Trusses, ditrusses, weak trusses
Alberto Facchini
TL;DR
The paper develops left skew trusses as a natural generalization of left skew rings by introducing a left skew distributive law mediated by an additional unary map $\sigma$, and then reinterprets these trusses as left ditrusses via the difference $\circ-\cdot=\sigma\pi_1$. It proves canonical category isomorphisms, notably between left skew trusses with a constant semigroup morphism and associative interchange near-rings, and establishes a duality exchanging $\sigma$ and $\tau$ in the constant-endomorphism setting. By unpacking the interplay among $+,\circ,\cdot,\sigma$ and endomorphisms, the authors derive first structural results (endomorphism properties, decompositions, and ideals/congruences) and connect distributivity properties to associativity. The work unifies several algebraic structures—left near-rings, left skew rings, left ditrusses, and interchange near-rings—under a common categorical and universal-algebraic framework, offering new tools for studying semidirect-product decompositions and potential links to Yang-Baxter equation constructions. Overall, it provides a rigorous bridge between truss-like systems and interchange near-rings, enabling systematic translation between operational frameworks and revealing symmetries between key endomorphisms.
Abstract
In this paper we extend to left skew trusses $(T,+,\circ,σ)$ 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 $π_1\colon N\times N\to N$. Here we extend this idea to the left skew trusses introduced in 2019 by Brzeziński, replacing the operation $π_1$ with the binary operation $σπ_1\colon T\times T\to T$. The case where the semigroup morphism $λ^T\colon T\to \End_\Gp(T,+)$ 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 $(T,+,\circ,σ)$ with $λ^T\colon (T,\circ)\to \End_\Gp(T,+)$ a constant semigroup morphism and $σ,λ^T_0$ image-commuting idempotent endomorphisms and the category of all associative interchange near-rings. Interchange near-rings were introduced by Edmunds in 2016. When $σ$ is an idempotent group endomorphism of the group $(T,+)$ and $λ^T\colon (T,\circ)\to \End_\Gp(T,+)$ is a semigroup morphism constantly equal to a group endomorphism $τ$, we also get a sort of duality exchanging the mappings $σ$ and $τ$.