Lynn Scow (California State University, San Bernardino)

Semi-retractions and preservation of the Ramsey property

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For structures $A$ and $B$ in possibly different languages we define what it means for $A$ to be a semi-retraction of $B$. An injection $f:A \rightarrow B$ is quantifier-free type respecting if tuples from $A$ that share the same quantifier-free type in $A$ are mapped by $f$ to tuples in $B$ that share the same quantifier-free type in $B$. We say that $A$is a semi-retraction of$B$ if there are quantifier-free type respecting injections $g: A \rightarrow B$ and $f: B \rightarrow A$ such that $f \circ g : A \rightarrow A$ is an embedding.

We will talk about examples of semi-retractions and give conditions for when the Ramsey property for (the age of) $B$ is inherited by a semi-retraction $A$ of $B$.