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Trusses: Between braces and rings

Tomasz Brzezinski Orcid Logo

Transactions of the American Mathematical Society, Volume: 372, Issue: 6, Pages: 4149 - 4176

Swansea University Author: Tomasz Brzezinski Orcid Logo

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DOI (Published version): 10.1090/tran/7705

Abstract

In an attempt to understand the origins and the nature of the law binding two group operations together into a skew brace, introduced in [L. Guarnieri & L. Vendramin, Math. Comp. 86 (2017), 2519–2534] as a non-Abelian version of the brace distributive law of [W. Rump, J. Algebra 307 (2007), 153–...

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Published in: Transactions of the American Mathematical Society
ISSN: 0002-9947 1088-6850
Published: American Mathematical Society (AMS) 2018
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URI: https://cronfa.swan.ac.uk/Record/cronfa44243
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spelling 2020-06-26T18:26:00.6021170 v2 44243 2018-09-15 Trusses: Between braces and rings 30466d840b59627325596fbbb2c82754 0000-0001-6270-3439 Tomasz Brzezinski Tomasz Brzezinski true false 2018-09-15 SMA In an attempt to understand the origins and the nature of the law binding two group operations together into a skew brace, introduced in [L. Guarnieri & L. Vendramin, Math. Comp. 86 (2017), 2519–2534] as a non-Abelian version of the brace distributive law of [W. Rump, J. Algebra 307 (2007), 153–170] and [F. Cedo, E. Jespers & J. Okninski, Commun. Math. Phys. 327 (2014), 101–116], the notion of a skew truss is proposed. A skew truss consists of a set with a group operation and a semigroup operation connected by a modified distributive law that interpolates between that of a ring and a brace. It is shown that a particular action and a cocycle characteristic of skew braces are already present in a skew truss; in fact the interpolating function is a 1-cocycle, the bijecitivity of which indicates the existence of an operation that turns a truss into a brace. Furthermore, if the group structure in a two-sided truss is Abelian, then there is an associated ring – another feature characteristic of a two-sided brace. To characterise a morphism of trusses, a pith is defined as a particular subset of the domain consisting of subsets termed chambers, which contains the kernel of the morphism as a group homomorphism. In the case of both rings and braces piths coincide with kernels. In general the pith of a morphism is a sub-semigroup of the domain and, if additional properties are satisfied, a pith is an N+-graded semigroup. Finally, giving heed to [I. Angiono, C. Galindo & L. Vendramin, Proc. Amer. Math. Soc. 145 (2017), 1981–1995] we linearise trusses and thus define Hopf trusses and study their properties, from which, in parallel to the set-theoretic case, some properties of Hopf braces are shown to follow. Journal Article Transactions of the American Mathematical Society 372 6 4149 4176 American Mathematical Society (AMS) 0002-9947 1088-6850 truss; brace 21 11 2018 2018-11-21 10.1090/tran/7705 http://dx.doi.org/10.1090/tran/7705 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University 2020-06-26T18:26:00.6021170 2018-09-15T16:09:01.5968927 Tomasz Brzezinski 0000-0001-6270-3439 1 0044243-15092018160949.pdf truss_tams_final.pdf 2018-09-15T16:09:49.1770000 Output 384629 application/pdf Accepted Manuscript true 2018-09-15T00:00:00.0000000 Distributed under the terms of a CC BY-NC-ND Creative Commons License. true eng
title Trusses: Between braces and rings
spellingShingle Trusses: Between braces and rings
Tomasz Brzezinski
title_short Trusses: Between braces and rings
title_full Trusses: Between braces and rings
title_fullStr Trusses: Between braces and rings
title_full_unstemmed Trusses: Between braces and rings
title_sort Trusses: Between braces and rings
author_id_str_mv 30466d840b59627325596fbbb2c82754
author_id_fullname_str_mv 30466d840b59627325596fbbb2c82754_***_Tomasz Brzezinski
author Tomasz Brzezinski
author2 Tomasz Brzezinski
format Journal article
container_title Transactions of the American Mathematical Society
container_volume 372
container_issue 6
container_start_page 4149
publishDate 2018
institution Swansea University
issn 0002-9947
1088-6850
doi_str_mv 10.1090/tran/7705
publisher American Mathematical Society (AMS)
url http://dx.doi.org/10.1090/tran/7705
document_store_str 1
active_str 0
description In an attempt to understand the origins and the nature of the law binding two group operations together into a skew brace, introduced in [L. Guarnieri & L. Vendramin, Math. Comp. 86 (2017), 2519–2534] as a non-Abelian version of the brace distributive law of [W. Rump, J. Algebra 307 (2007), 153–170] and [F. Cedo, E. Jespers & J. Okninski, Commun. Math. Phys. 327 (2014), 101–116], the notion of a skew truss is proposed. A skew truss consists of a set with a group operation and a semigroup operation connected by a modified distributive law that interpolates between that of a ring and a brace. It is shown that a particular action and a cocycle characteristic of skew braces are already present in a skew truss; in fact the interpolating function is a 1-cocycle, the bijecitivity of which indicates the existence of an operation that turns a truss into a brace. Furthermore, if the group structure in a two-sided truss is Abelian, then there is an associated ring – another feature characteristic of a two-sided brace. To characterise a morphism of trusses, a pith is defined as a particular subset of the domain consisting of subsets termed chambers, which contains the kernel of the morphism as a group homomorphism. In the case of both rings and braces piths coincide with kernels. In general the pith of a morphism is a sub-semigroup of the domain and, if additional properties are satisfied, a pith is an N+-graded semigroup. Finally, giving heed to [I. Angiono, C. Galindo & L. Vendramin, Proc. Amer. Math. Soc. 145 (2017), 1981–1995] we linearise trusses and thus define Hopf trusses and study their properties, from which, in parallel to the set-theoretic case, some properties of Hopf braces are shown to follow.
published_date 2018-11-21T03:58:13Z
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