  A zig-zag order is like a directed path, only with alternating directions. A
generating set of minimal size for the semigroup of all full transformations on
a finite set preserving the zig-zag order was determined by Fenandes et al. in
2019. This paper deals with generating sets of the semigroup $F_{\mathbb{N}}$
of all full transformations on the set of all natural numbers preserving the
zig-zag order. We prove that $F_{\mathbb{N}}$ has no minimal generating sets
and present two particular infinite decreasing chains of generating sets of
$F_{\mathbb{N}}$.
