Work

Decompositions of the positive real numbers into disjoint sets closed under addition and multiplication

Year: 2024

Type: article

Abstract: The general problem studied in this paper is to decompose the positive elements R + of a totally ordered ring R into the disjoint union of sets that are closed under addition and multiplication. We mainly investigate the case when R is a subfield of the reals R . We prove that for any n ∈ N the positive real numbers can be decomposed into n disjoint pieces which are also closed under addition and multiplication. We construct infinitely many different n -decompositions ( n ∈ N ) for all fields containing at least two algebraically independent transcendental numbers. We characterise all possible decompositions into two pieces for fields of transcendence degree one over Q . Further, we prove that the positive elements of real algebraic extensions of the rational numbers are indecomposable into finitely many pieces.

Source: Journal of Algebra

DOI: 10.1016/j.jalgebra.2024.10.044

Authors: Gergely Kiss, Gábor Somlai, Tamás Terpai

Institutions: HUN-REN Alfréd Rényi Institute of Mathematics, Corvinus University of Budapest, Eötvös Loránd University, The Graduate Center, CUNY, City University of New York

Language: English

Topic: Advanced Topology and Set Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,140