Square root meadows

TL;DR

This paper introduces an extension of the rational numbers meadow that includes a total sign function and guarantees a unique square root for every element, enhancing the algebraic structure of the rational meadow.

Contribution

It presents a novel extension of the rational meadow by adding a total sign function and ensuring unique square roots for all elements.

Findings

Q_0(s,√) extends Q_0 with total sign and square root functions

Every element in Q_0(s,√) has a unique square root

The structure enhances algebraic properties of the rational meadow

Abstract

Let Q_0 denote the rational numbers expanded to a meadow by totalizing inversion such that 0^{-1}=0. Q_0 can be expanded by a total sign function s that extracts the sign of a rational number. In this paper we discuss an extension Q_0(s ,\sqrt) of the signed rationals in which every number has a unique square root.