On cross-sections of partial wreath product of inverse semigroups

We classify R - and L -cross-sections of partial wreath product of inverse semigroups. As a corollary, we get the description of R - and L -cross-sections of the semigroup PAut T of partial automorphisms of finite regular rooted tree T and compute also the number of different R - ( L -) cross-sections in this semigroup.