# Fields, Meadows and Abstract Data Types

• Jan Bergstra
• Yoram Hirshfeld
• John Tucker
Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4800)

## 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].

## Keywords

Rational Number Inverse Function Commutative Ring Equational Theory Division Ring
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• Jan Bergstra
• 1
• Yoram Hirshfeld
• 2
• John Tucker
• 3
1. 1.University of Amsterdam
2. 2.Tel Aviv University
3. 3.University of Wales, Swansea