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diff --git a/mjo-convex.tex b/mjo-convex.tex
index 823f1f4..a51011f 100644
--- a/mjo-convex.tex
+++ b/mjo-convex.tex
@@ -22,5 +22,9 @@
 % The ``lineality'' operator.
 \newcommand*{\lin}[1]{ \operatorname{lin} \of{{#1}} }
 
-% The ``Exteme vectors'' operator.
+% The ``Extreme vectors'' (or ``Extreme directions'') operator.
 \newcommand*{\Ext}[1]{ \operatorname{Ext} \of{{#1}} }
+
+% The "is a face of" and "is a proper face of" relations.
+\newcommand*{\faceof}{ \trianglelefteq }
+\newcommand*{\properfaceof}{ \triangleleft }