Fracpairs: fractions over a reduced commutative ring

In the well-known construction of the field of fractions, 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 (that is, a commutative ring that has no nonzero nilpotent elements) and find that these constitute a "common meadow" (a field equipped with a multiplicative inverse and an additional element "a" that is the inverse of zero and propagates through all operations). We prove that fracpairs over Z constitute a homomorphic pre-image of the common meadow Qa, the field Q of rational numbers expanded with an a-totalized inverse. Moreover, we characterize the initial common meadow as an initial algebra of fracpairs. Next, we define canonical term algebras (and therewith normal forms) for fracpairs over the integers and some related structures that model the rational numbers, and we provide negative results concerning their associated term rewriting properties. Then we define "rational fracpairs" that constitute an initial algebra that is isomorphic to Qa. Finally, we express some negative conjectures concerning alternative specifications for these (concrete) datatypes.