  Let $G$ be a countable cancellative amenable semigroup and let $(F_n)$ be a
(left) F{\o}lner sequence in $G$. We introduce the notion of an $(F_n)$-normal
element of $\{0,1\}^G$. When $G$ = $(\mathbb N,+)$ and $F_n = \{1,2,...,n\}$,
the $(F_n)$-normality coincides with the classical notion. We prove that:
  $\bullet$ If $(F_n)$ is a F{\o}lner sequence in $G$, such that for every
$\alpha\in(0,1)$ we have $\sum_n \alpha^{|F_n|}<\infty$, then almost every
$x\in\{0,1\}^G$ is $(F_n)$-normal.
  $\bullet$ For any F{\o}lner sequence $(F_n)$ in $G$, there exists an
Cham\-per\-nowne-like $(F_n)$-normal set.
  $\bullet$ There is a natural class of "nice" F{\o}lner sequences in $(\mathbb
N,\times)$. There exists a Champernowne-like set which is $(F_n)$-normal for
every nice F{\o}lner \sq.
  $\bullet$ Let $A\subset\mathbb N$ be a classical normal set. Then, for any
F{\o}lner sequence $(K_n)$ in $(\mathbb N,\times)$ there exists a set $E$ of
$(K_n)$-density $1$, such that for any finite subset
$\{n_1,n_2,\dots,n_k\}\subset E$, the intersection $A/{n_1}\cap
A/{n_2}\cap\ldots\cap A/{n_k}$ has positive upper density in $(\mathbb N,+)$.
As a consequence, $A$ contains arbitrarily long geometric progressions, and,
more generally, arbitrarily long "geo-arithmetic" configurations of the form
$\{a(b+ic)^j,0\le i,j\le k\}$.
  $\bullet$ For any F{\o}lner \sq\ $(F_n)$ in $(\mathbb N,+)$ there exist
uncountably many $(F_n)$-normal Liouville numbers.
  $\bullet$ For any nice F{\o}lner sequence $(F_n)$ in $(\mathbb N,\times)$
there exist uncountably many $(F_n)$-normal Liouville numbers.
