Trusses: Between braces and rings

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.