On the strength of some topological lattices

Marcus Tressl (University of Manchester)

Frank Adams 1,

I will be looking into model theoretic properties of lattices (in the order theoretic sense) occurring naturally in topology, such as closed
sets of topological spaces and variations hereof (piecewise linear sets, or convex sets, or curves in affine space).

Given such a lattice \(L\), we are interested in the strength of the first order structure \( (L,\le) \), i.e., we want to know what is interpretable in
the partially ordered set (L,≤). There are at least two classical results to be named here:

-- The first one is Grzegorczyk's work on 'Undecidability of some topological theories', where a topological Robinson arithmetic is created.
-- The second classics in this context is from Staudt (~1850), saying (in model theoretic terms) that for each field K, the lattice of
subvectorspaces of \( K^3 \) is bi-interpretable with the field. So the theme has contact with classical questions on coordinatization.

Besides coordinatization, the upshot is (roughly):
-- If the lattice can be represented on a 1-dimensional space, or, if L has few connected elements, then it is interpretable in the monadic
second order logic of total orders, which is known to be tame by Rabin and Shelah.
-- If there is no such representation and L has many connected elements, then a significant fragment of polyadic second order logic of total
orders is definable in the lattice and L interprets the ring of integers.
(The logical terms will be explained in the talk.)

As a concrete example: the lattice of convex polytopes in the euclidean plane, interprets the real field expanded by a predicate for
integers, which is commonly seen as the wildest first order structure.

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