The GrassmannParity command computes the Grassmannian parity of expression, that is, 0, 1 or undefined, according to whether expression is commutative, anticommutative or noncommutative. In this sense, the parity here is equivalent to the type.

$\mathrm{with}\left(\mathrm{Physics}\right)\:$

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\right]$

$\mathrm{Setup}\left(\mathrm{anticommutativepre}=\mathrm{\theta }\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{anticommutativepre}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{anticommutativeprefix}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{anticommutativeprefix}=\left\{\mathrm{\theta }\right\}\right]$

$a\mathrm{\theta }\left[1\right]\mathrm{\theta }\left[2\right]+b$

$a{\mathrm{\theta }}_1{\mathrm{\theta }}_2+b$

$\mathrm{GrassmannParity}\left(\right)$

$0$

$F\left(x\,y\,\mathrm{\theta }\left[1\right]\,\mathrm{\theta }\left[2\right]\right)$

$F\left(x\,y\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\right)$

$\mathrm{GrassmannParity}\left(\right)$

$0$

$\mathrm{Gtaylor}\left(\,\mathrm{\theta }\left[1\right]\right)$

$F\left(x\,y\,0\,{\mathrm{\theta }}_2\right)+{\mathrm{D}}_3\left(F\right)\left(x\,y\,0\,{\mathrm{\theta }}_2\right){\mathrm{\theta }}_1$

$\mathrm{ToFieldComponents}\left(\right)$

$\mathrm{\_F1}\left(x\,y\right)-{\mathrm{\theta }}_1\mathrm{\_Q1}\left(x\,y\right)-{\mathrm{\theta }}_2\mathrm{\_Q2}\left(x\,y\right)+\mathrm{\_F2}\left(x\,y\right){\mathrm{\theta }}_1{\mathrm{\theta }}_2$

$\mathrm{GrassmannParity}\left(\right)$

$\mathrm{undefined}$

$\mathrm{GrassmannParity}\left(\right)$

$0$

$\mathrm{Coefficients}\left(=\,\mathrm{\theta }\left[1\right]\,0\right)$

$F\left(x\,y\,0\,{\mathrm{\theta }}_2\right)=\mathrm{\_F1}\left(x\,y\right)-{\mathrm{\theta }}_2\mathrm{\_Q2}\left(x\,y\right)$

$\mathrm{map}\left(\mathrm{GrassmannParity}\,\right)$

$0=0$

$\mathrm{Coefficients}\left(=\,\mathrm{\theta }\left[1\right]\,1\right)$

${\mathrm{D}}_3\left(F\right)\left(x\,y\,0\,{\mathrm{\theta }}_2\right)=-\mathrm{\_Q1}\left(x\,y\right)+\mathrm{\_F2}\left(x\,y\right){\mathrm{\theta }}_2$

$\mathrm{map}\left(\mathrm{GrassmannParity}\,\right)$

$0=1$