Division by zero in non-involutive meadows

TL;DR

This paper explores non-involutive meadows, variants of algebraic structures where the inverse of zero is not zero, focusing on cases where it equals one, to expand the theoretical framework of meadows.

Contribution

It introduces and analyzes non-involutive meadows, particularly those with inverse of zero equal to one, extending the algebraic theory beyond traditional involutive meadows.

Findings

Non-involutive meadows generalize traditional meadows.

The inverse of zero can be consistently defined as one.

New algebraic properties emerge in non-involutive variants.

Abstract

Meadows have been proposed as alternatives for fields with a purely equational axiomatization. At the basis of meadows lies the decision to make the multiplicative inverse operation total by imposing that the multiplicative inverse of zero is zero. Thus, the multiplicative inverse operation of a meadow is an involution. In this paper, we study `non-involutive meadows', i.e.\ variants of meadows in which the multiplicative inverse of zero is not zero, and pay special attention to non-involutive meadows in which the multiplicative inverse of zero is one.