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On The Axioms Of Common Meadows: Fracterm Calculus, Flattening And Incompleteness

Jan A Bergstra, John Tucker Orcid Logo

The Computer Journal, Volume: 66, Issue: 7

Swansea University Author: John Tucker Orcid Logo

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DOI (Published version): 10.1093/comjnl/bxac026

Abstract

Common meadows are arithmetic structures with inverse or division, made total on 0 by a flag⊥ for ease of calculation. We examine some axiomatizations of common meadows to clarify theirrelationship with commutative rings and serve different theoretical agendas. A common meadowfracterm calculus is a...

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Published in: The Computer Journal
ISSN: 0010-4620 1460-2067
Published: Oxford University Press (OUP) 2022
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URI: https://cronfa.swan.ac.uk/Record/cronfa60587
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spelling v2 60587 2022-07-21 On The Axioms Of Common Meadows: Fracterm Calculus, Flattening And Incompleteness 431b3060563ed44cc68c7056ece2f85e 0000-0003-4689-8760 John Tucker John Tucker true false 2022-07-21 SCS Common meadows are arithmetic structures with inverse or division, made total on 0 by a flag⊥ for ease of calculation. We examine some axiomatizations of common meadows to clarify theirrelationship with commutative rings and serve different theoretical agendas. A common meadowfracterm calculus is a special form of the equational axiomatization of common meadows, originallybased on the use of division on the rational numbers. We study axioms that allow the basic processof simplifying complex expressions involving division. A useful axiomatic extension of the commonmeadow fracterm calculus imposes the requirement that the characteristic of common meadows bezero (using a simple infinite scheme of closed equations). It is known that these axioms are completefor the full equational theory of common cancellation meadows of characteristic 0. Here, we showthat these axioms do not prove all conditional equations which hold in all common cancellationmeadows of characteristic 0. Journal Article The Computer Journal 66 7 Oxford University Press (OUP) 0010-4620 1460-2067 Arithmetic structures; rational numbers; division by zero; meadows; common meadows; fracterm calculus; equational specification; initial algebra semantics 5 4 2022 2022-04-05 10.1093/comjnl/bxac026 COLLEGE NANME Computer Science COLLEGE CODE SCS Swansea University SU Library paid the OA fee (TA Institutional Deal) 2023-09-04T16:58:31.6547132 2022-07-21T23:42:58.1291472 Faculty of Science and Engineering School of Mathematics and Computer Science - Computer Science Jan A Bergstra 1 John Tucker 0000-0003-4689-8760 2 60587__24713__5c1e17e3ad134693b1ef31dcb1e4b3c5.pdf 60587.pdf 2022-07-22T14:41:35.4100610 Output 306419 application/pdf Version of Record true © The Author(s) 2022. This is an Open Access article distributed under the terms of the Creative Commons Attribution License true eng http://creativecommons.org/licenses/by/4.0/
title On The Axioms Of Common Meadows: Fracterm Calculus, Flattening And Incompleteness
spellingShingle On The Axioms Of Common Meadows: Fracterm Calculus, Flattening And Incompleteness
John Tucker
title_short On The Axioms Of Common Meadows: Fracterm Calculus, Flattening And Incompleteness
title_full On The Axioms Of Common Meadows: Fracterm Calculus, Flattening And Incompleteness
title_fullStr On The Axioms Of Common Meadows: Fracterm Calculus, Flattening And Incompleteness
title_full_unstemmed On The Axioms Of Common Meadows: Fracterm Calculus, Flattening And Incompleteness
title_sort On The Axioms Of Common Meadows: Fracterm Calculus, Flattening And Incompleteness
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 Computer Journal
container_volume 66
container_issue 7
publishDate 2022
institution Swansea University
issn 0010-4620
1460-2067
doi_str_mv 10.1093/comjnl/bxac026
publisher Oxford University Press (OUP)
college_str Faculty of Science and Engineering
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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
document_store_str 1
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description Common meadows are arithmetic structures with inverse or division, made total on 0 by a flag⊥ for ease of calculation. We examine some axiomatizations of common meadows to clarify theirrelationship with commutative rings and serve different theoretical agendas. A common meadowfracterm calculus is a special form of the equational axiomatization of common meadows, originallybased on the use of division on the rational numbers. We study axioms that allow the basic processof simplifying complex expressions involving division. A useful axiomatic extension of the commonmeadow fracterm calculus imposes the requirement that the characteristic of common meadows bezero (using a simple infinite scheme of closed equations). It is known that these axioms are completefor the full equational theory of common cancellation meadows of characteristic 0. Here, we showthat these axioms do not prove all conditional equations which hold in all common cancellationmeadows of characteristic 0.
published_date 2022-04-05T16:58:33Z
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score 11.017731

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