Simplicial monoid actions and associated monoid constructions

For \(X\) a pointed simplicial set with a simplicial monoid action, we construct a simplicial monoid and give a sufficient condition for its classifying space to be the homotopy cofiber of the inclusion of \(X\) into its reduced Borel construction. The known formulae for the respective classifying spaces of the four classical constructions of James, Milnor, Carlsson and Wu are special cases of this result. Thus, we unify categorially the four above-mentioned constructions. We also give the classical constructions and an example as the applications of the main result.