School of Mathematical Sciences

External Seminar: Tomasz Brzezinski (Swansea University)

Physics B21
Wednesday 14th February 2018 (14:00-15:00)
Steffen Gielen

[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.

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