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.Comment: A complete overhaul of the paper with some new results on left
semi-trusse
