The images of multilinear non-associative polynomials evaluated on a rock-paper-scissors algebra with unit over an arbitrary field and its subalgebras

Abstract

Let ${\mathbb F}$ be an arbitrary field. We consider a  commutative, non-associative, $4$-dimensional algebra ${\mathfrak M}$ of the rock, the paper and the scissors with unit  over ${\mathbb F}$ and we prove that the image over ${\mathfrak M}$ of every non-associative multilinear polynomial over ${\mathbb F}$ is a vector space. The same question we consider for two subalgebras: an algebra of the rock, the paper and the scissors without unit, and an algebra of trace zero elements with zero scalar part. Moreover in this paper we consider the questions of possible eveluations of homogeneous polynomials on these algebras.