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Fracterm Calculus for Signed Common Meadows

Jan Aldert Bergstra, John Tucker Orcid Logo

Transmathematica, Volume: 2024

Swansea University Author: John Tucker Orcid Logo

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DOI (Published version): 10.36285/tm.97

Abstract

A common meadow is an enrichment of a field with a division operator and an error value to make division total. A signed common meadow enriches a common meadow with a sign function that can be equationally axiomatised; the sign function can simulate an ordering on the underlying field but is not lim...

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Published in: Transmathematica
ISSN: 2632-9212
Published: Transmathematica 2024
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URI: https://cronfa.swan.ac.uk/Record/cronfa66862
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spelling v2 66862 2024-06-23 Fracterm Calculus for Signed Common Meadows 431b3060563ed44cc68c7056ece2f85e 0000-0003-4689-8760 John Tucker John Tucker true false 2024-06-23 MACS A common meadow is an enrichment of a field with a division operator and an error value to make division total. A signed common meadow enriches a common meadow with a sign function that can be equationally axiomatised; the sign function can simulate an ordering on the underlying field but is not limited to orderings. In particular, of mathematical interest are the weakly signed common meadows. The prime example of a weakly signed common meadow is an expansion of a common meadow ofcomplex numbers with a weak sign function. We show that all common meadows may be enlarged to a weakly signed common meadow. A special case are the 4-signed common meadows, which are precisely the enlargements of ordered fields. To illustrate the equational calculus for signed common meadows, we use it as a foundation for building a probability calculus and derive some classical formulae. Journal Article Transmathematica 2024 Transmathematica 2632-9212 quational calculus, fracterm calculus, ordered fields, common meadow, sign function, equational specification, initial algebra semantics 14 6 2024 2024-06-14 10.36285/tm.97 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University Other 2024-07-23T11:07:53.1408165 2024-06-23T18:31:23.3045477 Faculty of Science and Engineering School of Mathematics and Computer Science - Computer Science Jan Aldert Bergstra 1 John Tucker 0000-0003-4689-8760 2 66862__30940__a3ef3bdb4371430a82f9f66246bde295.pdf 66862.VoR.pdf 2024-07-23T11:04:44.6188579 Output 403919 application/pdf Version of Record true Copyright 2024 Jan Aldert Bergstra, John V Tucker. This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. true eng https://creativecommons.org/licenses/by-sa/4.0/
title Fracterm Calculus for Signed Common Meadows
spellingShingle Fracterm Calculus for Signed Common Meadows
John Tucker
title_short Fracterm Calculus for Signed Common Meadows
title_full Fracterm Calculus for Signed Common Meadows
title_fullStr Fracterm Calculus for Signed Common Meadows
title_full_unstemmed Fracterm Calculus for Signed Common Meadows
title_sort Fracterm Calculus for Signed Common Meadows
author_id_str_mv 431b3060563ed44cc68c7056ece2f85e
author_id_fullname_str_mv 431b3060563ed44cc68c7056ece2f85e_***_John Tucker
author John Tucker
author2 Jan Aldert Bergstra
John Tucker
format Journal article
container_title Transmathematica
container_volume 2024
publishDate 2024
institution Swansea University
issn 2632-9212
doi_str_mv 10.36285/tm.97
publisher Transmathematica
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 A common meadow is an enrichment of a field with a division operator and an error value to make division total. A signed common meadow enriches a common meadow with a sign function that can be equationally axiomatised; the sign function can simulate an ordering on the underlying field but is not limited to orderings. In particular, of mathematical interest are the weakly signed common meadows. The prime example of a weakly signed common meadow is an expansion of a common meadow ofcomplex numbers with a weak sign function. We show that all common meadows may be enlarged to a weakly signed common meadow. A special case are the 4-signed common meadows, which are precisely the enlargements of ordered fields. To illustrate the equational calculus for signed common meadows, we use it as a foundation for building a probability calculus and derive some classical formulae.
published_date 2024-06-14T11:07:52Z
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score 11.035634

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