Classifying locally compact semitopological polycyclic monoids

We present a complete classification of Hausdorff locally compact polycyclic monoids up to a topological isomorphism. A {\em polycyclic monoid} is an inverse monoid with zero, generated by a subset $\Lambda$ such that $xx^{-1}=1$ for any $x\in\Lambda$ and $xy^{-1}=0$ for any distinct $x,y\in\Lambda$. We prove that any non-discrete Hausdorff locally compact topology with continuous shifts on a polycyclic monoid $M$ coincides with the topology of one-point compactification of the discrete space $M\setminus\{0\}$.