John Clemens: Relative primeness of equivalence relations

Wednesday, November 11 from 3 to 4pm
Room: MP 207
Speaker: John Clemens (BSU)

Title: Relative primeness of equivalence relations

Abstract: Let $E$ and $F$ be equivalence relations on the spaces $X$ and $Y$. We say that $E$ is prime to $F$ if: whenever $\varphi: X \rightarrow Y$ is a homomorphism from $E$ to $F$, there is a continuous embedding $\rho$ from $E$ to itself so that the range of $\varphi \circ \rho$ is contained in a single $F$ class. That is to say, $\varphi$ is constant (up to $F$-equivalence) on a set on which $E$ maintains its full complexity with respect to Borel reducibility. When $E$ is prime to $F$, $E$ fails to be Borel-reducible to $F$ in a very strong way. I will discuss this notion and show that many non-reducibility results in the theory of Borel equivalence relations can be strengthened to produce primeness results. I will also discuss the possibility of new types of dichotomies involving the notion of primeness.