  In the paper we consider a generalizations of the notion of Poisson process
to the case when classical convolution is replaced by generalized convolution
in the sense of K. Urbanik [16] following two classical definitions of Poisson
process. First, for every generalized convolution $\diamond$ we define
$\diamond$-generalized Poisson process type I as a Markov process with the
$\diamond$-generalized Poisson distribution. Such processes have stationary
independent increments in the sense of generalized convolution, but usually
they do not live on $\mathbb{N}_0$. The $\diamond$-generalized Poisson process
type II is defined as a renewal process based on the sequence $S_n$, which is a
Markov process with the step with the lack of memory property. Such processes
take values in $\mathbb{N}_0$, however they do not have to be Markov processes,
do not have to have independent increments, even in generalized convolution
sense. It turns out that the second construction is possible only for monotonic
generalized convolutions which admit the existence of distributions with lack
of memory, thus we also study these properties.
