  We prove that the fully asynchronous dynamics of a Boolean network
$f:\{0,1\}^n\to\{0,1\}^n$ without negative loop can be simulated, in a very
specific way, by a monotone Boolean network with $2n$ components. We then use
this result to prove that, for every even $n$, there exists a monotone Boolean
network $f:\{0,1\}^n\to\{0,1\}^n$, an initial configuration $x$ and a fixed
point $y$ of $f$ such that: (i) $y$ can be reached from $x$ with a fully
asynchronous updating strategy, and (ii) all such strategies contains at least
$2^{\frac{n}{2}}$ updates. This contrasts with the following known property: if
$f:\{0,1\}^n\to\{0,1\}^n$ is monotone, then, for every initial configuration
$x$, there exists a fixed point $y$ such that $y$ can be reached from $x$ with
a fully asynchronous strategy that contains at most $n$ updates.

Discrete Mathematics;Combinatorics