  These notes present a compact and self-contained approach to iterated forcing
with a particular emphasis on semiproper forcing. We tried to make our
presentation accessible to any scholar who has some familiarity with forcing
and boolean valued models and full details of all proofs are given.
  We focus our presentation using the boolean algebra language and defining an
iteration system as a directed and commutative system of complete and injective
homomorphisms between complete and atomless boolean algebras.
  It is well known that the boolean algebra approach to forcing and iterations
is fully equivalent to the standard one. While there are several monographs
where forcing is introduced by means of boolean valued models, to our knowledge
no detailed account of iterated forcing following a boolean algebraic approach
has yet appeared. We believe that this different approach is fruitful since the
richness of the algebraic language simplifies many calculations and
definitions, among which that of RCS-limits. Some of the advantages of this
approach have been already outlined by Donder and Fuchs in
http://arxiv.org/abs/math/9207204.
  The first part of these notes present the general framework needed to develop
the notion of limit of an iterated system of forcings in the boolean algebraic
language. The second part contains a proof of the main result of Shelah on
semiproper iterations, i.e. that RCS-limit of semiproper iterations are
semiproper.
