p'$ depends on $s$, then we have one single monoid and ${\mathrel{\downarrow}}$ is then a right action of that monoid on $S$.)

To help explain the operations and laws, we sketch in Fig. \[fig:dcontainer\] a data-structure with nested sub-data-structures.

$$\xymatrix@C=1.45em@R=1em{
& & & & & & & & \bullet \ar@(l,u)@{->}[]^(0.25){{\mathsf{o}}{\{s\}}} \ar@{-}[ddddddddllllllll] \ar@{-}[ddddddddrrrrrrrr]_>>>>>>{s = s {\mathrel{\downarrow}}{\mathsf{o}}{\{s\}}} \ar@{->}[dd]^(0.6){p = p {\mathrel{\oplus}}{\mathsf{o}}{\{s'\}} = {\mathsf{o}}{\{s\}} {\mathrel{\oplus}}p : P s} \ar@{->}@/_5pc/[dddd]_{p {\mathrel{\oplus}}p'}  \ar@{->}@/_8.5pc/[ddddddd]_{(p {\mathrel{\oplus}}p') {\mathrel{\oplus}}p'' = p {\mathrel{\oplus}}(p' {\mathrel{\oplus}}p'')} \\
& & & & & & & & & \\
& & & & & & & &  \bullet \ar@(l,u)@{.>}[]^(0.25){{\mathsf{o}}{\{s'\}}}  \ar@{.>}[dd]^(0.6){p' : P s'} \ar@{.}[ddddddllllll] \ar@{.}[ddddddrrrrrr]_>>>>>>{s' = s {\mathrel{\downarrow}}p} \ar@{.>}@/^4.5pc/[ddddd]^(0.4){p' {\mathrel{\oplus}}p''} & &  \\
& & & & & & & & & & & \\
& & & & & & & & \bullet \ar@{-->}[ddd]^(0.65){p'' : P s''} \ar@{--}[ddddllll]^>>>>>>{s'' = s {\mathrel{\downarrow}}(p {\mathrel{\oplus}}p') = s' {\mathrel{\downarrow}}p'} \ar@{--}[ddddrrrr]& & & &  \\
& & & & & & & & & & & & & \\
& & & & & & & & & & & & & \\
& & & & & & & & \bullet & & & & & & \\
\ar@{-}[rrrrrrrrrrrrrrrr] & & & &  & & & & 