general notion of compositions of two structures M and N in a relational language that is a generalization of the known notion of lexicographic order if M and N are posets, or the lexicographic product of graphs if M and N are graphs.
In general, this construction need not be eliminating quantifiers, but we will see how, assuming $\cM$ and $\cN$ eliminate quantifiers, the composition “almost” eliminates quantifiers.
We will use this to get a general result regarding elementarily indivisible structures and answer the two questions above.
[HKO11] Assaf Hasson, Menachem Kojman and Alf Onshuus. On symmetric indivisibility of countable structures. Available at http://www.math.bgu.ac.il/~hassonas/papers/final.pdf


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