Definition: Antibracket

For the frozen generators,

Then $(S,{\Phi }^A)=-{\delta }^{R}S/\delta {{\Phi }^{\star }}_A$. The blanket identity $(S,\Phi )={\delta }^{L}S/\delta {\Phi }^{\star }$ is false for odd $\Phi$. The antifield rule is $s{{\Phi }^{\star }}_A=(S,{{\Phi }^{\star }}_A)$, not the reverse bracket.