Generalizations of two theorems of Ritt on decompositions of polynomial maps

Two theorems of J. F. Ritt on decompositions of polynomials maps are generalized to a more general situation: for, so-called, reduction monoids ($(K[x], \circ)$ and $(K[x^2]x, \circ)$ are examples of reduction monoids). In particular, analogues of the two theorems of J. F. Ritt hold for the monoid $(K[x^2]x, \circ)$ of odd polynomials. It is shown that, in general, the two theorems of J. F. Ritt fail for the cusp $(K+K[x]x^2, \circ)$ but their analogues are still true for decompositions of maximal length of regular elements of the cusp.