Fracpairs and fractions over a reduced commutative ring

TL;DR

This paper introduces fracpairs over reduced commutative rings, extending the concept of fractions to include division by zero, and explores their algebraic properties and connections to common meadows and rational numbers.

Contribution

It defines fracpairs over reduced rings, shows they form a common meadow, and establishes their algebraic structure and relation to rational numbers with totalized inverses.

Findings

Fracpairs form a common meadow with a propagating error element.

Cc-fractions over integers model the rational numbers with totalized inverse.

Initial algebra of cc-fractions is isomorphic to the meadow Qa.

Abstract

In the well-known construction of the field of fractions of an integral domain, division by zero is excluded. We introduce "fracpairs" as pairs subject to laws consistent with the use of the pair as a fraction, but do not exclude denominators to be zero. We investigate fracpairs over a reduced commutative ring (a commutative ring that has no nonzero nilpotent elements) and provide these with natural definitions for addition, multiplication, and additive and multiplicative inverse. We find that modulo a simple congruence these fracpairs constitute a "common meadow", which is a commutative monoid both for addition and multiplication, extended with a weak additive inverse, a multiplicative inverse except for zero, and an additional element "a" that is the image of the multiplicative inverse on zero and that propagates through all operations. Considering "a" as an error-value supports the intuition. The equivalence classes of fracpairs thus obtained are called common cancellation fractions (cc-fractions), and cc-fractions over the integers constitute a homomorphic pre-image of the common meadow Qa, the field Q of rational numbers expanded with an a-totalized inverse. Moreover, the initial common meadow is isomorphic to the initial algebra of cc-fractions over the integer numbers. Next, we define canonical term algebras for cc-fractions over the integers and some meadows that model the rational numbers expanded with a totalized inverse, and provide some negative results concerning their associated term rewriting properties. Then we consider reduced commutative rings in which the sum of two squares plus one cannot be a zero divisor: by extending the equivalence relation on fracpairs we obtain an initial algebra that is isomorphic to Qa. Finally, we express negative conjectures concerning alternative specifications for these (concrete) datatypes.