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Seminar

Boolean Algebras and Semi-Retractions August 02, 2022 (03:30 PM PDT - 04:15 PM PDT)
Parent Program:
Location: SLMath: Eisenbud Auditorium, Online/Virtual
Speaker(s) Lynn Scow (California State University, San Bernardino)
Description No Description
Keywords and Mathematics Subject Classification (MSC)
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
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.

93849?type=thumb Boolean Algebras And Semi-Retractions 1 395 KB application/pdf

Boolean Algebras And Semi-Retractions