Patrick Speissegger
Let ${\cal M}$ be an o-minimal expansion of a dense linear order $(M,\lt)$. Let $f:I \longrightarrow M$ be definable, with $I = (a,b)$ an interval in $M$. We start by translating the condition “$f$ is strictly increasing” into a condition involving two subsets of $I^2$: on the one hand, we have the triangle $\displaystyle \Delta(I):=…
Let ${\cal M}$ be an o-minimal expansion of a dense linear order $(M,\lt)$. Let $f:I \longrightarrow M$ be definable, with $I = (a,b)$ an interval in $M$. Definition We call $f$ strictly monotone if $f$ is either constant, or strictly increasing, or strictly decreasing. Our first goal is to prove the Monotonicity Theorem There are…
Let ${\cal M}$ be an o-minimal expansion of a dense linear order $(M,\lt)$. The first big question about o-minimality is the following: is o-minimality an elementary property, that is, given ${\cal N} \equiv {\cal M}$, is ${\cal N}$ necessarily o-minimal? Lemma The following are equivalent: every ${\cal N} \equiv {\cal M}$ is o-minimal; for every…
Let ${\cal M}$ be an expansion of a dense linear order $(M,\lt)$. We call ${\cal M}$ o-minimal if every definable subset of $M$ is a finite union of points and intervals. Examples (without details) By quantifier elimination, every dense linear order without endpoints is o-minimal. Let ${\cal V} = (V,\lt,+,(\lambda_k)_{k \in K})$ be an ordered…
Let ${\cal M}$ be an expansion of a dense linear order $(M,