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1991•International Journal of Algebra and ComputationRequires access

TERM REWRITE RULES FOR FINITE FIELDS

Stanley N. Burris, John Lawrence

Open publisher page 6 citations

Abstract

Let F1, …, Fk be finite fields with distinct characteristics. We give a finite set of equations which axiomatize the equational theory of F1, …, Fk and then use these axioms to find a finite set of AC-term rewrite rules which is complete for this theory. This gives finite sets of complete AC-term rewrite rules for most instances of xm ≈ x rings by adding new rules to the usual AC-term rewrite rules for commutative rings. The first case for which we do not find a complete set of AC-term rewrite rules is x22 ≈ x, and we doubt that such rules can be found. If R is a set of AC-term rewrite rules from which one can derive x(y + z) → xy + xz, then we show R cannot be complete for x22 ≈ x rings.

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What this paper is about

Let F1, …, Fk be finite fields with distinct characteristics. We give a finite set of equations which axiomatize the equational theory of F1, …, Fk and then use these axioms to find a finite set of AC-term rewrite rules which is complete for this theory. This gives finite sets of complete AC-term rewrite rules for most instances of xm ≈ x rings by adding new rules to the usual AC-term rewrite rules for commutative rings. The first case for which we do not find a complete set of AC-term rewrite rules is x22 ≈ x, and we doubt that such rules can be found. If R is a set of AC-term rewrite rules from which one can derive x(y + z) → xy + xz, then we show R cannot be complete for x22 ≈ x rings.

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Available abstract

Let F1, …, Fk be finite fields with distinct characteristics. We give a finite set of equations which axiomatize the equational theory of F1, …, Fk and then use these axioms to find a finite set of AC-term rewrite rules which is complete for this theory. This gives finite sets of complete AC-term rewrite rules for most instances of xm ≈ x rings by adding new rules to the usual AC-term rewrite rules for commutative rings. The first case for which we do not find a complete set of AC-term rewrite rules is x22 ≈ x, and we doubt that such rules can be found. If R is a set of AC-term rewrite rules from which one can derive x(y + z) → xy + xz, then we show R cannot be complete for x22 ≈ x rings.

Key concepts: Term (time), Mathematics, Axiom, Finite set, Commutative property, Set (abstract data type), Finite field, Discrete mathematics

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