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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.23185 |
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Table of Contents:
- 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 $τ$.