  The standard Artin approximation (AP) is not readily applicable to the
left-right equivalence of morphisms Maps((k^n,o),(k^m,o)). Moreover, the
na\"ive extension does not hold in the k-analytic case, because of
Osgood-Gabrielov-Shiota examples.
  The left-right version of Artin approximation (LRAP) was established by M.
Shiota for morphisms that are either Nash or [real-analytic and of finite
singularity type]. We establish LRAP and its stronger version of P{\l}oski
(LRAPP) for Maps(X,Y) where X,Y are k-analytic/k-Nash germs of schemes of any
characteristic.
  More precisely:
  *(k-algebraic power series, k<x>/J_X and k<y>/J_Y) LRAP, LRAPP, the inverse
Artin approximation (and its P{\l}oski's version) hold for any morphism.
  *(k-analytic power series, k{x}/J_X and k{y}/J_Y) LRAP, LRAPP, the inverse
Artin approximation (and its P{\l}oski's version) hold for finite morphisms.
  * LRAP holds for analytic morphisms of weakly-finite singularity type.
  This later class of morphisms of ``weakly-finite singularity type" (which we
introduce) is of separate importance. It extends naturally the class of
morphisms of ``finite singularity type", while preserving their
non-pathological behaviour. The definition goes via the higher critical loci of
morphisms with singular target. (The critical locus Crit_X f is defined via the
Zariski-Lipman sequence.)
  We establish basic properties of these critical loci. In particular: any map
is finitely (right) determined by its higher singular loci.
