Algebraic specifications of computable and semicomputable data types
Theoretical Computer Science
Handbook of theoretical computer science (vol. B)
Handbook of logic in computer science (vol. 1)
Journal of the ACM (JACM)
The rational numbers as an abstract data type
Journal of the ACM (JACM)
Division Safe Calculation in Totalised Fields
Theory of Computing Systems
Elementary algebraic specifications of the rational function field
CiE'06 Proceedings of the Second conference on Computability in Europe: logical Approaches to Computational Barriers
A Process Calculus with Finitary Comprehended Terms
Theory of Computing Systems
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The rational, real and complex numbers with their standard operations, including division, are partial algebras specified by the axiomatic concept of a field. Since the class of fields cannot be defined by equations, the theory of equational specifications of data types cannot use field theory in applications to number systems based upon rational, real and complex numbers. We study a new axiomatic concept for number systems with division that uses only equations: a meadow is a commutative ring with a total inverse operator satisfying two equations which imply 0^-^1=0. All fields and products of fields can be viewed as meadows. After reviewing alternate axioms for inverse, we start the development of a theory of meadows. We give a general representation theorem for meadows and find, as a corollary, that the conditional equational theory of meadows coincides with the conditional equational theory of zero totalized fields. We also prove representation results for meadows of finite characteristic.