Meadows and the equational specification of division

TL;DR

This paper introduces meadows, a new algebraic structure using only equations to model number systems with division, including fields and their products, enabling equational specification of division in various number systems.

Contribution

It defines meadows as a novel axiomatic framework for number systems with division, providing representation theorems and linking their theory to zero totalized fields.

Findings

Meadows include all fields and their products.

The conditional equational theory of meadows matches that of zero totalized fields.

Representation results are established for finite characteristic meadows.

Abstract

The rational, real and complex numbers with their standard operations, including division, are partial algebras specified by the axiomatic concept of a field. Since the class of fields cannot be defined by equations, the theory of equational specifications of data types cannot use field theory in applications to number systems based upon rational, real and complex numbers. We study a new axiomatic concept for number systems with division that uses only equations: a meadow is a commutative ring with a total inverse operator satisfying two equations which imply that the inverse of zero is zero. All fields and products of fields can be viewed as meadows. After reviewing alternate axioms for inverse, we start the development of a theory of meadows. We give a general representation theorem for meadows and find, as a corollary, that the conditional equational theory of meadows coincides with the conditional equational theory of zero totalized fields. We also prove representation results for meadows of finite characteristic.