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Rings with common division, common meadows and their conditional equational theories

Jan A Bergstra Orcid Logo, John Tucker Orcid Logo

The Journal of Symbolic Logic, Pages: 1 - 27

Swansea University Author: John Tucker Orcid Logo

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DOI (Published version): 10.1017/jsl.2024.88

Abstract

We examine the consequences of having a total division operation xy on commutative rings. We consider two forms of binary division, one derived from a unary inverse, the other defined directly as a general operation; each are made total by setting 1/0 equal to an error value ⊥, which is added to the...

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Published in: The Journal of Symbolic Logic
ISSN: 0022-4812 1943-5886
Published: Cambridge University Press (CUP) 2024
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URI: https://cronfa.swan.ac.uk/Record/cronfa68637
Abstract: We examine the consequences of having a total division operation xy on commutative rings. We consider two forms of binary division, one derived from a unary inverse, the other defined directly as a general operation; each are made total by setting 1/0 equal to an error value ⊥, which is added to the ring. Such totalised divisions we call common divisions. In a field the two forms are equivalent and we have a finite equational axiomatisation E that is complete for the equational theory of fields equipped with common division, called common meadows. These equational axioms E turn out to be true of commutative rings with common division but only when defined via inverses. We explore these axioms E and their role in seeking a completeness theorem for the conditional equational theory of common meadows. We prove they are complete for the conditional equational theory of commutative rings with inverse based common division. By adding a new proof rule, we can prove a completeness theorem for the conditional equational theory of common meadows. Although, the equational axioms E fail with common division defined directly, we observe that the direct division does satisfies the equations in E under a new congruence for partial terms called eager equality.
College: Faculty of Science and Engineering
Start Page: 1
End Page: 27