2025/10/27 by Facchini, Alberto
#16Y99 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2510.23185
In this paper we extend to left skew trusses (T,+,∘,σ) previous work on left skew rings. We had presented a left skew ring as a group (N,+) with two binary operations ∘ and ⋅ with ∘ associative, ⋅ left distributive over the addition + of the group, and such that the difference of the two operations ∘ and ⋅ is the binary operation π1\colon N× N→ N. Here we extend this idea to the left skew trusses introduced in 2019 by Brzeziński, replacing the operation π1 with the binary operation σπ1\colon T× T→ T. The case where the semigroup morphism λT\colon T→ \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,+,∘,σ) with λT\colon (T,∘)→ \End_\Gp(T,+) a constant semigroup morphism and σ,λT0 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 λT\colon (T,∘)→ \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.