Concatenation of strings symbols

Question

In his 1989's Theorem for free! (doi: 10.1145/99370.99404, preprint in .dvi) paper, Wadler typeset the concatenation operation with two "plus" signs collapsed:

I asked write-math.com and Detexify to name this operator without success, and could not find anything apart from a not-so-elegant solution proposed by a blog, which consists in defining

\newcommand{\concat}{\ensuremath{+\!\!\!\!+\,}}

Is this operator defined in a package I'm not aware of, this notation not used at all?

Answer 1

I see nothing bad in the idea of partially superposing two plus signs. I've never seen such notation before, but it's not difficult to do it right, which is not the case with the macro you found out.

\documentclass{article}
\usepackage{amsmath}

\newcommand{\concat}{%
  \mathbin{{+}\mspace{-8mu}{+}}%
}

\newcommand{\starop}[1]{\mathop{#1^*}}

\begin{document}

\begin{gather*}
({\concat}): \forall X.\ X^* \to X^* \to X^* \\
\starop{a}(xs\concat_{A} ys)=(\starop{a} xs)\concat_{A'}(\starop{a} ys)
\end{gather*}

\end{document}

Compare with the original:

They seem to differ only for better spacing in my output. ;-) Maybe the distance between the vertical bars is slightly different, adjust it to suit.

And yes, the original uses two plus signs, judging from the DVI file that I transformed into PDF.

Alternative version

You might prefer a different symbol, as high and wide as the standard plus sign.

The use of color is to show precisely that the two symbols have the same width (red or black never overspills). Again, adjust the distance between the vertical bars to suit.

\documentclass{article}
\usepackage{amsmath,pict2e,color}

\makeatletter
\newcommand{\concat}{\mathbin{\mathpalette\conc@t\relax}}
\newcommand{\conc@t}[2]{%
  \vcenter{\hbox{%
    \sbox\z@{$\m@th#1-$}%
    \setlength{\unitlength}{\wd\z@}%
    \begin{picture}(1,1)
    \roundcap
    \put(0.1,0.5){\line(1,0){0.8}}
    \put(0.35,0.1){\line(0,1){0.8}}
    \put(0.65,0.1){\line(0,1){0.8}}
    \end{picture}%
  }}%
}
\makeatother

\newcommand{\starop}[1]{\mathop{#1^*}}

\begin{document}

\begin{gather*}
\rlap{\color{red}$+$}{\concat}{\concat}\llap{\color{red}$+$}\\
\rlap{\color{red}$\concat$}{+}{+}\llap{\color{red}$\concat$}\\
{+}{\concat}\quad a_{\concat_{\concat}}\\
{+}\\
{\concat}\\
({\concat}): \forall X.\ X^* \to X^* \to X^* \\
\starop{a}(xs\concat_{A} ys)=(\starop{a} xs)\concat_{A'}(\starop{a} ys)
\end{gather*}

\end{document}

Answer 2

Try +\kern -0.4em+ for example, the value of -0.4em provides some nice look, but this can be changed of course.

\documentclass[a4paper,10pt]{article}    
\usepackage{amsmath}

\DeclareMathOperator{\dplus}{+\kern -0.4em+}

\begin{document}
$ z_a \dplus z_b $
\end{document}

Answer 3

I think it's important that a binary symbol aligns correctly with another binary symbols. An obvious workaround is with \scalebox and brethren but I leave those for another answer if someone wants to show it. Here's a version that (surprinsingly) works.

I copied the code from egreg :)

\documentclass{scrartcl}

\usepackage{mathtools,calc}

\makeatletter
\newcommand*\concat
  {\mathbin{\mathmakebox[\widthof{${+}\m@th$}]{+\hskip-1emplus1fil+}}}
\newcommand*\starop[1]{\mathop{#1^*}}
\makeatother

\begin{document}

\begin{gather*}
({\concat})\colon \forall X.\ X^* \to X^* \to X^* \\
\starop{a}(xs\concat_{A} ys)=(\starop{a} xs)\concat_{A'}(\starop{a} ys)
\end{gather*}

\begin{align*}
  a + b + c + d \\
  a \concat b \concat c \concat d
\end{align*}

\end{document}