左半桁架的代数结构
The algebraic structure of left semi-trusses
摘要：
环论的分配律是代数学中的基本等式。然而，近年来在杨-巴克斯特方程的研究中，许多代数结构被定义为具有不同的“分配”律。在努力研究这些“左分配”的法律和相互作用，他们需要的代数结构，布热津斯基介绍了斜左桁架和左半桁架。特别是左半桁架类是非常广泛的，因为它包含所有的环，结合代数和分配格。在本文中，我们研究了左半桁架的子类，它的行为类似于在研究杨-巴克斯特方程中出现的代数结构。我们研究的相互作用的业务和这种相互作用需要对各自的半群。特别地，我们证明了在有限情形下，加法结构是完全正则半群。其次，我们应用我们的结果的一个特殊的情况下，左半桁架称为几乎左半支撑，介绍了Miccoli研究其代数结构。特别是，我们表明，一个左半括号可以关联到任何几乎左半括号。此外，我们证明了杨-巴克斯特方程的集合论解起源于几乎左半括号产生这种对应。
英文摘要：
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, Brzeziński 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.
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