Generalizations of two theorems of Ritt on decompositions of polynomial maps

In 1922, J. F. Ritt [13] proved two remarkable theorems on decompositions of polynomial maps of C[x] into irreducible polynomials (with respect to the composition ◦ of maps). Briefly, the first theorem states that in any two decompositions of a given polynomial function into irreducible polynomials the number of the irreducible polynomials and their degrees are the same (up to order). The second theorem gives four types of transformations of how to obtain all the decompositions from a given one. In 1941, H. T. Engstrom [7] and, in 1942, H. Levi [11] generalized respectively the first and the second theorem to polynomial maps over an arbitrary field K of characteristic zero. The aim of the paper is to generalize the two theorems of J. F. Ritt to a more general situation: for, so-called, reduction monoids ((K[x], ◦) and (K[x2]x, ◦) are examples of reduction monoids). In particular, analogues of the two theorems of J. F. Ritt hold for the monoid (K[x2]x, ◦) of odd polynomials. It is shown that, in general, the two theorems of J. F. Ritt fail for the cusp (K +K[x]x2, ◦) but their analogues are still true for decompositions of maximal length of regular elements of the cusp.