+% A four-tuple of things.
+\newcommand*{\quadruple}[4]{ \left({#1},{#2},{#3},{#4}\right) }
+
+% A five-tuple of things.
+\newcommand*{\quintuple}[5]{ \left({#1},{#2},{#3},{#4},{#5}\right) }
+
+% A six-tuple of things.
+\newcommand*{\sextuple}[6]{ \left({#1},{#2},{#3},{#4},{#5},{#6}\right) }
+
+% A seven-tuple of things.
+\newcommand*{\septuple}[7]{ \left({#1},{#2},{#3},{#4},{#5},{#6},{#7}\right) }
+
+% The "least common multiple of" function. Takes a nonempty set of
+% things that can be multiplied and ordered as its argument. Name
+% chosen for synergy with \gcd, which *does* exist already.
+\newcommand*{\lcm}[1]{ \operatorname{lcm}\of{{#1}} }
+\ifdefined\newglossaryentry
+ \newglossaryentry{lcm}{
+ name={\ensuremath{\lcm{X}}},
+ description={the least common multiple of the elements of $X$},
+ sort=l
+ }
+\fi
+
+% The factorial operator.
+\newcommand*{\factorial}[1]{ {#1}! }
+
+% Restrict the first argument (a function) to the second argument (a
+% subset of that functions domain). Abused for polynomials to specify
+% an associated function with a particular domain (also its codomain,
+% in the case of univariate polynomials).
+\newcommand*{\restrict}[2]{{#1}{\restriction}_{#2}}
+\ifdefined\newglossaryentry
+ \newglossaryentry{restriction}{
+ name={\ensuremath{\restrict{f}{X}}},
+ description={the restriction of $f$ to $X$},
+ sort=r
+ }
+\fi