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Meadows and the equational specification of division

J.A Bergstra, Y Hirshfeld, J.V Tucker, John Tucker Orcid Logo

Theoretical Computer Science, Volume: 410, Issue: 12-13, Pages: 1261 - 1271

Swansea University Author: John Tucker Orcid Logo

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DOI (Published version): 10.1016/j.tcs.2008.12.015

Abstract

The rational, real and complex numbers with their standard operations, including division, are partial algebras specified by the axiomatic concept of a field. Since the class of fields cannot be defined by equations, the theory of equational specifications of data types cannot use field theory in ap...

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Published in: Theoretical Computer Science
ISSN: 0304-3975
Published: Amsterdam Elsevier 2009
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URI: https://cronfa.swan.ac.uk/Record/cronfa7203
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spelling 2015-10-15T10:33:33.8896291 v2 7203 2012-02-23 Meadows and the equational specification of division 431b3060563ed44cc68c7056ece2f85e 0000-0003-4689-8760 John Tucker John Tucker true false 2012-02-23 SCS The rational, real and complex numbers with their standard operations, including division, are partial algebras specified by the axiomatic concept of a field. Since the class of fields cannot be defined by equations, the theory of equational specifications of data types cannot use field theory in applications to number systems based upon rational, real and complex numbers. We study a new axiomatic concept for number systems with division that uses only equations: a meadow is a commutative ring with a total inverse operator satisfying two equations which imply 0−1=0. All fields and products of fields can be viewed as meadows. After reviewing alternate axioms for inverse, we start the development of a theory of meadows. We give a general representation theorem for meadows and find, as a corollary, that the conditional equational theory of meadows coincides with the conditional equational theory of zero totalized fields. We also prove representation results for meadows of finite characteristic. Journal Article Theoretical Computer Science 410 12-13 1261 1271 Elsevier Amsterdam 0304-3975 Totalized fields; Meadow; Division-by-zero; Total versus partial functions; Representation theorems; Initial algebras; Equational specifications; von Neumann regular ring; Finite meadows; Finite fields 31 12 2009 2009-12-31 10.1016/j.tcs.2008.12.015 http://www.sciencedirect.com/science/article/pii/S0304397508008967 COLLEGE NANME Computer Science COLLEGE CODE SCS Swansea University 2015-10-15T10:33:33.8896291 2012-02-23T17:01:48.0000000 Faculty of Science and Engineering School of Mathematics and Computer Science - Computer Science J.A Bergstra 1 Y Hirshfeld 2 J.V Tucker 3 John Tucker 0000-0003-4689-8760 4
title Meadows and the equational specification of division
spellingShingle Meadows and the equational specification of division
John Tucker
title_short Meadows and the equational specification of division
title_full Meadows and the equational specification of division
title_fullStr Meadows and the equational specification of division
title_full_unstemmed Meadows and the equational specification of division
title_sort Meadows and the equational specification of division
author_id_str_mv 431b3060563ed44cc68c7056ece2f85e
author_id_fullname_str_mv 431b3060563ed44cc68c7056ece2f85e_***_John Tucker
author John Tucker
author2 J.A Bergstra
Y Hirshfeld
J.V Tucker
John Tucker
format Journal article
container_title Theoretical Computer Science
container_volume 410
container_issue 12-13
container_start_page 1261
publishDate 2009
institution Swansea University
issn 0304-3975
doi_str_mv 10.1016/j.tcs.2008.12.015
publisher Elsevier
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 http://www.sciencedirect.com/science/article/pii/S0304397508008967
document_store_str 0
active_str 0
description The rational, real and complex numbers with their standard operations, including division, are partial algebras specified by the axiomatic concept of a field. Since the class of fields cannot be defined by equations, the theory of equational specifications of data types cannot use field theory in applications to number systems based upon rational, real and complex numbers. We study a new axiomatic concept for number systems with division that uses only equations: a meadow is a commutative ring with a total inverse operator satisfying two equations which imply 0−1=0. All fields and products of fields can be viewed as meadows. After reviewing alternate axioms for inverse, we start the development of a theory of meadows. We give a general representation theorem for meadows and find, as a corollary, that the conditional equational theory of meadows coincides with the conditional equational theory of zero totalized fields. We also prove representation results for meadows of finite characteristic.
published_date 2009-12-31T03:08:56Z
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score 11.017731

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