On the structures of a monoid of triangular vector- permutation polynomials, its group of units and its induced group of permutations

Let n>1 and let R be a commutative ring with identity the 1≠0. A vector-polynomials f⃗ =(f0,f1+x2g1,…,fn−1+xngn−1), where f0∈R[x1] permutes the elements of R and fi,gi∈R[x1,…,xi] with gi mapping Ri into the units of R (i=1,…,n−1); permutes the elements of Rn. The set of all such vector-polynomial MTn is a monoid with respect to composition. Further, f⃗ is invertible in MTn if and only if f0 is an R automorphism of R[x1] and gi is invertible in R[x1,…,xi] for i=1,…,n−1. When R is finite, the monoid MTn induces a finite group of permutations of Rn. Moreover, we decompose the MTn into an iterated semi-direct product of n monoids. Such a decomposition allows us to obtain similar decompositions of its group of units, and when R is finite, of its induced group of permutations. Furthermore, the decomposition of the induced group helps us to characterize some of its properties