The algae or cyanobacta can and does live separately from the fungus but the fungus needs the algae. Some species can live with algae or cyanobacta others with one or the other. read more
Lichens are organisms made of both fungi and algae. This is a mutualist relationship. Both partners have almost equal benefits. While algae can photosynthesise useable carbon, fungi can establish grounded mycelia on almost any surface with limited nutrients. Together, they make a nice pair. Not all fungi and algae need to do this, however. read more