# Should the dots used in lambda-calculus notation be typographically treated the way binary relation and binary operation symbols are?

Within this question we find the notation $$\lambda f . \lambda g . \lambda x.f(g\ x)$$, and that causes me to wonder about a point of LaTeX and MathJax usage.

The code a.b.c. gets rendered as $$a.b.c,$$ with the period, or "full stop" if you like, closer to the letter to its left than the one to its right.

As one would expect, the code a{.}b{.}c gives more symmetrical results: $$a{.}b{.}c$$.

But if we put \newcommand{\.}{\mathbin{.}} $$\newcommand{\.}{\mathbin{.}}$$at the begining (or above the \begin{document} in $$\LaTeX$$) then the code a\.b\.c gives us $$a\.b\.c,$$ with an amount of horizontal space appropriate to a binary operation or binary relation symbol (e.g. the code 3+5 will normally result in $$3+5$$ and not in $$3{+}5$$).

\begin{align} & a.b.c \\[5pt] & a{.}b{.}c \\[5pt] & a\.b\.c \end{align}

Which version should be used in this context?

• It's not a binary symbol. Semantically, "punctuation" is probably the closest match among TeX symbol categories, so it's appropriate to format it as just \lambda x.f. And anyway, just about everybody does that, so you can consider it the typographical standard. Aug 31 at 5:42
• @EmilJeřábek : How is it done in the Pacific Journal of Mathematics? Aug 31 at 18:19
• I don't know. Why don't you check it? Aug 31 at 19:12
• @EmilJeřábek : Via search engines I couldn't quickly find any lambda-calculus notation in PJoM. I mentioned them because they are fastidious about normative rules of typesetting when it comes to details that most people would never notice. Sep 1 at 21:13
• If this is for submission to a journal, then I would heavily discourage use of the macro name \.. Journals use their own macros, and uncommon single-character macro names like this seem particularly prone to unexpected conflict. (I agree with being discomfited by treating the . as punctuation when, say, the $\mapsto$ in the mathematician's version $f \mapsto g \mapsto x \mapsto f(g(x))$ is not so treated, but here I agree with @EmilJeřábek's comment that one should probably regard this as established convention nonetheless.) Oct 23 at 17:25
• @LSpice Good point. No need to hypothesize about journals’ own macros: \. already is a standard LaTeX command (for the text-mode dot-above diacritic), and overloading standard commands in a submission is a bad idea indeed. Oct 23 at 17:39
• @EmilJeřábek, re, ah, thanks. I remembered the spacing macros \!, \,, \ , \;, and \: (and probably others), but forgot the existing meaning of \.. Oct 23 at 17:43