  A combinatorial property of prositive group presentations, called
completeness, is introduced, with an effective criterion for recognizing
complete presentations, and an iterative method for completing an incomplete
presentation. We show how to directly read several properties of the associated
monoid and group from a complete presentation: cancellativity or existence of
common multiples in the case of the monoid, or isoperimetric inequality in the
case of the group. In particular, we obtain a new criterion for recognizing
that a monoid embeds in a group of fractions. Typical presentations eligible
for the current approach are the standard presentations of the Artin groups and
the Heisenberg group.
