  This paper establishes the fundamental properties of the $s$-closures, a
recently introduced family of closure operations on ideals of rings of positive
characteristic. The behavior of the $s$-closure of homogeneous ideals in graded
rings is studied, and criteria are given for when the $s$-closure of an ideal
can be described exactly in terms of its tight closure and rational powers.
Sufficient conditions are established for the weak $s$-closure to equal to the
$s$-closure. A generalization of the Briancon-Skoda theorem is given which
compares any two different $s$-closures applied to powers of the same ideal.
