  Positroids are a family of matroids introduced by Postnikov in the study of
non-negative Grassmannians. Postnikov identified several combinatorial objects
in bijections with positroids, among which are bounded affine permutations. On
the other hand, the notion of essential sets, introduced for permutations by
Fulton, was used by Knutson in the study of the special family of interval rank
positroids. We generalize Fulton's essential sets to bounded affine
permutations. The bijection of the latter with positroids, allows study of the
relationship between them. From the point of view of positroids, essential sets
are maximally dependent cyclic interval. We define connected essential sets and
prove that they give a facet description of the positroid polytope, as well as
equations defining the positroid variety. We define a subset of essential sets,
called core, which contains minimal rank conditions to uniquely recover a
positroid. We provide an algorithm to retrieve the positroid satisfying the
rank conditions in the core or any compatible rank condition on cyclic
intervals.
