How do I create a custom math mode binary operation symbol that is a combination of two binary operation symbols?

Question

I want to create two math mode binary operation symbols:

(1) A combination of - (minus symbol) and * (asterisk)

(2) A combination of / (slash symbol) and * (asterisk)

The desired result would be something that looks like the following but is kerned and centered properly and has the left-right spacing of a typical binary operation:

\documentclass[12pt]{article}

\begin{document}
\Huge\[a \mathbin{-\mkern-11.5mu*} b = c \mathbin{/\mkern-8.875mu*} d.\] % my makeshift solution
\end{document}

\[a \minusast b = c \slashast d.\] % desired macro names and syntax

My question is:

How do I create the custom math mode binary relation symbols (1) and (2)?

It would also be nice if there's a version of (2) where the / symbol is adjusted to be a diagonal axis of symmetry of the * symbol.

If it's also possible, how do I generally create a custom math mode binary operation symbol that is a combination of two binary operation symbols?

Thank you.

Answer 1

Something like this? (updated to take into account the OP's additional requirement)

\documentclass{article} 

\usepackage{graphicx} % for '\rotatebox' macro
\newcommand\myslash{\rotatebox[origin=c]{-30}{$\vert$}}

\newcommand\minusast{\mathbin{\ooalign{\hss$*$\hss\cr$-$}}}
\newcommand\slashast{\mathbin{\ooalign{\hss$*$\hss\cr$\myslash$}}}

\begin{document}
$a\minusast b = c \slashast d$
\end{document}

Answer 2

The following will work in scriptstyle and scriptscriptstyle.

\documentclass{article}

\usepackage{graphicx}

\makeatletter
\newcommand{\asth}{\mathbin{\mathpalette\ast@h\relax}}
\newcommand{\ast@h}[2]{%
  \ooalign{%
    \hfil$\m@th#1*$\hfil\cr
    \hfil\rotatebox[origin=c]{-30}{$\m@th#1\mid$}\hfil\cr
  }%
}
\newcommand{\astd}{\mathbin{\mathpalette\ast@d\relax}}
\newcommand{\ast@d}[2]{%
  \ooalign{%
    \hfil$\m@th#1*$\hfil\cr
    $\m@th#1-$\cr
  }%
}
\makeatother

\begin{document}

$a*b\asth c\astd a$

\end{document}

Answer 3

The idea is to use \mathpalette and \ooalign. Some tricks are necessary, though: a \vphantom{*} ensures the correct bounding box, because \ooalign gets the height from the first item and the depth from the second one.

For the slash, one can use \centernot from the eponymous package.

\documentclass{article}
\usepackage{amsmath}
\usepackage{centernot}

\makeatletter
\newcommand{\astminus}{\mathbin{\mathpalette\astminus@\relax}}
\newcommand{\astminus@}[2]{%
  \vphantom{*}%
  \ooalign{\hidewidth$\m@th#1*$\hidewidth\cr$\m@th#1-$\cr}%
}
\makeatother
\newcommand{\astslash}{\mathbin{\centernot{*}}}

\begin{document}

$a\astminus b \astslash c$

$\scriptstyle a\astminus b \astslash c$

$\scriptscriptstyle a\astminus b \astslash c$

\end{document}

A different version where the \astslash symbol is obtained by rotating 60 degrees \astminus. Don't ask too much about how this is obtained. 😊

\documentclass{article}
\usepackage{amsmath}
\usepackage{graphicx}

% syntactic sugar
\newcommand{\xmathpalette}[2]{%
  \mathchoice
    {#1\displaystyle\textfont{#2}}
    {#1\textstyle\textfont{#2}}
    {#1\scriptstyle\scriptfont{#2}}
    {#1\scriptscriptstyle\scriptscriptfont{#2}}
}

\makeatletter
\newcommand{\astminus}{\mathbin{\mathpalette\astminus@\relax}}
\newcommand{\astminus@}[2]{%
  \vphantom{*}%
  \ooalign{\hidewidth$\m@th#1*$\hidewidth\cr$\m@th#1-$\cr}%
}

\newcommand{\astslash}{\mathbin{\xmathpalette\astslash@\relax}}
\newcommand{\astslash@}[3]{%
  \begingroup\settowidth\dimen@{$\m@th#1*$}%
  \makebox[\dimen@][s]{%
    \hspace{0pt minus 1fil}%
    \raisebox{0pt}[0.75\height][0.1\depth]{%
      \rotatebox[origin=c]{60}{%
        \raisebox{\dimexpr2\fontdimen22 #2 2 - \height}{%
          $\m@th#1\astminus$%
        }%
      }%
    }%
    \hspace{0pt minus 1.8fil}%
  }%
  \endgroup
}
\makeatother

\begin{document}

\fboxsep=0pt\fboxrule=0.1pt
\fbox{$\astslash$}\fbox{$*$}

$a\astminus b \astslash c$

$\scriptstyle a\astminus b \astslash c$

$\scriptscriptstyle a\astminus b \astslash c$

\end{document}

A lot of visual formatting, but the main idea is simple. I lower the \astminus symbol so it lies on the baseline, then rotate it. The bounding box has been set by eye.