×

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

MSC:

16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
PDF BibTeX XML Cite
Full Text: EuDML

References:

[1] Hahn H.: Über die nichtarchimedischen Grössensysteme. Sitzungsberichte der Österreichischen Akademie, Wien, 116, (1907), 601-655. · JFM 38.0501.01
[2] Kuratowski K., Mostowski A.: Set theory. Amsterdam-Warszawa 1967. · Zbl 0165.01701
[3] Zelinski D.: Non-associative valuations. Bull. Amer. Math. Soc., 54 (1948), 175-183.
[4] Zelinski D.: On ordered loops. Amer. Journal Math., 70 (1948), 681-697. · Zbl 0035.01201
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.