The rational numbers as an abstract data type

onal Numbers as an Abstract Data Type1 J A Bergstra2 University of Amsterdam, Informatics Institute, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands J V Tucker3 Department of Computer Science, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, United Kingdom We give an equational specification of the field operations on the rational numbers under initial algebra semantics using just total functions and 12 equations. A consequence of this specification is that 0−1 = 0, an interesting equation consistent with the ring axioms and many properties of division. The existence of an equational specification of the rationals without hidden functions was an open question of L Moss. We also give a though axiomatic examination of the divisibility operator, in which some interesting new axioms and models are discovered, and equational specifications of some other algebras of rationals are given, including one with the modulus function. We state some open problems, including: D...

• Type: Journal
• Year: 2007
• Where: JACM
• Authors: Jan A. Bergstra, J. V. Tucker
• Keywords: Equational Specification, Initial Algebra Semantics, JACM 2007, field Operations