  We study equilibrium properties of catalytically-activated $A + A \to
\oslash$ reactions taking place on a lattice of adsorption sites. The particles
undergo continuous exchanges with a reservoir maintained at a constant chemical
potential $\mu$ and react when they appear at the neighbouring sites, provided
that some reactive conditions are fulfilled. We model the latter in two
different ways: In the Model I some fraction $p$ of the {\em bonds} connecting
neighbouring sites possesses special catalytic properties such that any two
$A$s appearing on the sites connected by such a bond instantaneously react and
desorb. In the Model II some fraction $p$ of the adsorption {\em sites}
possesses such properties and neighbouring particles react if at least one of
them resides on a catalytic site. For the case of \textit{annealed} disorder in
the distribution of the catalyst, which is tantamount to the situation when the
reaction may take place at any point on the lattice but happens with a finite
probability $p$, we provide an exact solution for both models for the interior
of an infinitely large Cayley tree - the so-called Bethe lattice. We show that
both models exhibit a rich critical behaviour: For the annealed Model I it is
characterised by a transition into an ordered state and a re-entrant transition
into a disordered phase, which both are continuous. For the annealed Model II,
which represents a rather exotic model of statistical mechanics in which
interactions of any particle with its environment have a peculiar Boolean form,
the transition to an ordered state is always continuous, while the re-entrant
transition into the disordered phase may be either continuous or discontinuous,
depending on the value of $p$.
