nLab Ribenboim power series

Idea

A Ribenboim power series group (or ring, or field) is a generalization of a formal power series ring whose exponents can belong to almost any partially ordered monoid. It includes formal power series, formal Laurent series?, polynomials, Laurent polynomials, and Hahn series as special cases.

Definition

Let $P$ be a poset; say that a subset of $P$ is artinian if it contains no strictly decreasing infinite chain, and narrow if it contains no infinite antichain. Let $A$ be an abelian group. The space of Ribenboim power series from $P$ with coefficients in $A$, denoted $G(P,A)$, is the set of functions $f:P\to A$ whose support $supp(f) = \{ p\in P \mid f(p)\neq 0 \}$ is artinian and narrow. This is an abelian group under pointwise addition.

Now let $M$ be a strict poset monoid, i.e. a monoid object in the monoidal category $Pos_{str}$ of posets and strict functions (i.e. morphisms that preserve the strict ordering as well as the non-strict one) whose monoidal structure is the cartesian product of $Pos$ (which is not the cartesian product of $Pos_{str}$). More explicitly, this means $M$ is a monoid and a poset and the multiplication preserves both orders $\le$ and $\lt$ on both sides.

If $M$ is such a “strict pomonoid” and $R$ a ring, then $G(M,R)$ is again a ring, with multiplication

$$(f\cdot g)(m) = \sum_{(m_1,m_2) \in X_m(f,g)} f(m_1)\cdot g(m_2)$$

where $X_m(f,g)$ is the set of pairs $(m_1,m_2)\in supp(f) \times supp(g)$ such that $m_1\cdot m_2 = m$. (The nontrivial fact to verify is that $X_m(f,g)$ is always finite.)

Examples

• If $M=\mathbb{N}$ with the usual order, we obtain the usual ring of formal power series with coefficients in $R$.

• If $M=\mathbb{Z}$ with the usual order, we obtain the usual ring of formal Laurent series?.

• If $M=\mathbb{N}$ with the discrete order, we obtain the usual ring of polynomials.

• Similarly, for $\mathbb{Z}$ with the discrete order, we obtain the ring of Laurent polynomials.

• If $M$ is a linearly ordered abelian group and $R$ a field, then $G(M,R)$ is the field of Hahn series with value group $M$.

Related pages

Other rings of generalized power series include:

• Puiseux series
• Novikov field
• Hahn series

Hahn series are a special kind of Ribenboim power series, but Puiseux and Novikov series are not. However, they are all instances of the linearization of a finiteness space. For Puiseux and Ribenboim series, this is shown in BJCS.

References

• P. Ribenboim, Rings of generalized power series: Nilpotent elements, Abh. Math. Semin. Univ. Hambg. 61, pp. 15–3 (1991)

• P. Ribenboim, Noetherian rings of generalized power series, J. Pure Appl. Algebra 79, pp. 293–312 (1992)

• P. Ribenboim, Rings of generalized power series II: Units and zero-divisors, Journal of Algebra 168, pp. 71–89 (1994)

• Richard Blute, Robin Cockett, Pierre-Alain Jacqmin, and Philip Scott, Finiteness spaces and generalized power series, doi, arXiv:1805.09836