The countable condensation on linear orders

Abstract

The countable condensation on a linear order $L$ is the equivalence relation $\sim_\omega$ defined by declaring $x \sim_\omega y$ when the set of points between $x$ and $y$ is countable. We characterize the linear orders $L$ that condense to $1$ under the countable condensation by constructing a linear order $U$ that is universal for the order types $L$ such that $L/\!\!\sim_\omega\, \cong 1$. We define a multiplication operation $\cdot_\omega$ on the class of linear orders by setting $M \cdot_\omega L$ to be the order type of $(ML)/\!\!\sim_\omega$ (where $ML$ denotes the lexicographic product), and show that the right identities for $\cdot_\omega$ are exactly the uncountable suborders of $U$. The order types of these uncountable suborders of $U$ form a left regular band under $\cdot_\omega$, and the order types of all suborders of $U$ form a semigroup.