## Trusses: Between braces and rings

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- by Tomasz Brzeziński PDF
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**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 Cedó, Jespers, and Okniński, 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 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 Brzeziński**- Affiliation: Department of Mathematics, Swansea University, Swansea University Bay Campus, Fabian Way, Swansea SA1 8EN, United Kingdom; and Department of Mathematics, University of Białystok, K. Ciołkowskiego 1M, 15-245 Białystok, 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