  We exhibit a Hamel basis for the concrete $*$-algebra $\mathfrak{M}_o$
associated to monotone commutation relations realised on the monotone Fock
space, mainly composed by Wick ordered words of annihilators and creators. We
apply such a result to investigate spreadability and exchangeability of the
stochastic processes arising from such commutation relations. In particular, we
show that spreadability comes from a monoidal action implementing a dissipative
dynamics on the norm closure $C^*$-algebra $\mathfrak{M} =
\overline{\mathfrak{M}_o}$. Moreover, we determine the structure of spreadable
and exchangeable monotone stochastic processes using their correspondence with
sp\-reading invariant and symmetric monotone states, respectively.
