  Partial dynamical systems (X,alpha) arise naturally when dealing with
commutative C*-dynamical system (A,delta). We associate with every pair
(X,alpha), or (A,delta), a covariance C*-algebra C*(X,alpha)=C*(A,delta) which
agrees with a partial crossed product - in case alpha is injective, and a
crossed product by a monomorphism - in case alpha is onto. The relevance
between (X,alpha) and C*(X,alpha) is deeply investigated. In particular, the
notions of topological freedom and invariance of a set are generalized, and as
a consequence a version of Isomorphism Theorem and a description of ideals of
C*(X,alpha) are obtained.
