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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
Online Access: Check full text

URI: https://cronfa.swan.ac.uk/Record/cronfa68637
first_indexed 2025-01-13T20:35:02Z
last_indexed 2025-01-13T20:35:02Z
id cronfa68637
recordtype SURis
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spelling 2025-01-13T15:15:33.4939147 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 1 27 Cambridge University Press (CUP) 0022-4812 1943-5886 23 12 2024 2024-12-23 10.1017/jsl.2024.88 https://doi.org/10.1017/jsl.2024.88 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University 2025-01-13T15:15:33.4939147 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
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
container_start_page 1
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
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
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
url https://doi.org/10.1017/jsl.2024.88
document_store_str 0
active_str 0
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 2024-12-23T08:37:22Z
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score 11.364387

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