A note on the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded identities of $E\otimes E$ over a finite field

Abstract

Let $F$ be a finite field of $charF=p$ and size $|F|=q$. Let $E$ be the unitary infinity dimensional Grassmann algebra. In this short note, we describe the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded identities of $E_{k^{\ast }}\otimes E$, where $E_{k^{\ast }}$ is the Grassmann algebra with a specific ${\mathbb{Z}}_2$-grading. In the end, we discuss about the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded GK-dimension of $E_{k^{\ast }}\otimes E$ in $m$ variables.