  A countable family of $*$-commuting surjective, non-injective local
homeomorphisms of a compact Hausdorff space $X$ gives rise to an action
$\theta$ of a countably generated, free abelian monoid $P$. For such a triple
$(X,P,\theta)$, which we call an irreversible $*$-commutative dynamical system,
we construct a universal $C^*$-algebra $\mathcal{O}[X,P,\theta]$. Within this
setting we show that the following four conditions are equivalent:
$(X,P,\theta)$ is topologically free, $C(X) \subset \mathcal{O}[X,P,\theta]$
has the ideal intersection property, the natural representation of
$\mathcal{O}[X,P,\theta]$ on $\ell^2(X)$ is faithful, and $C(X)$ is a masa in
$\mathcal{O}[X,P,\theta]$. As an application, we characterise simplicity of
$\mathcal{O}[X,P,\theta]$ by minimality of $(X,P,\theta)$. We also show that
$\mathcal{O}[X,P,\theta]$ is isomorphic to the Cuntz-Nica-Pimsner algebra of a
product system of Hilbert bimodules naturally associated to $(X,P,\theta)$.
Moreover, we find a close connection between $*$-commutativity and independence
of group endomorphisms, a notion introduced by Cuntz and Vershik. This leads to
the observation that, for commutative irreversible algebraic dynamical systems
of finite type $(G,P,\theta)$, the dual model $(\hat{G},P,\hat{\theta})$ is an
irreversible $*$-commutative dynamical system and
$\mathcal{O}[\hat{G},P,\hat{\theta}]$ is canonically isomorphic to
$\mathcal{O}[G,P,\theta]$. This allows us to conclude that minimality of
$(G,P,\theta)$ is not only sufficient, but also necessary for simplicity of
$\mathcal{O}[G,P,\theta]$ if $(G,P,\theta)$ is commutative and of finite type.
