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Transactions of the American Mathematical Society

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Trusses: Between braces and rings

Author: Tomasz Brzeziński
Journal: Trans. Amer. Math. Soc. 372 (2019), 4149-4176
MSC (2010): Primary 16Y99, 16T05
Published electronically: November 21, 2018
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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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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

Keywords: Truss, skew brace, Hopf brace, distributive law
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.
Article copyright: © Copyright 2018 American Mathematical Society