Metric Semigroups and Groups of Multisets

We investigate the algebraic and topological properties of sets of complex multisets associated with Banach spaces having symmetric bases. We consider algebraic structures on the sets of multisets and compare some natural metrics on the (semi)groups of multisets. Also, we construct nonlinear analogs of the weighted backward shift operator on metric spaces of multisets, establish conditions of topological transitivity, and prove an analog of the Topological Transitivity Criterion for metric semigroups.