Cleanest implementation for Unicode U+2AEC (Double-Stroke Not Sign) using overlapping \lnot's?

Question

I'm using the Unicode character "Double-Stroke Not Sign" (U+2AEC) to represent bitwise logical not operation (a.k.a. one's complement). I've implemented a control sequence called \blnot which accomplishes this, but my solution feels kludgey and I suspect there must be a best-practices solution which is much cleaner. Here is how my output looks with $-$, \lnot, and \blnot, respectively:

My question is two parts:

1. What is the cleanest way to stack two \lnot operators as shown above? I made an \hbox and used \raisebox to adjust the base of each instance, but I don't like how I used \kern-.666666em to back up 2/3 em after the first instance. I feel like there should be a way to cleanly go back the exact width of \lnot without having to know it, or better yet simply to not advance at all between the two.

2. I would also like to insert approximately .11em after both \lnot and \blnot because it looks too tight otherwise. (I'll be using \blnot mostly with multi-character identifiers in italics.) I've added the space this by saying \kern.11em, but again that feels kludgey to me. Ideally, I'd like to inuit the width of $-$ and simply use that, without hard-coding any width.

Here's an MWE:

\documentclass{article}

\newcommand{\blnot}{
  \mathord{%
    \hbox{%
      \raisebox{.19ex}{$\lnot$}\kern-.666666em%
      \raisebox{-.21ex}{$\lnot$}%
    }%
  }%
}

\begin{document}
$-a$                  \vskip -1ex
$\lnot\kern.11em a$   \vskip -1ex
$\blnot\kern.11em a$  \par
\end{document}

p.s. I know I could easily implement this using TikZ, but stacking two \lnots seems easier to make work at multiple font sizes. (I won't probably be using this at any size other than 10pt, but I might use it in a footnote or a caption.)

Answer 1

This is a job for \ooalign:

\newcommand{\blnot}{\mathord{\ooalign{%
  \relax\cr
  \noalign{\vskip-.2ex}
  $\lnot$\cr
  \noalign{\vskip.4ex}
  $\lnot$\cr
  \noalign{\vskip-.2ex}}%
  }}

Here's a complete example, where the \fbox show the effect on the bounding box; the first is \lnot, the second my proposed definition, the third is from Todd Lehman's self answer.

\documentclass{article}

\newcommand{\blnot}{\mathord{\ooalign{%
  \relax\cr
  \noalign{\vskip-.2ex}
  $\lnot$\cr
  \noalign{\vskip.4ex}
  $\lnot$\cr
  \noalign{\vskip-.2ex}}%
  }}

\newcommand{\TLblnot}{
  \mathord{%
    \hbox{%
      \raisebox{.18ex}{\rlap{$\lnot$}}%
      \raisebox{-.22ex}{\rlap{$\lnot$}}%
      \hphantom{$-$}%
    }%
  }%
}\begin{document}
\fbox{$\lnot a$}\fbox{$\blnot a$}\fbox{$\TLblnot a$}
\end{document}

For a size changing symbol, you can do the more complicated

\newcommand{\blnot}{\mathord{\mathpalette\xblnot\relax}}

\newcommand{\xblnot}[2]{
  \sbox2{$#1\lnot$}\vrule height 1.1\ht2 width0pt \ooalign{%
  \relax\cr
  \noalign{\vskip-.2\ht2}
  $#1\lnot$\cr
  \noalign{\vskip.4\ht2}
  $#1\lnot$\cr
  \noalign{\vskip-.2\ht2}}%
  }

and here's the result of

\fbox{$\lnot$}\fbox{$\blnot$}\fbox{$\blnot_{\blnot_{\blnot}}$}

It's quite simple also to get the symbol's bounding box as wide as a minus sign:

\newcommand{\blnot}{\mathord{\mathpalette\xblnot\relax}}

\newcommand{\xblnot}[2]{
  \sbox2{$#1\lnot$}\vrule height 1.1\ht2 width0pt \ooalign{%
  \hphantom{$#1-$}\relax\cr
  \noalign{\vskip-.2\ht2}
  \hfil$#1\lnot$\hfil\cr
  \noalign{\vskip.4\ht2}
  \hfil$#1\lnot$\hfil\cr
  \noalign{\vskip-.2\ht2}}%
  }

Some information on \ooalign can be found in this answer of mine

Answer 2

(Answering my own question here.) I ended up solving this myself, after Mikael Vejdemo-Johansson recommended that I use \llap or \rlap. Here's what I did. I am very happy with this solution:

\newcommand{\blnot}{
  \mathord{%
    \hbox{%
      \raisebox{.18ex}{\rlap{$\lnot$}}%
      \raisebox{-.22ex}{\rlap{$\lnot$}}%
      \hphantom{$-$}%
    }%
  }%
}

But taking it one step further, I defined it in terms of a \stacktwo command (for lack of a better name) that takes five parameters...

1. First character (e.g., \lnot)
2. Distance to raise first character (e.g., .18ex)
3. Second character (e.g., \lnot)
4. Distance to raise second character (e.g., -.22ex)
5. Phantom character to define width of composite character sequence (e.g., $-$)

...and goes like this:

\newcommand{\stacktwo}[5]{%
  \mathord{%
    \hbox{%
      \raisebox{#2}{\rlap{#1}}%
      \raisebox{#4}{\rlap{#3}}%
      \hphantom{#5}%
    }
  }
}

The final definition then is very simple:

\newcommand{\xblnot}{\stacktwo{$\lnot$}{.18ex}{$\lnot$}{-.22ex}{$-$}}

This does exactly what I want for all of \normalsize, \footnotesize, \scriptsize, and \tiny, and I can re-use \stacktwo for other composites like Unicode character U+2A59 ("Logical Or Overlapping Logical And").

A minor flaw here is that the widths of the two characters must be exactly the same, because I haven't yet figured out how to vertically align two stacked items of different widths, but this solution works fine for the characters I need.

Answer 3

This forum thread: https://groups.google.com/group/comp.text.tex/browse_thread/thread/8ed2ea2326858f8f?pli=1 provides several suggestions on how to implement a negative \phantom command which would allow you to step backwards with exactly your symbol's width.

In particular, this thread points out the LaTeX commands \llap and \rlap, both built to allow the construction of complex shapes by overlapping glyphs with sensible alignment. The example given is a construction of ≠ by either of the commands /\llap{=} or \rlap{=}/.

Thus, for your case, it would seem as if \rlap{\raisebox{-0.21em}{\lnot}}\raisebox{0.19em}{\lnot} might work.