The countable condensation on linear orders

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 $L\mathrm{/}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\sim }_{\omega }\cong 1$. 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 $(ML)\mathrm{/}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\sim }_{\omega }$ (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.