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The Boolean space of higher level orderings

Katarzyna Osiak (2007)

Fundamenta Mathematicae

Let K be an ordered field. The set X(K) of its orderings can be topologized to make it a Boolean space. Moreover, it has been shown by Craven that for any Boolean space Y there exists a field K such that X(K) is homeomorphic to Y. Becker's higher level ordering is a generalization of the usual concept of ordering. In a similar way to the case of ordinary orderings one can define a topology on the space of orderings of fixed exact level. We show that it need not be Boolean. However, our main theorem...

The ring of arithmetical functions with unitary convolution: Divisorial and topological properties

Jan Snellman (2004)

Archivum Mathematicum

We study ( 𝒜 , + , ⊕ ) , the ring of arithmetical functions with unitary convolution, giving an isomorphism between ( 𝒜 , + , ⊕ ) and a generalized power series ring on infinitely many variables, similar to the isomorphism of Cashwell-Everett [NumThe] between the ring ( 𝒜 , + , · ) of arithmetical functions with Dirichlet convolution and the power series ring [ [ x 1 , x 2 , x 3 , ⋯ ] ] on countably many variables. We topologize it with respect to a natural norm, and show that all ideals are quasi-finite. Some elementary results on factorization into atoms...

Théorèmes de préparation Gevrey et étude de certaines applications formelles

Augustin Mouze (2003)

Annales Polonici Mathematici

We consider subrings A of the ring of formal power series. They are defined by growth conditions on coefficients such as, for instance, Gevrey conditions. We prove preparation theorems of Malgrange type in these rings. As a consequence we study maps F from ℂ s to ℂ p without constant term such that the rank of the Jacobian matrix of F is equal to 1. Let be a formal power series. If F is a holomorphic map, the following result is well known: ∘ F is analytic implies there exists a convergent power series...

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