Recent zbMATH articles in MSC 06F20https://www.zbmath.org/atom/cc/06F202022-05-16T20:40:13.078697ZWerkzeugOn the Riesz structures of a lattice ordered abelian grouphttps://www.zbmath.org/1483.060232022-05-16T20:40:13.078697Z"Lenzi, Giacomo"https://www.zbmath.org/authors/?q=ai:lenzi.giacomoSummary: A Riesz structure on a lattice ordered abelian group \(G\) is a real vector space structure where the product of a positive element of \(G\) and a positive real is positive. In this paper we show that for every cardinal \(k\) there is a totally ordered abelian group with at least \(k\) Riesz structures, all of them isomorphic. Moreover two Riesz structures on the same totally ordered group are partially isomorphic in the sense of model theory. Further, as a main result, we build two nonisomorphic Riesz structures on the same \(l\)-group with strong unit. This gives a solution to a problem posed by \textit{P. Conrad} in 1975 [J. Aust. Math. Soc., Ser. A 20, 332--347 (1975; Zbl 0317.06014)]. Finally we apply the main result to MV-algebras and Riesz MV-algebras.On Rayner structureshttps://www.zbmath.org/1483.130392022-05-16T20:40:13.078697Z"Krapp, Lothar Sebastian"https://www.zbmath.org/authors/?q=ai:krapp.lothar-sebastian"Kuhlmann, Salma"https://www.zbmath.org/authors/?q=ai:kuhlmann.salma"Serra, Michele"https://www.zbmath.org/authors/?q=ai:serra.micheleThe article ``On Rayner structures'' by Lothar Sebastian Krapp, Salma Kuhlmann and Michele Serra explores the algebraic and combinatorial properties of generalised power series fields. More specifically, given the field \(k((G))\) of \(k\)-valued power series in a totally ordered abelian group \(G\) (which can be realised as the space of k-valued functions on \(G\) with well ordered support), and a set \(\mathcal{F}\) of well ordered subsets of \(G\), the article explores the \emph{\(k\)-hull} of \(\mathcal{F}\), and establishes necessary conditions for this to satisfy appropriate algebraic properties.
The paper begins by giving a list of algebraic and set theoretic properties (Conditions 2.1) labelled (S1)--(S6), (A1)--(A5) that can be satisfied by the set \(\mathcal{F}\), and recalls from a previous work of Rayner that the \(k\)-hull \(k((\mathcal{F}))\) of \(\mathcal{F}\) is a subfield of \(k((G))\) in the event that \(\mathcal{F}\) satisfies an appropriate collection of these conditions. Explicitly, \(k((\mathcal{F}))\) is an additive subgroup when it satisfies conditions (S2), (S3), and (S5), it is a subring when it also satisfies (A3) and (A4), and it is a subfield when it also satisfies (A1). However, these are merely sufficient conditions, and the aim of the paper is to establish necessary conditions for these properties to hold.
The first result of the article, Proposition 3.4, states that provided \(k\neq\mathbb{F}_2\), \(k((\mathcal{F}))\) is indeed an additive subgroup of \(k((G))\) if and only if \(\mathcal{F}\) is closed under taking unions and subsets, and contains the singleton {0}, i.e. if and only if \(\mathcal{F}\) satisfies (S2), (S3) and (S5). This strengthens Rayner's original result (Theorem 3.1(i)), showing that the original condition is indeed necessary. The result does not hold if \(k=\mathbb{F}_2\), as demonstrated by Example 3.6.
The authors go on to demonstrate a necessary and sufficient condition for \(k((\mathcal{F}))\) to be a subring of \(k((G))\) in Proposition 3.9, provided k is a sufficiently large field. This result gives a stronger condition than that originally stated by Rayner, since it only requires \(\mathcal{F}\) to satisfy (S2), (S3), (S5) and (A2), while (A3) and (A4) are unnecessary.
The remainder of the paper focuses on field structure. The aim is to find necessary and sufficient conditions for \(k((\mathcal{F}))\) to be a field, a \emph{Hahn field} and a \emph{Rayner field}. Briefly, a Hahn field is a subfield of \(k((G))\) containing all polynomials, and a Rayner field is a \(k\)-hull \(k((\mathcal{F}))\) where \(\mathcal{F}\) satisfies (S2), (S3), (S5), (A1), (A3) and (A4), which is a subfield of \(k((G))\) by Rayner's original theorem.
The final main results of the paper are Proposition 3.15 and Theorem 3.18. Proposition 3.15 gives a necessary and sufficient condition for \(k((\mathcal{F}))\) to be a subfield and a Hahn field in terms of the Conditions 2.1, at least in the case where \(k\) has characteristic 0. Theorem 3.18 states that if \(k((\mathcal{F}))\) is a Raynor field then it is a Hahn field, and that the converse holds when \(k\) has characteristic 0. The article concludes by stating that the \(k\)-hull \(k((\mathcal{F}))\) is a Rayner field if and only if it is a Hahn field, if and only if it satisfies all of Conditions 2.1.
Overall, this paper should be of interest to anyone who is concerned with power series and generalisations thereof, but since it is a short article with very understandable proofs, I would say that it is accessible to anyone algebraically minded, and certainly worth reading.
Reviewer: Adam Jones (Manchester)