  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.
