Trusses, ditrusses, weak trusses

Abstract

In this paper we extend to left skew trusses $(T,+,\circ,\sigma)$ previous work on left skew rings. We had presented a left skew ring as a group $(N,+)$ with two binary operations $\circ$ and $\cdot$ with $\circ$ associative, $\cdot$ left distributive over the addition $+$ of the group, and such that the difference of the two operations $\circ$ and $\cdot$ is the binary operation $\pi_1\colon N\times N\to N$. Here we extend this idea to the left skew trusses introduced in 2019 by Brzezi\'nski, replacing the operation $\pi_1$ with the binary operation $\sigma\pi_1\colon T\times T\to T$. The case where the semigroup morphism $\lambda^T\colon T\to \End_\Gp(T,+)$ is constant turns out to be particular interesting. We get several canonical category isomorphisms. For instance, we get a category isomorphism between the category of all left skew trusses $(T,+,\circ,\sigma)$ with $\lambda^T\colon (T,\circ)\to \End_\Gp(T,+)$ a constant semigroup morphism and $\sigma,\lambda^T_0$ image-commuting idempotent endomorphisms and the category of all associative interchange near-rings. Interchange near-rings were introduced by Edmunds in 2016. When $\sigma$ is an idempotent group endomorphism of the group $(T,+)$ and $\lambda^T\colon (T,\circ)\to \End_\Gp(T,+)$ is a semigroup morphism constantly equal to a group endomorphism $\tau$, we also get a sort of duality exchanging the mappings $\sigma$ and $\tau$.