Presentations for subsemigroups of
Let be an -chain. We give presentations for the following transformation semigroups: the semigroup of full order-decreasing mappings of , the semigroup of partial one-to-one order-decreasing mappings of , the semigroup of full order-preserving and order-decreasing mappings of , the semigroup of partial one-to-one order-preserving and order-decreasing mappings of , and the semigroup of partial order-preserving and order-decreasing mappings of .