Division by zero in common meadows

TL;DR

This paper studies common meadows, algebraic structures extending fields with a total inverse, focusing on their properties and providing a basis theorem for a specific subclass called common cancellation meadows of characteristic zero.

Contribution

It introduces a basis theorem for common cancellation meadows of characteristic zero, advancing the algebraic understanding of these structures with total division including division by zero.

Findings

Established a basis theorem for common cancellation meadows of characteristic zero

Characterized the propagation of the error element 'a' in operations

Extended the algebraic framework of fields with total inverse

Abstract

Common meadows are fields expanded with a total inverse function. Division by zero produces an additional value denoted with "a" that propagates through all operations of the meadow signature (this additional value can be interpreted as an error element). We provide a basis theorem for so-called common cancellation meadows of characteristic zero, that is, common meadows of characteristic zero that admit a certain cancellation law.