
Originally Posted by
UsulDaNeriak
The word you look for is Asia. The romans used Asia for the continent Asia or the entire East or for just modern Turkey. Sometimes you read Asia minor, but Asia alone is also very popular. The 3rd meaning of Asia is just the province of Asia. Armenia does not belong to Asia in a narrower sense from a roman point of view afaik. But a pragmatic roman would not care.
Finding the right grammar for your approach is not that easy, because legionary cohorts were never named this way. But perhaps we can find a hint.
A legion could be named like a region, e.g. Legio II Britannica
A cohort of this legion now was called e.g. Cohors V Legionis II Britannica (or Britannicae?)
Auxiliary cohorts were never named according to regions but often according to the tribes, e.g. not Cohors I Asia but Cohors I Asianorum.
I guess you like to have a different naming for legionary cohorts, than for auxilia cohorts and no numbers? In this case I would use simply the regional name, e.g. Cohors Asia, Cohors Gallia, Cohors Hispania, even if that sounds a bit weird.