Trusses: between braces and rings

TL;DR

This paper introduces skew trusses, a generalization of braces and rings, exploring their algebraic structure, morphisms, and linearizations into Hopf trusses, revealing new insights into their properties and relationships.

Contribution

It proposes the concept of skew trusses, linking group and semigroup operations via a cocycle, and extends to Hopf trusses, enriching the algebraic framework of braces.

Findings

Skew trusses interpolate between rings and braces.

A cocycle characterizes the transition from truss to brace.

Hopf trusses generalize the set-theoretic structures.

Abstract

In an attempt to understand the origins and the nature of the law binding together two group operations into a {\em skew brace}, introduced in [L.\ Guarnieri \& L.\ Vendramin, Math.\ Comp.\ \textbf{86} (2017), 2519--2534] as a non-Abelian version of the {\em brace distributive law} of [W.\ Rump, J.\ Algebra {\bf 307} (2007), 153--170] and [F.\ Ced\'o, E.\ Jespers \& J.\ Okni\'nski, Commun.\ Math.\ Phys.\ {\bf 327} (2014), 101--116], the notion of a {\em 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 {\em pith} is defined as a particular subset of the domain consisting of subsets termed {\em 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 a $\mathbb{N}_+$-graded semigroup. Finally, giving heed to [I.\ Angiono, C.\ Galindo \& L.\ Vendramin, Proc.\ Amer.\ Math.\ Soc.\ {\bf 145} (2017), 1981--1995] we linearise trusses and thus define {\em Hopf trusses} and study their properties, from which, in parallel to the set-theoretic case, some properties of Hopf braces are shown to follow.