## Franklin's Notes

### Condensation

In order theory, a condensation (also called a splitting) of a linearly ordered set $A$ consists of a set of nonempty intervals $I_j$ indexed by $j\in J$ such that so that the intervals $I_j$ partition $A$. The intervals $I_j$ are ordered by letting $I_j < I_{j'}$ if and only if $j<j'$ in $J$, so that $J$ is isomorphic to the condensation in question. Every condensation is induced by a surjective monotone function $f:A\to J$, and vice versa. If there exists such a function $A\to J$, we say that $J$ is a homomorphic image of $A$ and equivalently write $J\precsim A$. (A monotone function $A\to J$ is also called a homomorphism , so the only additional requirement for $J$ to be a homomorphic image of $A$ is that the homomorphism $A\to J$ be surjective.)

If ${I_j}{j\in J}$ comprises a condensation of $A$, then there is a function $c:A\to {I_j}{j\in J}$ mapping each element of $A$ to the interval containing it. Conversely, we can specify a condensation by defining a mapping $c:A\to {I_j}_{j\in J}$ sending each element of $A$ to an interval, so long as we verify that $c$ is such that $c(a)=c(a')$ whenever $a\in c(a')$. Here are a couple of examples:

• The finite condensation is specified by letting $c_F(x)$ be the interval consisting of all $y$ such that there are finitely many elements between $x$ and $y$.

• If $\kappa$ is an infinite cardinal such that $\kappa'+\kappa=\kappa$ for all $\kappa'\leq \kappa$, we can define a $\kappa$-condensation by letting $c_\kappa(x)$ be the interval consisting of all $y$ such that the set of elements between $x$ and $y$ has cardinality $\kappa$. For instance, $c_{\aleph_0}$ defines the countable condensation.

• The well-ordering condensation is specified by letting $c_W(x)$ be the interval consisting of all $y$ such that the interval of elements between $x$ and $y$ is well-ordered .