Order Automorphism on Well-Ordered Class is Forward Moving/Linguistic Note

Linguistic Note on Order Automorphism on Well-Ordered Class is Forward Moving
The concept is introduced by and  briefly in their  of $2010$ as a stepping-stone to the stronger result Order Automorphism on Well-Ordered Class is Identity Mapping.

They do not actually give a name to the concept, but merely characterise it as a mapping under which no element moves backwards.

It is worth comparing with the concept of a progressing mapping.