[Mathematical Physics Seminar]
Tomasz Brzezinski (Swansea University)
How rigid is the distributive law?
Recently, the notion of a brace was introduced by Rump in the context of solving set-theoretic Yang-Baxter equation. In a formulation of Cedo, Jespers and Okninski, a (left) brace consists of a set A with two binary operations ◦ and +, such that (A, +) is an abelian group, (A, ◦) is a group, and operations are connected by the following brace distributive law:
(1) a ◦ (b + c) = a ◦ b + a ◦ c − a.
In this talk we probe a possibility of modifying the brace distributive law in a way that connects it with the usual distributive law for rings.
Thus we study a set A with two operations connected by
(2) a ◦ (b + c) = a ◦ b + a ◦ c − σ(a),
where σ is any function A → A. We study the restrictions that need to be put on binary operations and on σ, and show that the both brace and (usual) ring distributive laws are characterised by a particular robustness. We place this discussion in a more general context of skew braces introduced by Guarnieri and Vendramin and skew or near-rings, and show that the ad-hoc modification of the distributive law such as in (2) has in fact a very natural (and quite far from the ad-hoc feel) formulation.