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Topics in the theory of trusses / BERNARD RYBOLOWICZ

Swansea University Author: BERNARD RYBOLOWICZ

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DOI (Published version): 10.23889/SUthesis.58273

Abstract

In 2017 a truss was defined. Thus one can say that the theory of trusses is new and not yet well-established. In recent years trusses start to gain attention due to their connections to ring theory and braces. Braces are closely related to solutions of set-theoretic Yang-Baxter equations, which can l...

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Published: Swansea 2021
Institution: Swansea University
Degree level: Doctoral
Degree name: Ph.D
Supervisor: Brzezinski, Tomasz
URI: https://cronfa.swan.ac.uk/Record/cronfa58273
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first_indexed 2021-10-08T12:59:59Z
last_indexed 2021-10-09T03:23:17Z
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spelling 2021-10-08T14:12:36.0430310 v2 58273 2021-10-08 Topics in the theory of trusses 5767cf3c129843fa32a0383037b1fecc BERNARD RYBOLOWICZ BERNARD RYBOLOWICZ true false 2021-10-08 In 2017 a truss was defined. Thus one can say that the theory of trusses is new and not yet well-established. In recent years trusses start to gain attention due to their connections to ring theory and braces. Braces are closely related to solutions of set-theoretic Yang-Baxter equations, which can lead to applications of trusses in physics. In this thesis, we study connections among groups, heaps, rings, modules, braces and trusses. In the beginning, one can find a description in details of free heaps and coproducts of Abelian heaps. Both constructions are applied to describe a functor from the category of heaps to the category of groups. We establish a connection between unital near-trusses and skew left braces. We show that for a specific choice of congruence on a unital near-truss the quotient is a brace. We also prove that if one localises a regular unital near-truss without an absorber, the result is a skew left brace. In this thesis, one can find many small results on categories of heaps, trusses and modules over a truss. Methods to extend trusses to unital trusses and rings are presented. Then first one allows us to show that a category of modules over a truss is isomorphic with the category of modules over its extension to the unital truss. The second method establishes a deep connection between rings and trusses, i.e. every truss is an equivalence class of some congruence on some specific ring. We present the ring construction. Using this result, we introduce the definition of a minimal extension of a truss into a ring. We construct tensor product and free modules over trusses. The Eilenberg-Watts theorem for modules over trusses is stated and proven. Thus the Morita theory for modules over trusses is developed. The thesis is concluded with results on projectivity and decompositions through a product of the modules. E-Thesis Swansea 8 10 2021 2021-10-08 10.23889/SUthesis.58273 COLLEGE NANME COLLEGE CODE Swansea University Brzezinski, Tomasz Doctoral Ph.D 2021-10-08T14:12:36.0430310 2021-10-08T13:57:33.9410744 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics BERNARD RYBOLOWICZ 1 58273__21120__dc860e9696b34cccb294866e8b40c441.pdf Rybolowicz_Bernard_PhD_Thesis_Final_Redacted_Signature.pdf 2021-10-08T14:06:33.5692884 Output 2415288 application/pdf E-Thesis – open access true Copyright: The author, Bernard Rybolowicz, 2021. true eng
title Topics in the theory of trusses
spellingShingle Topics in the theory of trusses
BERNARD RYBOLOWICZ
title_short Topics in the theory of trusses
title_full Topics in the theory of trusses
title_fullStr Topics in the theory of trusses
title_full_unstemmed Topics in the theory of trusses
title_sort Topics in the theory of trusses
author_id_str_mv 5767cf3c129843fa32a0383037b1fecc
author_id_fullname_str_mv 5767cf3c129843fa32a0383037b1fecc_***_BERNARD RYBOLOWICZ
author BERNARD RYBOLOWICZ
author2 BERNARD RYBOLOWICZ
format E-Thesis
publishDate 2021
institution Swansea University
doi_str_mv 10.23889/SUthesis.58273
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 - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description In 2017 a truss was defined. Thus one can say that the theory of trusses is new and not yet well-established. In recent years trusses start to gain attention due to their connections to ring theory and braces. Braces are closely related to solutions of set-theoretic Yang-Baxter equations, which can lead to applications of trusses in physics. In this thesis, we study connections among groups, heaps, rings, modules, braces and trusses. In the beginning, one can find a description in details of free heaps and coproducts of Abelian heaps. Both constructions are applied to describe a functor from the category of heaps to the category of groups. We establish a connection between unital near-trusses and skew left braces. We show that for a specific choice of congruence on a unital near-truss the quotient is a brace. We also prove that if one localises a regular unital near-truss without an absorber, the result is a skew left brace. In this thesis, one can find many small results on categories of heaps, trusses and modules over a truss. Methods to extend trusses to unital trusses and rings are presented. Then first one allows us to show that a category of modules over a truss is isomorphic with the category of modules over its extension to the unital truss. The second method establishes a deep connection between rings and trusses, i.e. every truss is an equivalence class of some congruence on some specific ring. We present the ring construction. Using this result, we introduce the definition of a minimal extension of a truss into a ring. We construct tensor product and free modules over trusses. The Eilenberg-Watts theorem for modules over trusses is stated and proven. Thus the Morita theory for modules over trusses is developed. The thesis is concluded with results on projectivity and decompositions through a product of the modules.
published_date 2021-10-08T04:14:40Z
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score 11.01409

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