On generalized formal power series. (English) Zbl 0635.16001

The author studies a generalization of Laurent series: D(T,L) denotes the set of mappings \(f: L\to T\), where L is a linearly ordered loop (of “exponents”), where \((T,+,\cdot)\) is a structure such that \((T,+)\) is a commutative group with 0 as neutral element, where (T\(\setminus 0,\cdot)\) is a quasi-group annihilated by 0, such that \(\sup p(f)=\{x\in L|\) f(x)\(\neq 0\}\) is a well-ordered subset of L. She shows that addition and multiplication (defined as usually when f is viewed as a formal power series \(f=\sum_{\ell \in L}f(\ell)^{\ell})\) are well-defined in D(T,L). She shows that D(T,L) is also a quasi-group and indicates that many usual properties of T are shared by D(T,L): associativity, commutativity, distributivity, neutral element, ring, division ring.
Reviewer: C.Reutenauer


16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
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