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

Linear induction algebra and a normal form for linear operators