Definition: The algebra

${\mathfrak{A}}_W$ is the graded commutative algebra over $\mathbb{C}$ of local functionals

where $\mathcal{F}$ is a polynomial in finite jets of the fields and antifields whose coefficients are smooth functions of $x$ and of the backgrounds and are compactly supported in ${M_T}^{\circ }$, evaluated on ${\mathfrak{F}}_W$. Parity and ghost number are those of the integrand. Antifield degree is polynomial degree in the ${\Phi }_A^{\ast }$.