Linear induction algebra and a normal form for linear operators

TL;DR

This paper explores the linear analogues of natural number structures within modules over a ring, establishing free induction algebra and monoid structures, and introduces a unique normal form for linear endomorphisms as formal power series.

Contribution

It defines the free linear induction algebra and monoid over the integers in the module category and proves a unique normal form for linear endomorphisms.

Findings

Established free induction algebra and monoid structures in modules.

Proved existence and explicit construction of a unique normal form for linear endomorphisms.

Represented linear endomorphisms as non-commutative formal power series.

Abstract

The set of natural integers is fundamental for at least two reasons: it is the free induction algebra over the empty set (and at such allows definitions of maps by primitive recursion) and it is the free monoid over a one-element set, the latter structure being a consequence of the former. In this contribution, we study the corresponding structure in the linear setting, i.e., in the category of modules over a commutative ring rather than in the category of sets, namely the free module generated by the integers. It also provides free structures of induction algebra and of monoid (in the category of modules). Moreover we prove that each of its linear endomorphisms admits a unique normal form, explicitly constructed, as a non-commutative formal power series.