A Complete Finite Axiomatisation of the Equational Theory of Common Meadows

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 bot 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 = bot. 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.