Why are eukarya and archaea bacteria related?

Archaea = Archaebacteria = 'archaea' + bacteria = Ancient Bacteria
Eukarya= Eubacteria = 'eu' + bacteria = New bacteria

Archaea and Eukarya both are prokaryotic, are unicellular, cell wall is made up of peptidoglycan, lack a definite nucleus, have a single circular chromosome .

So, both are Monerans yet Archaea and Eukarya are named different because archaea has some special features that let them survive in unfavourable conditions where other organisms (even Eubacteria) cannot survive,therefore makes them different from all organisms. Some of them are :

1) They have lipid-ether linked cell membrane.

2) Some of the archaebacterias have ability to survive in highly concentrated marine water called Halophiles .
- They tolerate increased salt concentration due to accumulation of Potassium salts present in them, that makes them Isotonic to the marine surrounding.

3) Some of them are Barophilic i.e. tolerates hydrostatic pressure by a chain of chemical reactions.

4) They have a different cell wall structure- as cell wall lacks Peptiodoglycan. * *Here, cell wall contain proteins and have precipitation of S * which makes them survive in hot springs, thus called Thermoacidophiles* .