Fields, meadows and abstract data types

Authors:
Jan Bergstra;Yoram Hirshfeld;John Tucker
Affiliations:
University of Amsterdam;Tel Aviv University;University of Wales, Swansea
Venue:
Pillars of computer science
Year:
2008

Citing 5
Cited 0

Algebraic specification

Handbook of theoretical computer science (vol. B)
Universal algebra

Handbook of logic in computer science (vol. 1)
Specification of abstract data types

Specification of abstract data types
Algebraic specification of abstract data types

Handbook of logic in computer science
The rational numbers as an abstract data type

Journal of the ACM (JACM)

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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 ċ 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].