  We study the behavior of the chemical reactions $A+A\to A+S$ and $A+A\to S+S$
(where the reactive species $A$ and the inert species $S$ are both assumed to
be immobile) embedded on Bethe lattices of arbitrary coordination number $z$
and on a two-dimensional (2D) square lattice. For the Bethe lattice case, exact
solutions for the coverage in the $A$ species in terms of the initial condition
are obtained. In particular, our results hold for the important case of an
infinite one-dimensional (1D) lattice ($z=2$). The method is based on an
expansion in terms of conditional probabilities which exploits a Markovian
property of these systems. Along the same lines, an approximate solution for
the case of a 2D square lattice is developed. The effect of dilution in a
random initial condition is discussed in detail, both for the lattice coverage
and for the spatial distribution of reactants.
