Which Arithmetical Data Types Admit Fracterm Flattening?

The formal theory of division in arithmetical algebras reconstructs fractions as syntactic objects called fracterms. Basic to calculation, is the simplification of fracterms to fracterms with one division operator, a process called fracterm attening. We consider the equational axioms of a calculus f...

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Bibliographic Details
Published in:	Scientific annals of computer science Vol. XXXII; no. 1; pp. 87 - 107
Main Authors:	Bergstra, Jan A., Tucker, John V.
Format:	Journal Article
Language:	English
Published:	Iasi Alexandru Ioan Cuza University of Iasi 01-01-2022
Subjects:	Algebra Axioms Calculus Dividing (mathematics) Flattening Fractions Isomorphism Mathematical analysis Methods Multiplication & division Numbers Operators (mathematics) Semantics
Online Access:	Get full text
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Summary:	The formal theory of division in arithmetical algebras reconstructs fractions as syntactic objects called fracterms. Basic to calculation, is the simplification of fracterms to fracterms with one division operator, a process called fracterm attening. We consider the equational axioms of a calculus for calculating with fracterms to determine what is necessary and sufficient for the fracterm calculus to allow fracterm flattening. For computation, arithmetical algebras require operators to be total for which there are several semantical methods. It is shown under what constraints up to isomorphism, the unique total and minimal enlargement of a field Q(\div) of rational numbers equipped with a partial division operator \div has fracterm attening.
ISSN:	1843-8121 2248-2695
DOI:	10.7561/SACS.2022.1.87