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

Bergstra, Jan A; Tucker, John

doi:10.1017/jsl.2024.88

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

Jan A Bergstra

, John Tucker

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...

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Published in:	The Journal of Symbolic Logic
ISSN:	0022-4812 1943-5886
Published:	Cambridge University Press (CUP) 2024
Online Access:	Check full text
URI:	https://cronfa.swan.ac.uk/Record/cronfa68637

first_indexed	2025-01-13T20:35:02Z
last_indexed	2025-04-17T04:44:34Z
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spelling	2025-04-16T13:49:54.4177476 v2 68637 2025-01-02 Rings with common division, common meadows and their conditional equational theories 431b3060563ed44cc68c7056ece2f85e 0000-0003-4689-8760 John Tucker John Tucker true false 2025-01-02 MACS 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. Journal Article The Journal of Symbolic Logic 0 1 27 Cambridge University Press (CUP) 0022-4812 1943-5886 23 12 2024 2024-12-23 10.1017/jsl.2024.88 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University Not Required 2025-04-16T13:49:54.4177476 2025-01-02T15:00:41.4630453 Faculty of Science and Engineering School of Mathematics and Computer Science - Computer Science Jan A Bergstra 0000-0003-2492-506x 1 John Tucker 0000-0003-4689-8760 2 68637__33307__c313beea590348799dc03f36a714e431.pdf 68637.pdf 2025-01-13T15:14:50.6411174 Output 519773 application/pdf Accepted Manuscript true Author accepted manuscript document released under the terms of a Creative Commons CC-BY licence using the Swansea University Research Publications Policy (rights retention). true eng https://creativecommons.org/licenses/by/4.0/deed.en
title	Rings with common division, common meadows and their conditional equational theories
spellingShingle	Rings with common division, common meadows and their conditional equational theories John Tucker
title_short	Rings with common division, common meadows and their conditional equational theories
title_full	Rings with common division, common meadows and their conditional equational theories
title_fullStr	Rings with common division, common meadows and their conditional equational theories
title_full_unstemmed	Rings with common division, common meadows and their conditional equational theories
title_sort	Rings with common division, common meadows and their conditional equational theories
author_id_str_mv	431b3060563ed44cc68c7056ece2f85e
author_id_fullname_str_mv	431b3060563ed44cc68c7056ece2f85e_***_John Tucker
author	John Tucker
author2	Jan A Bergstra John Tucker
format	Journal article
container_title	The Journal of Symbolic Logic
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publishDate	2024
institution	Swansea University
issn	0022-4812 1943-5886
doi_str_mv	10.1017/jsl.2024.88
publisher	Cambridge University Press (CUP)
college_str	Faculty of Science and Engineering
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hierarchy_top_title	Faculty of Science and Engineering
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hierarchy_parent_title	Faculty of Science and Engineering
department_str	School of Mathematics and Computer Science - Computer Science{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Computer Science
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description	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.
published_date	2024-12-23T08:29:23Z
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score	11.501582

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

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