Fracpairs and fractions over a reduced commutative ring

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• 作者：Jan A. Bergstra ; Alban Ponse ; ^{A.Ponse@uva.nl" class="auth_mail" title="E-mail the corresponding author} ;
• 关键词：Fraction as a pair ; Common meadow ; Division by zero ; Abstract datatype ; Rational numbers ; Term rewriting
• 刊名：Indagationes Mathematicae
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：27
• 期：3
• 页码：727-748
• 全文大小：330 K

文摘

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 (that is, 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$\text{<bold>a</bold>}$ that is the image of the multiplicative inverse on zero and that propagates through all operations. Considering a$\text{<bold>a</bold>}$ 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 Q_a${\mathbb{Q}}_{\text{<bold>a</bold>}}$, the field Q$\mathbb{Q}$ of rational numbers expanded with an a$\text{<bold>a</bold>}$-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 (and therewith normal forms) for cc-fractions over the integers and some meadows that model the rational numbers expanded with a totalized inverse, and we 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${\mathbb{Q}}_{\text{<bold>a</bold>}}$. Finally, we express some negative conjectures concerning alternative specifications for these (concrete) datatypes.