The Clowder Project

The collage of $R$¹ is the poset $\mathbf{Coll}\left(R\right)\stackrel{\text{def}}{=}(\mathrm{Coll}\left(R\right),{⪯}_{\mathbf{Coll}\left(R\right)})$ consisting of

• The Underlying Set. The set $\mathrm{Coll}\left(R\right)$ defined by
  $$\mathrm{Coll}\left(R\right)\stackrel{\text{def}}{=}A\coprod B.$$

• The Partial Order. The partial order
  $${⪯}_{\mathbf{Coll}\left(R\right)}:\mathrm{Coll}\left(R\right)\times \mathrm{Coll}\left(R\right)\to \{\mathsf{true},\mathsf{false}\}$$

  on $\mathrm{Coll}\left(R\right)$ defined by

  $$⪯(a,b)\stackrel{\text{def}}{=}\{\begin{array}{ll}\mathsf{true}& \text{if }a=b\text{ or }a{\sim }_{R}b\text{,}\\ \mathsf{false}& \text{otherwise.}\end{array}$$