  Polynomial reduction is one of the main tools in computational algebra with
innumerable applications in many areas, both pure and applied. Since many years
both the theory and an efficient design of the related algorithm have been
solidly established.
  This paper presents a general definition of polynomial reduction structure,
studies its features and highlights the aspects needed in order to grant and to
efficiently test the main properties (noetherianity, confluence, ideal
membership).
  The most significant aspect of this analysis is a negative reappraisal of the
role of the notion of term order which is usually considered a central and
crucial tool in the theory. In fact, as it was already established in the
computer science context in relation with termination of algorithms, most of
the properties can be obtained simply considering a well-founded ordering,
while the classical requirement that it be preserved by multiplication is
irrelevant.
  The last part of the paper shows how the polynomial basis concepts present in
literature are interpreted in our language and their properties are
consequences of the general results established in the first part of the paper.
