Ten (dasha) monastic orders (sampradaya) founded by Shankara which have their monasteries (Amnaya Matha/Pitham) in four cities.
East Puri – Order Vana, Aranya
South Sringeri – Order Puri, Sarasvati, Bharati
West Dwaraka – Order Ashrama, Tirtha
North Jyotirmath – Order Giri, Parvata, Sagara
Vana – living in the forest
Aranya – living in the wilderness
Puri – living in the city
Sarasvati – named after the Goddess of wisdom
Bharati – a name of Sarasvati
Ashrama – living in a hermitage
Tirtha – living near a sacred water
Giri – living on a hill
Parvata – living in the mountain
Sagara – living near the ocean
The initiation name is followed by the order name.
More information you can find here
https://www.sringeri.net/history/amnaya-peethams