 Author
 Date
 62016
 Title
 Fracpairs and fractions over a reduced commutative ring
 Journal
 Indagationes Mathematicae
 Volume  Issue number
 27  3
 Pages (fromto)
 727748
 Document type
 Article
 Faculty
 Faculty of Science (FNWI)
 Institute
 Informatics Institute (IVI)
 Abstract

In the wellknown 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 errorvalue supports the intuition.
The equivalence classes of fracpairs thus obtained are called common cancellation fractions (ccfractions), and ccfractions over the integers constitute a homomorphic preimage of the common meadow Q_{a}, the field Q of rational numbers expanded with an atotalized inverse. Moreover, the initial common meadow is isomorphic to the initial algebra of ccfractions over the integer numbers. Next, we define canonical term algebras for ccfractions 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 Q_{a}. Finally, we express negative conjectures concerning alternative specifications for these (concrete) datatypes.  URL
 go to publisher's site
 Language
 English
 Permalink
 http://hdl.handle.net/11245.1/9aa904d996424ddab7d216a0f997e783
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