TY - GEN

T1 - Fields, meadows and abstract data types

AU - Bergstra, Jan

AU - Hirshfeld, Yoram

AU - Tucker, John

PY - 2008

Y1 - 2008

N2 - 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].

AB - 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].

UR - http://www.scopus.com/inward/record.url?scp=49949098306&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-78127-1_10

DO - 10.1007/978-3-540-78127-1_10

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AN - SCOPUS:49949098306

SN - 3540781269

SN - 9783540781264

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 166

EP - 178

BT - Pillars of Computer Science - Essays Dedicated to Boris (Boaz) Trakhtenbrot on the Occasion of His 85th Birthday

PB - Springer Verlag

ER -