A classical result of Ore says that, if M is a cancellative monoid and
any two elements of M admit a least common multiple, that every
element of the enveloping group U(M) of M can be represented by a
unique irreducible fraction on M. We extend this result by showing
that, when common multiples need not exist but a certain "3-Ore
condition" is satisfied, every elements of U(G) can be represented by
a unique irreducible iterated fraction, leading to a solution of the
Word Problem reminiscent of the Dehn algorithm for hyperbolic groups.
This applies in particular to Artin-Tits groups of FC-type and,
conjecturally, to all Artin-Tits groups.