Let $A_X$ be a free non-associative algebra over a field $k$ on a set $X$ of free generators, where $X = X_0 \cup X_1$ and $X_0 \cap X_1 = \phi$. We will think of the elements of $X_0$ as even, and those of $X_1$ as odd. Then how to see that there is an induced $\mathbb Z_2$-grading on $A_X$?
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2$\begingroup$ See Bourbaki, A.III.3.1 Exemple 3. $\endgroup$– Fred RohrerCommented Jun 24, 2019 at 8:03
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1$\begingroup$ More generally there's a unique grading in $\mathbf{Z}^{(X)}$, the free abelian group with basis $(e_x)_{x\in X}$, for which each generator $x$ is homogeneous of degree $e_x$. One gets your grading by projecting to $\mathbf{Z}/2\mathbf{Z}$, mapping $e_x$ to $1$ for $x\in X_1$ and to $0$ for $x\in X_0$. $\endgroup$– YCorCommented Jun 24, 2019 at 8:45
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$\begingroup$ @YCor Thanks. What is (X)? $\endgroup$– GA316Commented Jun 24, 2019 at 11:07
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$\begingroup$ $X$ is your set and I already said what I mean by $\mathbf{Z}^{(X)}$. $\endgroup$– YCorCommented Jun 24, 2019 at 13:26
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