Strange spacing when putting a math macro before a binary operation

Question

I found there to be very odd spacing when using a math macro just before a binary operation. Here is a minimal example:

\documentclass[11pt]{article}
\usepackage{amsmath}
\DeclareMathOperator{\id}{id}
\begin{document}

$x\otimes \id$

$\id \otimes x$

\end{document}

produces

I am aware that sometimes a macro "eats up" the space after it, requiring that we put it in braces or put a \ after it, but I wouldn't have expected the space after the operation to be what is affected. Can someone explain what is going on, and how I could set this up differently so that the spacing will turn out correctly?

Answer 1

Those are TeX's rules. You're defining \id as an Op atom (math operator), whereas \otimes is a Bin (binary operation) and x is an Ord (ordinary symbol). Thus $\id\otimes x$ would be interpreted as

Op Bin Ord

but this is not accepted by the rules, so the Bin atom is changed into a Ord (think to $\sin -x$, which is not what is usually printed, but would be surely wrong if it was "sine minus x").

Therefore the final sequence of math atoms is

Op Ord Ord

and the spaces inserted are

Op (thin space) Ord (no space) Ord

If your \id macro is for denoting the identity map, it shouldn't be a math operator, but an ordinary symbol:

\newcommand{\id}{\mathrm{id}}

In this way $\id\otimes x$ would be

Ord (medium space) Bin (medium space) Ord

Answer 2

For correct spacing around \otimes - a binary operator, I would make sure that the operator \id resembles an ordinal symbol by wrapping it inside a group {}:

\documentclass[11pt]{article}
\usepackage{amsmath}% http://ctan.org/pkg/amsmath
\DeclareMathOperator{\id}{id}
\begin{document}

$x \otimes {\id}$

${\id} \otimes x$

\end{document}