Journal article 16 views
Rings with common division, common meadows and their conditional equational theories
The Journal of Symbolic Logic, Pages: 1 - 27
Swansea University Author: John Tucker
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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...
Published in: | The Journal of Symbolic Logic |
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ISSN: | 0022-4812 1943-5886 |
Published: |
Cambridge University Press (CUP)
2024
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Online Access: |
Check full text
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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. |
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College: |
Faculty of Science and Engineering |
Start Page: |
1 |
End Page: |
27 |