A Complete Finite Axiomatisation of the Equational Theory of Common Meadows

Abstract

We analyse abstract data types that model numerical structures with a concept of error. Specifically, we focus on arithmetic data types that contain an error value $\perp$ whose main purpose is to always return a value for division. To rings and fields, we add a division operator $x/y$ and study a class of algebras called common meadows wherein $x/0=\perp$. The set of equations true in all common meadows is named the equational theory of common meadows. We give a finite equational axiomatisation of the equational theory of common meadows and prove that it is complete and that the equational theory is decidable.