How to fix the size of variable-size operators

Question

When using variable-size operators such as \prod, \sum, \bigcup, and \bigcap, their sizes are usually not quite predictable, which makes me pretty annoyed. I just want to make their sizes fixed throughout the document, e.g., the same vertical size as X_{1}^{1}.

I tried to renew the definition of \prod using \renewcommand and scalerel package:

\renewcommand{\prod}{{\scalerel*{\prod}{X_{1}^{1}}}}

but it throws an error:

TeX capacity exceeded, sorry [grouping levels=255]. \prod

It should be allowed to use \limits to put limits on the top and bottom of the symbol in display mode.

Any comment on this issue?

Edit:

I prepared an example demonstrating the issue:

\documentclass{article}
\usepackage{amsmath}
\usepackage{scalerel}
\newcommand{\set}[2]{\left{{#1}~\middle|~{#2}\right}}
\begin{document}
    \begin{equation}
        \prod_{\alpha \in A} X_{\alpha}
        =
        \set{x \in A\to \bigcup_{\alpha\in A} X_{\alpha}}{\forall \alpha \in A: x_{\alpha} = x(\alpha) \in X_{\alpha}}
    \end{equation}
\begin{equation*}
    \begin{array}{r@{~}l}
        \prod_{\alpha \in A} X_{\alpha}
        &=
        \set{x \in A\to \bigcup_{\alpha\in A} X_{\alpha}}{\forall \alpha \in A: x_{\alpha} = x(\alpha) \in X_{\alpha}}\\
        &=
        \set{x \in A\to \bigcup_{\alpha\in A} X_{\alpha}}{\forall \alpha \in A: x_{\alpha} = x(\alpha) \in X_{\alpha}}
    \end{array}
\end{equation*}

The above two equations show the inconsistent sizing of operators.

\newcommand{\prode}{\mathop{\scalerel*{\prod}{X_{1}^{1}}}}
\newcommand{\bigcupe}{\mathop{\scalerel*{\bigcup}{X_{1}^{1}}}}

The desired result is
\begin{equation*}
    \begin{array}{r@{~}l}
        \prode\limits_{\alpha \in A} X_{\alpha}
        =
        \set{x \in A\to \bigcupe\limits_{\alpha\in A} X_{\alpha}}{\forall \alpha \in A: x_{\alpha} = x(\alpha) \in X_{\alpha}}
    \end{array}
\end{equation*}

\end{document}

In the last equation, I defined \prode and \bigcupe. But I just want to renew \prod and \bigcup. I think the error is caused by infinite loops.

Found solution:

I solved the problem by using \let found in Can I redefine a command to contain itself?

\let\oldprod\prod
\renewcommand{\prod}{\mathop{\scalerel*{\oldprod}{X_{1}^{1}}}}
\let\oldbigcup\bigcup
\renewcommand{\bigcup}{\mathop{\scalerel*{\oldbigcup}{X_{1}^{1}}}}

Thank you!

Answer 1

The size is predictable and it follows the typographic traditions.

Anyway, here's a way to reduce the size in display style to be the same as in text style.

I take advantage from the fact that big operators are defined in a uniform way by amsmath, using a common template and the operator \xyz refers internally to \xyz@.

\documentclass{article}
\usepackage{amsmath}
\ExplSyntaxOn
\NewDocumentCommand{\reduceoperators}{m}
 {
  \clist_map_function:nN { #1 } \hermis_reduceoperator:n
 }
\cs_new_protected:Nn \hermis_reduceoperator:n
 {
  \cs_set_eq:cc { latex@#1@ } { #1@ }
  \cs_set_protected:cpx { #1@ }
   {
    \mathop
     {
      \mathchoice
      {\textstyle\use:c{latex@#1@}}
      {\use:c{latex@#1@}}
      {\use:c{latex@#1@}}
      {\use:c{latex@#1@}}
     }
   }
 }
\ExplSyntaxOff
\reduceoperators{sum,prod,bigcup,bigcap}
\begin{document}
\begin{gather}
\tag{samples}
\sum\prod\bigcup\bigcap
\[4ex]
\tag{good}
\prod_{\alpha \in A} X_{\alpha} =
\bigl{x \in A\to \bigcup_{\alpha\in A} X_{\alpha}
  \bigm| \forall \alpha \in A: x_{\alpha} = x(\alpha) \in X_{\alpha}\bigr}
\
\tag{bad}
\prod_{\alpha \in A} X_{\alpha} =
\Bigl{x \in A\to \bigcup_{\alpha\in A} X_{\alpha}
  \Bigm| \forall \alpha \in A: x_{\alpha} = x(\alpha) \in X_{\alpha}\Bigr}
\
\tag{ugly}
\prod_{\alpha \in A} X_{\alpha} =
\left{x \in A\to \bigcup_{\alpha\in A} X_{\alpha}
  ;\middle|; \forall \alpha \in A: x_{\alpha} = x(\alpha) \in X_{\alpha}\right}
\end{gather}
\end{document}

The idea is to save the meaning of \sum@ into \latex@sum and then redefining \sum@ to mean

\mathop\mathchoice{\textstyle\latex@sum@}{\latex@sum@}{\latex@sum@}{\latex@sum@}

so in display style we actually use the \textstyle version. With the help of expl3 trickery, we're able to do all the needed redefinitions at once with no code duplication.

I also take the occasion to show how your definition of \set with \left and \right is not really good. For the expression you have \big size is more than enough: already \Big size is too much.

Answer 2

The above two equations show the inconsistent sizing of operators.

The inconsistency is due to the misuse of the array environment which is not designed for displayed equations, but for typesetting matrices: it sets its content in textstyle not displaystyle.

If you use a display alignment then the operators use a consistent style.

However the delimiters here are far too large. You can modify the set command to take an optional size.

\documentclass{article}
\usepackage{mathtools}
\newcommand{\set}[3][]{\mathopen{#1{}{#2}~#1|~{#3}\mathclose{#1}}}
\begin{document}
    \begin{equation}
        \prod_{\alpha \in A} X_{\alpha}
        =
        \set[\Big]{x \in A\to \bigcup_{\alpha\in A} X_{\alpha}}{\forall \alpha \in A: x_{\alpha} = x(\alpha) \in X_{\alpha}}
    \end{equation}
\begin{equation*}
    \begin{aligned}
        \prod_{\alpha \in A} X_{\alpha}
        &=
        \set[\Big]{x \in A\to \bigcup_{\alpha\in A} X_{\alpha}}{\forall \alpha \in A: x_{\alpha} = x(\alpha) \in X_{\alpha}}\\
        &=
        \set[\Big]{x \in A\to \bigcup_{\alpha\in A} X_{\alpha}}{\forall \alpha \in A: x_{\alpha} = x(\alpha) \in X_{\alpha}}
    \end{aligned}
\end{equation*}


\end{document}