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 0 and R[x1,…,xn]n the set of all n-tuples of polynomials of the form (f1,…,fn), where f1,…,fn∈ R[x1,…,xn]. We call these n-tuples vector-polynomials. We define composition on R[x1,…,xn]n by g f=(g1(f1, … ,fn), … ,gn (f1, … ,fn)), where f=(f1, … ,fn), g=(g1, … ,gn). In this paper, we investigate vector-polynomials of the form f=(f0,f1 +x2g1,…, fn-1 +xn gn-1), where f0∈ R[x1] permutes the elements of R and fi ,gi∈ R[x1,…,xi] such that each gi maps Ri into the units of R (i=1,…, n-1). We show that each such vector-polynomial permutes the elements of Rn and that the set of all such vector-polynomials MTn is a monoid with respect to composition. We also show that f 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 monoid 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.
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