Fields, meadows and abstract data types

Jan Bergstra, Yoram Hirshfeld, John Tucker

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Fields and division rings are not algebras in the sense of "Universal Algebra", as inverse is not a total function. Mending the inverse by any definition of 0-∈1 will not suffice to axiomatize the axiom of inverse x -∈1•x∈=∈1, by an equation. In particular the theory of fields cannot be used for specifying the abstract data type of the rational numbers. We define equational theories of Meadows and of Skew Meadows, and we prove that these theories axiomatize the equational properties of fields and of division rings, respectively, with 0 -∈1=∈0 . Meadows are then used in the theory of Von Neumann regular ring rings to characterize strongly regular rings as those that support an inverse operation that turns it into a skew meadow. To conclude, we present in this framework the specification of the abstract type of the rational numbers, as developed by the first and third authors in [2].

Original languageEnglish
Title of host publicationPillars of Computer Science - Essays Dedicated to Boris (Boaz) Trakhtenbrot on the Occasion of His 85th Birthday
PublisherSpringer Verlag
Pages166-178
Number of pages13
ISBN (Print)3540781269, 9783540781264
DOIs
StatePublished - 2008

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4800 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

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