Strong representations of the polycyclic inverse monoids: Cycles and atoms

This paper was inspired by a monograph by Bratteli and Jorgensen, and the work of Kawamura. We introduce two new semigroups: a wide inverse submonoid of the polycyclic inverse monoid, called the gauge inverse monoid, and a Zappa-Szép product of an arbitrary free monoid with the free monoid on one generator. Both these monoids play an important role in studying arbitrary, not necessarily transitive, strong actions of polycyclic inverse monoids. As a special case of such actions, we obtain some new results concerning the strong actions of P ₂ on ℤ determined by the choice of one positive odd number. We explain the role played by Lyndon words in characterising these repesentations and show that the structure of the representation can be explained by studying the binary representations of the numbers $\frac{1} {p},\frac{2} {p}, \ldots \frac{{p - 1}} {p}$\frac{1} {p},\frac{2} {p}, \ldots \frac{{p - 1}} {p}. We also raise some questions about strong representations of the polycyclic monoids on free abelian groups.