  The algebraic monoid structure of an incidence algebra is investigated. We
show that the multiplicative structure alone determines the algebra
automorphisms of the incidence algebra. We present a formula that expresses the
complexity of the incidence monoid with respect to the two sided action of its
maximal torus in terms of the zeta polynomial of the poset. In addition, we
characterize the finite (connected) posets whose incidence monoids have
complexity $\leq 1$. Finally, we determine the covering relations of the
adherence order on the incidence monoid of a star poset.
