The collage of $R$1 is the poset $\smash {\mathbf{Coll}\webleft (R\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\mathrm{Coll}\webleft (R\webright ),\preceq _{\mathbf{Coll}\webleft (R\webright )}\webright )}$ consisting of
- The Underlying Set. The set $\mathrm{Coll}\webleft (R\webright )$ defined by
\[ \mathrm{Coll}\webleft (R\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B. \]
- The Partial Order. The partial order
\[ \preceq _{\mathbf{Coll}\webleft (R\webright )}\colon \mathrm{Coll}\webleft (R\webright )\times \mathrm{Coll}\webleft (R\webright )\to \{ \mathsf{true},\mathsf{false}\} \]
on $\mathrm{Coll}\webleft (R\webright )$ defined by
\[ \mathord {\preceq }\webleft (a,b\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \mathsf{true}& \text{if $a=b$ or $a\sim _{R}b$,}\\ \mathsf{false}& \text{otherwise.}\end{cases} \]
1Further Terminology: Also called the cograph of $R$.
