  Let $[n]=\{1,\ldots,n\}$ be the $n$-chain. We give presentations for the
following transformation semigroups: the semigroup of full order-decreasing
mappings of $[n]$, the semigroup of partial one-to-one order-decreasing
mappings of $[n]$, the semigroup of full order-preserving and order-decreasing
mappings of $[n]$, the semigroup of partial one-to-one order-preserving and
order-decreasing mappings of $[n]$, and the semigroup of partial
order-preserving and order-decreasing mappings of $[n]$.
