Say we have the theory of DLO, which is not complete, but if we extend it to DLO+NE, where NE is the theory of no endpoint. (aka no maximum and no minimum, for example, the set R), then it is complete.
The question says that there is 5 such kind of minimal extension, I find another one which is DLO + the theory: "there is only one element".
However, I met some problems to find the other 3 extension Xs to make DLO + X be complete. Any clues?
Thanks.