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 $R[x_1,\ldots,x_n]^n$ the set of all $n$-tuples of polynomials of the form $(f_1,\ldots,f_n),$ where $f_1,\ldots,f_n\in R[x_1,\ldots,x_n]$. We call these $n$-tuples vector-polynomials. We define composition on $R[x_1,\ldots,x_n]^n$ by $$\vec{g}\circ \vec{f}=(g_1(f_1, \ldots ,f_n), \ldots ,g_n (f_1, \ldots ,f_n)), \text{ where }\vec{f}=(f_1, \ldots ,f_n), \vec{g}=(g_1, \ldots ,g_n).$$ In this paper, we investigate vector-polynomials of the form $$ \vec{f}=(f_0,f_1 +x_2g_1,\ldots, f_{n-1} +x_n g_{n-1}),$$ where $f_0\in R[x_1]$ permutes the elements of $R$ and $f_i ,g_i\in R[x_1,\ldots,x_i]$ such that each $g_i$ maps $R^i$ into the units of $R$ ($i=1,\ldots, n-1$). We show that each such vector-polynomial permutes the elements of $R^n$ and that the set of all such vector-polynomials $\mathcal{MT}_n$ is a monoid with respect to composition. We also show that $ \vec{f} $ is invertible in $\mathcal{MT}_n$ if and only if $f_0$ is an $R$-automorphism of $R[x_1]$ and $g_i$ is invertible in $R[x_1,\ldots,x_i]$ for $i=1,\ldots, n-1$. When $R$ is finite, the monoid $\mathcal{MT}_n$ induces a finite group of permutations of $R^n$. Moreover, we decompose the monoid $\mathcal{MT}_n$ 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.