  We define, for any special matching of a finite graded poset, an idempotent,
regressive and order preserving function. We consider the monoid generated by
such functions. The idempotents of this monoid are called special idempotents.
They are interval retracts. Some of them realize a kind of parabolic map and
are called special projections. We prove that, in Eulerian posets, the image of
a special projection, and its complement, are graded induced subposets. In a
finite Coxeter group, all projections on right and left parabolic quotients are
special projections, and some projections on double quotients too. We extend
our results to special partial matchings.
