Algebraic structures arising from the finite condensation on linear orders

The finite condensation $\sim_F$ is an equivalence relation defined on a linear order $L$ by $x \sim_F y$ if and only if the set of points lying between $x$ and $y$ is finite. We define an operation $\cdot_F$ on linear orders $L$ and $M$ by $L \cdot_F M = \operatorname{o.t.}\left((LM)/\!\sim_F\right)$; that is, $L \cdot_F M$ is the order type of the lexicographic product of $L$ and $M$ modulo the finite condensation. The infinite order types $L$ such that $L / \! \sim_F\, \cong 1$ are $\omega, \omega^*,$ and $\zeta$ (where $\omega^*$ is the reverse ordering of $\omega$, and $\zeta$ is the order type of $\mathbb{Z}$). We show that under the operation $\cdot_F$, the set $R=\{1, \omega, \omega^*, \zeta\}$ forms a left regular band. Further, each of the ordinal elements of $R$ defines, via left or right multiplication modulo the finite condensation, a weakly order-preserving map on the class of ordinals. We study these maps' effect on the ordinals of finite degree in Cantor normal form. In particular, we examine the extent to which one of these maps, sending $\alpha$ to the order type of $\alpha$ modulo the finite condensation, behaves similarly to a derivative operator on the ordinals of finite degree in Cantor normal form.