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The Transrational Numbers as an Abstract Data Type

Jan Aldert Bergstra, John Tucker Orcid Logo

Transmathematica, Volume: 2020, Pages: 1 - 29

Swansea University Author: John Tucker Orcid Logo

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DOI (Published version): 10.36285/tm.47

Abstract

In an arithmetical structure one can make division a total function by defining 1/0 to be an element of the structure, or by adding a new element, such as an error element also denoted with a new constant symbol, an unsigned infinity or one or both signed infinities, one positive and one negative. W...

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Published in: Transmathematica
ISSN: 2632-9212
Published: Transmathematica 2020
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URI: https://cronfa.swan.ac.uk/Record/cronfa56723
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spelling 2021-09-16T11:20:23.7625530 v2 56723 2021-04-22 The Transrational Numbers as an Abstract Data Type 431b3060563ed44cc68c7056ece2f85e 0000-0003-4689-8760 John Tucker John Tucker true false 2021-04-22 SCS In an arithmetical structure one can make division a total function by defining 1/0 to be an element of the structure, or by adding a new element, such as an error element also denoted with a new constant symbol, an unsigned infinity or one or both signed infinities, one positive and one negative. We define an enlargement of a field to a transfield, in which division is totalised by setting 1/0 equal to the positive infinite value and -1/0 equal to the negative infinite value , and which also contains an error element to help control their effects. We construct the transrational numbers as a transfield of the field of rational numbers and consider it as an abstract data type. We give it an equational specification under initial algebra semantics. Journal Article Transmathematica 2020 1 29 Transmathematica 2632-9212 Fields, Meadows, Rational numbers, Infinity, Errors 16 12 2020 2020-12-16 10.36285/tm.47 COLLEGE NANME Computer Science COLLEGE CODE SCS Swansea University 2021-09-16T11:20:23.7625530 2021-04-22T22:42:06.9929862 Faculty of Science and Engineering School of Mathematics and Computer Science - Computer Science Jan Aldert Bergstra 1 John Tucker 0000-0003-4689-8760 2 56723__19739__aebc3bb3387b488fba0e454a691d6dbb.pdf Bergstra & Tucker-Transrational numbers as an ADT.pdf 2021-04-22T22:51:00.4675550 Output 333158 application/pdf Version of Record true Released under the terms of a Creative Commons Attribution Share Alike 4.0 license true eng http://creativecommons.org/licenses/by-sa/4.0
title The Transrational Numbers as an Abstract Data Type
spellingShingle The Transrational Numbers as an Abstract Data Type
John Tucker
title_short The Transrational Numbers as an Abstract Data Type
title_full The Transrational Numbers as an Abstract Data Type
title_fullStr The Transrational Numbers as an Abstract Data Type
title_full_unstemmed The Transrational Numbers as an Abstract Data Type
title_sort The Transrational Numbers as an Abstract Data Type
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 2020
container_start_page 1
publishDate 2020
institution Swansea University
issn 2632-9212
doi_str_mv 10.36285/tm.47
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
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description In an arithmetical structure one can make division a total function by defining 1/0 to be an element of the structure, or by adding a new element, such as an error element also denoted with a new constant symbol, an unsigned infinity or one or both signed infinities, one positive and one negative. We define an enlargement of a field to a transfield, in which division is totalised by setting 1/0 equal to the positive infinite value and -1/0 equal to the negative infinite value , and which also contains an error element to help control their effects. We construct the transrational numbers as a transfield of the field of rational numbers and consider it as an abstract data type. We give it an equational specification under initial algebra semantics.
published_date 2020-12-16T04:11:54Z
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