Is there a difference between \bot and \perp, when they are used in exponent?

Question

I am used to write $M^\bot$ for the orthogonal complement of the set M. Is this incorrect?

I have seen this question, the answer there says that these two symbols have different spacing. (One of them is treated as relation. Does this difference manifest in some way when the symbols are used at exponent (without any other symbols around them).

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I have tried to search a little to find out whether there are other people which use \bot in the same way I do. You can judge for yourself, but I found some occurrences of this.

• "\bot" "\oplus"
• "\bot" "\oplus" filetype:tex
• "V^\bot" complement
• "S^\bot" complement

Here are the same searches with perp instead of bot:

• "\perp" "\oplus"
• "\perp" "\oplus" filetype:tex
• "V^\perp" complement
• "S^\perp" complement

Answer 1

Ah well it all depends.

In the default setup as commented in the linked question they are the same symbol but with different mathclass (so different spacing) \bot is a mathord (like an ordinary letter) and \perp is a relation (like <). Relations get more space either side if used between two symbols but in M^{\perp} the mathlist that forms the superscript only has one atom, so there is no additional spacing applied. That means that M^\bot and M^\perp produce identical output.

You could argue that \bot was better as it is naturally a symbol and a relation is not intended here.

Or you could argue that \perp is better as bot refers to a logical notion of bottom/false whereas perp refers to perpendicular which is somehow semantically related to orthogonal.

Or you could argue that they make identical output so it makes no difference and you can use either.

Answer 2

The definitions of \perp and \bot are, in Plain TeX and LaTeX,

% Plain TeX
\mathchardef\perp="323F
\mathchardef\bot="023F

% LaTeX
\DeclareMathSymbol{\perp}{\mathrel}{symbols}{"3F}
\DeclareMathSymbol{\bot}{\mathord}{symbols}{"3F}

that amount to the same thing. This means that the glyph used is the same, but with different spacings:

$X\bot Y$

will typeset the three math symbols without any space between them, while

$X\perp Y$

will have thick spaces around the central symbol.

When a relation symbol such as \perp is the only object in a math list, for example in a superscript, for instance your $M^{\perp}$ (with or without the braces is irrelevant), there will be no space added; also the result of

$X{\perp}Y$

would be identical to $X\bot Y$ because {\perp} makes the symbol the only object in a math list and a braced subformula is considered as an ordinary symbol (like a letter) as far as math spacing is concerned.

It's convenient to use \perp when orthogonality is concerned: you can write

$v\perp w$

to say that the vectors v and w are orthogonal to each other, and $U^\perp$ to denote the orthogonal complement of the subspace U. Using \bot for the latter would hide the meaning.

Technical note. The result of the (meaningless) formula $X^{v\perp w}$ would be identical to $X^{v\bot w}$ because TeX doesn't insert spaces around relation symbols in superscripts/subscripts. This doesn't mean that $\bot$ and $\perp$ are interchangeable. It's like using | or \mid in a math formula: both use the same glyph, but with different “math kind” (there are thirteen of them): the former is “ordinary”, the latter is ”relation”.