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Transactions of the American Mathematical Society

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ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Trusses: Between braces and rings
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by Tomasz Brzezi艅ski PDF
Trans. Amer. Math. Soc. 372 (2019), 4149-4176 Request permission

Abstract:

In an attempt to understand the origins and the nature of the law binding two group operations together into a skew brace, introduced by Guarnieri and Vendramin as a non-Abelian version of the brace distributive law of Rump and Ced贸, Jespers, and Okni艅ski, 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鈥攁nother feature characteristic of a two-sided brace. To characterize 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 subsemigroup of the domain and, if additional properties are satisfied, pith is an ${\mathbb N}_+$-graded semigroup. Finally, giving heed to ideas of Angiono, Galindo, and Vendramin, we linearize 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.
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Additional Information
  • Tomasz Brzezi艅ski
  • Affiliation: Department of Mathematics, Swansea University, Swansea University Bay Campus, Fabian Way, Swansea SA1 8EN, United Kingdom; and Department of Mathematics, University of Bia艂ystok, K. Cio艂kowskiego 1M, 15-245 Bia艂ystok, Poland
  • Email: T.Brzezinski@swansea.ac.uk
  • Received by editor(s): January 30, 2018
  • Received by editor(s) in revised form: August 13, 2018
  • Published electronically: November 21, 2018
  • Additional Notes: The research presented in this paper is partially supported by Polish National Science Centre grant 2016/21/B/ST1/02438.
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 4149-4176
  • MSC (2010): Primary 16Y99, 16T05
  • DOI: https://doi.org/10.1090/tran/7705
  • MathSciNet review: 4009388