The algebraic structure of left semi-trusses

TL;DR

This paper explores the algebraic structure of left semi-trusses, especially those related to the Yang-Baxter equation, revealing their properties and connections to solutions of the equation.

Contribution

It characterizes the structure of left semi-trusses in the context of the Yang-Baxter equation and introduces a correspondence with left semi-braces.

Findings

In the finite case, the additive structure is a completely regular semigroup.

Any almost left semi-brace can be associated with a left semi-brace.

Set-theoretic solutions from almost left semi-braces originate from this association.

Abstract

The distributive laws of ring theory are fundamental equalities in algebra. However, recently in the study of the Yang-Baxter equation, many algebraic structures with alternative "distributive" laws were defined. In an effort to study these "left distributive" laws and the interaction they entail on the algebraic structures, Brzezi\'nski introduced skew left trusses and left semi-trusses. In particular the class of left semi-trusses is very wide, since it contains all rings, associative algebras and distributive lattices. In this paper, we investigate the subclass of left semi-trusses that behave like the algebraic structures that came up in the study of the Yang-Baxter equation. We study the interaction of the operations and what this interaction entails on their respective semigroups. In particular, we prove that in the finite case the additive structure is a completely regular semigroup. Secondly, we apply our results on a particular instance of a left semi-truss called an almost left semi-brace, introduced by Miccoli to study its algebraic structure. In particular, we show that one can associate a left semi-brace to any almost left semi-brace. Furthermore, we show that the set-theoretic solutions of the Yang-Baxter equation originating from almost left semi-braces arise from this correspondence.