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Trusses: Between braces and rings
Transactions of the American Mathematical Society, Volume: 372, Issue: 6, Pages: 4149 - 4176
Swansea University Author: Tomasz Brzezinski
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In an attempt to understand the origins and the nature of the law binding two group operations together into a skew brace, introduced in [L. Guarnieri & L. Vendramin, Math. Comp. 86 (2017), 2519–2534] as a non-Abelian version of the brace distributive law of [W. Rump, J. Algebra 307 (2007), 153–...
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In an attempt to understand the origins and the nature of the law binding two group operations together into a skew brace, introduced in [L. Guarnieri & L. Vendramin, Math. Comp. 86 (2017), 2519–2534] as a non-Abelian version of the brace distributive law of [W. Rump, J. Algebra 307 (2007), 153–170] and [F. Cedo, E. Jespers & J. Okninski, Commun. Math. Phys. 327 (2014), 101–116], 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 characterise 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 sub-semigroup of the domain and, if additional properties are satisfied, a pith is an N+-graded semigroup. Finally, giving heed to [I. Angiono, C. Galindo & L. Vendramin, Proc. Amer. Math. Soc. 145 (2017), 1981–1995] we linearise 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.