Compressing and summarizing. Part 4.

[Disclaimer: this a blog-post consciously written in the style adumbrated in Compressing and summarizing. Part 0. For the original, see Compression ratio here is high. ]

This reports on this recent preprint by Siddharth Baskar and Alex Kruckman.

  • Loc. cit. proves that for any purely relational signature \Sigma, and for any class \mathbb{K} consisting of finite \Sigma-structures,

\mathbb{K} has the strict order property

if and only if

\mathbb{K} uniformly interprets the natural numbers.

  • I recall that a logical formula \psi having the strict order property relative to a theory \mathbb{T} means that for every model \mathfrak{A} of \psi, the poset P := \{ (a_i,a_j)|  \models\exists a\in \lvert \mathfrak{A}\rvert,\, (  \neg\psi(a,a_i)\wedge\psi(a,a_j)     )                                    \} contains arbitrarily long finite chains.

(to be continued)

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