  We establish elements of a new approch to ellipticity and parametrices within
operator algebras on a manifold with higher singularities, only based on some
general axiomatic requirements on parameter-dependent operators in suitable
scales of spaces. The idea is to model an iterative process with new
generations of parameter-dependent operator theories, together with new scales
of spaces that satisfy analogous requirements as the original ones, now on a
corresponding higher level.
  The "full" calculus is voluminous; so we content ourselves here with some
typical aspects such as symbols in terms of order reducing families, classes of
relevant examples, and operators near a corner point.
