2017/10/08 by Brzeziński, Tomasz · 4 citations
#16T05 #16Y99 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1710.02870
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. 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. Cedó, E. Jespers & J. Okniński, 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 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.