  A quasi-automatic semigroup is defined by a finite set of generators, a
rational (regular) set of representatives, such that if a is a generator or
neutral, then the graph of right multiplication by a on the set of
representatives is a rational relation. This class of semigroups contains
previously considered semigroups and groups (Sakarovitch, Epstein et al.,
Campbell et al.). Membership of a semigroup to this class does not depend on
the choice of the generators. These semigroups are rationally presented.
Representatives may be computed in exponential time. Their word problem is
decidable in exponential time. They enjoy a property similar to the so-called
Lipschitz property, or fellow traveler property. If graded, they are automatic.
In the case of groups, they are finitely presented with an exponential
isoperimetric inequality and they are characterized by the weak Lipschitz
property.
