PI-álgebras e multiplicidades limitadas por 1

Abstract

A variety $\mathcal{L}$ is distributive if $(\mathcal{V} \cap \mathcal{U}) \cup \mathcal{W} = (\mathcal{V} \cup \mathcal{W}) \cap (\mathcal{U} \cup \mathcal{W})$, for any subvarieties $\mathcal{V}, \mathcal{U, \mathcal{W}}$ of $\mathcal{L}$. In 1976, Ananin and Kemer classified distributive varieties of algebras based on a specific polynomial identity of the variety. The distributivity relation follows if, and only if, the multiplicities in the associated $n$-th cocharacter are bounded by 1. Thus, classifying varieties with multiplicities bounded by 1 is equivalent to require a specific polynomial identity satisfied by the variety. In the recent years, the class of algebras satisfying this property has been studied in differents contexts. Our main goal is to provide an original proof of Ananin and Kemer's result.