Special idempotents and projections

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