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Journal article 1184 views

Non-commutative orders. A preliminary study

T Brzeziński, Tomasz Brzezinski Orcid Logo

Acta Physica Polonica B Proceedings Supplement, Volume: 4, Issue: 3, Pages: 273 - 286

Swansea University Author: Tomasz Brzezinski Orcid Logo

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DOI (Published version): 10.5506/APhysPolBSupp.4.273

Abstract

The first steps towards linearisation of partial orders and equivalence relations are described. The definitions of partial orders and equivalence relations (on sets) are formulated in a way that is standard in category theory and that makes the linearisation (almost) automatic. The linearisation is...

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Published in: Acta Physica Polonica B Proceedings Supplement
Published: 2011
Online Access: http://th-www.if.uj.edu.pl/acta/sup4/abs/s4p0273.htm
URI: https://cronfa.swan.ac.uk/Record/cronfa7448
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spelling 2011-10-01T00:00:00.0000000 v2 7448 2012-02-23 Non-commutative orders. A preliminary study 30466d840b59627325596fbbb2c82754 0000-0001-6270-3439 Tomasz Brzezinski Tomasz Brzezinski true false 2012-02-23 SMA The first steps towards linearisation of partial orders and equivalence relations are described. The definitions of partial orders and equivalence relations (on sets) are formulated in a way that is standard in category theory and that makes the linearisation (almost) automatic. The linearisation is then achieved by replacing sets by coalgebras and the Cartesian product by the tensor product of vector spaces. As a result, definitions of orders and equivalence relations on coalgebras are proposed. These are illustrated by explicit examples that include relations on coalgebras spanned by grouplike elements (or linearised sets), the diagonal relation, and an order on a three-dimensional non-cocommutative coalgebra. Although relations on coalgebras are defined for vector spaces, all the definitions are formulated in a way that is immediately applicable to other braided monoidal categories. Journal Article Acta Physica Polonica B Proceedings Supplement 4 3 273 286 31 12 2011 2011-12-31 10.5506/APhysPolBSupp.4.273 http://th-www.if.uj.edu.pl/acta/sup4/abs/s4p0273.htm COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University 2011-10-01T00:00:00.0000000 2012-02-23T17:01:55.0000000 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics T Brzeziński 1 Tomasz Brzezinski 0000-0001-6270-3439 2
title Non-commutative orders. A preliminary study
spellingShingle Non-commutative orders. A preliminary study
Tomasz Brzezinski
title_short Non-commutative orders. A preliminary study
title_full Non-commutative orders. A preliminary study
title_fullStr Non-commutative orders. A preliminary study
title_full_unstemmed Non-commutative orders. A preliminary study
title_sort Non-commutative orders. A preliminary study
author_id_str_mv 30466d840b59627325596fbbb2c82754
author_id_fullname_str_mv 30466d840b59627325596fbbb2c82754_***_Tomasz Brzezinski
author Tomasz Brzezinski
author2 T Brzeziński
Tomasz Brzezinski
format Journal article
container_title Acta Physica Polonica B Proceedings Supplement
container_volume 4
container_issue 3
container_start_page 273
publishDate 2011
institution Swansea University
doi_str_mv 10.5506/APhysPolBSupp.4.273
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 - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
url http://th-www.if.uj.edu.pl/acta/sup4/abs/s4p0273.htm
document_store_str 0
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description The first steps towards linearisation of partial orders and equivalence relations are described. The definitions of partial orders and equivalence relations (on sets) are formulated in a way that is standard in category theory and that makes the linearisation (almost) automatic. The linearisation is then achieved by replacing sets by coalgebras and the Cartesian product by the tensor product of vector spaces. As a result, definitions of orders and equivalence relations on coalgebras are proposed. These are illustrated by explicit examples that include relations on coalgebras spanned by grouplike elements (or linearised sets), the diagonal relation, and an order on a three-dimensional non-cocommutative coalgebra. Although relations on coalgebras are defined for vector spaces, all the definitions are formulated in a way that is immediately applicable to other braided monoidal categories.
published_date 2011-12-31T03:09:15Z
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