# Mathematical Sciences Research Institute

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# Seminar

Boolean Algebras and Semi-Retractions August 02, 2022 (03:30 PM PDT - 04:15 PM PDT)
Parent Program: Definability, Decidability, and Computability in Number Theory, part 2 MSRI: Simons Auditorium, Online/Virtual
Speaker(s) Lynn Scow (California State University, San Bernardino)
Description No Description
Video

#### Boolean Algebras And Semi-Retractions

Abstract/Media

Say that an injection $f:\mathcal{A} \rightarrow \mathcal{B}$ is \emph{quantifier-free type-respecting} if finite tuples from $\mathcal{A}$ that share the same quantifier-free type in $\mathcal{A}$ are mapped by $f$ to tuples in $\mathcal{B}$ that share the same quantifier-free type in $\mathcal{B}$. For structures $\mathcal{A}$ and $\mathcal{B}$ in possibly different languages we say that \emph{$\mathcal{A}$ is a semi-retract of $\mathcal{B}$} if there are quantifier-free type-respecting injections $g: \mathcal{A} \rightarrow\mathcal{B}$ and $f: \mathcal{B} \rightarrow \mathcal{A}$ such that $f \circ g : \mathcal{A} \rightarrow \mathcal{A}$ is an embedding.  Quantifier-free interdefinable structures are an example of a pair of semi-retracts.

In joint work with Dana Barto\v{s}ov\'a, we showed that the Ramsey property transfers to semi-retracts $\mathcal{A}$ of structures $\mathcal{B}$ with the Ramsey property under certain assumptions -- that $\mathcal{A}$ is locally finite and that the age of $\mathcal{B}$ consists of rigid structures. One key example is the case where $\mathcal{B}$ is the class of all finite Boolean algebras with natural orders and $\mathcal{A}$ is the class of all finite ordered simple graphs. Counterexamples of an algebraic nature show that these assumptions are necessary to make the transfer work,  which I will highlight in my talk.

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