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Two theorems of J. F. Ritt on decompositions of polynomials maps aregeneralized to a more general situation: for, so-called, reduction monoids($(K[x], \circ)$ and $(K[x^2]x, \circ)$ are examples of reduction monoids). Inparticular, 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 twotheorems of J. F. Ritt fail for the cusp $(K+K[x]x^2, \circ)$ but theiranalogues are still true for decompositions of maximal length of regularelements of the cusp.

Generalizations of two theorems of Ritt on decompositions of polynomial maps