  Conditionals play a key role in different areas of logic and probabilistic
reasoning, and they have been studied and formalized from different angles. In
this paper we focus on the de Finetti's notion of conditional as a three-valued
object, with betting-based semantics, and its related approach as random
quantity as mainly developed by two of the authors. Compound conditionals have
been studied in the literature, but not in full generality. In this paper we
provide a natural procedure to explicitly attach conditional random quantities
to arbitrary compound conditionals that also allows us to compute their
previsions. By studying the properties of these random quantities, we show
that, in fact, the set of compound conditionals can be endowed with a Boolean
algebraic structure. In doing so, we pave the way to build a bridge between the
long standing tradition of three-valued conditionals and a more recent proposal
of looking at conditionals as elements from suitable Boolean algebras.
