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

Abstract

Let $n>1$ and let R be a commutative ring with identity $1\ne 0$ and $R{[x_1,\dots ,x_n]}^n$ the set of all n-tuples of polynomials of the form $(f_1,\dots ,f_n)$, where $f_1,\dots ,f_n\in R[x_1,\dots ,x_n]$. We call these n-tuples vector-polynomials. We define composition on $R{[x_1,\dots ,x_n]}^n$ by$\overrightarrow{g}\circ \overrightarrow{f}=\left(g_1\right(f_1,\dots ,f_n),\dots ,g_n(f_1,\dots ,f_n\left)\right),\text{ where }\overrightarrow{f}=(f_1,\dots ,f_n),\overrightarrow{g}=(g_1,\dots ,g_n).$ In this paper, we investigate vector-polynomials of the form$\overrightarrow{f}=(f_0,f_1+x_2{g}_1,\dots ,f_{n-1}+x_n{g}_{n-1}),$ where $f_0\in R\left[x_1\right]$ permutes the elements of R and $f_i,g_i\in R[x_1,\dots ,x_i]$ such that each $g_i$ maps $R^i$ into the units of R ($i=1,\dots ,n-1$). We show that each such vector-polynomial permutes the elements of $R^n$ and that the set of all such vector-polynomials ${\mathrm{MT}}_n$ is a monoid with respect to composition. We also show that $\overrightarrow{f}$ is invertible in ${\mathrm{MT}}_n$ if and only if $f_0$ is an R-automorphism of $R\left[x_1\right]$ and $g_i$ is invertible in $R[x_1,\dots ,x_i]$ for $i=1,\dots ,n-1$. When R is finite, the monoid ${\mathrm{MT}}_n$ induces a finite group of permutations of $R^n$. Moreover, we decompose the monoid ${\mathrm{MT}}_n$ 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.