  We describe some connections between three different fields: combinatorics
(umbral calculus), functional analysis (linear functionals and operators) and
harmonic analysis (convolutions on group-like structures). Systematic usage of
cancellative semigroup, their convolution algebras, and tokens between them
provides a common language for description of objects from these three fields.
  Keywords: cancellative semigroups, umbral calculus, harmonic analysis, token,
convolution algebra, integral transform
