Equations for formally real meadows

TL;DR

This paper provides complete axiomatizations for the equational theories of real and complex numbers within the framework of meadows, using signatures that include operations and inverses, with applications to formal realness and orderings.

Contribution

It introduces two comprehensive axiomatizations of real numbers as meadows, extending existing theories and applying them to complex numbers.

Findings

Complete axiomatization of real numbers with formal realness

Axiomatization presupposing an ordering for real numbers

Application to axiomatize complex numbers

Abstract

We consider the signatures $\Sigma_m=(0,1,-,+, \cdot, \ ^{-1})$ of meadows and $(\Sigma_m, {\mathbf s})$ of signed meadows. We give two complete axiomatizations of the equational theories of the real numbers with respect to these signatures. In the first case, we extend the axiomatization of zero-totalized fields by a single axiom scheme expressing formal realness; the second axiomatization presupposes an ordering. We apply these completeness results in order to obtain complete axiomatizations of the complex numbers.