The order was:

Chattur'gha beat Xello'tath
Xel'lotath beath Ulyaoth
Ulyaoth beat Chattur'gha

Some websites theorize that Mantorok was the one who manipulated Pious into finding the Essence so that he'd become loyal to one of the Ancients and then Mantorok could manipulate them all into destroying each other.

Honestly, I don't know if I can buy that theory. For one thing, some people say that the first voice that calls out to Pious is Mantorok's and that's the proof of this. However, the second voice that calls out is clearly Xello'tath's (her whispering child-like voice). And I'm not completely convinced the first voice is Mantorok's unless someone from Nintendo or Silicon Knights specifically says it is.

Also, it doesn't seem to me like Mantorok would've wanted to get purposefully binded for all of eternity by Pious. Of course, maybe that wasn't part of his plan and was something that he didn't think would happen. But it seems to me that Mantorok had such a good thing going on Earth that it would've been best had he just left the other Ancients imprisoned.
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I sailed on the wings of time to be with you
I followed this dream of mine
And I'll see it through
It was a dream and a promise that I intend to keep
Right here within my soul so deep

"The universe is a yawning chasm, filled with emptiness and the purile meandering of sentience." - Ulyaoth