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Totalising Partial Algebras

Jan Aldert Bergstra, John Tucker Orcid Logo

Transmathematica, Volume: 2022, Pages: 1 - 22

Swansea University Author: John Tucker Orcid Logo

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DOI (Published version): 10.36285/tm.57

Abstract

We will examine totalising a partial operation in a general algebra by using an absorbtive element, bottom, such as an error flag. We then focus on the simplest example of a partial operation, namely subtraction on the natural numbers: n - m is undefined whenever n < m. We examine the use of bott...

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Published in: Transmathematica
ISSN: 2632-9212
Published: Reading Transmathematica 2022
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URI: https://cronfa.swan.ac.uk/Record/cronfa61536
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first_indexed 2022-10-11T21:09:29Z
last_indexed 2023-01-13T19:22:20Z
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spelling 2022-10-28T11:24:08.9859826 v2 61536 2022-10-11 Totalising Partial Algebras 431b3060563ed44cc68c7056ece2f85e 0000-0003-4689-8760 John Tucker John Tucker true false 2022-10-11 SCS We will examine totalising a partial operation in a general algebra by using an absorbtive element, bottom, such as an error flag. We then focus on the simplest example of a partial operation, namely subtraction on the natural numbers: n - m is undefined whenever n < m. We examine the use of bottom in algebraic structures for the natural numbers, especially semigroups and semirings. We axiomatise this totalisation process and introduce the algebraic concept of a team, being an additive cancellative semigroup with totalised subtraction. Also, with the natural numbers in mind, we introduce the property of being generated by an iterative function, which we call a splinter. We prove a number of theorems about the algebraic specification of datatypes of natural numbers. Journal Article Transmathematica 2022 1 22 Transmathematica Reading 2632-9212 partiality, meadows, teams, splinters, absorptive elements 28 3 2022 2022-03-28 10.36285/tm.57 COLLEGE NANME Computer Science COLLEGE CODE SCS Swansea University Other 2022-10-28T11:24:08.9859826 2022-10-11T21:58:41.8223782 Faculty of Science and Engineering School of Mathematics and Computer Science - Computer Science Jan Aldert Bergstra 1 John Tucker 0000-0003-4689-8760 2 61536__25418__383e00de73e14eb5af53cdee7a057b55.pdf Totalising Partial Algebras-Teams and Splinters.pdf 2022-10-11T22:09:04.5785956 Output 349288 application/pdf Version of Record true Copyright 2022 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 Totalising Partial Algebras
spellingShingle Totalising Partial Algebras
John Tucker
title_short Totalising Partial Algebras
title_full Totalising Partial Algebras
title_fullStr Totalising Partial Algebras
title_full_unstemmed Totalising Partial Algebras
title_sort Totalising Partial Algebras
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 2022
container_start_page 1
publishDate 2022
institution Swansea University
issn 2632-9212
doi_str_mv 10.36285/tm.57
publisher Transmathematica
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
document_store_str 1
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description We will examine totalising a partial operation in a general algebra by using an absorbtive element, bottom, such as an error flag. We then focus on the simplest example of a partial operation, namely subtraction on the natural numbers: n - m is undefined whenever n < m. We examine the use of bottom in algebraic structures for the natural numbers, especially semigroups and semirings. We axiomatise this totalisation process and introduce the algebraic concept of a team, being an additive cancellative semigroup with totalised subtraction. Also, with the natural numbers in mind, we introduce the property of being generated by an iterative function, which we call a splinter. We prove a number of theorems about the algebraic specification of datatypes of natural numbers.
published_date 2022-03-28T04:20:25Z
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