  We study algebraic structure of the $\lambda$-polycyclic monoid $P_{\lambda}$
and its topologizations. We show that the $\lambda$-polycyclic monoid for an
infinite cardinal $\lambda\geqslant 2$ has similar algebraic properties so has
the polycyclic monoid $P_n$ with finitely many $n\geqslant 2$ generators. In
particular we prove that for every infinite cardinal $\lambda$ the polycyclic
monoid $P_{\lambda}$ is a congruence-free combinatorial $0$-bisimple
$0$-$E$-unitary inverse semigroup. Also we show that every non-zero element $x$
is an isolated point in $(P_{\lambda},\tau)$ for every Hausdorff topology
$\tau$ on $P_{\lambda}$, such that $(P_{\lambda},\tau)$ is a semitopological
semigroup, and every locally compact Hausdorff semigroup topology on
$P_\lambda$ is discrete. The last statement extends results of the paper [33]
obtaining for topological inverse graph semigroups. We describe all feebly
compact topologies $\tau$ on $P_{\lambda}$ such that
$\left(P_{\lambda},\tau\right)$ is a semitopological semigroup and its Bohr
compactification as a topological semigroup. We prove that for every cardinal
$\lambda\geqslant 2$ any continuous homomorphism from a topological semigroup
$P_\lambda$ into an arbitrary countably compact topological semigroup is
annihilating and there exists no a Hausdorff feebly compact topological
semigroup which contains $P_{\lambda}$ as a dense subsemigroup.
