Special idempotents and projections

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. We call special idempotent any idempotent element of this monoid. 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.