Formal power series with cyclically ordered exponents
An abelian group \(C\) is said to be cyclically-ordered if there is a ternary operation \((.,.,.)\) defined on \(C\) such that for all \(a,b,c\in C\) (1) \( (a,b,c)\) implies that \(a,b,c\) are all distinct (2) \((a,b,c)\) implies \((b,c,a) \) (3) \((c,.,.)\) is a strict total order on \(C\backslash \{c\}\) (for each \( c\in C)\) and (4) \((.,.,.)\) is compatible with the group operation of \(C\). For a brief treatment of cyclically-ordered groups see pages 61--65 of the book by \textit{L. Fuchs} [Partially ordered algebraic systems (1963; Zbl 0137.02001)]. Fuchs refers to Rieger's pioneer work on cyclically ordered group. Let \(k\) be a ring with \(1\) and let \(C\) be a cyclically-ordered group. For \( f\in k^{C}\) let Supp\((f)=\{c\in C:f(c)\neq 0\}\) be the support of \(f.\) The authors denote by \(kC\) (respectively \(k[C])\) the set of all elements of \( k^{C}\) whose support is well-ordered (respectively finite), calling it the ring of formal power series (respectively polynomial ring) over \(C.\) Then they discuss the units and zero divisors of these rings and indicate when \(kC\) (respectively \( k[C]\)) is a field (respectively integral domain). They show that \(kC\) is a quotient ring of a subring of a power series ring over a suitable totally ordered group. It turns out that if the cyclically-ordered group \(C\) is totally-ordered then \(kC\) is precisely the formal power series ring over a totally ordered group. The authors also discuss cyclically-ordered valuations and delve briefly into logic, in keeping with Rieger's tradition perhaps.
scientific article; zbMATH DE number 2155976