Opposite product signs in the sense that black and white are interchanged

Question

In a special algebra I use with amssymb the products

\newcommand{\bDiamond}{\mathbin{\Diamond}}
\newcommand{\bLozenge}{\mathbin{\blacklozenge}}
\newcommand{\bCircledast}{\mathbin{\circledast}}
\newcommand{\bOdot}{\mathbin{\odot}}

See also the two earlier answered questions sign for multiple $\Diamond$-product and Filled \Diamond-product sign. The two product signs \bDiamond and \bLozenge are opposite or partners in the sense that black and white are interchanged.

Now I would like to have

• a partner sign for \bCircledast, i.e. a white asterisk on a black disk (the shape should be the same as \bCircledast) and

• a partner sign for \bOdot, i.e. a white dot on a black disk (the shape should be the same as \bOdot)

Thanks in advance for your help!

Answer 1

Print a big black circle and overprint to it a white asterisk or a white dot.

\documentclass{article}
\usepackage{amsmath,amssymb,graphicx,color}

\newcommand{\invcircledast}{%
  \mathbin{\vphantom{\circledast}\text{%
    \ooalign{\smash{\blackcircle}\cr
             \hidewidth\smash{\textcolor{white}{$*$}}\hidewidth\cr
            }%
  }}%
}
\newcommand{\invodot}{%
  \mathbin{\vphantom{\odot}\text{%
    \ooalign{\smash{\blackcircle}\cr
             \hidewidth\smash{\whitedot}\hidewidth\cr
            }%
  }}%
}

\newcommand{\blackcircle}{\raisebox{-.4ex}{\scalebox{1.66}{$\bullet$}}}
\newcommand{\whitedot}{\raisebox{.4ex}{\textcolor{white}{\scalebox{.3}{$\bullet$}}}}

\begin{document}
$a\circledast b\invcircledast c$

$a\odot b\invodot c$
\end{document}